Revisiting Deep Learning-Based Semantic Segmentation on Large-Scale Hydraulic-Structure LiDAR Point Clouds: A Spatial Surrogate Modeling Perspective

Highlights

What are the main findings?

• Spatial surrogate modeling improves point-wise segmentation accuracy by enhancing label propagation, an often overlooked step in large-scale 3D point cloud semantic segmentation. The post-processing strategy can be applied to the outputs of an existing pre-trained model without additional neural network training which is important in a specific practical application when a pre-trained model is provided by a research project but the original training data are unavailable and only limited labeled target-site data can be obtained, making it difficult for reliable fine-tunning.
• Different semantic classes benefit from diverse configurations of spatial surrogate modeling, indicating class-specific spatial-scale dependence during label propagation.
• The evaluated response surfaces support site-specific calibration when limited target-site labels are available and provide empirical starting settings for comparable bridge applications when target-site labels are unavailable.

What are the implications of the main findings?

• Label propagation for annotating points not directly predicted by a neural network should be treated as an important spatial modeling step rather than a simple technical completion procedure.
• Spatially informed label propagation can improve the accuracy and reliability of applications that depend on precise spatial measurement and object delineation using hydraulic-structure LiDAR point clouds.

Abstract

Geospatial Artificial Intelligence (GeoAI) and the rapid advancement of 3D data acquisition technologies (e.g., LiDAR) have enabled scalable semantic interpretation of georeferenced large-scale 3D point clouds across different applications. 3D deep learning-based semantic segmentation on large-scale 3D point clouds typically relies on data partitioning and sampling strategies to address computational constraints, resulting in a substantial proportion of points not being directly predicted by deep neural networks. These points not directly predicted by the model require a processing step for label propagation, which is commonly handled using simple spatial proximity-based rules. While this simplification may be acceptable for some applications, it becomes critical in tasks that require precise spatial measurements and accurate object delineation, where propagation errors can directly affect downstream analyses. To bridge this research gap, this study views this process as a spatial surrogate modeling problem, where predictions from deep learning models are used to infer labels for underrepresented points based on spatial relationships. We adopt inverse distance weighting (IDW) as a transparent, deterministic, and training-free spatial post-processing strategy to examine whether explicitly incorporating spatial relationships improves segmentation outcomes. We evaluate the proposed method within an existing 3D semantic segmentation workflow for bridge inspection, a practical application requiring accurate spatial measurement and reliable object delineation. In the experiments, we use self-collected terrestrial LiDAR point clouds of bridges and associated hydraulic structures and systematically assess how neighborhood size and distance-decay parameters affect model performance on the segmentation task. Results show that this spatial post-processing step provides a modest enhancement over the conventional nearest-neighbor propagation baseline, with notable gains especially on spatially sparse or geometrically complex classes. The results also reveal class-dependent spatial effects, suggesting that different semantic classes exhibit distinct spatial dependencies during label propagation. These findings highlight the practical importance of accounting for spatial context when propagating semantic information to those points not directly predicted by deep neural networks. This is particularly important for downstream applications that require highly accurate spatial measurement and object delineation. The practical value of this post-processing step becomes more apparent in data-scarce application domains such as hydraulic-structure inspection, where large labeled point-cloud datasets and public benchmarks remain limited. In such settings, users may rely on domain-specific pre-trained models, while the original training data may not be publicly available for retraining or extensive model modification. Using a practical case study in the hydraulic domain, this study shows that deterministic post-inference label propagation can improve complete point-wise prediction from an existing model, thereby supporting the reuse of available models for domain-specific applications. The resulting response surfaces support two practical uses: site-specific calibration when limited labeled target data are available, and empirically informed initial settings, a rule of thumb, for comparable bridge-LiDAR applications when target-site labels are unavailable.

1. Introduction

Geospatial Artificial Intelligence (GeoAI), also referred to as spatially explicit AI, integrates geospatial data, spatial science, and machine learning to extract knowledge from geospatial big data and to support downstream decision-making [1,2,3,4]. In recent years, there has been a rapid growth in 3D geospatial data acquisition (e.g., terrestrial, mobile, and airborne LiDAR (Light Detection and Ranging), SAR (Synthetic Aperture Radar), aerial photogrammetry, and RGB-D sensors), making large-scale, georeferenced point clouds increasingly accessible and enabling applications such as 3D Geographic Information System (GIS), digital twins, and world models [5]. Recent systematic reviews have further delineated the rapidly expanding GeoAI literature and highlighted emerging directions such as generative AI and geo-foundation models [2,6,7,8]. Within this context, deep learning on 3D point clouds has become a key paradigm for extracting spatial semantics, yet accurately modeling spatial relationships and contextual dependencies remains a fundamental challenge.

3D point cloud semantic segmentation is one of the core tasks in 3D deep learning, which plays a foundational role by converting raw geometry into structured spatial semantics (e.g., terrain, vegetation, and man-made components), which have the potential to be integrated into pipelines of spatial analytics, asset management, and digital twins. As a result, this task has attracted increasing interest not only in the domains of computer science (e.g., computer vision) and engineering (e.g., electrical and civil engineering) but also from spatial science [9,10,11,12,13,14] where the explicit modeling of spatial relationships is essential. Although many studies primarily focus on improving network architecture and model performance, the application of these deep learning models on large-scale point clouds is still constrained by GPU memory and computational throughput. Consequently, inference often relies on data partitioning and sampling strategies. These strategies can leave an observable portion of points not predicted directly by deep learning models. Therefore, they require a processing step to propagate semantic information to produce complete point-wise predictions. In this study, we refer to these points as underrepresented points, as they are not adequately captured and predicted during model inference. Recent surveys have provided comprehensive overviews of 3D deep learning advances in semantic segmentation [15], and new architectures continue to improve efficiency and contextual reasoning for urban-scale point clouds [16,17]. However, despite extensive advances in network architecture, relatively limited attention has been paid to the design and evaluation of such label propagation strategies in practical large-scale settings. Accordingly, this manuscript focuses on the spatial reconstruction and propagation stage after neural network inference, rather than on proposing a new segmentation backbone.

