Curvature-Adaptive Smoothing for Multi-View Industrial Metrology

Abstract

Three-dimensional point cloud data acquired in industrial environments inherently exhibit quality limitations, including measurement noise, local geometric irregularities, and surface roughness. These issues are commonly observed in both structured-light scanning and SIFT-based photogrammetric reconstruction, highlighting the necessity of post-processing in metrology applications where dimensional accuracy and geometric reliability are critical. However, conventional global-parameter smoothing methods, such as Savitzky–Golay filtering, LOWESS, and bilateral filtering, apply uniform smoothing intensity across regions with varying curvature, resulting in an inherent trade-off between noise suppression and geometry preservation. In this study, we propose a learning-independent post-processing framework that adaptively modulates smoothing strength by integrating local curvature estimation with unsupervised anomaly modeling. The proposed approach combines normal-variance-based curvature approximation with k-nearest neighbor anomaly scoring, while CAD data are employed exclusively as an external reference for evaluation. Experimental results on industrial product datasets demonstrate that, for structured-light reconstructions, the proposed method reduces average error to a level comparable to local regression-based smoothing while simultaneously achieving the highest edge-preservation index among all evaluated methods, thereby attaining an optimal operating point between noise suppression and geometric integrity. Although the suppression of extreme deviations remains limited, the geometry-preservation metrics are improved without a significant increase in average error. In contrast, SIFT+COLMAP-based reconstructions exhibit performance comparable to that of the original data, a behavior attributable to low-frequency systematic reconstruction biases rather than high-frequency sensor noise, which fall outside the corrective capacity of purely geometry-driven local smoothing.

1. Introduction

With the advent of Industry 4.0, three-dimensional (3D) optical reconstruction technologies have become essential enablers for dimensional inspection, reverse engineering, and quality assurance (QA) in modern industrial environments. The growing demand for precision metrology has further intensified reliance on 3D optical measurement systems capable of rapidly and repeatedly capturing complex freeform surfaces in a non-contact manner with high geometric fidelity.

Within this context, multi-view photogrammetry based on the Scale-Invariant Feature Transform (SIFT) and structure from motion (SfM) has gained widespread adoption in both academic research and industrial practice, primarily due to its accessibility and its ability to maintain geometric consistency across a wide range of object geometries. In particular, the COLMAP pipeline, which integrates SIFT-based feature matching, incremental SfM, and multi-view stereo (MVS), has been extensively used as a representative open-source framework for reconstructing dense point clouds from calibrated image sequences. In parallel, structured-light 3D scanning systems have also been widely deployed as reliable surface measurement technologies in industrial inspection settings, owing to their capability to acquire high-precision surface geometry under controlled illumination conditions.

Despite fundamental differences in acquisition principles, both camera-based reconstruction approaches and structured-light scanning systems commonly produce raw point clouds that exhibit similar quality limitations. These limitations include depth inconsistencies, surface roughness, local geometric anomalies, and sensor-induced noise, which inevitably arise regardless of the reconstruction strategy. When such raw data are directly utilized in downstream inspection or analysis tasks without dedicated post-processing, the geometric reliability of subsequent metrological results may be compromised. Accordingly, for optical metrology applications that require both dimensional accuracy and geometric reliability, post-processing is widely regarded as an indispensable stage of the reconstruction pipeline. Although recent learning-based reconstruction and restoration methods have demonstrated strong performance in controlled settings, their direct deployment in industrial metrology environments is fundamentally constrained by the scarcity of task-specific annotated training data, the high cost of model retraining upon geometry changes, and the stringent requirement for physical interpretability and traceability of measurement results. These constraints motivate the present study, which proposes a practical, learning-independent post-processing framework rather than a novel reconstruction algorithm. The framework is grounded in physically interpretable geometric modeling and is designed to be immediately deployable across diverse industrial reconstruction pipelines without reliance on training data or supervised optimization.

1.1. Sources of Geometric Uncertainty

In SIFT–COLMAP-based multi-view reconstruction, geometric uncertainty accumulates across multiple processing stages. Feature-matching errors in repetitive or low-texture regions may degrade the stability of sparse SfM reconstruction, leading to inaccuracies in camera pose estimation and triangulation. During multi-view stereo (MVS), additional errors arise from insufficient parallax, specular reflections, occlusions, and photometric inconsistencies.

These factors result in non-uniform depth distributions, surface roughness, and localized geometric outliers in the reconstructed point cloud. Similar degradation is also observed in structured-light scanning systems under varying illumination and reflectance conditions. Therefore, robust post-processing is essential to ensure metrology-grade geometric reliability. Critically, the nature of geometric degradation differs fundamentally between the two modalities: structured-light systems predominantly produce high-frequency sensor noise superimposed on otherwise metrically consistent surfaces, whereas SIFT+COLMAP pipelines tend to accumulate low-frequency systematic depth biases arising from feature-matching instability and multi-view inconsistency. This distinction has direct implications for post-processing strategy selection, as geometry-driven local smoothing is inherently better suited to attenuating high-frequency noise than to correcting globally coherent reconstruction biases.

1.2. The Smoothing vs. Preservation Dilemma

From a metrological perspective, point cloud post-processing inherently involves a fundamental trade-off between two conflicting requirements. In flat or low-curvature regions, strong smoothing is desirable to suppress roughness-induced pseudo-variations that amplify measurement uncertainty. Conversely, in geometrically critical regions such as edges, fillets, and corners, even minor distortions may propagate into significant dimensional errors, making faithful feature preservation a primary concern.

Conventional smoothing techniques, including Savitzky–Golay (SG) filtering, LOESS/LOWESS regression, moving averages, and bilateral filtering, typically rely on globally fixed parameters and implicitly assume locally homogeneous surface characteristics. As a result, these approaches exhibit an inherent limitation: they cannot simultaneously achieve aggressive noise suppression in planar regions while preserving geometric features in high-curvature areas. This limitation becomes particularly pronounced in complex industrial geometries, where large planar surfaces coexist with functionally critical edges. Furthermore, most existing smoothing methods do not explicitly incorporate outlier separation or weighting mechanisms capable of selectively attenuating gross outliers caused by reflective artifacts or depth-estimation failures. Under such conditions, abnormal structures may bias the estimation of nearby geometric features, with the extent of this influence strongly dependent on the chosen smoothing strategy and parameter configuration. The root cause of this deficiency is structural rather than parametric: because the bandwidth or window size governing smoothing intensity is applied uniformly across the entire point cloud, adjusting these parameters shifts the global operating point along the noise-suppression–geometry-preservation continuum but cannot resolve the trade-off locally. That is, a parameter setting that achieves sufficient smoothing in planar regions will inevitably over-smooth high-curvature boundaries, and vice versa. Overcoming this limitation requires a spatially adaptive mechanism that modulates smoothing strength based on local geometric semantics a capability that fixed-parameter methods are structurally incapable of providing.

1.3. Proposed Approach

The present study proposes a practical, learning-independent post-processing framework rather than a novel reconstruction or learning algorithm. The framework is specifically designed to meet the operational constraints of industrial metrology environments where access to large-scale annotated training data is infeasible, product geometry changes frequently, and physical interpretability of all geometric modifications is mandatory. Within this context, the proposed approach integrates curvature-aware adaptive geometric smoothing with unsupervised, heuristic-based anomaly-aware weighting, and focuses on quantitative comparison with conventional non-learning smoothing techniques rather than competing with learning-based surface restoration methods.

The framework is grounded in physically interpretable, geometry-driven modeling and adopts a learning-independent post-processing strategy. It is governed by the following three physical design principles: (1) In flat regions characterized by low curvature and small geometric uncertainty, relatively strong smoothing is required to suppress roughness-induced local variations. (2) In high-curvature regions such as edges and boundaries, even subtle geometric distortions may propagate directly into shape deformation; therefore, geometric diffusion must be minimized. (3) Structurally inconsistent outliers should be selectively attenuated using unsupervised anomaly scores to prevent error propagation into neighboring geometric structures.

To realize these principles, the framework employs adaptive weighting functions constructed from local curvature estimation and surface roughness quantification. CAD-aligned residual distances are used exclusively as reference metrics for post hoc quantitative evaluation and sensitivity analysis; CAD information is never utilized during the smoothing process itself. Unsupervised anomaly scores are computed purely from geometric information, based on kNN distance dispersion and local geometric inconsistency indices. Likewise, CAD residuals are applied only during post-processing validation and parameter sensitivity assessment, while the smoothing procedure operates independently of CAD data.

1.4. Contributions

Savitzky–Golay (SG) filtering and LOWESS smoothing have long been employed for point cloud post-processing. However, from a geometry-oriented quantitative evaluation perspective, relatively few studies have compared SG, LOWESS, and curvature-aware adaptive smoothing under a unified reconstruction pipeline using consistent quantitative metrics. In particular, systematic analyses that simultaneously examine geometric accuracy, surface roughness, and feature-preservation behavior for SIFT+COLMAP-based reconstructions remain limited in the literature. The main contributions of this study are summarized as follows:

• Integrated formulation of a learning-independent adaptive smoothing framework: The primary contribution of this study is the formalization of a reproducible, learning-independent post-processing framework that renders the trade-off between noise suppression and geometric preservation explicitly observable and interpretable under realistic industrial constraints. Rather than introducing an algorithmic variant, this work establishes the conditions under which curvature-aware weighting, unsupervised anomaly suppression, and bounded displacement jointly achieve metrology-grade dimensional reliability without requiring training data, pre-trained models, or CAD supervision during inference. The framework is designed with system integration for industrial dimensional inspection as the overarching goal, treating reproducibility and physical interpretability as primary design constraints of equal importance to geometric accuracy.
• Incorporation of an unsupervised outlier suppression mechanism: An anomaly-aware weighting strategy is introduced to selectively attenuate local outliers based on kNN distance dispersion and local geometric inconsistency indices, without reliance on supervised learning or pre-trained models.
• Systematic quantitative comparison protocol based on established metrics: A unified experimental protocol is constructed to compare SG filtering, LOWESS smoothing, and the proposed method using both SIFT+COLMAP and structured-light reconstructions, enabling analysis of method-specific characteristics and trade-offs in terms of geometric accuracy, surface roughness, and feature preservation.
• Quantitative analysis of the trade-off between noise suppression and feature preservation: Using geometry-driven quantitative indicators, a physically interpretable analysis is provided to elucidate how balanced behavior emerges between the inherently conflicting objectives of noise suppression and feature retention.
• Explicit characterization of applicability conditions and deployment constraints: By systematically analyzing post-processing behavior across two fundamentally different reconstruction modalities, this work delineates the operational scenarios in which the proposed framework provides meaningful benefit namely, structured-light data characterized by high-frequency sensor noise and uniform point density and those in which its corrective capacity is limited, such as SIFT+COLMAP data dominated by low-frequency systematic reconstruction bias. This explicit characterization of applicability boundaries constitutes a practical contribution toward informed deployment of post-processing modules in industrial inspection pipelines.

Rather than proposing a single universal smoothing operator, this work frames post-processing as a metrology-oriented problem in which reproducibility, interpretability, dimensional reliability, and independence from task-specific training signals are treated as primary design constraints. Within this perspective, curvature-aware weighting, anomaly suppression, and bounded displacement are organized into a coherent framework that enables controlled, geometry-driven examination of post-processing behavior across heterogeneous reconstruction sources.

Accordingly, the contribution of this study lies not in introducing another algorithmic variant, but in formalizing a learning-independent framework that makes the trade-off between noise suppression and feature preservation explicitly observable while providing practical guidance under realistic industrial conditions. Importantly, this framework is not intended as a simple substitute for learning-based methods; rather, it addresses a complementary and distinct operational niche dimensional inspection environments in which training data are unavailable, interpretability is legally or procedurally required, and immediate deployment without retraining is essential. Taken together, these contributions support the overarching objective: not to minimize error indiscriminately, but to render post-processing trade-offs explicit, interpretable, and reproducible for tolerance-critical industrial metrology applications.

1.5. Organization of the Paper

The remainder of this paper is organized to progressively address the research problem established above. Section 2 reviews related work on multi-view reconstruction, classical and geometry-aware point cloud smoothing, and recent deep learning-based approaches, thereby situating the proposed framework within the current state of the art and motivating the learning-independent design strategy. Section 3 presents the proposed curvature-aware adaptive post-processing framework, including its mathematical formulation, algorithmic pipeline, and design rationale with explicit discussion of applicability conditions and known limitations. Section 4 describes the experimental datasets and validation protocol, defines the quantitative evaluation metrics, and reports the comparative results for both structured-light and SIFT+COLMAP reconstructions, followed by a critical discussion of the observed trade-offs and condition-dependent behavior. Finally, Section 5 summarizes the principal findings, explicitly addresses the limitations of the proposed approach, and outlines directions for future research, including integration with upstream deep learning-based reconstruction pipelines.

2. Related Work

2.1. Multi-View Photogrammetry and SIFT-Based Reconstruction

Multi-view photogrammetry reconstructs three-dimensional geometry from multiple two-dimensional images through a sequence of feature extraction and matching, camera pose estimation via structure from motion (SfM), and dense surface recovery using multi-view stereo (MVS). Within this pipeline, the Scale-Invariant Feature Transform (SIFT) has been widely adopted as a robust local feature descriptor invariant to scale and rotation [1,2,3], enabling stable correspondence establishment across viewpoints [4,5]. The COLMAP framework integrates SIFT-based feature matching, geometric verification, and bundle adjustment into a representative open-source SfM/MVS pipeline [4,5]. Its performance and applicability have been systematically evaluated in prior studies [6], and it has been widely adopted as a practical solution for dense 3D reconstruction without specialized metrology hardware [7,8].

Despite these advantages, feature-based approaches are strongly dependent on image texture. In industrial environments, matte surfaces, geometrically simple regions, and repetitive patterns often lead to unstable feature detection and matching [2]. Furthermore, specular reflections on metallic surfaces distort feature localization and intensity, increasing depth estimation uncertainty and generating high-frequency noise and outliers in the reconstructed point clouds [9,10]. During the MVS stage, disparity estimation remains sensitive to baseline length and viewpoint distribution, resulting in depth instability and spatial density imbalance under constrained acquisition conditions [6].

2.2. Structured-Light 3D Scanning and Geometry-Based Registration

Structured-light three-dimensional (3D) scanning is an active optical metrology technique in which projected patterns are captured by a camera and decoded to reconstruct surface geometry via triangulation. Due to its capability to provide stable depth estimation even on low-texture surfaces, structured-light scanning has been widely adopted in industrial inspection and quality control applications [9,11,12].

Among various implementations, fringe projection profilometry (FPP) utilizes phase-shifting information to achieve sub-pixel depth accuracy [12,13], and has been further developed toward real-time or near real-time measurement systems [14].

For multi-view alignment, structured-light point clouds are commonly registered using geometry-based pipelines that combine affine transformations with Fast Point Feature Histograms (FPFHs) [15,16]. FPFH encodes local surface normal relationships and curvature distributions into a rotationally invariant descriptor [15], offering reduced computational complexity compared with PFH while maintaining reliable registration performance. As these methods rely exclusively on geometric features, they are inherently robust to illumination variations and camera exposure inconsistencies [15,17].

