Virtual Reassembly Method for Cultural Relic Fragments Based on Multi-Feature Extraction

Abstract

The virtual reassembly of fragmented cultural relics remains a challenging task due to incomplete contours, complex fracture geometries, and the lack of reliable accuracy evaluation when ground-truth models are unavailable. To address these issues, this study proposes an automated virtual reassembly framework based on multi-feature extraction and hierarchical fragment matching. First, contour points are extracted from fragment point clouds using neighborhood roughness analysis and further refined through a Cylinder Box-based completion strategy to recover missing contour segments. Then, multiple complementary features, including Fast Point Feature Histograms (FPFHs), Heat Kernel Signatures (HKSs), and a spatial cube-based contour shape descriptor, are jointly constructed to characterize both local geometric details and global structural properties of fragments. To improve matching efficiency and robustness, a tree-based fragment retrieval strategy combined with a coarse-to-fine registration scheme is employed to identify adjacent fragments while reducing computational complexity. In addition, a pseudo-ground-truth accuracy evaluation method is introduced to quantitatively assess cumulative reassembly errors in the absence of reliable reference data. Experiments conducted on the public Buddha head dataset demonstrate that the proposed method achieves stable and visually consistent reassembly results, with a cumulative error as low as 1.58%, while significantly reducing retrieval computations compared with exhaustive matching strategies. These results indicate that the proposed framework provides a practical and verifiable solution for the automated digital restoration of fragmented cultural relics.

1. Introduction

Cultural heritage serves as a vital testament to the development of human civilization, carrying the historical memory, cultural traditions, and spiritual values of nations. As the processes of globalization and modernization accelerate, cultural heritage preservation faces unprecedented challenges. Historical artifacts are continuously damaged under the influence of multiple factors, including natural environmental erosion [1], human destruction [2], and natural disasters [3]. With the deepening of archaeological excavations, a large number of cultural relic fragments have been unearthed, necessitating urgent and effective restoration and protection methods. The reassembly and restoration of cultural relic fragments concern not only the integrity of the artifact itself but also the accurate restoration of historical information and the inheritance of cultural value.

Traditional manual reassembly methods rely on the experience and intuition of experts, which is not only time-consuming and laborious but also heavily influenced by subjective factors, making it difficult to guarantee the accuracy and reliability of the reassembly. With the development of computer technology, image processing, and artificial intelligence, digital-based reassembly methods for cultural relic fragments have gradually become a research hotspot in the field of cultural heritage preservation, providing new technical support and methodological guidance for restoration work [4,5,6].

The emergence of 3D point cloud technology has provided an important tool for the precise recording and reassembly of cultural relic fragments [7]. Point cloud data can completely record the geometric features of the artifact surface, providing a foundation for subsequent feature extraction and matching. Liu et al. [8] pointed out that with the development of 3D laser scanning technology, its application in cultural heritage protection has gradually become mainstream, effectively solving complex structural problems of cultural relics that are difficult for traditional methods to handle. Against this backdrop, automated reassembly technology based on point clouds has emerged and become an important research direction for digital cultural heritage protection.

The development of cultural relic fragment reassembly technology can be traced back to the 1980s, with early research primarily focusing on fragment matching and reassembly based on 2D images. With the development of computer vision and graphics, researchers began exploring fragment reassembly methods based on 3D point clouds [9]. Early methods mainly relied on geometric feature matching, such as curvature-based feature point extraction and contour-based matching [10,11]. Papaioannou and Karabassi [12] proposed a method for the automated assembly of arbitrarily fractured solid artifacts by analyzing the geometric features of the artifact surface to achieve automatic matching. As technology progressed, researchers introduced more complex feature description methods. For instance, Yu et al. [13] utilized heat kernel methods to model fractured skulls, achieving automatic reassembly by analyzing the geometric properties of point cloud data. These early studies laid the foundation for subsequent cultural relic fragment reassembly technologies.

Entering the 21st century, with the rise of deep learning technology, cultural relic fragment reassembly has entered a new stage of development. Gao and Geng [14] proposed a 3D Terracotta Warrior fragment classification method based on deep learning and template guidance, utilizing convolutional neural networks to automatically extract features and avoid the tedious process of manual feature extraction. Ming et al. [7] conducted a systematic review of the application of virtual reality technology in ancient ceramic restoration, noting that virtual reality can effectively assist restoration work and improve precision and efficiency. Wang et al. [15] introduced a deep learning architecture combining a Local Geometric Perception Mechanism and a convolutional–Transformer hybrid module to effectively address the irregularity in cultural heritage data. Pinho et al. [16] utilized independent network branches to predict rotation and translation parameters, enabling them to accurately move a sherd from its canonical position to the expected normalized position relative to the vessel’s coordinate system. Despite these technical breakthroughs, the efficacy of deep learning in this field is currently limited by the high requirement for large volumes of training data. Due to incomplete archaeological records, real-world cultural heritage data is often scarce, leading researchers to turn to virtual shattering procedures to generate sufficient synthetic data to meet training demands.