In practice, predictions are generated on blocks with a fixed number of points and a regularized size as a sampling strategy to satisfy the requirement (i.e., regularized batch) for the input of deep learning neural networks. As a result, the remaining points are often labeled by a predicted nearest neighbor based on spatial proximity. From a geospatial perspective, this practice is not merely a computational workaround; it can alter the spatial distribution of predicted labels, potentially influencing subsequent spatial analysis and decision-making. For example, in terrain classification, mispropagation of labels near class boundaries (e.g., between ground and vegetation) can overestimate or underestimate surface representations, leading to errors in derived elevation models and consequently affecting local flood risk assessment and hydrological modeling.

Motivated by this observation, we revisit label propagation as a spatial surrogate modeling problem. The surrogate model, also referred to as metamodeling, has been widely used to approximate outcomes that are costly or inaccessible to obtain directly [18,19,20]. This concept is also widely used in the domains of environmental science and geographic information science (GIScience) [21,22]. Viewed through this lens, conventional nearest-neighbor propagation in 3D semantic segmentation workflows can be understood as a simple surrogate model, in which labels for underrepresented points are approximated from nearby predicted outputs. A spatial surrogate model extends this idea by explicitly modeling spatial dependence among points, estimating an unknown value at a spatial location based on surrounding known observations. In this study, we use inverse distance weighting (IDW) as an interpretable, computationally efficient, and effective spatial surrogate model to examine whether explicitly incorporating spatial context during label propagation can improve full-scene label reconstruction without retraining the segmentation model. By treating class scores produced by deep neural networks as spatially autocorrelated variables, we investigate how interpolation parameters (i.e., neighborhood size and distance decay) interact with class-specific spatial dependencies within a 3D semantic segmentation workflow for bridge segmentation. This perspective complements recent efforts that embed spatial awareness directly into network architectures [9,10,17,23], as both approaches address spatial consistency from different stages of the pipeline. This scope is intentionally application-oriented where the goal is to evaluate a simple spatial method that can be attached to existing hydraulic-structure segmentation workflows.

Overall, this study contributes to the application-oriented field of hydraulic-structure LiDAR point cloud segmentation in three ways:

(1)

This study reframes the label propagation for underrepresented points in large-scale hydraulic-structure 3D semantic segmentation as a spatial problem, providing a spatial perspective for improving label propagation in applications where accurate spatial measurement and object delineation are required and where domain-specific pre-trained models may need to be reused without retraining.

(2)

We examine spatial surrogate modeling as a spatially aware post-inference label propagation strategy and use IDW as an interpretable proof-of-concept to reveal class-dependent spatial context effects.

(3)

Through experiments on field-collected, georeferenced hydraulic-structure and bridge LiDAR point clouds, this study provides empirically informed parameter guidance. The identified favorable and stable parameter ranges can support site-specific calibration when limited labels are available and serve as initial reference settings for comparable bridge applications.

The rest of this paper is organized as follows: Section 2 reviews the recent literature using 3D semantic segmentation on LiDAR point clouds and discusses label propagation challenges in large-scale semantic segmentation. Section 3 presents the conceptual and methodological framework, including spatial surrogate modeling, data collection and processing, and experimental design. Section 4 reports the experimental results, including baseline prediction coverage, IDW response surfaces, class-wise patterns, and the additional DeepHyd bridge-component evaluation. Section 5 discusses the spatial meaning, practical value, relationship to related methods, and limitations of the study. Section 6 draws conclusions based on the results and discussion.

2. Related Work in 3D Semantic Segmentation

2.1. Observed Research Gap in Semantic Segmentation on 3D Point Cloud

Large-scale LiDAR scenes can contain millions to billions of points, posing significant challenges for storage, memory management, and computational efficiency in deep learning pipelines. Consequently, scalable 3D deep learning frameworks for point clouds commonly adopt partitioning, sampling, or hierarchical subsampling strategies to make data processing tractable. Early point-based architectures, such as PointNet++ [24], process unordered point sets but rely on block partitioning and local neighborhood sampling when applied to large-scale scenes. Similarly, convolution-based networks, including ConvPoint [14] and KPConv [25], operate on local regions and typically require fixed size subsets of points for efficient training and inference. More recent efficient architectures, such as RandLA-Net [26], explicitly employ random sampling to reduce computational complexity while maintaining segmentation performance. Despite architectural differences, the strategy of sampling and partitioning remains prevalent in recent state-of-the-art models. For example, transformer-based methods, including Point Transformer [27], OneFormer3D [28], PointGA [29], and CSP-Former [30], introduce attention mechanisms to capture long-range dependencies but still depend on hierarchical sampling or window-based processing to ensure scalability.

While these strategies are effective for managing computational resources and enabling large-scale learning, they introduce an underexplored issue in the reconstruction of scene-level predictions. Some points may be underrepresented and receive inconsistent predictions across overlapping partitions depending on the sampling configuration. This issue is particularly evident near partition boundaries or in sparsely sampled regions and persists across both convolution-based and transformer-based deep neural network architectures. Although practical workflows often rely on projection, averaging, voting, or nearest-neighbor assignment to recover complete point-wise labels, this final reconstruction step is not always evaluated as a main spatial modeling component. This gap is especially important in application domains such as bridge and hydraulic-structure inspection, where complete and spatially coherent point-wise labels are needed for object delineation, asset monitoring, and spatial measurement.