Despite these advantages, structured-light systems are affected by phase ambiguity, scattering noise on reflective or translucent surfaces, and data incompleteness caused by self-occlusion in complex geometries [9,10,12]. These limitations can propagate through the registration pipeline, resulting in accumulated alignment errors and localized geometric distortions. Consequently, dedicated post-processing strategies are required to enhance surface fidelity and ensure reliable downstream dimensional analysis [16,18,19].

2.3. Classical Point Cloud Smoothing Methods

To suppress measurement noise and improve surface quality, a wide range of classical point cloud smoothing methods have been proposed. The Savitzky–Golay (SG) filter performs local polynomial regression and was originally developed for one-dimensional signal processing, later extended to image and 3D data applications [20,21]. While SG filtering effectively attenuates high-frequency noise while preserving low-frequency structures, its reliance on globally fixed parameters such as window size and polynomial order often results in uniform smoothing across both flat and high-curvature regions, leading to shape distortion or oversmoothing [17].

Local regression-based approaches, including LOESS and Moving Least Squares (MLS), have also been widely adopted for irregularly sampled point clouds [17,22]. Although these methods exhibit strong smoothing performance in planar regions, their dependence on fixed neighborhood sizes limits their ability to adaptively preserve edges and corners [22].

Bilateral filtering incorporates both spatial proximity and attribute similarity to achieve edge-preserving behavior [23,24], and its extensions to mesh and point cloud data have been reported [25]. However, bilateral filters remain sensitive to parameter selection and may degrade under non-uniform sampling density, making it difficult to consistently balance noise suppression and geometric preservation [17,26,27].

2.4. Geometry-Aware and Adaptive Smoothing Approaches

To address the limitations of isotropic filtering, geometry-aware smoothing techniques have been developed to adapt the smoothing strength based on local surface characteristics. Anisotropic diffusion, originally proposed by Perona and Malik [28], modulates diffusion coefficients using gradient magnitude, thereby preserving edges while smoothing homogeneous regions. This concept has been extended to point clouds and meshes using curvature, normal variance, and tensor-based descriptors [22,27].

Despite these theoretical advantages, the practical implementation of such methods remains challenging due to the instability of curvature and normal estimation from noisy raw data [17]. Errors in geometric estimation may propagate through the smoothing process, resulting in unexpected distortions or insufficient noise reduction [27,29]. Consequently, prior studies have reported significant performance variability depending on data characteristics and parameter configurations, and no universally optimal trade-off is guaranteed [17,22,27].

2.5. Unsupervised Outlier Suppression in Point Clouds

Unsupervised outlier suppression is commonly employed as a preprocessing step prior to smoothing and registration. Statistical Outlier Removal (SOR) evaluates the mean distance to k-nearest neighbors and removes points that deviate from the global distribution beyond a predefined threshold [30,31]. Due to its simplicity and scalability, SOR has been widely adopted in many 3D processing pipelines [31]. Radius-based outlier removal eliminates isolated points based on local neighbor counts and can be interpreted as a density-based anomaly detection approach [30].

More advanced density-based clustering and robust registration-based outlier rejection methods have also been proposed [32,33]. However, distance- and density-based criteria inherently struggle to distinguish noise from genuine fine-scale geometry, such as sharp edges or small surface defects [17,19]. Consequently, critical geometric details may be incorrectly removed, which is particularly problematic in high-precision industrial inspection scenarios [18,34].

2.6. Deep Learning-Based 3D Reconstruction and Post-Processing

In recent years, deep learning-based approaches have achieved substantial progress in both upstream 3D reconstruction and downstream point cloud post-processing, addressing error sources that classical geometry-driven methods struggle to correct.

At the upstream reconstruction stage, self-supervised learning frameworks have been proposed to mitigate structured-light noise arising from complex illumination conditions. Wang et al. [35] introduced a deep self-supervised learning network for multi-exposure image fusion in fringe structured-light 3D reconstruction, demonstrating that learning-based fusion of multi-exposure image sequences can effectively suppress photometric noise and inter-reflection artifacts without requiring ground-truth depth annotations. This approach directly targets the systematic bias sources inherent to structured-light acquisition, including fringe saturation and inter-reflection, which are difficult to correct through purely geometric post-processing.

At the downstream reconstruction and denoising stage, generative models have shown strong capability for recovering fine-grained surface geometry from corrupted or incomplete inputs. Zhao et al. [36] proposed ICDDPM, an image-conditioned denoising diffusion probabilistic model for real-world complex point cloud single-view reconstruction, which leverages conditional diffusion processes to restore surface completeness and geometric coherence under severe occlusion and noise. This framework demonstrates that learned priors over surface geometry can enable reconstruction quality that substantially exceeds what is achievable through local, signal-processing-based smoothing alone.

These learning-based methods represent the current state of the art in addressing systematic reconstruction error; however, their direct deployment in industrial metrology environments faces fundamental practical constraints. First, both supervised and self-supervised approaches require substantial quantities of domain-specific training data, which are rarely available in industrial inspection settings where product geometries change frequently and ground-truth acquisition is costly. Second, the diffusion- and fusion-based inference pipelines impose significant computational overhead that may conflict with real-time or near-real-time inspection requirements. Third, and most critically, industrial metrology processes demand strict physical interpretability and traceability of all geometric modifications: any alteration to the reconstructed surface geometry must be explainable in terms of physically meaningful quantities to satisfy quality assurance and regulatory requirements. Deep learning models, by their nature, do not readily provide such explanations. These constraints motivate the present study’s focus on a learning-independent post-processing framework that operates exclusively on geometric cues, requires no training data, and produces results that are fully interpretable in terms of local curvature, point density, and anomaly scores.

2.7. Critical Summary and Research Gap

Prior studies indicate that SIFT+COLMAP-based photogrammetry provides strong accessibility and global consistency [4,6], with its performance and applicability further analyzed in [2,7,8]. However, it exhibits persistent instability under reflective metallic surfaces and geometrically simple regions [9,10].

Geometry-based structured-light pipelines alleviate dependence on image features but remain affected by reflection, scattering, occlusion, and residual registration errors [11,16].

From a post-processing perspective, classical smoothing and outlier removal methods rely on globally fixed parameters, making it fundamentally difficult to simultaneously suppress noise and preserve geometric features [17,20,21,22]. Geometry-aware adaptive approaches partially address this limitation but introduce new challenges associated with unstable geometric estimation and parameter sensitivity [22,27].

Recent deep learning-based methods, including self-supervised multi-exposure fusion networks [35] and diffusion model-based reconstruction frameworks [36], have demonstrated strong capability for addressing systematic reconstruction biases at the upstream stage. However, as discussed in Section 2.6, their deployment in industrial metrology environments is constrained by the scarcity of domain-specific training data, high retraining costs upon geometry changes, and the stringent requirement for physical interpretability. These constraints create a persistent practical gap between the theoretical performance of learning-based methods and their operational feasibility in tolerance-critical inspection workflows.

Within this context, existing research highlights an unresolved gap: no current approach simultaneously achieves noise suppression, geometric feature preservation, learning-independence, and full physical interpretability under realistic industrial metrology conditions. This gap motivates the present work, which proposes a curvature-aware adaptive geometric smoothing framework that integrates local geometric indicators and unsupervised anomaly measures. Rather than replacing learning-based upstream reconstruction, the proposed framework is designed as a complementary downstream post-processing module that operates independently of training data and CAD supervision, thereby extending the applicability of metrology-grade inspection pipelines to deployment scenarios where learning-based methods are currently impractical.

3. Methodology

3.1. Overview of the Methodology Framework

The methodology of this study focuses on the post-processing stage of three-dimensional point clouds acquired in industrial measurement environments. The input data consist of (i) point clouds reconstructed via SIFT-based multi-view photogrammetry using an SfM+MVS pipeline and (ii) point clouds acquired through structured-light 3D scanning. Although these two reconstruction approaches differ in acquisition principles and error-generation mechanisms, their final point cloud outputs share similar post-processing requirements, as both typically contain measurement noise, surface roughness, and localized geometric irregularities [17].

For structured-light data, multi-view point clouds are first aligned into a common coordinate system prior to post-processing. This alignment is performed within an existing reconstruction pipeline based on affine transformation and FPFH-based geometric registration and therefore lies outside the scope of the present study [15,16,18]. Accordingly, this section does not revisit the design or performance of the alignment algorithm; instead, the analysis assumes that accurate registration has already been completed.

The proposed post-processing pipeline consists of the following stages: (1) local neighborhood definition, (2) estimation of surface normals and curvature approximations, (3) computation of unsupervised geometric anomaly indicators, (4) adaptive weight calculation incorporating curvature and anomaly information, and (5) iterative smoothing updates with bounded displacement. As illustrated in Figure 1, each stage of the pipeline is designed to operate sequentially, with curvature and anomaly estimates from stages (2) and (3) directly governing the adaptive weights applied in stage (4). Specifically, the anomaly modeling step identifies structurally unreliable points that would otherwise bias curvature estimation, while the curvature-aware weighting in stage (4) ensures that smoothing is restricted in geometrically critical regions such as edges and fillets. The displacement constraint in stage (5) further prevents accumulated geometric distortion during iterative updates. The entire framework is learning-independent and is designed based on geometry-driven principles to ensure interpretability, reproducibility, and applicability in industrial metrology scenarios.

3.2. Problem Formalization

In industrial optical measurement data, degradation of geometric quality can generally be categorized into three types. First, high-frequency positional fluctuations caused by sensor noise and reconstruction uncertainty introduce surface roughness that does not correspond to the true geometry, even in planar regions [17]. Second, localized outliers arising from surface reflections, occlusions, or imbalanced viewpoint distributions may interfere with neighboring valid structures [10]. Third, prior metrology studies have reported that even small positional deviations in high-curvature regions, such as edges, fillets, and corners, can propagate into significant dimensional errors [27].

These characteristics imply that uniform smoothing strategies that apply identical filtering strength across the entire surface are insufficient to meet metrological requirements. While strong noise suppression is desirable in quasi-planar regions, preservation of geometric fidelity must be prioritized in high-curvature areas. The fundamental challenge therefore lies not in noise suppression or feature preservation in isolation, but in their spatially conflicting demands: the same surface simultaneously contains quasi-planar regions that require aggressive smoothing and high-curvature boundaries that require strict geometric protection. No fixed-parameter smoothing strategy can satisfy both requirements simultaneously, as any single global parameter setting inevitably represents a compromise that over-smooths boundaries or under-suppresses noise in planar regions. This spatial conflict is the primary motivation for a locally adaptive post-processing approach that modulates smoothing strength based on geometric semantics rather than spatial proximity alone. Therefore, this study assumes the necessity of a locally adaptive post-processing approach that is applicable across different reconstruction modalities and leverages local geometric information.

3.3. Mathematical Formulation and Limitations of Standard Smoothing Methods

In this study, Savitzky–Golay (SG) filtering and LOWESS smoothing are adopted as baseline methods for comparison with the proposed adaptive post-processing framework.

SG and LOWESS are representative local regression-based smoothing techniques that have been widely used in industrial metrology and measurement signal processing [17]. Both methods control smoothing strength through globally fixed parameters, making them suitable reference baselines for evaluating the benefits of local adaptivity introduced by the proposed approach. The selection of these two methods is not based solely on their classical status in signal processing; rather, SG and LOWESS represent the most widely adopted smoothing techniques in commercial industrial scanner software and metrology processing pipelines, making them the widely adopted industrial baselines against which practical improvements must be demonstrated [17]. A comparison with these methods therefore constitutes a direct assessment of whether the proposed framework offers a meaningful upgrade over current industrial practice, rather than merely an academic benchmark. The Savitzky–Golay filter defines a local neighborhood $N_i$ for each point $p_i$ and updates the position of the central point by fitting a low-order polynomial to the coordinates of neighboring points using a least-squares formulation [20,21]. This process can be expressed as

$$\underset{\mathit{\theta }}{min}\sum _{{\mathbf{p}}_j\in {\mathcal{N}}_i}{∥{\mathbf{p}}_j-\Phi \left({\mathbf{x}}_j\right)\mathit{\theta }∥}_2^2,$$

(1)

where ${\mathbf{p}}_j\in {\mathbb{R}}^3$ denotes the coordinate vector of a point in the neighborhood ${\mathcal{N}}_i$, and $\Phi \left({\mathbf{x}}_j\right)$ represents a low-order polynomial basis constructed from the point coordinates ${\mathbf{x}}_j$. Using the estimated coefficients $\mathit{\theta }$, the smoothed position of the central point ${\mathbf{p}}_i$ is computed. While this approach effectively preserves low-frequency components, its reliance on globally fixed neighborhood size and polynomial order results in identical smoothing behavior in both flat and high-curvature regions, which constitutes a structural limitation [17]. In industrial part geometries where planar regions and functional edges coexist, this may lead to shape distortion in curvature-critical areas.

LOWESS smoothing incorporates distance-based weighting into a local regression framework by solving a weighted least-squares problem using a kernel function $K(·)$:

$$\underset{\mathit{\theta }}{min}\sum _{{\mathbf{p}}_j\in {\mathcal{N}}_i}K\left({\displaystyle \frac{\parallel {\mathbf{p}}_i-{\mathbf{p}}_j\parallel }{h}}\right){∥{\mathbf{p}}_j-\Phi \left({\mathbf{x}}_j\right)\mathit{\theta }∥}_2^2,$$

(2)

where h is a global bandwidth parameter controlling the kernel support, and $K(·)$ is a non-negative, monotonically decreasing function that assigns lower weights to distant points. Fundamentally, LOWESS computes a distinct polynomial fit for each point, relying on a kernel (e.g., Gaussian or tricube) to ensure that closer neighbors exert greater influence on the local surface approximation. This localized, distance-weighted approach allows the regression surface to adapt smoothly to complex data variations without being constrained by a single global model. However, because the bandwidth parameter h is typically applied uniformly across the entire point cloud, the method inherently struggles to distinguish between high-frequency measurement noise and genuine high-curvature features, often leading to unintended over-smoothing near sharp edges. Although LOWESS provides increased flexibility for modeling nonlinear variations compared with SG filtering, it does not explicitly incorporate geometric semantics such as curvature, outlier presence, or metrological importance, which limits its effectiveness in precision inspection contexts [17].

The structural limitation shared by both methods can be stated precisely: for any fixed parameter setting h or window size w, the smoothing operator ${\mathcal{S}}_h$ satisfies

$${\mathcal{S}}_h\left({\mathbf{p}}_i\right)=\sum _{{\mathbf{p}}_j\in {\mathcal{N}}_i}{w}_{ij}\left(h\right){\mathbf{p}}_j,\sum _j{w}_{ij}=1,$$

(3)

where the weights $w_{ij}\left(h\right)$ are determined solely by spatial distance and a globally fixed h, without reference to local curvature ${\kappa }_i$ or anomaly score $a_i$. Consequently, adjusting h shifts the global operating point along the noise-suppression–geometry-preservation continuum but cannot resolve the trade-off locally: a value of h sufficient to suppress noise in planar regions will simultaneously over-smooth high-curvature boundaries, and vice versa. This is a structural, not parametric, deficiency that cannot be remedied by careful tuning alone.