2.2. Explicit and Implicit Label Propagation, Refinement, and Upsampling Strategies

Building on this gap, this subsection reviews existing explicit and implicit strategies for recovering spatial detail, propagating information across sampling levels, and refining point-wise predictions in point cloud segmentation workflows. Existing point cloud segmentation studies address spatial consistency and resolution recovery through several related routes. CRF-based refinement methods can refine raw-point predictions after coarse scores or probabilities have been projected or interpolated back to the original point cloud; for example, SEGCloud combines coarse voxel prediction, trilinear interpolation to raw points, and fully connected CRF refinement [31]. Graph-based methods, such as Superpoint Graph, first partition large point clouds into superpoints and then train graph-based message-passing models [32]. Consistency-based approaches, such as HybridCR, are usually training-stage strategies for weak or semi-supervised semantic segmentation [33]. Feature propagation and upsampling are also related to the present problem. PointNet++ and variants use inverse-distance interpolation within learned feature-propagation layers, and point-cloud upsampling networks learn to generate denser point sets from sparse inputs [34,35,36,37]. These approaches demonstrate the importance of propagating information across sampling levels, but they are generally part of trainable neural architectures or require additional optimization/training. In contrast, the IDW-based spatial interpolation step evaluated in this study is applied after inference and does not require modifying or retraining the segmentation model. This study isolates that final post-inference propagation step in hydraulic-structure LiDAR applications and evaluates IDW as a spatially interpretable alternative.

Recent geospatial and remote-sensing segmentation studies continue to improve feature representation, context modeling, and multimodal fusion, including density-based grid decimation, transformer-based road LiDAR segmentation, and CNN-Mamba fusion for remote-sensing image segmentation [38,39]. These studies are complementary to this work because they mainly address network-level representation, whereas this manuscript focuses on a training-free spatial post-processing step for full-scene point-wise label reconstruction.

2.3. Related Work of Surrogate Model

Surrogate modeling (a.k.a. metamodel, a “model of the model” as mentioned by [18,19] has been used across different domains such as engineering and computer science to approximate the results of interest, which are too costly to access or cannot be directly accessed. Kleijnen [19] defined “[m]etamodels as being fitted to the I/O data produced by the experiment with the simulation model” (p. 707). Instead of directly evaluating a complex system, a surrogate model learns the relationship between inputs and outputs from sampled data and provides an efficient approximation for prediction or analysis [20]. Various methods have been used as surrogate modeling techniques in previous studies, such as polynomial response surfaces [40,41], deep neural networks [42,43], multivariate adaptive regression splines [44], and radial basis functions [45,46].

From an implementation perspective, a surrogate model has four objectives: understanding the problem, estimating the unknown values of interest, optimizing the configuration, and validating the model performance [47]. When it comes to the GIScience domain, this idea is often extended to spatially explicit prediction problems, where the primary objective is to estimate unknown target values across geographic or geometric space. For addressing spatial problems specifically, spatial surrogate models, such as Kriging and IDW [48], have been widely adopted because they explicitly incorporate spatial location and neighborhood relationships into the estimation process. For example, [49] applied ordinary kriging to bridge-scour-related point cloud data (i.e., point clouds used to analyze sediment erosion around bridge foundations) collected from terrestrial LiDAR scans. Moreover, previous research [50] used IDW, ordinary kriging, and geographically weighted regression to generate reference precipitation surfaces from rain-gauge observations for evaluating precipitation measurement. These studies demonstrate that spatial surrogate models can transform discrete, incomplete, or irregularly sampled observations into spatially continuous estimates for downstream analysis. This perspective is directly relevant to large-scale 3D point cloud semantic segmentation, where partitioning and sampling strategies allow only a subset of points to receive direct predictions from the deep neural network. The remaining underrepresented points still require semantic labels to support complete scene-level interpretation, spatial measurement, and object delineation.

3. Materials and Methods

3.1. Spatial Surrogate Model for Labeling Underrepresented Points

In this study, we investigate whether the identified research gap can be addressed by treating label propagation as a spatial surrogate modeling problem. Here, a spatial surrogate model is used in a narrow, application-specific sense, denoting a spatial approximation that estimates class-score vectors at underrepresented point locations from the outputs of neighboring points after neural network inference. In this context, the primary deep learning model provides known outputs for a subset of points, represented by class scores or probabilities. The role of the surrogate model is then to approximate the corresponding class-score vector for underrepresented points that were not directly inferred by the neural network. We demonstrate the problem of deep learning-based semantic segmentation on large-scale point cloud in an illustrative way in Figure 1. Partitioning and sampling are required in the preprocessing of the semantic segmentation workflow for large-scale scenes. This step is done by using a fixed size of 3D window (a.k.a. block) to go across the space of the scene. For each block, a fixed number of points are sampled for downstream prediction using a deep learning-based model. Both fixed size block and the fixed number of points are empirical workflows necessitated by the heterogeneous spatial distribution of point clouds and the need for regularized input for concurrent deep neural network architectures.

Because LiDAR point clouds are spatially explicit and georeferenced, this surrogate modeling problem should further account for spatial relationships among points. In this study, we treat a spatial surrogate model as a model that estimates missing semantic information at underrepresented points by using the model outputs of nearby predicted points and their spatial relationships, as shown in Figure 2. This formulation is consistent with the spatial principle that nearby points located on the same object or surface are more likely to share similar semantic labels than distant points. Therefore, a spatial surrogate model provides a natural implementation of the spatial surrogate modeling perspective in large-scale 3D point cloud semantic segmentation.

This study adopts IDW as an interpretable and computationally efficient spatial surrogate model. Compared with kriging, which requires modeling spatial covariance or variogram structures, IDW provides a simpler distance-decay-based formulation that is suitable for evaluating whether spatially informed label propagation can provide a training-free alternative to the conventional nearest-neighbor baseline. In addition, nearest-neighbor propagation can be viewed as a limiting case of distance-based spatial estimation because it assigns each underrepresented point using only the closest predicted point as shown in Figure 2. IDW generalizes this strategy by incorporating multiple neighboring predicted points and weighting their contributions according to distance. Unlike trainable refinement networks, IDW is only a post-inference operation on already generated score vectors; it introduces no new trainable parameters and does not require re-training the existing model. In this study, training-free specifically means that the architecture and learned weights of the neural network are not modified. Once k and p have been specified, the execution of IDW requires only point coordinates and the class-score outputs of the fixed model. Accuracy-based selection of k and p, including site-specific calibration, requires labeled reference data. Nevertheless, this remains distinct from model training because it is a lightweight, low-dimensional parameter-selection step involving only two deterministic post-processing parameters, rather than optimization of the neural network’s learned weights. Moreover, the availability of reference labels for this purpose does not imply that the data are sufficiently large, diverse, or representative for reliable model fine-tuning, which supports its distinct usefulness in practice.

Having established IDW as the spatial surrogate model, the next step is to define the target variable to be estimated at underrepresented point locations. The output of a deep learning-based 3D point cloud semantic segmentation model not only provides a predicted label for each point but also reflects the uncertainty and probability of these predictions, represented as scores and probabilities. Initially, the model outputs a score for each class that it is trained to recognize for a given point. A higher score indicates greater confidence that the point belongs to that class. These scores are subsequently normalized using a softmax function, converting them into probability values that sum to one and represent the relative confidence of the model across different classes. For instance, in a model trained to differentiate between three classes, a clear sample from one class may yield a highly peaked probability distribution, while an ambiguous sample may produce a more uncertain distribution. This view based on either scores or probabilities provides a natural interface for a spatially informed surrogate model to estimate the unknown, where class scores or probabilities can be propagated to underrepresented points based on spatial context. In this study, we use pre-softmax class scores rather than post-softmax probabilities. This choice is motivated by both methodological and practical considerations. Guo et al., [51] state that softmax probabilities from modern neural networks are often poorly calibrated, meaning that the reported probability values do not necessarily correspond well to the true likelihood of correctness. Moreover, Muller [52] claimed that pre-softmax outputs can preserve useful inter-class relational information, suggesting that pre-normalization outputs retain a structure that may be useful for downstream processing. Therefore, in this study, we used scores for spatial surrogate modeling. The raw scores can be denoted as Equation (1):

$$z\left(x_i\right)=f_{\theta \left(x_i\right)},\hspace{1em}\mathrm{z}\left(x_i\right)\in {\mathbb{R}}^C,x_i\in {\mathbb{R}}^3$$

(1)

where $z\left(x_i\right)\in {\mathbb{R}}^C$ denotes the vector of class scores at point $x_i$ in a 3-dimentional space, and $C$ is the number of classes. $f_{\theta }(\cdot )$ is the neural network with learned parameters $\theta$.

Based upon this understanding, this study examines IDW as a spatial surrogate model for the post-inference label-propagation challenge in large-scale hydraulic-structure point cloud segmentation. Moreover, by manually perturbing the parameter configurations (e.g., k-nearest neighbors and power), we investigate how the performance of 3D semantic segmentation correlates with these parameters, reflecting the underlying spatial dependencies. In this context, the formula for estimating the score of a specific class of an underrepresented point based on the output (i.e., scores) of surrounding predicted points from a multi-class classification is as given by Equations (2) and (3):

$${\widehat{z}}_c\left(x_0\right)={\displaystyle \frac{{\sum }_{i=1}^N{w}_i ·{ z}_c\left(x_i\right)}{{\sum }_{i=1}^N{w}_i}},c\in \left\{1,\dots ,C\right\}$$

(2)

$$w_i={\displaystyle \frac{1}{{h\left({x_0,x}_i\right)}^p}}$$

(3)

Here, c = 1, …, C indexes semantic classes, and N denotes the number of neighboring points used. In Equations (2) and (3), ${\widehat{z}}_c\left(x_0\right)$ is estimated score for class $c$ at target point $x_0$. $z_c\left(x_i\right)$ is the raw output score, produced by the neural network for class $c$ at point $x_i$. $w_i$ is the distance-based weight of surrounding neighbor at point $x_i$. $h\left(x_0,x_i\right)$ is distance between $x_0$ and $x_i$. $p$ is the power parameter controlling distance decay, and $N$ is the number of neighboring points used.

Then the final semantic label is assigned by taking the class with the highest interpolated score as shown in Equation (4):

$$\widehat{y}\left(x_0\right)=\underset{c\in \left\{1,\dots ,C\right\}}{\mathrm{Argmax}}\left({\widehat{z}}_c\left(x_0\right)\right)$$

(4)

where $\widehat{y}\left(x_0\right)$ is the predicted class of the target point $x_0$ obtained by selecting the class with the highest interpolated score.

3.2. Experiment Baseline: A Pracitcal Study Case in Bridge Inspection

The identified research gap is particularly consequential in application domains that require complete and reliable point-wise predictions for downstream analysis, such as transportation infrastructure monitoring, which is directly related to public safety. Bridges and associated hydraulic structures are critical assets in transportation networks, where routine inspections are required at regular intervals (i.e., 12 months to 72 months as required by federal regulations (Code of federal regulations: https://www.ecfr.gov/current/title-23/chapter-I/subchapter-G/part-650/subpart-C/section-650.311 accessed on 17 July 2026) and can be resource-intensive when solely relying on on-site fieldwork. 3D sensing technologies provide a scalable alternative for capturing as-is conditions of the assets, but turning raw point clouds into actionable information still requires not only expert knowledge in this domain, but also accurate spatial measurements and semantic understanding. Recent bridge point cloud understanding research has advanced semantic segmentation using transformers [53]. Moreover, there are domain knowledge-enhanced and foundation model-based pipelines that reduce dependence on large, labeled datasets for model training [54,55,56]. Meanwhile, bridge-specific datasets and domain gap analyses continue to highlight generalization challenges across bridge types, sensors, and environments, which is particularly relevant for real-world deployment [57]. In this study, we used a practical deep learning-based workflow in bridge inspection, DeepHyd (https://rosap.ntl.bts.gov/view/dot/62502 accessed on 17 July 2026), as a case study to investigate the research gap identified above. DeepHyd is a two-tier deep learning framework originally designed to automate the classification of hydraulic structures from multimodal remotely sensed data as shown in Figure 3 [58]. In this study, hydraulic structures refer to bridges and associated transportation-waterway infrastructure, for which complete and spatially coherent point-wise labeling is required for inspection, measurement, and asset monitoring.