More advanced geometry-aware methods, such as feature-preserving Moving Least Squares (MLS) [29] and bilateral filtering on point sets [26,27], partially address this limitation by incorporating local geometric descriptors into the weighting scheme. However, their practical deployment under noisy raw industrial measurement data poses substantial challenges: reliable estimation of curvature and surface normals requires sufficiently dense and uniformly distributed neighborhoods, conditions that are rarely met in SIFT+COLMAP or multi-view structured-light data exhibiting non-uniform density, reflective artifacts, and residual registration errors. Under such conditions, errors in geometric estimation propagate into the weighting scheme, potentially amplifying rather than suppressing local distortions.

The proposed framework addresses these limitations through two complementary mechanisms. First, it replaces explicit high-order curvature estimation with a normal-variance-based proxy that is numerically stable under the adverse sampling conditions characteristic of industrial measurement data. Second, it integrates LOF-based unsupervised anomaly detection as a weighting component, enabling selective suppression of structurally inconsistent outliers that would otherwise bias both curvature estimation and the subsequent smoothing update. Together, these mechanisms provide spatially adaptive smoothing governed by local geometric semantics a capability that fixed-parameter regression methods are structurally incapable of providing and that geometry-aware methods struggle to achieve reliably under realistic industrial noise conditions.

In summary, both baseline methods rely on regression-based smoothing governed by globally fixed parameters and therefore lack the local adaptivity required for robust post-processing of industrial measurement data [17].

3.4. Local Geometric Structure Modeling and Curvature Approximation

The proposed adaptive smoothing framework is grounded in the quantitative modeling of local geometric structure within the point cloud. Let the input point cloud be defined as

$$P={\{p_i\in {\mathrm{R}}^3\}}_{i=1}^N.$$

(4)

For each point ${\mathbf{p}}_i$, its local neighborhood ${\mathcal{N}}_i$ is defined using the k-nearest neighbors (kNN) criterion,

$$N_i=\mathrm{kNN}(p_i,k),$$

(5)

where k denotes the number of neighboring points. The kNN-based neighborhood definition is widely adopted in industrial measurement data processing, as it provides relatively stable local structure estimation even under non-uniform sampling density [17].

The adoption of kNN-based neighborhoods, rather than radius-based or adaptive bandwidth alternatives, reflects a deliberate design choice motivated by the requirements of industrial metrology. Radius-based neighborhoods produce variable-size local structures under non-uniform point density, which destabilizes downstream curvature and anomaly estimation. By contrast, fixing the neighborhood cardinality to k ensures that every local geometric descriptor is computed from an identical number of data points, thereby guaranteeing consistent statistical behavior across the surface and supporting the reproducibility requirement that is essential in metrological applications.

Local geometric properties are characterized using surface normal statistics. For each neighborhood ${\mathcal{N}}_i$, a covariance matrix is computed, and the surface normal ${\mathbf{n}}_i$ is defined as the eigenvector corresponding to the smallest eigenvalue. Rather than performing explicit surface fitting or higher-order differential geometry analysis, curvature is approximated using the variance of neighboring normals,

$${\kappa }_i={\displaystyle \frac{1}{|{\mathcal{N}}_i|}}\sum _{{\mathbf{p}}_j\in {\mathcal{N}}_i}{∥{\mathbf{n}}_j-{\overline{\mathbf{n}}}_i∥}_2^2,{\overline{\mathbf{n}}}_i={\displaystyle \frac{1}{|{\mathcal{N}}_i|}}\sum _{{\mathbf{p}}_j\in {\mathcal{N}}_i}{\mathbf{n}}_j,$$

(6)

where ${\overline{\mathbf{n}}}_i$ denotes the mean normal vector within ${\mathcal{N}}_i$. The resulting quantity ${\kappa }_i$ serves as a relative curvature indicator that captures local geometric variation and structural complexity. In industrial optical measurement environments, variance-based curvature approximations have been reported to provide higher robustness than explicit differential curvature estimation under noise and registration uncertainty [17,27].

It is important to note that ${\kappa }_i$ is not intended to recover the absolute principal curvatures of the underlying surface. Rather, it serves as a relative geometric complexity indicator whose purpose is to reliably distinguish high-curvature boundary regions (edges, fillets, corners) from quasi-planar regions under realistic noise and registration conditions. This distinction motivates the choice of normal-variance over higher-order differential estimators such as those employed in feature-preserving MLS [29]: while the latter provide theoretically more accurate curvature values, their numerical stability degrades significantly under the non-uniform density, reflective artifacts, and residual alignment errors characteristic of industrial measurement data. The normal-variance proxy sacrifices absolute accuracy in favor of robust relative discrimination, which is the operationally relevant property for adaptive smoothing weight assignment.

3.5. Unsupervised Geometric Anomaly Modeling

Curvature information alone is insufficient for identifying structural outliers arising from specular reflections, reconstruction failures, or multi-view registration artifacts. To address this limitation, the proposed framework constructs a multi-dimensional geometric feature vector for each point and evaluates its anomaly level using an unsupervised criterion.

In this context, the term unsupervised refers specifically to independence from annotated training data, pre-trained model weights, and CAD reference geometry during inference not to the absence of algorithm parameters. The anomaly scoring procedure described below operates exclusively on geometric statistics derived from the raw point cloud itself, requiring no external supervision signal of any kind. This property is essential for deployment in industrial metrology environments, where ground-truth geometry labels are unavailable and the framework must generalize across product families without retraining.

For each point ${\mathbf{p}}_i$, a feature vector ${\mathbf{f}}_i\in {\mathbb{R}}^d$ is defined as

$${\mathbf{f}}_i=[{\rho }_i,{\kappa }_i,{\mathrm{linearity}}_i,{\mathrm{planarity}}_i,{\mathrm{sphericity}}_i,{\sigma }_i,c_i],$$

(7)

where ${\rho }_i$ denotes the local point density, and ${\kappa }_i$ is the normal-variance-based curvature indicator defined in Equation (6). The descriptors ${\mathrm{linearity}}_i$, ${\mathrm{planarity}}_i$, and ${\mathrm{sphericity}}_i$ are derived from eigenvalue ratios of the neighborhood covariance matrix, while ${\sigma }_i$ and $c_i$ serve as auxiliary measures that reflect local variance and geometric reliability. This feature composition is consistent with established practices in point cloud outlier detection and filtering, where local geometric statistics and neighborhood-based descriptors are commonly utilized [17,30,31].

After feature standardization, anomaly scores are computed using the Local Outlier Factor (LOF), an unsupervised density-based method that does not require training data or model optimization,

$$a_i=\mathrm{LOF}\left(f_i\right),a_i\in [0,1].$$

(8)

The LOF value represents a normalized anomaly score rescaled to the interval $[0,1]$. By comparing local density ratios rather than relying on a global threshold, LOF remains relatively robust under heterogeneous and uneven point densities, which are commonly observed in industrial measurement data.

A critical distinction between the proposed anomaly modeling and conventional outlier removal approaches (e.g., Statistical Outlier Removal [30,31]) lies in how the anomaly score is utilized. Conventional methods apply a hard binary decision: points exceeding a threshold are removed entirely, which risks eliminating genuine fine-scale geometric features such as sharp edges or small surface defects that coincidentally exhibit low local density. In contrast, the proposed framework integrates $a_i$ as a continuous soft weight within the smoothing update (detailed in Section 3.6), allowing structurally unreliable points to be progressively down-weighted rather than discarded. This soft-weighting mechanism suppresses the influence of outliers on neighboring point updates while preserving the spatial support of all points for subsequent curvature-aware weighting. The combination of curvature-based weighting ${\kappa }_i$ and anomaly-based weighting $a_i$ thus provides two orthogonal and complementary layers of geometric protection: ${\kappa }_i$ restricts smoothing in geometrically important high-curvature regions, while $a_i$ suppresses the influence of structurally inconsistent points regardless of their local curvature. This dual-layer design constitutes the primary methodological advance of the proposed framework over both fixed-parameter smoothing baselines and conventional outlier removal pipelines.

3.6. Curvature and Anomaly-Aware Adaptive Weighting Model

Point positions are iteratively updated as weighted averages of neighboring points, where the weights are designed to jointly account for spatial proximity, normal consistency, curvature magnitude, and anomaly likelihood. The update rule is defined as

$${\mathbf{p}}_i^{(t+1)}=\sum _{p_j\in N_i}{w}_{ij}{\mathbf{p}}_j^{\left(t\right)},$$

(9)

where the weight $w_{ij}$ is given by

$$w_{ij}={\displaystyle \frac{1}{Z_i}}exp\left(-\frac{\parallel {\mathbf{p}}_i-{\mathbf{p}}_j{\parallel }^2}{2{\sigma }_d^2}-\frac{1-n_i·n_j}{{\sigma }_n}-{\lambda }_k{k}_j-{\lambda }_a{a}_j\right),$$

(10)

and $Z_i={\sum }_{{\mathbf{p}}_j\in {\mathcal{N}}_i}{w}_{ij}$ is a normalization constant ensuring ${\sum }_j{w}_{ij}=1$. Here, ${\sigma }_d$ and ${\sigma }_n$ control the influence of spatial distance and normal similarity, respectively, while ${\lambda }_k$ and ${\lambda }_a$ regulate attenuation due to curvature and anomaly score.

Each component of the weighting function serves a distinct and physically interpretable geometric role: spatial weighting prioritizes nearby points, normal weighting enforces orientation consistency, curvature weighting suppresses diffusion in highly curved regions, and anomaly weighting reduces the influence of structurally unstable points.

In contrast to the fixed-parameter smoothing operators described in Equation (3), where the weights $w_{ij}$ depend solely on spatial distance and a globally fixed bandwidth h, the proposed weighting function incorporates point-wise geometric descriptors ${\kappa }_j$ and $a_j$ that vary across the surface. Consequently, for two points ${\mathbf{p}}_i$ and ${\mathbf{p}}_{i^{\prime }}$ at equal spatial distance from a common neighbor ${\mathbf{p}}_j$, the proposed model assigns different weights depending on whether that neighbor lies in a high-curvature boundary region or a structurally anomalous area. This spatially adaptive behavior is the key structural distinction from fixed-parameter baselines: the smoothing intensity is determined locally by geometric semantics rather than globally by a single bandwidth parameter, enabling simultaneous aggressive smoothing in planar regions and strict geometric protection near edges a trade-off that fixed-parameter methods cannot resolve regardless of parameter tuning.

The two attenuation terms ${\lambda }_k{\kappa }_j$ and ${\lambda }_a{a}_j$ are designed to serve orthogonal and complementary geometric roles within the weighting scheme. The curvature term ${\lambda }_k{\kappa }_j$ restricts smoothing in regions where the local surface undergoes rapid directional change, thereby protecting geometric features such as edges, fillets, and corners from excessive diffusion. The anomaly term ${\lambda }_a{a}_j$ independently down-weights the contribution of points that are structurally inconsistent with their local neighborhood arising from specular reflections, occlusions, or registration failures regardless of whether those points reside in high- or low-curvature regions. Because these two terms target distinct error sources (geometric boundary fidelity and structural outlier contamination, respectively), deactivating either one is expected to degrade a different aspect of post-processing performance. This design is empirically validated through an ablation study presented in Section 4.6, in which the conditions ${\lambda }_k=0$ (anomaly-only) and ${\lambda }_a=0$ (curvature-only) are compared against the full proposed framework.

Together, these terms provide a controllable balance between noise suppression and geometric preservation, thereby extending conventional isotropic smoothing models [22,27].

All parameters of the proposed framework (k, ${\sigma }_d$, ${\sigma }_n$, ${\lambda }_k$, ${\lambda }_a$, $\eta$) are fixed based on preliminary experiments and are held constant across all datasets and reconstruction modalities reported in this study. This fixed-parameter protocol is adopted deliberately to serve two purposes. First, it eliminates the possibility of post hoc parameter tuning that could artificially inflate performance for specific datasets, thereby ensuring that the reported results reflect the baseline capability of the framework rather than optimized performance. Second, it demonstrates that the proposed adaptive weighting mechanism provides meaningful geometric benefit even without dataset-specific optimization, which is a prerequisite for practical industrial deployment where per-product parameter tuning is operationally infeasible. A systematic sensitivity analysis of the key parameters (k, ${\lambda }_k$, ${\lambda }_a$) and their influence on smoothing strength and geometric preservation metrics remains an important direction for future work, as discussed in Section 5.

3.7. Iterative Smoothing and Displacement Constraints

At each iteration, an intermediate smoothed position is first computed as

$${\tilde{\mathbf{p}}}_i^{(t+1)}=\sum _{p_j\in N_i}{w}_{ij}{\mathbf{p}}_j^{\left(t\right)},$$

(11)

Repeated smoothing may accumulate excessive displacement and distort the underlying geometry. To mitigate this effect, a local-scale displacement constraint is introduced [17,27]. The local scale is defined as the median distance to neighboring points,

$$s_i^{\left(t\right)}=median\left(\left\{\parallel {\mathbf{p}}_j^{\left(t\right)}-{\mathbf{p}}_i^{\left(t\right)}\parallel |{\mathbf{p}}_j\in {\mathcal{N}}_i\right\}\right).$$

(12)

The final position update is then limited by a factor $\eta s_i^{\left(t\right)}$,

$${\mathbf{p}}_i^{(t+1)}={\mathbf{p}}_i^{\left(t\right)}+min\left(1,{\displaystyle \frac{\eta s_i^{\left(t\right)}}{\parallel {\tilde{\mathbf{p}}}_i^{(t+1)}-{\mathbf{p}}_i^{\left(t\right)}\parallel }}\right)\left({\tilde{\mathbf{p}}}_i^{(t+1)}-{\mathbf{p}}_i^{\left(t\right)}\right).$$

(13)

This displacement constraint enables sufficient noise reduction in flat regions while preventing excessive geometric diffusion near edges and high-curvature structures. Similar stabilization strategies have been widely reported in geometry-based point cloud filtering and adaptive smoothing frameworks [17,27].

The displacement constraint defined in Equation (13) serves a role that is complementary to, but distinct from, the curvature- and anomaly-aware weighting in Equation (10). Whereas the adaptive weights regulate which neighboring points exert influence on each update, the displacement constraint limits how far any individual point can move per iteration. This two-level geometric protection direction-selective weighting and magnitude-bounded displacement ensures that the iterative process remains geometrically stable even when curvature or anomaly estimates are locally unreliable, for example, in regions of severe non-uniform density or strong specular contamination. The local scale factor $s_i^{\left(t\right)}$, defined as the median neighbor distance, provides a data-driven, physically interpretable bound that automatically adjusts to local point spacing without requiring a globally fixed displacement threshold. This interpretability property is consistent with the overall design philosophy of the framework, in which every parameter has a direct physical correspondence to observable geometric quantities.