This workflow packages data preprocessing, model training and validation, and inference into reusable scientific workflows using Jupyter notebooks and containerized execution. The project collected data from field sites using a combination of LiDAR, sonar, and GPS (Global Positioning System), and produced 41 labeled point cloud scenes for model training and validation. The labeled training pool contains approximately 200 million points for the scene-level model and 100 million points for the component-level model, which supports the two-tier design. In practice, this workflow provides two fine-tuned models (see Figure 3): the first model detects bridges from outdoor scenes represented by large-scale point clouds, and the second model further segments bridge components from the detected bridge point clouds. In this study, we evaluate this as an application-level support experiment.

DeepHyd project adopted ConvPoint as the backbone architecture and was fine-tuned on labeled scenes with detailed hyperparameter tuning (e.g., learning rate, number of iterations, block size, and points per block) [14,58,59]. ConvPoint is a continuous convolutional neural network designed for classification and segmentation on irregular and unordered point clouds. By selecting relevant neighboring points within a given radius for convolution, ConvPoint can handle varying point densities and distributions and has demonstrated competitive performance on large-scale outdoor point cloud benchmarks (e.g., Semantic3D from https://semantic3d.ethz.ch/ accessed on 17 July 2026). Semantic3D, introduced by Hackel and Savinov, stands as a large-scale dataset for outdoor scene analysis and provides diverse terrestrial laser scans with multiple semantic categories. For the experiments in this manuscript, the trained DeepHyd models are kept fixed; the proposed IDW step uses their inference outputs and does not involve additional model training.

In this study, we used the DeepHyd workflow as an application baseline and focused on the investigation of how spatial surrogate model is able to bridge the gap between the demand for complete point-wise prediction for specific use cases (i.e., hydraulic-structure and bridge inspection) and the limitations of the traditional nearest-neighbor method for propagating labels to underrepresented points. Accordingly, we retain the core workflow and deep learning models adopted in the workflow. This design makes the comparison application-oriented: the conventional nearest-neighbor step is compared with IDW while the underlying neural network predictions remain unchanged.

3.3. Experiment Design

To evaluate the effectiveness of a spatial surrogate model as a label propagation strategy in the DeepHyd workflow, we designed a set of controlled experiments to examine how interpolation parameters affect segmentation performance. In this study, we collected a total of 10 scans for 7 bridges using terrestrial LiDAR (see Figure 4 for detail), where 1–2 scans were acquired for each bridge depending on the fieldwork environment and accessibility. Moreover, we manually labeled the scans following the DeepHyd label settings—the dataset is prepared with labels including bridge, vegetation, and ground, as shown in Figure 4c.

The experiments are designed with a focus on how neighborhood size and distance decay influence results across classes with different spatial characteristics. A key design choice in IDW is how neighboring points are defined. The neighbor searching method can be fixed radius or k-nearest neighbor (kNN) but we chose to use the latter. Due to the unevenly distributed nature of 3D point cloud data, especially LiDAR-generated point clouds where points become denser closer to the scanning center, inadequate settings for a fixed-radius search may return too few neighbors in sparse regions and overly dense local neighborhoods in dense regions. In some cases, it may even fail to provide valid neighboring values for interpolation at a target point. Accordingly, kNN guarantees that each target point is associated with a specified number of neighbors, making it more robust for complete point-wise label propagation in irregularly distributed point clouds. Moreover, kNN has been broadly used in 3D deep learning on point clouds [12,14].

To systematically evaluate the parameter sensitivity on segmentation performance, we designed a set of controlled experiment treatments that vary the neighborhood size and distance decay in IDW. For the treatments shown in

Table 1. Parameter ranges for experiment treatments used to configure the spatial surrogate model for label propagation in the large-scale point cloud semantic segmentation workflow.

Parameter	Values
k (number of neighbors)	1–32 (step = 1)
p (power of distance decay)	0.1–3.0 (step = 0.1)

, we set the number of nearest neighbors with a minimum value of 1 (i.e., the nearest neighbor) and a maximum value of 32, which follows an empirical setting (e.g., 16–32 neighbors) when local context is considered for a point during the deep learning process. The power parameter is varied from 0.1 to 3.0 with 0.1 as the interval. We assume that the maximum value of 3.0 is sufficient because of the nature of IDW where the weights of nearby neighbors as opposed to distant ones, increase exponentially with the increase in power. Moreover, our results also support this setting in Section 4. Since the DeepHyd outputs are fixed and IDW is deterministic for a specified neighborhood size k and distance-decay power p, repeating the same propagation configuration would return identical labels. The treatment grid is therefore designed to evaluate spatial-parameter sensitivity, rather than random training-seed variability.

The performance measurements used in this study are overall accuracy (OA), mean Intersection of Union (mean IoU), and IoU for each class. IoU and OA are two measurements commonly used for semantic segmentation [14,60]. IoU has a similar principle to the Jaccard index, which is used for similarity measurements in the geography domain [59]. The equations of the three metrics are shown in Equations (5)–(7).

The IoU for class i is defined as the ratio of true positives to the union of true positives, false negatives, and false positives:

$${IoU}_i={\displaystyle \frac{T{P}_i}{T{P}_i+F{N}_i+F{P}_i}}$$

(5)

$$mIoU={\displaystyle \frac{1}{C}}\sum \left({IoU}_i\right)$$

(6)

$$OA={\displaystyle \frac{T{P}_{all}}{T{P}_{all}+F{P}_{all}}}$$

(7)

where TP, FP, and FN are true positive, false positive, and false negative; subscript all represents across all classes, and the subscript i is for each individual class. C is the number of classes.

Once the results for all treatments were obtained, we used a response surface approach to characterize how neighborhood size $k$ and distance-decay power $p$ affect OA, mIoU, and class-wise IoU within the study datasets. The primary purpose of this analysis is to examine parameter sensitivity and class-dependent spatial dependence in the post-inference label-propagation stage. This analysis enables us to quantify how different parameter combinations influence model outcomes, reveal class-dependent spatial context patterns, and provide practical guidance for selecting appropriate IDW settings in real-world applications.

4. Results

4.1. Baseline Prediction and Underrepresented-Point Coverage

In this section, we use the direct prediction results generated by the command-line implementation of the DeepHyd workflow as a baseline. In this setting, nearest-neighbor propagation is applied during label propagation to assign labels to points that are not directly predicted by the model. We demonstrate the issue of underrepresented points in Figure 5. The existing sampling strategy adopted by the DeepHyd workflow is to use an 8 m block to sample 8192 points per block (Sampling configuration from DeepHyd: https://connect.ncdot.gov/projects/research/RNAProjDocs/RP2019-03%20Final%20Report.pdf accessed on 17 July 2026). Consequently, a substantial proportion of the original points are not directly represented during model prediction. These underrepresented points tend to cluster near the scanning center, where the point cloud is denser because LiDAR produces more closely spaced measurements at shorter ranges. They also frequently occur on objects with vertical surfaces, such as trees or shrubs on slopes, which tend to generate dense LiDAR returns. Combined with the fixed-block sampling strategy illustrated in Figure 1, these variations in local point density increase the likelihood that many points are excluded from direct model prediction. To quantify the extent of this issue,

Table 2. Statistics of the total number of points and the number of points directly predicted by neural network for each collected LiDAR scan in the large-scale point cloud semantic segmentation workflow.

Scan Name	Number of Total Points	Number of Predicted	Predicted Rate
Scan 1	719,106	350,558	48.75%
Scan 2	1,085,824	242,730	22.35%
Scan 3	8,009,476	2,539,892	31.71%
Scan 4	3,382,589	1,381,216	40.83%
Scan 5	4,239,833	1,289,660	30.42%
Scan 6	1,313,687	782,698	59.58%
Scan 7	11,061,405	2,790,427	25.23%
Scan 8	6,253,579	1,044,304	16.70%
Scan 9	4,435,961	1,214,976	27.39%
Scan 10	5,705,619	921,896	16.16%
Average	4,620,708	1,255,836	31.91%

reports the total number of points, the number of directly predicted points, and the corresponding prediction rate for each scan. The prediction rate is the proportion of points receiving direct model predictions. It varies substantially across sites, as suggested by Table 2. This variation is influenced by settings such as scene characteristics (e.g., bridge size and distance from the scanning position), preprocessing configurations (e.g., sampling strategies), and label propagation settings (e.g., step size). The prediction rate ranges from 16.16% to 59.58%, with an average of 31.91%, indicating that a significant portion of points are not directly predicted by the neural network but instead assigned labels through nearest-neighbor propagation. These results show that label propagation affects a substantial portion of the final full-scene labels in the evaluated hydraulic-structure LiDAR dataset.

We demonstrate the performance measurements in

Table 3. Model performance using traditional label propagation on the collected large-scale LiDAR point cloud datasets for bridges.

Measurement	Values
OA	86.71%
mIoU	73.08%
Bridge IoU	94.11%
Vegetation IoU	51.49%
Ground IoU	73.65%

OA: overall accuracy. mIoU: mean intersection over union.

. This performance serves as a baseline for comparison purposes. The model performance on the collected dataset reaches 86.71% and 73.08% for OA and mIoU, which is relatively good for outdoor 3D semantic segmentation datasets, compared to model performance in large-scale outdoor classification benchmarks (e.g., Semantic3D). The IoU for the bridge class is 94.11%, which means the bridge class is well segmented. These results support the effectiveness of the DeepHyd model in segmenting bridges from a 3D scene.

4.2. Scene-Level Analysis of IDW-Based Label Propagation

The overall model performance with IDW for label propagation is demonstrated in Figure 6. Using only the nearest neighbor for label propagation is a special case of IDW with k = 1. Therefore, the baseline performance is also included in the figure, where it is located on the far-left end of the x axis. The OA and mIoU range from 86.72% to 86.88% and from 73.04% to 73.22%, respectively. The variations are modest, with improvements of 0.16% in OA and 0.18% in mIoU, and should not be overstated as large global accuracy gains (see Figure 5). Nevertheless, the increasing trend indicates that spatial propagation parameters can influence full-scene label reconstruction. This is important in practice because even small global metric changes may affect local object boundaries and spatial measurements in hydraulic-structure point clouds. The increasing pattern along the x-axis indicates that the overall performance benefits from more nearest neighbors. Moreover, a lower power results in higher performance, suggesting the accuracy of points relies more on the local context rather than on only the nearest ones. Since the post-processing step is deterministic for fixed inputs and parameters, the small global improvement is interpreted as an application-level effect of spatial propagation rather than as a stochastic training improvement.

When it comes to class-wise performance, we can see diverse patterns for the response surfaces. Generally, there is an increase in the IoU of different classes when we apply spatial surrogate model, where the increases are separately 0.10%, 0.25%, and 0.45% for bridge, vegetation, and ground, respectively. The IoU of the bridge appears to benefit more with a smaller number of nearest neighbors and higher power values (see Figure 7a). The pattern of the response surface points out that the identification of bridge data depends more on the relatively local neighbors. We can tell that when there is an increase in the number of nearest neighbors, a higher power is preferred, indicating more weight is given to the near neighbors than distant ones. There is a fluctuation in the trend of the IoU for vegetation along with the increase in the number of nearest neighbors (see Figure 7b), where the maximum values are located on the left end of the x-axis. Moreover, the IoU is more sensitive to the number of neighbors when the number is around 1 and 2. The IoU metric for ground classification in point cloud data, as indicated by a range from 73.65% to 74.1%, suggests a noteworthy level of accuracy in distinguishing ground points from non-ground points. This range is similar to the pattern observed in the mIoU across different classes, implying that the accuracy of point classification is significantly influenced by the local context of points, rather than being solely dependent on the nearest neighbors. This observation underscores the importance of considering a broader neighborhood and spatial relationships among points to capture the underlying structure and features of the data effectively. These class-wise response patterns are more informative for the spatial objective of this manuscript than the small magnitude of the global OA and mIoU changes.