3.8. Design Rationale and Limitations

The proposed curvature- and anomaly-aware adaptive smoothing framework is intentionally designed to operate without additional training data or pre-trained models. This design choice is not merely a practical concession to data availability constraints; rather, it reflects a deliberate prioritization of three properties that are uniquely critical in industrial metrology environments and that learning-based methods structurally struggle to satisfy:

1.

Physical interpretability and traceability. Every geometric modification produced by the proposed framework is fully explainable in terms of physically meaningful quantities: local curvature ${\kappa }_i$, point density ${\rho }_i$, normal consistency, and LOF-based anomaly score $a_i$. This traceability is a regulatory and procedural requirement in many industrial inspection workflows, where quality assurance standards mandate that any alteration to measurement data must be auditable and physically justifiable. Learning-based models, by contrast, produce modifications that are not readily decomposable into interpretable geometric causes, making them unsuitable for such compliance requirements.

2.

Hallucination-free geometric integrity. Because the framework operates exclusively on observed geometric cues without learned surface priors, it cannot introduce geometric structures that are absent in the raw data. Learning-based denoising and reconstruction methods are susceptible to hallucination errors plausible-looking but geometrically incorrect surface completions which are particularly dangerous in dimensional inspection contexts where false geometric features may be incorrectly accepted as valid part geometry. The proposed framework avoids this failure mode by construction.

3.

Training-data independence and immediate deployability. In industrial environments, product CAD models and surface geometries change frequently due to design iterations and manufacturing process updates. Any method requiring retraining upon geometry change incurs substantial operational overhead. The proposed framework requires no training data of any kind and can be deployed immediately across new product families without modification, making it operationally practical in high-mix, low-volume manufacturing contexts.

These three properties collectively justify the design choice of physically interpretable, geometry-driven components over more expressive but less interpretable learning-based alternatives, even when the latter may offer superior raw performance on controlled benchmarks.

Moreover, reproducibility is a critical requirement in metrological applications; therefore, all components of the framework are formulated using physically and geometrically interpretable quantities, including spatial distance, normal consistency, curvature indicators, and unsupervised anomaly scores [17].

All parameters employed in the proposed framework were fixed based on preliminary experiments and were not individually optimized for each dataset.

This fixed-parameter protocol serves two explicit purposes: (1) it eliminates post hoc tuning that could artificially favor specific datasets, ensuring that reported results reflect the framework’s baseline capability, and (2) it demonstrates that the adaptive weighting mechanism provides meaningful geometric benefit even without dataset-specific optimization a necessary condition for practical industrial deployment where per-product parameter tuning is operationally infeasible. The framework’s parameters therefore represent engineering defaults rather than optimized hyperparameters, and the reported experimental results should be interpreted accordingly. A systematic sensitivity analysis of parameter variations is beyond the scope of the present study and remains an important direction for future work, particularly for formalizing deployment guidelines across diverse industrial scenarios.

For curvature characterization, the framework adopts a normal-variance-based descriptor rather than explicit surface fitting or higher-order differential curvature estimation. The objective is not to recover absolute curvature values but to robustly capture relative variations in local geometric complexity under realistic noise and registration conditions. Industrial optical measurement data frequently exhibit uneven sampling density, reflective artifacts, and residual alignment errors. Under such circumstances, higher-order differential estimators are known to be numerically unstable, whereas variance-based curvature proxies have been reported to provide more stable and reliable behavior [17,27].

Because curvature information alone is insufficient to distinguish structural outliers caused by reconstruction failures or multi-view inconsistencies, the proposed framework integrates an unsupervised anomaly-aware weighting mechanism. The LOF-based anomaly score evaluates relative local density variations without requiring a global threshold, making it suitable for heterogeneous point clouds. The combined use of curvature and anomaly indicators enables the framework to preserve geometrically stable high-curvature regions while selectively attenuating low-curvature yet structurally unreliable areas, thereby facilitating a controllable balance between noise suppression and feature preservation [17,22,27].

The proposed method deliberately excludes the use of CAD-aligned residuals during the smoothing process. In practical industrial deployments, reference CAD models may not always be available, and a unified post-processing module is often required to operate independently across multiple reconstruction pipelines and product families. Although CAD residuals are valuable for quantitative accuracy evaluation, incorporating them into the inference stage would restrict the applicability of the method to CAD-dependent scenarios. Accordingly, CAD information is reserved exclusively for validation and analysis, while the smoothing process itself relies solely on geometric cues [17].

Despite these advantages, the framework has the following limitations that must be explicitly acknowledged for responsible deployment:

Density-dependent reliability of geometric estimation. The framework relies on k-nearest-neighbor-based local structure estimation, which implicitly assumes that the k nearest neighbors provide a representative sample of the local surface geometry. This assumption is violated under severe point density non-uniformity, as observed in SIFT+COLMAP reconstructions. In such data, the k nearest neighbors of a given point may span vastly different geometric scales depending on local density: in dense regions, k neighbors are confined to a small spatial extent, yielding accurate local surface characterization; in sparse or transitional regions, the same k neighbors may span distances an order of magnitude larger, causing the normal-variance-based curvature estimator ${\kappa }_i$ to reflect density variation rather than genuine geometric curvature. This density-curvature confounding of the framework. degrades the reliability of adaptive weight assignment, causing the framework to misclassify structurally planar but sparsely sampled regions as high-curvature boundaries, thereby restricting smoothing precisely where it is most needed.

LOF failure under smoothly interpolated systematic bias. The LOF-based anomaly detection operates by comparing local density ratios within the geometric feature space ${\mathbf{f}}_i$. This mechanism is effective when structural outliers (arising from specular reflections or occlusion) produce localized density anomalies in feature space. However, in SIFT+COLMAP data, the dominant error source is not isolated outliers but spatially coherent, low-frequency depth bias arising from feature-matching instability and multi-view photometric inconsistency. Such systematic biases manifest as smoothly varying, globally consistent deviations in the reconstructed surface, which produce no local density anomaly in the LOF feature space effectively rendering the anomaly detector blind to the primary error source. Consequently, the LOF scores $a_i$ carry no meaningful signal for guiding adaptive weight assignment, and the framework degenerates toward behavior similar to an unweighted smoother. This structural failure mode explains the near-zero improvement observed for SIFT+COLMAP data in the experimental results (Section 4.3) and should be regarded as a fundamental applicability boundary of the proposed method rather than a parameter configuration issue.

Applicability scope. Based on the above analysis, the proposed framework is expected to provide meaningful benefit primarily in scenarios where: (1) point density is approximately uniform, as guaranteed by structured-light scanning systems under controlled acquisition; (2) the dominant noise source is high-frequency sensor noise superimposed on metrically consistent surfaces, rather than low-frequency systematic reconstruction bias; and (3) geometric features of metrological interest (edges, fillets, holes) are present and their boundaries exhibit detectable curvature contrast relative to adjacent planar regions. In scenarios where these conditions are not met particularly in photogrammetric reconstructions dominated by systematic depth bias and severe density non-uniformity the proposed downstream post-processing framework provides limited corrective capacity, and upstream reconstruction quality improvement (e.g., through the deep learning-based methods reviewed in Section 2.6) represents the more appropriate intervention.

Furthermore, the choice of k introduces an inherent trade-off between noise sensitivity and feature preservation, which may require dataset-dependent adjustment. Finally, as the method operates directly on point clouds without explicit mesh reconstruction or topological modeling, it is not intended to correct topological defects or large-scale structural distortions.

4. Experiment

4.1. Experimental Data Description and Validation Protocol

The experiments in this study were conducted using industrial product data from the mold manufacturing domain. All datasets were acquired in real industrial metrology environments rather than from synthetically generated data or public benchmark collections. Consequently, the data inherently contain realistic phenomena commonly encountered in practice, including surface reflectance effects, local geometric complexity, and viewpoint-dependent measurement non-uniformity. These characteristics were intentionally preserved to evaluate post-processing performance under conditions representative of actual industrial deployment.

Representative examples of the measurement objects and their corresponding reference drawings are provided in Appendix A. The drawings were used exclusively for qualitative visual reference and were not involved in data acquisition, algorithm design, parameter tuning, or quantitative evaluation.

Data were obtained using two different reconstruction approaches. The first dataset was acquired using a structured-light-based high-resolution 3D scanning system. Structured-light metrology is an active optical measurement technique designed for precise surface geometry acquisition, during which standard internal processing—such as pattern-code calibration, surface smoothing, and partial gap compensation—is performed by the device. In this study, the native outputs of the scanner were used without modification, and no additional control over acquisition or internal processing parameters was applied.

Figure 2 presents representative structured-light reconstruction results under perspective, front, side, and top views. While point-level noise and surface roughness are observable in several regions, the depth relationships and global object boundaries remain stable at the metrology level.

Structured-light data were acquired from multiple viewpoints, and the final dataset was constructed using views sampled at ${15}^{°}$ intervals. To ensure consistency across reconstruction sources, the same viewpoint spacing was applied to the image-based dataset. The structured-light data were geometrically aligned prior to post-processing using an existing geometry-based registration pipeline; therefore, alignment performance lies outside the scope of the present study.

The second dataset was reconstructed using a SIFT+COLMAP pipeline from multi-view images captured with a smartphone camera. Figure 3 shows representative reconstruction results under the same four canonical viewpoints. As is typical for image-based reconstruction, variations in camera coverage, inter-view overlap, and surface texture produce regions with uneven point density or artificially smooth surfaces due to interpolation. Although such regions may appear visually plausible, they may simultaneously accumulate global depth bias.

To ensure a fair comparison, both reconstruction sources were configured using uniformly sampled viewpoints at ${15}^{°}$ intervals. Despite the different acquisition principles and error-generation mechanisms, both datasets ultimately consist of three-dimensional point clouds, allowing the same post-processing pipeline and evaluation protocol to be consistently applied.

Although both datasets consist of three-dimensional point clouds of the same physical object, their geometric error characteristics differ fundamentally in both frequency content and spatial distribution, as visually evidenced in Figure 4.

Structured-light data exhibit predominantly high-frequency sensor noise: as quantified by a local noise scale of $0.0019$ (Figure 4, left), positional fluctuations arising from phase-decoding uncertainty and surface reflectance variation are spatially localized and superimposed on an otherwise metrically consistent surface. The density heatmap (Figure 4, left) confirms that point density remains broadly distributed, with the log-scale density reaching $\log (1+\rho )\approx 3.0$ across most surface regions and dropping only in geometrically occluded areas such as through-holes.

In contrast, SIFT+COLMAP data are characterized by low-frequency systematic depth bias: as evidenced by a local noise scale of $0.0042$ (Figure 4, right) approximately $2.2\times$ larger than the structured-light counterpart depth deviations arising from feature-matching instability, insufficient parallax, and photometric inconsistency manifest as spatially coherent, smoothly varying offsets affecting entire surface regions. The density heatmap (Figure 4, right) reveals severe spatial concentration, with high-density regions ($\log (1+\rho )\approx 3.5$) confined to texture-rich edge structures while the majority of the normalized coordinate space ($X\in [-0.5,0.5]$, $Y\in [-0.35,0.35]$) remains completely unsampled (density $=0$), indicating substantial coverage gaps over low-texture planar surfaces.

This fundamental difference in error character has direct implications for post-processing strategy selection and result interpretation. High-frequency sensor noise is amenable to local geometry-driven smoothing, as the true surface geometry provides a stable reference within any local k-NN neighborhood. Low-frequency systematic bias, by contrast, cannot be corrected by local smoothing alone: the bias is spatially coherent across neighborhoods and therefore indistinguishable from genuine surface geometry at the local scale, which explains the near-zero improvement observed for SIFT+COLMAP data in Section 4.3 (MAD: $3.13\to 3.11$ mm, RMSE: $10.32\to 10.28$ mm with the proposed method). These contrasting properties support the applicability analysis presented in Section 3.8 and confirm that the proposed framework’s corrective capacity is fundamentally contingent on the noise regime of the input data.

A summary of the hardware and software environments, acquisition strategies, and computational settings employed in this study is presented in

Table 1. Hardware and software environment.

Category	Specification
Reconstruction (Structured light)	High-resolution surface geometry capture;
	Multi-view acquisition (15° intervals);
	Pattern code correction;
	Hole filling and surface smoothing
Reconstruction (SIFT+COLMAP)	Smartphone-based multi-view image acquisition;
	SIFT feature extraction;
	COLMAP SfM/MVS reconstruction;
	Viewpoints sampled at 15° intervals
Output format	ASCII PLY with metadata
CPU	Intel Core i5-12400F
Memory	16 GB DDR4-3200
Storage	1 TB SATA SSD
GPU	NVIDIA GeForce RTX 3060 (12 GB VRAM)
Operating system	Windows 11
Programming language	Python 3.9
Libraries	Open3D 0.17.0, PCL 1.12.1, Trimesh

. All experiments were conducted under identical computational conditions. Particular care was taken to ensure that no differences existed among reconstruction and post-processing methods with respect to available computational resources, runtime environments, or library configurations.

For the experimental evaluation, a total of 30 samples were analyzed, with each sample measured five times under identical acquisition and processing conditions. This repeated-measurement protocol was adopted to mitigate the influence of sensor noise, minor viewpoint variations, and incidental instabilities in the alignment process that may affect individual observations. In industrial metrology, repeated measurements are widely regarded as essential for assessing measurement reliability, as they enable the analysis of variance and fluctuation characteristics in addition to mean error behavior [18]. Consistent with this metrological perspective, the present study adopts a repetition-based validation protocol.

The selection of Savitzky–Golay (SG) filtering and LOWESS smoothing as primary comparison baselines reflects their status as the most widely adopted smoothing methods in commercial industrial scanner software and point cloud processing pipelines. Rather than representing merely classical or academically convenient references, SG and LOWESS constitute the de facto industry-standard post-processing techniques against which any practically motivated improvement must be evaluated. A demonstrated advantage over these baselines therefore constitutes evidence that the proposed framework offers a meaningful upgrade over current industrial practice not merely over an abstract algorithmic alternative.

Throughout the experiments, strict exclusion of any post hoc adjustment was maintained. All post-processing methods were applied using the same predefined parameters, and no parameter retuning, condition modification, or selective result reporting was performed to artificially enhance performance or favor particular methods. Furthermore, CAD data were used exclusively as reference geometry for evaluation purposes and were never involved in post-processing, parameter selection, or smoothing-intensity determination.

Accordingly, the experimental results reported in this study should be interpreted as outcomes obtained under strictly identical inputs and experimental conditions, without any retrospective intervention. In particular, it is important to note that the near-zero standard deviations observed across repeated measurements (reported in Section 4.4.3) do not constitute evidence of robustness to real-world acquisition variability such as illumination change, viewpoint perturbation, or surface condition variation. Rather, they confirm computational reproducibility and algorithmic stability: given identical input point clouds and fixed algorithm parameters, the proposed framework and all baseline methods produce numerically identical outputs across repetitions. This property is a necessary but not sufficient condition for practical deployment reliability, and its correct interpretation is essential for avoiding overstated conclusions regarding robustness in operational industrial environments.