These observations highlight the varying characteristics of spatial context within each class, illustrating the complex interaction between spatial context and classification performance in point cloud analysis. Understanding these dynamics is essential for the development of more efficient and accurate point cloud processing workflows for hydraulic-structure applications.

4.3. Component-Level Evaluation Using the DeepHyd Bridge-Component Model

To further evaluate whether the spatial post-processing behavior is specific to the first DeepHyd bridge-detection model or can also support another hydraulic-structure segmentation task, we added an additional experiment using the second tier of the DeepHyd project. This second DeepHyd model is a separately trained bridge-component segmentation model applied to bridge-only point clouds, with classes including deck, retaining wall, pillar, and railing. The same IDW label-propagation procedure was applied after this fixed model produced class-score outputs; no new neural network training or architectural modification was introduced for this additional evaluation.

For the DeepHyd bridge-component model, OA ranges from 70.3% to 70.9% and mIoU ranges from 41.0% to 42.2% across the evaluated IDW settings. Compared with the first DeepHyd model, the bridge-component task shows a slightly larger response to the IDW post-processing step, suggesting that component-level segmentation may benefit more from incorporating broader spatial neighborhoods during full-scene label reconstruction, as shown in Figure 8.

The class-wise results further support the interpretation that label propagation is class-dependent (see Figure 9). The observed improvements are approximately 0.2 percentage points for railing, 0.35 percentage points for deck, 1.0 percentage point for retaining wall, and 3.5 percentage points for pillar. The larger improvement for pillar is consistent with the geometric complexity and spatial discontinuity of bridge-component point clouds, where nearby contextual evidence can help refine labels assigned to underrepresented points.

The optimal response for different components emerges at different neighborhood sizes. Performance begins to stabilize around 12 nearest neighbors for pillar, 16 for deck, and 29 for retaining wall. Railing shows a different trend: its performance decreases as more neighbors are included and favors higher distance-decay power values, indicating that elongated and thin components may depend more strongly on very local neighborhoods. These results are consistent with the first DeepHyd model in showing that different semantic classes respond differently to spatial neighborhood and distance-decay settings.

Taken together, the DeepHyd bridge-component experiment provides additional application-level support for the proposed spatial post-processing perspective. It does not establish generalization to all public point-cloud benchmarks, but it shows that the same deterministic and training-free IDW propagation procedure can be applied to a second hydraulic-structure segmentation task using a separately trained DeepHyd model.

5. Discussion

5.1. Label Propagation as a Spatial Post-Processing Problem

The results in Section 4 show that label propagation is not a minor implementation detail in the evaluated hydraulic-structure LiDAR workflow. Many raw points are not directly predicted by the trained model, and the final full-scene labels therefore depend on the post-inference propagation strategy. This supports treating label propagation as an explicit spatial post-processing problem rather than only as a heuristic completion step.

Recent progress in bridge point cloud understanding is rapidly moving toward domain-knowledge integration, efficient transformer backbones for large-scale infrastructure segmentation, and training-free or low-label pipelines that leverage foundation models [55,56,61]. Nevertheless, many large-scale inference workflows still require partitioning and sampling; therefore, a robust label propagation strategy remains necessary to ensure full-scene labeling coverage. From this scope, the proposed interpolation-based surrogate can be treated as a post-processing step that can complement fully supervised models as well as emerging zero-shot pipelines by providing spatially informed label propagation for underrepresented points. Overall, these findings support the interpretation of label propagation as a spatial surrogate modeling problem rather than a purely heuristic assignment step. This statement is not intended to imply that every modern architecture relies on the same propagation mechanism. Rather, it highlights the practical need to make full-scene reconstruction steps explicit in application workflows, especially in data-scarce domains such as hydraulic-structure inspection, where users may need to reuse available pre-trained models while still producing complete and spatially reliable point-wise labels for site-specific LiDAR data.

5.2. Practical Value for Hydraulic-Structure LiDAR Applications

The practical value of the proposed strategy lies in its ability to improve full-scene label reconstruction without retraining the segmentation model in real-world applications. A trained model may already be publicly available from an earlier project or from an existing inspection workflow, for example, DeepHyd project mentioned in this study. However, the original training data of the project may not be published due to confidentiality considerations; moreover, acquiring and labeling sufficient data for training a new deep learning-based model is costly. Therefore, retraining the model for application in hydraulic structures is often difficult. In this situation, a training-free spatial post-processing step can provide a practical way to improve the final full-resolution labeling while preserving the original trained model.

This point is especially important for terrestrial LiDAR scans of bridges and hydraulic structures. Point density is often highly uneven because it depends on scanner position, range, incidence angle, occlusion, and the accessibility of different structural components. As a result, points near the scanner may be much denser than points on distant decks, piers, railings, retaining walls, or other hydraulic-structure elements. When a block-wise or sampling-based segmentation workflow is used, the final point-wise map may depend strongly on how labels are propagated to points that were not directly predicted by the neural network. Therefore, label propagation is not only a technical implementation step; it can affect object delineation, boundary consistency, and downstream spatial measurements.

The results from the bridge-level model and the component-level DeepHyd model support this application-oriented interpretation. The first model evaluates full-scene bridge-related semantic segmentation, while the second model evaluates bridge-component segmentation for classes such as deck, railing, retaining wall, and pillar. The component-level results show that the IDW-based strategy can also improve or stabilize selected structural-component labels. This suggests that the value of the method is not limited to a single model output but is relevant to practical hydraulic-structure workflows where complete point-wise labels are needed for inspection, measurement, and object-level analysis. The favorable and stable parameter regions observed in these experiments can also provide empirically informed starting settings for comparable bridge and hydraulic-structure applications. When a limited labeled subset is available at a new site, it can be used to calibrate k and p for that specific application. When no target-site labels are available, the parameter patterns identified in this study can serve as practical rules of thumb, although their site-specific optimality cannot be guaranteed without reference data.