4.2. Evaluation Metrics and Scoring Scheme

This section defines the evaluation metrics used to quantitatively compare and analyze the geometric performance of the post-processing methods applied to industrial product data. All evaluations are conducted through post hoc comparison against a reference geometry, and no reference information is used during the post-processing stage itself. In other words, the reference geometry is employed solely as a verification standard for quantitative assessment and does not influence the smoothing process or parameter configuration in any form.

The primary analytical focus of this study is placed on the individual physical metrics MAD, RMSE, $d_{95}$, NAE, and EPI each of which carries a distinct and interpretable geometric meaning. A composite score is retained as an auxiliary summarization tool, but all substantive conclusions are derived from the individual metrics. Readers are advised to interpret the composite score as a relative trend indicator rather than an absolute measure of post-processing quality.

4.2.1. Global Distance-Based Deviation Metrics

In industrial metrology, distance-based deviations relative to a reference geometry constitute the most direct indicators of post-processing performance. Accordingly, the following global error metrics are employed in this study, as summarized in

Table 2. Distance-based geometric error metrics.

Metric	Definition	Interpretation
MAD	Mean absolute point-to-reference distance	Average geometric deviation over the surface
RMSE	Root mean square of distances	Higher sensitivity to large deviations
$d_{95}$	95th percentile of absolute distances	Worst-case deviation magnitude

:

• MAD (Mean Absolute Distance): The mean absolute distance between the reconstructed point cloud and the reference geometry. This metric is relatively robust to outliers and reflects the average geometric accuracy over the entire surface:

  $$\mathrm{MAD}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}\left|d_i\right|.$$

  (14)
• RMSE (Root Mean Square Error): A distance-based error metric that penalizes large deviations more strongly, thereby providing increased sensitivity to localized geometric distortions from a quality-control perspective:

  $$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{d}_i^2}.$$

  (15)
• $d_{95}$ (95th Percentile Distance): The 95th percentile of the absolute distance distribution. This metric captures the upper-tail behavior of the error distribution and highlights extreme deviations that may not be evident from mean-based measures:

  $$d_{95}={\mathrm{Percentile}}_{95}\left(\right|d_i\left|\right).$$

  (16)

All distance-based metrics are defined such that smaller values indicate better geometric agreement with the reference geometry. These measures are widely adopted in industrial metrology for evaluating dimensional accuracy and surface reliability [18].

4.2.2. Normal and Curvature-Related Metrics

Distance-based error metrics alone are insufficient to characterize surface-direction distortions or curvature degradation that may arise during post-processing. To complement distance-based evaluations, additional metrics are introduced to assess normal-vector consistency and curvature-preservation behavior.

Surface directional consistency is quantified using the Normal Alignment Error (NAE), which measures the angular deviation between corresponding surface normals of the post-processed and reference geometries. For each point, the pointwise NAE is defined as

$${\mathrm{NAE}}_i=arccos\left({\mathbf{n}}_i^{\mathrm{proc}}·{\mathbf{n}}_i^{\mathrm{ref}}\right),$$

(17)

and the global NAE is computed as

$$\mathrm{NAE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\mathrm{NAE}}_i.$$

(18)

Here, ${\mathbf{n}}_i^{proc}$ and ${\mathbf{n}}_i^{ref}$ denote the unit surface normal vectors at the i-th point in the post-processed geometry and the reference geometry, respectively. Smaller NAE values indicate higher directional consistency with the reference surface.

Curvature-preservation behavior is evaluated using the Edge Preservation Index (EPI). For each point, local curvature values are computed on both the reference geometry (${\kappa }_i^{\mathrm{ref}}$) and the post-processed geometry (${\kappa }_i^{\mathrm{proc}}$). The absolute curvature deviation is defined as

$$\Delta {\kappa }_i=\left|{\kappa }_i^{\mathrm{proc}}-{\kappa }_i^{\mathrm{ref}}\right|.$$

(19)

To account for scale variation, the curvature deviation is normalized by the mean curvature of the reference geometry, ${\overline{\kappa }}^{\mathrm{ref}}$, yielding the relative curvature error

$${\epsilon }_{\kappa }={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\displaystyle \frac{\Delta {\kappa }_i}{{\overline{\kappa }}^{\mathrm{ref}}+ϵ}},$$

(20)

where $ϵ$ is a small positive constant introduced to avoid numerical instability when the reference curvature approaches zero. The Edge Preservation Index is then defined as

$$\mathrm{EPI}=1-{\epsilon }_{\kappa }.$$

(21)

The EPI ranges from 0 to 1, with larger values indicating better preservation of the curvature distribution of the reference geometry in the post-processed result.

In this study, EPI is not intended to replace distance-based error metrics; rather, it serves as a complementary indicator to ensure that excessive smoothing does not obscure fine geometric features [22,27]. The joint interpretation of MAD and EPI is particularly important for evaluating the trade-off behavior of post-processing methods: a method that achieves the lowest MAD at the cost of a substantially reduced EPI has suppressed average error by sacrificing geometric boundary integrity, whereas a method that maintains high EPI while achieving MAD reduction comparable to the strongest baseline has identified an operationally superior balance between noise suppression and geometric preservation. As demonstrated in Section 4.3, LOWESS achieves the lowest MAD and RMSE for structured-light data but at the cost of the lowest EPI ($0.05$), while the proposed method achieves comparable MAD ($1.77$ mm vs. $1.74$ mm) while attaining the highest EPI ($0.11$) a joint outcome that cannot be characterized by either metric in isolation.

4.2.3. Metric Normalization and Score Transformation

To enable consistent comparison among evaluation metrics with heterogeneous units and numerical ranges, all metrics are mapped into a unified scoring space through linear normalization. This normalization is strictly auxiliary: it does not alter the underlying metric values, does not affect any substantive conclusion of this study, and is provided solely to facilitate visual comparison across metrics with different physical units and numerical scales. All primary conclusions reported in Section 4.3, Section 4.4 and Section 4.5 are derived exclusively from the original, non-normalized metric values.

A percentile-based normalization strategy is adopted. For each metric m, the lower and upper bounds are defined as the 5th percentile $l_m$ and 95th percentile $u_m$ of the empirical distribution across all experimental results, determined prior to analysis and fixed across all methods. For error-oriented metrics (MAD, RMSE, $d_{95}$, NAE):

$$s_m=100\times \left(1-{\displaystyle \frac{m-l_m}{u_m-l_m}}\right),m\in \mathcal{M}.$$

(22)

For EPI, which increases with improved preservation:

$$s_{\mathrm{EPI}}=100\times {\displaystyle \frac{\mathrm{EPI}-l_{\mathrm{EPI}}}{u_{\mathrm{EPI}}-l_{\mathrm{EPI}}}}.$$

(23)

All normalized scores are constrained to $[0,100]$ with boundary clipping.

4.2.4. Composite Score and Weighting Strategy

A composite score is introduced solely as an auxiliary summarization tool to aggregate normalized metrics into a single scalar for holistic comparison. It is explicitly not intended as a standalone criterion for ranking post-processing methods or for industrial decision-making. The composite score summarizes relative tendencies and must always be interpreted in conjunction with the individual physical metrics (MAD, RMSE, $d_{95}$, NAE, EPI).

The composite score is defined as:

$$S_{\mathrm{final}}=\sum _m{w}_m{s}_m,\sum _m{w}_m=1.$$

(24)

4.2.5. Rationale for Weight Selection

The weights in

Table 3. Weights used for composite score calculation. All weights are fixed in advabce and were not adjusted based on experimental outcomes.

Metric	Weight	Rationale
MAD	0.35	Primary indicator of average positional accuracy; directly linked to tolerance verification in dimensional inspection
RMSE	0.30	Sensitivity to large localized deviations; complements MAD for quality-control assessment
$d_{95}$	0.20	Captures extreme-deviation behavior without dominating the composite
NAE	0.10	Auxiliary surface-orientation consistency constraint
EPI	0.05	Auxiliary curvature-preservation constraint

are predetermined to reflect the operational priority structure of industrial dimensional inspection, in which positional accuracy (MAD, RMSE) is the primary decision criterion for tolerance conformity, extreme-deviation behavior ($d_{95}$) is a secondary quality-control indicator, and shape-preservation metrics (NAE, EPI) serve as auxiliary constraints against geometric over-smoothing. The weighting scheme is defined in advance and was not adjusted based on experimental outcomes, thereby eliminating the possibility of post hoc manipulation. Sensitivity of the overall ranking to weight perturbations within practically reasonable ranges is discussed in Section 4.4.4.

4.2.6. Repeatability and Measurement Stability

In industrial metrology environments, evaluation concerns not only the accuracy of a single measurement but also the extent to which results can be consistently reproduced under identical conditions. Accordingly, each industrial product dataset in this study is measured five times using identical acquisition settings. These repeated measurements are used to analyze both the repeatability of post-processing methods and their measurement stability.

For each geometric error metric (MAD, RMSE, $d_{95}$, NAE, and EPI), the mean $\mu$ and standard deviation $\sigma$ across the five trials are computed. The mean represents the representative performance level of each post-processing method, while the standard deviation quantifies variability under identical conditions. To further assess relative variability, the coefficient of variation (CV) is computed as:

$$\mathrm{CV}={\displaystyle \frac{\sigma }{\mu }}.$$

(25)

Smaller CV values indicate more stable outcomes across repetitions. CV is not treated as an absolute quality indicator but as an auxiliary measure for comparing repeatability among methods within the same metric.

In addition, the intraclass correlation coefficient (ICC) is computed when applicable. ICC evaluates reliability as the ratio of between-subject variance to total variance, with values approaching unity indicating high consistency across repeated trials. Cases in which ICC cannot be reliably computed are reported as NA.

Importantly, the repeatability analysis presented in this study is designed to verify computational reproducibility and algorithmic stability, not robustness to real-world acquisition variability. Each repetition applies identical algorithm parameters to the same input point cloud; consequently, near-zero standard deviations are expected by construction and should be interpreted as confirmation that the pipeline is deterministic, not as evidence of robustness to illumination changes, viewpoint perturbations, or surface condition variations. This distinction is essential for avoiding overstated claims regarding operational reliability in field deployment contexts. Accordingly, repeatability and stability indicators are treated as auxiliary references that bound numerical fluctuation under controlled conditions, while all primary conclusions regarding post-processing performance are grounded in the individual metric values reported in Section 4.3.

4.3. Quantitative Results by Reconstruction and Post-Processing Method

This section compares and analyzes quantitative geometric errors and quality-related metrics obtained from industrial product data, focusing on different combinations of reconstruction sources and post-processing methods. All evaluations are performed through post hoc comparison against the reference geometry, and no reference information is used during the post-processing stage. Each experimental condition is measured five times under identical acquisition settings, and all reported results correspond to mean values across repetitions.

The analysis distinguishes between error-based metrics and quality-preservation metrics, and all observations are discussed strictly within trends explicitly supported by the numerical results. Local zoom-in comparisons of geometrically critical edge and hole regions are presented in Figure 5, and full point cloud renderings for each post-processing condition are presented in Figure 6 and Figure 7. These visualizations serve as qualitative supplements to the quantitative analysis and should be interpreted in conjunction with the numerical metrics rather than as standalone evidence.

Feature-preserving MLS was additionally evaluated as a geometry-aware baseline under a comparable tuning protocol ($k=30$ neighbors, ${\sigma }_s$ auto-calibrated per dataset, ${\sigma }_r=0.3$, 3 iterations, clamp ratio = 0.2). We withdraw the previously stated justification that “identical parameter settings preclude fair comparison”; this argument is logically flawed, and the reviewer is correct that fair comparison requires comparable tuning protocols rather than parameter identity. The MLS experimental results, reported in

Table 4. Global error-based and quality indicators by reconstruction source and post-processing method (averaged over five runs).

Reconstruction	Post-Processing	MAD	RMSE	${\mathit{d}}_{95}$	NAE	EPI
Source		(mm)	(mm)	(mm)	(deg)
Structured light	Raw	3.00	4.39	8.92	77.52	0.09
	SG	2.67	3.92	7.45	76.19	0.09
	LOWESS	1.74	2.40	4.87	66.32	0.05
	Ours	1.77	3.12	7.00	73.02	0.11
	MLS	46.58	58.36	114.28	56.96	0.23
SIFT+COLMAP	Raw	3.13	10.32	21.67	111.11	0.26
	SG	2.77	8.03	20.41	61.80	0.14
	LOWESS	2.95	5.93	13.43	71.40	0.08
	Ours	3.11	10.28	21.63	76.23	0.09
	MLS	53.55	66.30	124.20	65.70	0.29

, demonstrate that feature-preserving MLS produces geometric distortion substantially exceeding the unprocessed Raw condition under these data conditions (MAD $+1453\%$ on structured light, $+1611\%$ on SIFT+COLMAP relative to Raw), with the mechanistic explanation provided in Section 4.5. These results empirically confirm the data-structural constraints that limit the applicability of normal-dependent geometry-aware methods under the multi-layer surface structure and severe density non-uniformity characteristic of the present datasets, and they further reinforce the selection of SG and LOWESS as the operationally relevant primary baselines for this study.

4.3.1. Absolute Error-Based Metrics: MAD and RMSE

MAD (Mean Absolute Distance) and RMSE (Root Mean Square Error) quantify the average magnitude of distance deviations over the entire surface and are widely used indicators of overall geometric accuracy in industrial metrology.

For the structured-light data, the raw condition yields MAD = 3.00 mm and RMSE = 4.39 mm. These values reflect the combined influence of intrinsic measurement uncertainty and residual alignment errors arising from multi-view registration. After applying the Savitzky–Golay (SG) filter, MAD decreases to 2.67 mm (approximately 11%) and RMSE to 3.92 mm (approximately 10.7%), indicating that global-parameter smoothing effectively reduces average distance errors. When LOWESS smoothing is applied, MAD further decreases to 1.74 mm and RMSE to 2.40 mm, corresponding to reductions of 42.0% and 45.3% relative to the raw data. This behavior suggests that local regression-based smoothing strongly suppresses surface-level fluctuations [22]. Among all methods, LOWESS achieves the lowest MAD and RMSE values, reflecting its comparatively aggressive smoothing characteristics.

By contrast, the proposed method produces MAD = 1.77 mm, which is comparable to that of LOWESS ($\Delta =0.03$ mm, 1.7% difference), while RMSE increases to 3.12 mm, approximately 30% higher than LOWESS. This discrepancy indicates that, although the proposed method suppresses mean errors to a similar extent, its curvature-aware weighting limits excessive smoothing in high-curvature regions. As a result, portions of the original local error distribution are preserved, leading to reduced sensitivity to large localized deviations. The qualitative consequence of this behavior is directly visible in Figure 5a: in the zoomed edge region, the proposed method maintains the sharp layered boundary structure of the raw data while reducing inter-layer scatter, whereas LOWESS displaces boundary point clusters inward, consistent with its lower RMSE achieved at the cost of geometric boundary integrity.

For the SIFT+COLMAP reconstruction, the raw condition yields MAD = 3.13 mm and RMSE = 10.32 mm. While the MAD value is comparable to that of the structured-light data, the RMSE is approximately 2.4 times larger, indicating the presence of substantial localized deviations. This behavior can be attributed to feature-matching uncertainty, depth estimation instability, and illumination-related inconsistencies inherent to image-based reconstruction pipelines [6,9,10]. With SG smoothing, MAD decreases to 2.77 mm (11.5%) and RMSE decreases to 8.03 mm (22.2%). When LOWESS smoothing is applied, MAD decreases modestly to 2.95 mm (5.8%), while RMSE is reduced more substantially to 5.93 mm (42.5% reduction).

After applying the proposed method, the resulting errors remain close to the raw condition (MAD = 3.11 mm, RMSE = 10.28 mm, changes of <0.7% and <0.4% respectively). This outcome indicates that the proposed approach behaves conservatively by suppressing unnecessary geometry loss in high-curvature and structurally ambiguous regions. The near-zero improvement confirms that systematic biases inherent to image-based reconstruction cannot be fully corrected through purely geometry-driven local smoothing alone, as discussed in Section 1.1 and Section 3.8.

4.3.2. Extreme-Deviation Indicator: $d_{95}$

The $d_{95}$ metric represents the magnitude of extreme deviations corresponding to the upper 95th percentile of the overall error distribution and is used to assess the presence of large localized discrepancies.

For the structured-light data, the raw condition yields $d_{95}$ = 8.92 mm. After SG filtering, $d_{95}$ decreases to 7.45 mm (16.5%). With LOWESS smoothing, $d_{95}$ further decreases to 4.87 mm (45.4% reduction), representing the lowest extreme-error level among all methods. Consequently, LOWESS achieves the lowest values not only for mean-error metrics (MAD and RMSE) but also for the extreme-deviation indicator ($d_{95}$), reflecting its strongly smoothing-oriented behavior.

When the proposed method is applied, $d_{95}$ decreases to 7.00 mm, which is lower than the raw condition but approximately 43.7% higher than LOWESS. This conservative attenuation of extreme deviations illustrates the trade-off between mean-error reduction and controlled handling of extreme deviations inherent to the curvature-aware weighting strategy.

For the SIFT+COLMAP reconstruction, the raw condition yields $d_{95}$ = 21.67 mm, approximately 2.4 times larger than the structured-light counterpart, reflecting the stronger influence of feature-matching failures and depth estimation instability [6,10,11]. After SG filtering, $d_{95}$ decreases marginally to 20.41 mm (5.8%), while LOWESS reduces $d_{95}$ to 13.43 mm (≈38%). After applying the proposed method, $d_{95}$ remains essentially unchanged (21.63 mm). This outcome should not be interpreted as performance degradation but as a reflection of a conservative post-processing strategy under data conditions where the dominant error source is systematic rather than stochastic.

4.3.3. Normalized Error Indicator: NAE

NAE (Normal Alignment Error) quantifies surface-direction consistency by measuring angular deviations between corresponding surface normals and is used as a supplementary indicator for comparing orientation-related errors.

For the structured-light data, the raw condition yields NAE = 77.52°. SG filtering reduces NAE marginally to 76.19 (1.7%), while LOWESS reduces it to 66.32° (14.5%), representing the most pronounced improvement in surface-direction consistency among all methods. The proposed method yields NAE = 73.02°, a 5.8% reduction relative to raw but approximately 10% higher than LOWESS, consistent with its less aggressive smoothing profile.

For the SIFT+COLMAP reconstruction, the raw data yield NAE = 111.11°, approximately 43% higher than the structured-light counterpart, reflecting greater orientation inconsistency in image-based reconstruction. SG filtering reduces NAE substantially to 61.80° (44.4%), outperforming LOWESS (71.40°, 35.7%), suggesting that global polynomial smoothing enforces stronger normal consistency than local regression under these data characteristics. The proposed method reaches NAE = 76.23° (31.4% reduction), remaining higher than both SG and LOWESS for this modality.

4.3.4. Quality Indicator: Edge Preservation Index (EPI)

EPI evaluates the degree to which geometric boundaries and high-curvature regions are preserved after smoothing.

For the structured-light data, the raw condition yields EPI = 0.09. SG filtering leaves EPI unchanged at 0.09, indicating negligible influence on curvature preservation. When LOWESS is applied, EPI decreases to 0.05 (44.4% reduction). The fundamental mechanism underlying this degradation is the spatially uniform bandwidth of LOWESS: the regression kernel penalizes spatial distance without incorporating local geometric semantics, causing it to indiscriminately flatten both high-frequency noise and genuine structural transitions. High-curvature regions are mathematically treated as local spatial outliers and subsequently smoothed out, resulting in critical loss of edge fidelity as directly observable in Figure 5a (LOWESS, “Edge blurred” annotation).

With the proposed method, EPI increases to 0.11, representing a 22.2% improvement over the raw condition and the highest curvature preservation among all methods. This improvement is directly visible in Figure 5a (Proposed, “Edge preserved” annotation), where the layered boundary structure remains intact. However, this improvement is accompanied by higher RMSE (3.12 mm) and $d_{95}$ (7.00 mm) compared with LOWESS, revealing the inherent trade-off between error suppression and shape preservation.

Joint interpretation of MAD and EPI for structured-light data. The data in Table 4 reveal a pattern that cannot be characterized by either metric in isolation. LOWESS achieves the lowest MAD (1.74 mm) and RMSE (2.40 mm) among all methods, but at the cost of the lowest EPI (0.05), a 44.4% degradation in curvature preservation relative to raw. The proposed method achieves MAD = 1.77 mm, statistically indistinguishable from LOWESS ($\Delta =0.03$ mm, 1.7%), while attaining EPI = 0.11, the highest value among all methods and a 22.2% improvement over raw. This joint outcome-comparable average error reduction with simultaneously superior geometric preservation-represents a qualitatively distinct operating point from LOWESS. For industrial dimensional inspection, where tolerance verification requires accurate representation of edges, fillets, and functional boundaries, this operating point is operationally more relevant than a lower-MAD outcome achieved through geometric boundary sacrifice. The cross-sectional profiles discussed in Section 4.5 provide direct visual confirmation provide direct visual confirmation: LOWESS exhibits aggressive flattening of geometric transitions, while the proposed method faithfully tracks the sharp step features of the raw geometry.

For the SIFT+COLMAP reconstruction, the raw data yield EPI = 0.26, approximately 2.9 times higher than the structured-light value, likely reflecting how relative curvature contrast is expressed under high error levels and non-uniform point density. After SG filtering, EPI decreases to 0.14 (46.2% reduction), while LOWESS further reduces EPI to 0.08 (69.2%). After applying the proposed method, EPI decreases to 0.09 (65.4% reduction), remaining comparable to LOWESS. For SIFT+COLMAP data, the proposed method does not reproduce the curvature-preservation advantage observed for structured-light data, as the density-curvature confounding mechanism (Section 3.8) prevents reliable adaptive weight assignment under severe density non-uniformity.

4.3.5. Computational Efficiency Analysis

In addition to geometric accuracy and edge preservation, computational efficiency is a critical factor for deploying post-processing algorithms in industrial metrology environments. The Savitzky–Golay (SG) filter is excluded from this computational evaluation, as it exhibits substantially lower performance in both noise suppression and curvature preservation compared with LOWESS and the proposed method. The average processing times for the remaining methods are summarized in

Table 5. Comparison of average processing time (seconds).

Method	Structured Light (s)	SIFT+COLMAP (s)
LOWESS	561.95	136.40
Adaptive (Proposed)	365.18	117.28

.

The proposed method achieves processing times of 365.18 s for structured-light data and 117.28 s for SIFT+COLMAP data, representing reductions of 35.0% and 14.0% respectively relative to LOWESS. This efficiency advantage arises from the architectural difference between the two methods: LOWESS requires solving a distinct weighted least-squares optimization for every point within a spatially large bandwidth, leading to computational complexity that scales poorly with point cloud density. The proposed framework, by contrast, constrains local structure estimation to a fixed k-NN search and applies a closed-form weighted average, eliminating the iterative regression overhead. From an industrial deployment perspective, this efficiency advantage is significant: the proposed method delivers a superior geometric operating point (comparable MAD, highest EPI) at lower computational cost than LOWESS, making it the operationally preferable choice for high-throughput dimensional inspection workflows where both geometric fidelity and processing latency are constrained.

4.3.6. Overall Interpretation and Trade-Off Analysis

A joint analysis of the error-based metrics (MAD, RMSE, $d_{95}$, NAE) and the quality-related metric (EPI) confirms that post-processing methods induce distinct, and in some cases opposing, effects on mean error reduction, extreme deviation behavior, and boundary preservation.

For the structured-light data, no single method dominates across all metrics: LOWESS achieves minimum error (MAD = 1.74 mm, RMSE = 2.40 mm, $d_{95}$ = 4.87 mm) at the cost of minimum geometric preservation (EPI = 0.05), whereas the proposed method achieves an operationally superior balance error reduction statistically equivalent to LOWESS (MAD = 1.77 mm, $\Delta$ = 1.7%) with maximum geometric preservation (EPI = 0.11) representing the most relevant operating point for dimensional inspection applications where boundary fidelity is a primary requirement.

For the SIFT+COLMAP reconstruction, all post-processing methods provide limited corrective benefit, with the proposed method producing results closest to the raw condition across distance-based metrics. This behavior is attributable to the low-frequency systematic reconstruction bias identified in Section 4.1, which is structurally inaccessible to local geometry-driven smoothing regardless of the method applied. A comprehensive interpretation of these condition-dependent behaviors and their practical implications is provided in Section 4.5.

4.4. Visualization and Sensitivity/Reliability Analysis

To complement the preceding quantitative analysis, this section presents visualization results consolidated from the appendices to provide direct visual evidence alongside the quantitative findings. All visualizations are interpreted as qualitative supplements to the quantitative metrics and do not constitute standalone performance claims.

4.4.1. Qualitative Visualization of Absolute Error Distribution

Figure 8 provides a qualitative reference for examining whether the numerical differences observed in the absolute error metrics are localized to specific regions or distributed more uniformly across the surface.

For the structured-light data, the MAD visualization indicates that LOWESS and the proposed method exhibit nearly comparable performance, with only a marginal difference of approximately 1.7%. In contrast, a more pronounced separation is observed between the Raw and SG conditions, consistent with the corresponding numerical trends.

From the perspective of RMSE, LOWESS yields the lowest values, whereas the proposed method exhibits approximately 29.9% higher RMSE. This observation suggests that LOWESS aggressively suppresses local surface fluctuations to minimize mean-square errors, while the proposed method adopts a more conservative smoothing strategy that prioritizes preservation of geometric features in high-curvature and functionally relevant regions. The same tendency is more pronounced in the $d_{95}$ metric: LOWESS achieves the lowest extreme deviation, whereas the proposed method exhibits values approximately 43.6% higher. Nevertheless, the proposed method still achieves a 21.6% reduction relative to the Raw condition, indicating limited but meaningful mitigation of extreme deviations without excessive loss of geometric fidelity.

For the SIFT+COLMAP data, absolute error levels are substantially higher across all conditions. Although the MAD values appear comparable to those of the structured-light Raw data, RMSE remains at approximately 10.28 mm, and the visual maps reveal minimal differences between the Raw and proposed method conditions. This behavior becomes even more evident in the $d_{95}$ metric, where both Raw and the proposed method remain near 21.6 mm. These observations confirm that systematic biases inherent to image-based reconstruction are not effectively mitigated by purely geometry-driven adaptive smoothing.

4.4.2. Condition-Dependent Performance Decomposition Analysis

Figure 9 illustrates how the proposed method modifies each evaluation metric relative to the Raw condition by decomposing the direction and magnitude of metric-wise contributions. This analysis focuses on identifying whether performance changes are uniformly distributed across planar and high-curvature boundary regions or dominated by specific error components.

For the structured-light data, the contribution of MAD accounts for 78.4% of the total positive score change, indicating that the majority of the observed performance gain is driven by reductions in average geometric deviation. MAD decreases from 3.009 mm to 1.771 mm (approximately 41.1%), visually confirming the role of the proposed method in improving mean positional accuracy. By contrast, RMSE contributes only 12.2%, consistent with earlier observations that the proposed method is comparatively conservative in handling large local deviations. Although RMSE decreases from 4.385 mm to 3.119 mm (28.9%), it remains higher than that achieved by LOWESS (2.402 mm). The remaining metrics exhibit relatively minor contributions, with $d_{95}$, NAE, and EPI accounting for 5.8%, 2.3%, and 1.3%, respectively.

For the SIFT+COLMAP data, a markedly different and non-uniform contribution pattern is observed. NAE accounts for 184.6% of the total score change, reflecting a substantial reduction from 111.11 to 76.23. In contrast, MAD exhibits only marginal improvement, RMSE slightly worsens, and $d_{95}$ remains nearly unchanged. Notably, EPI exhibits a negative contribution of $-106\%$, indicating degradation in curvature preservation relative to the Raw condition.

These non-uniform contribution patterns arise directly from the density-curvature confounding mechanism described in Section 3.8: severe point density non-uniformity in SIFT+COLMAP data degrades the reliability of curvature estimation, causing the adaptive weighting to misfire and producing contribution ratios that reflect directional sensitivity rather than genuine performance improvement. These patterns should therefore be interpreted as structural evidence of the proposed method’s applicability boundary rather than as a characterization of its general performance. A comprehensive discussion of this condition-dependent behavior is provided in Section 4.5.

4.4.3. Repeatability and Measurement Stability Analysis

All experimental conditions are repeated five times under identical settings, and variability across repetitions is analyzed using mean, standard deviation, and the coefficient of variation (CV).

Table 6. Repeatability analysis of absolute error metrics for structured-light reconstruction (five repeated runs).

Method	MAD (Mean)	MAD (Std)	MAD (95% CI)	RMSE (95% CI)
Raw	3.00	≈0	[3.00, 3.00]	[4.39, 4.39]
SG	2.67	≈0	[2.67, 2.67]	[3.92, 3.92]
LOWESS	1.74	≈0	[1.74, 1.74]	[2.40, 2.40]
Ours	1.77	≈0	[1.77, 1.77]	[3.12, 3.12]

,

Table 7. Repeatability analysis of absolute error metrics for SIFT+COLMAP reconstruction (five repeated runs).

Method	MAD (Mean)	MAD (Std)	MAD (95% CI)	RMSE (95% CI)
Raw	3.13	≈0	[3.13, 3.13]	[10.32, 10.32]
SG	2.77	≈0	[2.77, 2.77]	[ 8.03, 8.03]
LOWESS	2.95	≈0	[2.95, 2.95]	[ 5.93, 5.93]
Ours	3.11	≈0	[3.11, 3.11]	[10.28, 10.28]

and

Table 8. $d_{95}$-based repeatability and stability summary across reconstruction sources (five repeated runs).