The method is also easy to deploy once $k$ and $p$ have been specified. Applying IDW does not require changing the network architecture or retraining the original model. It requires only the spatial coordinates of the raw points and the class-score information from the fixed model output. Accuracy-based, site-specific calibration of $k$ and $p$ requires labeled reference data; however, this remains a lightweight, low-dimensional parameter-selection step involving only two deterministic post-processing parameters rather than optimization of the neural network’s learned weights. When target-site labels are unavailable, the empirically favorable parameter settings identified in this study can be used as initial rules of thumb for comparable applications. This makes the method suitable for applied workflows where practitioners have project-specific LiDAR data and an existing trained model, but do not have the time, sufficient labels, or computing resources needed to rebuild the segmentation model for each new site.

5.3. Limitations and Future Work

This study serves as an application-focused proof of concept for integrating a spatial surrogate model into the label propagation stage of 3D deep learning-based point cloud semantic segmentation. While the results demonstrate the effectiveness of IDW in improving label completeness and consistency, several limitations remain, especially regarding generalization beyond the evaluated hydraulic-structure workflow.

First, the current work adopts IDW as a relatively simple and computationally efficient spatial surrogate model. Although effective, IDW does not explicitly model spatial variability or covariance structures. More advanced geostatistical methods, such as kriging, could potentially provide more theoretically grounded estimations by incorporating spatial covariance structures. Exploring such methods represents an important direction for future research. Second, the current implementation uses kNN in Euclidean space to define the neighborhood. Although practical for irregular point densities, Euclidean proximity does not ensure that neighboring points lie on the same physical surface. For thin or columnar components, such as railings and pillars, neighbors may be selected from the opposite side of the structure or from an adjacent surface with a different orientation, potentially causing class scores to be propagated across surface boundaries. Future work could explore normal-constrained kNN, in which neighbors are filtered or weighted according to the similarity of their locally estimated normal vectors, to reduce cross-surface propagation for slender infrastructure components. Third, the selection of key IDW parameters, including the number of neighbors (k) and the distance decay parameter (p), is determined through controlled experiments in this study for investigation purposes. However, in real-world applications, optimal parameter settings are likely to depend on the characteristics of the point cloud, such as point density, spatial distribution, and scene complexity. Developing adaptive or data-driven strategies for parameter selection remains an open challenge. Finally, our analysis reveals that different semantic classes exhibit distinct spatial contexts, suggesting that a unified parameter configuration may not be optimal across all classes. In this study, a single set of parameters is applied globally for proof of concept. Future work could investigate class-specific interpolation strategies to better capture spatial context. We also acknowledge that CRF-based refinement, graph-based propagation, consistency-based learning, adaptive kernel regression, and local kriging are important related directions. Some of these methods require changing the segmentation architecture or training procedure, while others require additional optimization or raw-point unary score design. A fair comparison with these methods would therefore require a broader experimental design than the post-inference replacement studied here. Addressing these limitations will further enhance the robustness and adaptability of spatial surrogate model in point cloud processing and strengthen its integration within 3D deep learning workflows for specific geospatial applications. Future work should evaluate whether the same spatial post-processing idea transfers to other sensors, public benchmarks, and segmentation backbones, while maintaining the distinction between training-free post-processing and trainable segmentation models.

6. Conclusions

This study highlights a relatively overlooked but practically critical step in deep learning-based semantic segmentation on large-scale point clouds: label propagation for underrepresented points resulting from data partitioning and sampling in large-scale point cloud inference. Addressing this research gap is particularly important in applications that require complete point-wise labeling for accurate geospatial object delineation and spatial measurements, such as hydraulic-structure and bridge inspection. While prior research has primarily focused on improving deep neural network architectures, training strategies, and data representations, the generation of complete point-wise labeled scenes still depends heavily on effective label propagation.

To address this research gap, this study treats label propagation as a spatial surrogate modeling problem and adopts inverse distance weighting as an alternative to the conventional nearest-neighbor baseline. The results demonstrate that this training-free spatial post-processing step can modestly improve full-scene label propagation, especially on spatially sparse or geometrically complex classes. Although the overall gains are modest, they are practically important because in large-scale inference a substantial proportion of points may not be directly predicted and must rely on label propagation for semantic assignment. From this perspective, label propagation is not merely a technical completion step but an integral component of the full semantic segmentation workflow that affects the consistency and reliability of downstream geospatial analysis.

More broadly, this study helps to bridge the gap between the limitations (i.e., incomplete predictions) of existing 3D semantic segmentation workflows due to data sampling on large-scale datasets and the demand for spatially coherent point-wise labeling that better supports geospatial object delineation and spatial measurement in real-world applications. As a result, the proposed post-processing strategy improves the consistency and reliability of semantic segmentation outputs for downstream geospatial analysis. The bridge and hydraulic-structure setting examined in this study represents a practical class of data-scarce geospatial applications, where large labeled point-cloud datasets are difficult to obtain and public benchmarks remain limited. In such contexts, the reuse of available domain-specific pre-trained models can be important for real-world deployment, especially when retraining is impractical or the original training data are unavailable. Using DeepHyd as an example, this study demonstrates that spatially informed post-inference label propagation can improve complete point-wise prediction from an existing model for site-specific LiDAR data. When limited labeled target-site data are available, they can support site-specific calibration of the IDW parameters; when such labels are unavailable, the favorable parameter patterns identified in this study can provide empirically informed starting settings for comparable bridge applications, although they should not be interpreted as universally optimal values. Future work can further examine how this spatial post-processing perspective transfers to other large-scale 3D semantic segmentation tasks, including digital twin construction, infrastructure monitoring, and broader geospatial applications.