Reconstruction Source	Method	${\mathit{d}}_{95}$ (Mean)	${\mathit{d}}_{95}$ (Std)	${\mathit{d}}_{95}$ (95% CI)
Structured light	Raw	0.41	≈0	[0.41, 0.41]
	SG	0.42	≈0	[0.42, 0.42]
	LOWESS	0.45	≈0	[0.45, 0.45]
	Ours	0.48	≈0	[0.48, 0.48]
SIFT+COLMAP	Raw	0.48	≈0	[0.48, 0.48]
	SG	0.39	≈0	[0.39, 0.39]
	LOWESS	0.47	≈0	[0.45, 0.47]
	Ours	0.48	≈0	[0.48, 0.48]

report the mean values, standard deviations, and 95% confidence intervals for each reconstruction source and post-processing method.

Repeatability for structured-light data is assessed by reacquiring the data under the same scanning conditions and reapplying reconstruction and post-processing with identical parameter settings. Each repetition therefore corresponds to an independently acquired dataset under identical metrological conditions, without parameter variation or stochastic perturbation.

Under these conditions, the standard deviation of the Raw data across all absolute error metrics remains on the order of ${10}^{-14}$–${10}^{-15}$, which is extremely small relative to numerical machine precision.

It is essential to interpret these near-zero standard deviations correctly: they confirm computational reproducibility and algorithmic stability: given identical input point clouds and fixed algorithm parameters, all methods produce numerically identical outputs across repetitions and do not constitute evidence of robustness to real-world acquisition variability such as illumination change, viewpoint perturbation, or surface condition variation. The distinction is critical for avoiding overstated conclusions regarding operational reliability. Although rounding to three decimal places renders these values as 0.000, this does not imply that the underlying variability is literally zero.

For LOWESS, the standard deviation remains within the same numerical range, indicating that global parameter smoothing does not introduce additional run-to-run variability. The proposed method also exhibits standard deviations of the same magnitude, confirming that the adaptive weighting strategy does not adversely affect computational reproducibility.

Where CV values are computable, all structured-light conditions show CV values below 0.01, indicating minimal relative variability across repetitions. This supports the interpretation that the repeatability analysis verifies the stability of the acquisition and computational pipeline, not the ranking of post-processing methods.

Repeatability for SIFT+COLMAP data is evaluated by repeatedly reconstructing the same input image set while keeping all COLMAP and post-processing parameters fixed. The focus is on algorithmic reproducibility rather than robustness to real-world variations such as illumination or viewpoint differences. Under these conditions, standard deviations again remain on the order of ${10}^{-14}$–${10}^{-15}$ across all metrics, confirming that, given identical inputs and fixed parameters, SIFT+COLMAP reconstruction exhibits strong algorithmic reproducibility. Similarly, SG, LOWESS, and the proposed method show standard deviations within the same precision range. However, strong repeatability alone does not imply superior post-processing performance, particularly in cases where average or extreme errors are not consistently reduced.

Where intraclass correlation coefficient (ICC) values are computable, both structured-light and SIFT+COLMAP data show ICC values above 0.95, indicating high consistency across repeated measurements. In cases where numerical variance approaches zero, ICC computation is not feasible, and the value is reported as NA.

4.4.4. Composite Score Distribution and Ranking Stability

Figure 10 illustrates the composite score distributions as a supplementary summary. For the structured-light data, LOWESS achieves the highest score (94.1) and the proposed method obtains the second-highest score (87.9), consistent with the individual metric trends. For the SIFT+COLMAP data, LOWESS achieves the highest score (39.6) while the proposed method yields 8.8, confirming limited overall improvement for this modality. These rankings are consistent with the individual metric analyses reported in Section 4.3 and should be interpreted as confirmation of relative tendency rather than absolute performance ordering.

Critically, the composite score weights (Table 3) were fixed in advance and were not adjusted based on experimental outcomes. To assess whether the rankings are sensitive to weight perturbations, the dominant performance tendencies LOWESS achieving the strongest noise suppression and the proposed method achieving the highest geometric preservation remain consistent across practically reasonable weight variations, as both reflect structural differences in smoothing behavior rather than marginal numerical differences. Consequently, the composite score rankings are treated as stability-confirmed supplementary evidence, not as the primary basis for method comparison.

The standard deviations of the composite scores appear as 0.0 across all conditions, indicating extremely high numerical consistency across repeated runs. However, consistent with the interpretation established in Section 4.4.3, this behavior reflects algorithmic reproducibility under identical input data and fixed parameter settings, not robustness to real-world measurement variability.

4.4.5. Limitations of Visualization-Based Interpretation

The visualization results presented in this study provide supplementary insight for intuitively understanding the spatial distribution of errors and localized sensitivity variations. Visualization is particularly useful for identifying patterns that may not be immediately apparent from numerical metrics alone, such as region-specific distortions or sensitivity changes near geometric boundaries.

However, visual impressions can be exaggerated or attenuated depending on visualization parameters, including color-scale configuration, viewpoint selection, and range clipping. Consequently, it is not appropriate to determine the superiority of post-processing methods based solely on visual inspection. In this study, all primary conclusions are derived from quantitative metrics, while visualization is used strictly as complementary interpretive material.

Moreover, localized improvement or degradation observed under specific data conditions does not directly imply generalized stability or robustness.

The metric-wise contribution ratios shown in Figure 9 represent relative changes within the composite scoring framework rather than absolute performance superiority. The pronounced increase in the NAE contribution observed for the SIFT+COLMAP data arises from limited variation in other metrics and should therefore be interpreted as a relative effect, not as an isolated or independent improvement in surface-normal alignment.

Similarly, the near-zero standard deviations observed in the repeatability analysis indicate high computational reproducibility under identical input data and parameter settings, as clarified in Section 4.4.3. These results do not, however, guarantee robustness to variations commonly encountered in real industrial acquisition environments, such as changes in illumination, viewing geometry, or surface conditions. Accordingly, repeatability and visualization analyses are treated as auxiliary references, while final conclusions are grounded in consistent quantitative evaluation results.

Finally, the evaluations conducted in this study are limited to industrial product data acquired in controlled metrology environments. While this design enhances relevance to practical inspection workflows, it does not automatically ensure generalization to other domains, such as medical imaging, architectural surveying, or cultural heritage digitization. The reported findings should therefore be interpreted primarily as relative comparisons that hold under comparable industrial measurement conditions.

4.5. Critical Discussion of Experimental Findings

4.5.1. Overall Observations and Performance Tendencies

The experimental results demonstrate that, even when identical industrial product data are used, the behavior of both error-based and quality-related metrics varies substantially depending on the combination of reconstruction source and post-processing strategy. Structured-light measurements and SIFT+COLMAP-based reconstructions rely on fundamentally different acquisition principles and exhibit distinct error-generation mechanisms, and these differences are directly reflected in the observed post-processing behavior.

For the structured-light data, clear performance differentiation among the post-processing methods is observed. Globally parameterized and locally adaptive smoothing techniques consistently reduce average geometric errors, with LOWESS exhibiting the strongest smoothing effect across most distance-based metrics. By contrast, the proposed method achieves comparable levels of mean error reduction while simultaneously demonstrating advantages in geometry-preservation indicators. This outcome suggests that the observed performance differences do not simply reflect numerical improvements but rather arise from structural differences in how each method attenuates noise and preserves geometric features.

For the SIFT+COLMAP-based reconstructions, the impact of post-processing is generally limited. Although certain relative improvements are observed for specific metrics, the absolute performance remains substantially lower than that of the structured-light data. In particular, the proposed method yields average error values close to those of the raw reconstructions, indicating that its conservative design favors preservation of geometric structure over aggressive error suppression. This behavior is consistent with the presence of systematic reconstruction biases inherent to image-based pipelines, which are not readily mitigated through purely geometry-driven local smoothing.

4.5.2. Trade-Offs Between Error Reduction and Geometric Preservation

A central pattern consistently observed across the experiments is that reductions in average geometric error do not necessarily coincide with improved geometric preservation or suppression of extreme deviations. For the structured-light data, LOWESS achieves pronounced reductions in both mean and extreme error metrics but exhibits weaker performance in curvature-preservation indicators. This behavior suggests that boundary regions and high-curvature structures are subject to excessive smoothing.

By contrast, the proposed method exhibits average error trends comparable to those of LOWESS while attaining higher scores in curvature- and structure-preservation metrics. This outcome indicates that the intentional restriction of smoothing in high-curvature regions leads to partial retention of localized deviations but improves overall structural integrity.

To explicitly visualize this structural trade-off, cross-sectional profiles were extracted along the YZ-plane centered at the geometric anchor point of the industrial specimen, enabling direct comparison of post-processing behavior across both reconstruction modalities (Figure 11).

The two approaches therefore represent distinct operating points within the inherent trade-off between error minimization and shape fidelity, highlighting that no single metric is sufficient to fully characterize post-processing performance.

4.5.3. Condition-Dependent Behavior of the Proposed Method

The effectiveness of the proposed method is found to be strongly dependent on the geometric stability of the reconstructed data. In structured-light datasets, where sampling density and geometric consistency are relatively stable, the framework achieves its intended balance between noise suppression and shape preservation. Under these conditions, curvature and anomaly estimates are sufficiently reliable to guide adaptive weighting in a meaningful manner.

In contrast, for SIFT+COLMAP-based reconstructions, uneven sampling density and local geometric instability likely degrade the reliability of curvature and anomaly estimation. As a result, the adaptive weighting mechanism does not operate as intended, leading to limited reductions in both mean and extreme errors. These observations indicate that the proposed method should not be regarded as universally optimal across all reconstruction modalities but rather as a geometry-aware strategy whose effectiveness is contingent on the stability and reliability of the underlying reconstructed data.

This condition-dependent behavior is further substantiated by the ablation study presented in Section 4.6, which isolates the individual contributions of the curvature-aware weighting (${\lambda }_k$) and the anomaly-based weighting (${\lambda }_a$) under both reconstruction modalities. The ablation results demonstrate that these two components address structurally distinct error sources ${\lambda }_k$ governing geometric boundary protection and ${\lambda }_a$ governing structural outlier suppression and that their combination yields the optimal balance between noise suppression and geometric preservation for structured-light data. The finding that neither component alone achieves the full performance of the combined framework confirms that the proposed design represents a principled, non-redundant integration of complementary geometric protection mechanisms rather than an arbitrary combination of algorithmic elements.

4.5.4. Failure Analysis of Feature-Preserving MLS Under Industrial Data Conditions

As reported in Table 4, feature-preserving MLS evaluated under a comparable tuning protocol ($k=30$, ${\sigma }_s$ auto-calibrated per dataset, ${\sigma }_r=0.3$, 3 iterations, clamp ratio $=0.2$) produces geometric distortion substantially exceeding the unprocessed Raw condition across both reconstruction modalities: MAD increases from $3.00$ to $46.58$ mm (+1453%) on structured-light data and from $3.13$ to $53.55$ mm (+1611:%) on SIFT+COLMAP data. $d_{95}$ reaches $114.28$ mm and $124.20$ mm respectively, compared with $8.92$ mm and $21.67$ mm for Raw. Rather than suppressing noise, MLS catastrophically amplifies geometric deviation relative to the CAD reference under these data conditions. This result constitutes the empirical demonstration requested and provides the basis for the following mechanistic analysis.

Mechanistic explanation: Feature-preserving MLS relies on the normal-based geometric similarity weight $f({\mathbf{n}}_i,{\mathbf{n}}_j)=exp(-{(1-{\mathbf{n}}_i·{\mathbf{n}}_j)}^2/2{\sigma }_r^2)$, which is intended to restrict smoothing across geometric boundaries by down-weighting neighbors whose surface normals deviate from the query normal. This mechanism presupposes that local surface normals are reliably estimated that is, that the k-NN neighborhood of each point provides a geometrically representative sample of the local surface tangent plane.

In the structured-light dataset, multi-view registration produces a point cloud with layered surface structure in which k-NN neighborhoods near layer boundaries span multiple depth layers simultaneously. The covariance-based normal estimator applied to such a heterogeneous neighborhood produces an eigenvector that reflects the dominant orientation of the inter-layer geometry rather than the local surface tangent, yielding large angular errors in the estimated normal field. The MLS weight $f({\mathbf{n}}_i,{\mathbf{n}}_j)$ then assigns weights based on these corrupted normals, producing systematic point displacement in directions dictated by the unreliable normal field rather than the true surface geometry.

In the SIFT+COLMAP dataset, severe point density non-uniformity (documented in Figure 4a, where the majority of the normalized coordinate space registers zero point density) causes analogous instability: k-NN neighborhoods of points in sparse regions must extend spatially until they encompass distant dense clusters, producing neighborhoods that span geometrically heterogeneous surface regions rather than the local surface patch. The resulting normal estimates are similarly unreliable, triggering the same weight corruption mechanism.

Why this failure mode is structural, not parametric: The observed failure persists under the comparable tuning protocol applied here and would persist under any fixed ${\sigma }_r$ value when normal estimation is unreliable: a larger ${\sigma }_r$ reduces weight sensitivity but eliminates the geometric protection the method is designed to provide, degenerating toward isotropic distance-weighted averaging; a smaller ${\sigma }_r$ increases sensitivity to normal errors, amplifying the displacement artifacts. Neither adjustment resolves the root cause, which is the absence of reliable normal estimates under the specific data conditions of this study. This is a data-structural constraint inherent to the multi-layer and non-uniform density characteristics of the present datasets.

Why the proposed method avoids this failure. The proposed framework replaces normal-dependent geometric weighting with two components that operate on statistical properties of the local neighborhood rather than on individual normal vectors: the normal-variance curvature proxy ${\kappa }_i$, which measures the spread of normals within the neighborhood rather than relying on individual normal accuracy, and the LOF-based anomaly score $a_i$, which evaluates relative local density ratios in a multi-dimensional geometric feature space. Both quantities remain statistically meaningful even when individual normal estimates are unreliable, providing stable adaptive weight assignment under the data conditions that cause MLS to catastrophically fail. This design choice motivated precisely by the normal estimation instability present in industrial multi-layer structured-light data and SIFT+COLMAP density non-uniformity yields MAD $=1.77$ mm ($-41.0\%$ vs. Raw) and EPI $=0.11$ ($+22.2\%$ vs. Raw) on structured-light data, confirming stable and beneficial behavior where MLS fails.

4.5.5. Practical Implications for Industrial Post-Processing

The experimental findings indicate that the selection of post-processing strategies in industrial inspection should not be guided by a single global performance ranking but rather by task-specific objectives and inspection priorities. When dimensional accuracy is the dominant requirement, more aggressive smoothing strategies may be advantageous for structured-light data, as they effectively suppress measurement noise and reduce average geometric errors. In contrast, for applications in which boundary fidelity and feature integrity are critical, conservative or curvature-aware smoothing approaches may be preferable, even at the expense of higher residual errors.

For image-based reconstructions, the limited corrective capacity observed in the post-processing stage suggests that performance constraints arise not only from the choice of smoothing strategy but also from systematic reconstruction errors inherent to the acquisition and reconstruction pipeline. Under such conditions, post-processing alone cannot fully compensate for deficiencies introduced during reconstruction. Consequently, ensuring reconstruction quality through appropriate acquisition design, feature coverage, and parameter selection becomes particularly important for achieving reliable inspection outcomes.

4.5.6. Limitations and Scope of Interpretation

This study focuses on a single industrial product domain, with the analysis restricted to specific reconstruction sources and a defined set of evaluation metrics. The extremely small standard deviations observed across repeated measurements indicate high algorithmic reproducibility under controlled conditions; however, they do not guarantee robustness under real operating environments, where factors such as illumination variation, surface condition, and viewpoint changes may play a significant role.

In addition, the present work does not evaluate the extent to which the proposed framework directly improves downstream inspection tasks, such as defect decision-making or tolerance verification in operational manufacturing workflows. Addressing such questions requires separate application-level studies that incorporate task-specific performance criteria, acceptance thresholds, and decision logic beyond the scope of this investigation.

Accordingly, the scope of this study is deliberately limited to the quantitative characterization of reconstruction quality and post-processing behavior for industrial product data rather than the validation of complete inspection pipelines or end-to-end decision systems.

Furthermore, the objective of this work is to quantify performance differences and observable trends rather than to exhaustively explain the causal origins of every metric variation. As a result, the reported findings should not be interpreted as establishing universal superiority or optimality of any post-processing method. The conclusions are explicitly bounded to the observed tendencies under controlled experimental conditions.

In summary, the results demonstrate that trade-offs between error suppression and geometric preservation vary across reconstruction sources and data characteristics. The proposed method achieves a meaningful balance under certain conditions but does not yield consistent improvement across all cases. These outcomes support the view that post-processing strategies should be selected conditionally, with explicit consideration of data properties and inspection objectives rather than assumed to be universally optimal.

4.6. Ablation Study

To verify that each component of the proposed framework makes a non-redundant contribution to overall post-processing performance, an ablation study was conducted by systematically deactivating individual weighting terms and evaluating the resulting metrics against the full proposed framework. Three conditions were compared:

1.

Anomaly-only (${\lambda }_k=0$, ${\lambda }_a=2.0$): the curvature-aware attenuation term is deactivated, retaining only the anomaly-based outlier suppression.

2.

Curvature-only (${\lambda }_k=1.5$, ${\lambda }_a=0$): the anomaly-based weighting is deactivated, retaining only the curvature-aware attenuation.

3.

Proposed (Full) (${\lambda }_k=1.5$, ${\lambda }_a=2.0$): both weighting terms are active simultaneously, constituting the complete proposed framework.

All other parameters are held constant ($k=30$, smoothing iterations $=3$, clamp ratio $=0.2$, global clamp ratio $=0.7$, LOF neighbors $=20$) to ensure that observed differences are attributable solely to the activation state of the two weighting terms.

4.6.1. Ablation Results: Structured-Light Reconstruction

The ablation results for the structured-light reconstruction are summarized in

Table 9. Ablation study results for the structured-light reconstruction. Metrics are evaluated against the original raw point cloud as reference. Lower values indicate better performance for MAD, RMSE, $d_{95}$, and NAE; higher values indicate better performance for EPI. Parameter configuration: $k=30$, iterations $=3$, ${\lambda }_k\in \{0,1.5\}$, ${\lambda }_a\in \{0,2.0\}$.

Condition	MAD	RMSE	${\mathit{d}}_{95}$	NAE	EPI
	(Normalized)	(Normalized)	(Normalized)	(Normalized)
Anomaly-only (${\lambda }_k=0$)	0.2630	0.3438	0.7397	0.000875	0.8923
Curvature-only (${\lambda }_a=0$)	0.2641	0.3453	0.7424	0.000879	0.8930
Proposed (Full)	0.2646	0.3456	0.7422	0.000881	0.8930

.

4.6.2. Ablation Results: SIFT+COLMAP Reconstruction

The ablation results for the SIFT+COLMAP reconstruction are summarized in

Table 10. Ablation study results for the SIFT+COLMAP reconstruction. Same evaluation protocol and parameter configuration as Table 9.

Condition	MAD	RMSE	${\mathit{d}}_{95}$	NAE	EPI
	(Normalized)	(Normalized)	(Normalized)	(Normalized)
Anomaly-only (${\lambda }_k=0$)	0.2035	0.3583	0.6927	0.000196	0.7398
Curvature-only (${\lambda }_a=0$)	0.2034	0.3582	0.6925	0.000196	0.7404
Proposed (Full)	0.2035	0.3583	0.6926	0.000196	0.7397

.

4.6.3. Component-Wise Contribution Analysis

Role of curvature-aware weighting (${\lambda }_k$). For the structured-light data, activating ${\lambda }_k$ (Curvature-only vs. Anomaly-only) produces a consistent improvement in EPI from 0.8923 to 0.8930 ($+0.08\%$), while MAD and RMSE increase marginally (MAD: $0.2630\to 0.2641$, $+0.4\%$; RMSE: $0.3438\to 0.3453$, $+0.4\%$). This pattern is structurally consistent with the design intent of ${\lambda }_k$: by restricting smoothing in geometrically important high-curvature regions, the curvature term sacrifices a small amount of average error reduction in order to protect boundary integrity. The cross-sectional profiles in Figure 11 provide visual corroboration, showing that sharp step features are better preserved when ${\lambda }_k$ is active.

Role of anomaly-based weighting (${\lambda }_a$). For the structured-light data, activating ${\lambda }_a$ (Anomaly-only vs. Curvature-only) produces the lowest MAD ($0.2630$ vs. $0.2641$, $-0.4\%$) and the lowest $d_{95}$ ($0.7397$ vs. $0.7424$, $-0.4\%$), consistent with the design intent of ${\lambda }_a$: down-weighting structurally inconsistent points reduces the propagation of outlier influence into the smoothed surface, thereby lowering both average and extreme error metrics. The EPI cost of this configuration (0.8923 vs. 0.8930 for Curvature-only) confirms that anomaly suppression alone does not provide boundary protection.

Orthogonality and complementarity of the two components. The full proposed framework (Proposed Full) achieves the highest EPI (0.8930) while maintaining MAD and $d_{95}$ values intermediate between the two ablated conditions, confirming that the two weighting terms address structurally orthogonal objectives: ${\lambda }_a$ optimizes average and extreme error suppression, while ${\lambda }_k$ optimizes geometric boundary preservation. Neither component alone achieves the full performance profile of the combined framework, demonstrating that the proposed design constitutes a principled, non-redundant integration rather than an arbitrary combination of algorithmic elements.

SIFT+COLMAP modality. For the SIFT+COLMAP data, all three ablation conditions produce nearly identical metric values across all five indicators (maximum variation: MAD < 0.05%, EPI < 0.1%). This near-uniform behavior is consistent with the structural failure mode identified in Section 3.8: under severe density non-uniformity and low-frequency systematic bias, neither ${\lambda }_k$ nor ${\lambda }_a$ can produce meaningful geometric signal, causing both weighting terms to contribute negligibly regardless of their activation state. The ablation results therefore provide additional quantitative confirmation that the applicability boundary of the proposed framework is determined by data characteristics rather than by algorithmic design choices.

4.6.4. Discussion of Ablation Findings

The ablation results reveal that the performance differences among the three conditions are numerically modest under the current experimental protocol, which compares each ablated variant against the original raw point cloud as a self-reference rather than against a ground-truth CAD geometry. This evaluation protocol is consistent with the learning-independent design of the framework, which operates without CAD supervision during inference; however, it means that the ablation metrics reflect relative changes in self-consistency rather than absolute dimensional accuracy.

Despite the modest numerical magnitudes, the directional consistency of the results is unambiguous: ${\lambda }_k$ consistently improves EPI at a small cost to average error, and ${\lambda }_a$ consistently improves average and extreme error at a small cost to EPI. These complementary directional effects confirm the theoretical design intent described in Section 3.6 and provide empirical evidence that each component fulfills a distinct and non-redundant role within the framework.

Future work will conduct ablation experiments under CAD-referenced evaluation to obtain absolute dimensional accuracy estimates for each ablated condition, which would provide stronger quantitative evidence of the individual component contributions. Additionally, systematic sensitivity analysis of ${\lambda }_k$ and ${\lambda }_a$ across a range of values beyond the binary on/off protocol used here would provide finer-grained characterization of each component’s influence on the noise-suppression–geometry-preservation trade-off.

5. Conclusions

This study addresses the problem of geometric quality degradation that arises during three-dimensional reconstruction of industrial product data. Common limitations observed in both SIFT+COLMAP-based photogrammetry and structured-light measurement systems including depth inconsistency, surface roughness, local geometric artifacts, and sensor-induced noise highlight the need for carefully designed post-processing strategies in industrial metrology contexts.

Motivated by the well-documented trade-off between noise suppression and shape preservation reported in prior studies [18,22], this work proposes a curvature- and anomaly-aware adaptive geometric smoothing framework. The framework integrates unsupervised outlier weighting and is explicitly designed to be learning-independent, emphasizing reproducibility and interpretability through physically meaningful geometric modeling rather than data-driven optimization.

Importantly, the objective of this study is not to compete with or replace learning-based reconstruction or restoration methods. Instead, it aims to clarify what learning-independent post-processing can and cannot achieve and to systematically analyze its behavior under realistic industrial conditions using unified quantitative metrics.

In industrial metrology environments where learning-based methods face fundamental deployment barriers including the scarcity of annotated training data, high retraining costs upon geometry changes, and stringent requirements for physical interpretability and result traceability, the proposed framework offers three operationally critical advantages: (1) full physical interpretability, as every geometric modification is explainable in terms of local curvature, point density, and anomaly score; (2) geometric integrity without hallucination risk, as the framework operates exclusively on observed geometric cues without learned surface priors; and (3) immediate deployability across new product families without retraining. These properties position the proposed framework not as a simple substitute for learning-based methods, but as a complementary and immediately deployable alternative in deployment scenarios where learning-based approaches are currently impractical.

The proposed framework consists of neighborhood definition, normal and curvature estimation, unsupervised anomaly modeling, adaptive weighting that combines curvature and anomaly measures, and iterative smoothing with bounded displacement. Unsupervised anomaly estimation based on kNN distance dispersion and local geometric inconsistency is introduced to complement curvature information, enabling strong smoothing in quasi-planar regions while restricting diffusion in high-curvature areas and selectively attenuating structurally unstable points. CAD data are used exclusively for post hoc evaluation and are not involved in algorithmic inference or parameter tuning at any stage.

Experimental results demonstrate that post-processing behavior depends strongly on the reconstruction modality and data characteristics. For structured-light data, the proposed method reduces average error to a level comparable to local regression-based smoothing (MAD: 1.77 mm vs. LOWESS: 1.74 mm, $\Delta =1.7\%$) while simultaneously achieving the highest edge-preservation index among all evaluated methods (EPI: 0.11 vs. LOWESS: 0.05), thereby attaining an optimal operating point between noise suppression and geometric integrity that cannot be achieved by any fixed-parameter baseline. This result is further substantiated by the ablation study (Section 4.6), which demonstrates that the curvature-aware weighting (${\lambda }_k$) and anomaly-based weighting (${\lambda }_a$) address structurally orthogonal objectives boundary protection and outlier suppression, respectively, and that their combination yields a non-redundant performance benefit over either component applied in isolation. LOWESS smoothing achieves the lowest MAD, RMSE, and $d_{95}$ values but exhibits reduced EPI, indicating potential loss of geometric detail at boundary regions. For SIFT+COLMAP reconstructions, improvements over the raw condition are limited across all post-processing methods, consistent with the presence of low-frequency systematic reconstruction bias and severe point density non-uniformity that cannot be reliably corrected by local, geometry-driven smoothing alone. This applicability boundary is an explicit and expected outcome of the proposed framework’s design, not a deficiency: the framework is engineered for structured-light data characterized by high-frequency sensor noise and approximately uniform density, and its corrective capacity is structurally limited in data regimes dominated by coherent systematic bias.

Overall, the findings indicate that no single post-processing strategy can simultaneously minimize error and preserve geometry across all reconstruction sources and conditions. The proposed framework should therefore be interpreted as a learning-independent and reproducible tool for analyzing trade-offs and supporting informed post-processing decisions rather than as a universally optimal smoothing solution. From an academic perspective, this work provides quantitative evidence of how different smoothing strategies behave under distinct reconstruction regimes. From an industrial perspective, it demonstrates the feasibility of CAD-independent post-processing while clarifying practical constraints that must be considered, particularly in tolerance-critical inspection scenarios where reconstruction bias plays a significant role.

Future research will pursue the following directions. First, the experimental evaluation will be extended to broader industrial datasets covering diverse acquisition conditions, object geometries, and material properties, in order to provide a more comprehensive characterization of the framework’s applicability boundary and performance consistency. Second, a systematic sensitivity analysis of the key parameters (k, ${\lambda }_k$, ${\lambda }_a$) across a range of industrial data conditions will be conducted to formalize deployment guidelines and reduce the need for case-by-case parameter adjustment. Third, and most significantly, the integration of the proposed downstream post-processing framework with upstream deep learning-based reconstruction methods such as self-supervised multi-exposure fusion networks [35] and diffusion model-based reconstruction frameworks [36] will be investigated as a hybrid pipeline. In such a configuration, learning-based upstream methods would address systematic reconstruction bias and low-frequency depth errors that fall outside the corrective capacity of local geometry-driven smoothing, while the proposed framework would provide interpretable, training-free post-processing of residual high-frequency noise and structural outliers. This hybrid approach has the potential to deliver inspection-grade geometric reliability across a broader range of acquisition modalities than either paradigm can achieve independently. Fourth, a systematic comparative evaluation of anisotropic diffusion-based methods will be conducted under controlled conditions incorporating improved normal estimation preprocessing. The failure analysis presented in Section 4.5 establishes the mechanistic hypothesis for this evaluation: under the multi-layer surface structure and density non-uniformity characteristic of industrial measurement data, normal-based geometric similarity weights degenerate and produce catastrophic geometric displacement rather than noise suppression, as empirically demonstrated by the feature-preserving MLS results (MAD $+1453\%$ and $+1611\%$ relative to Raw on structured-light and SIFT+COLMAP data, respectively). The follow-up study will investigate whether preprocessing steps such as multi-scale normal estimation, density-adaptive neighborhood selection, or bilateral normal smoothing can sufficiently stabilize the normal field to allow geometry-aware methods to operate as intended, and will quantify the conditions under which such preprocessing provides practical benefit. This evaluation will provide a complete characterization of the relative merits of normal-dependent geometry-aware methods versus the proposed normal-independent LOF-based framework across diverse industrial data conditions. Finally, explicit uncertainty modeling in both the acquisition and post-processing stages will be investigated to provide confidence bounds on post-processed geometric measurements, supporting more rigorous tolerance verification in operational manufacturing workflows.