Feature extraction is the core component of the fragment reassembly process, directly influencing the precision and efficiency of the results. Feature extraction methods fall into three primary categories: contour feature extraction, surface feature extraction, and texture feature extraction. Contour feature extraction focuses on the boundary shapes of fragments, achieving matching by analyzing the geometric properties of the contour lines. Yang et al. [17] proposed a cultural heritage fragment matching method based on thickness and contour feature constraints, achieving precise matching by combining these two features. Surface feature extraction focuses on the geometric properties of the artifact surface, such as curvature and normal vectors, to describe local surface shapes. Wang et al. [18] proposed a method for generating cultural heritage feature lines from point clouds, achieving precise generation through point cloud segmentation and feature point extraction. Texture feature extraction focuses on visual information on the artifact surface, such as color and patterns. Zheng et al. [19] proposed an ancient ceramic restoration method based on image processing and texture stitching, achieving precise fragment reassembly by analyzing surface texture features. Yang et al. [20] proposed a fast laser point cloud registration algorithm that integrates normal vector preprocessing with 3D-SIFT feature extraction, utilizing fast point feature histogram (FPFH) descriptors and a point-to-plane ICP algorithm with a symmetric objective function to significantly enhance registration efficiency while maintaining high accuracy. Similarly, Yoo et al. [21] developed an incremental 3D reassembly framework called Structure-from-Sherds++ that employs multi-graph beam search and axis-based edge line descriptors to achieve robust, high-precision automated reassembly of mixed and unordered axially symmetric pottery fragments without requiring prior information such as the base.

Despite these advancements, existing reassembly methods still have several deficiencies. Regarding contour point extraction, existing methods often rely on global curvature calculations, making it difficult to identify local feature points accurately. This is particularly true for fragments with complex shapes and irregular contours, where problems such as false extraction or omission of feature points frequently occur. In terms of feature description, most existing methods employ single-feature descriptions, which fail to comprehensively describe the geometric properties of the fragments, thereby limiting matching accuracy. Liu et al. [22], in their study on bone fragment image stitching, pointed out that traditional contour matching methods have low precision when dealing with irregular contours and require the introduction of more complex feature description methods. Regarding the verification of reassembly accuracy, existing methods lack effective inspection mechanisms, making it difficult to accurately evaluate the reliability of the results. Zeng et al. [23] noted in their research on point cloud registration and fusion algorithms that accumulated errors during the reassembly process are difficult to quantify, which affects the credibility of the final results.

To address the above limitations, this study proposes an automated virtual reassembly method for cultural relic fragments based on multi-feature extraction and hierarchical matching. Unlike existing approaches that rely on incomplete contour information, single-feature descriptors, or qualitative evaluation, the proposed method integrates refined contour extraction, complementary feature representation, and quantitative accuracy verification into a unified framework.

This study proposes an automated virtual reassembly framework for cultural relic fragments based on multi-feature extraction and hierarchical matching. Compared with prior methods that address contour extraction, feature description, retrieval efficiency, and accuracy evaluation in isolation, the proposed framework unifies these four aspects into a coherent pipeline.

The main contributions of this study can be summarized as follows:

(1)

A refined contour point extraction strategy based on a Cylinder Box model is proposed. Unlike conventional methods that rely solely on curvature thresholding or normal vector angle analysis and frequently fail to recover missing contour segments, the proposed strategy explicitly identifies gap endpoints via resultant force analysis and recovers missing contour points by constructing cylindrical bounding boxes along supplementary line segments, yielding accurate and topologically complete contour representations for fragments with complex fracture geometries.

(2)

A multi-feature description framework is constructed by jointly integrating fracture-surface geometric features (FPFH), manifold-based texture features (HKS), and a novel spatial cube-based contour shape descriptor. In contrast to prior work that typically employs a single descriptor type—either local geometric features or global shape signatures—the complementary combination proposed here simultaneously captures local fracture geometry, surface manifold structure, and global contour shape, substantially improving matching discriminability in the presence of surface texture ambiguity and irregular fracture patterns.

(3)

A tree-based fragment retrieval strategy combined with a coarse-to-fine registration scheme is developed. Whereas exhaustive pairwise retrieval requires O(n²) computations and existing accelerated strategies such as AFSF reduce this only modestly, the proposed hierarchical tree structure dynamically narrows the candidate field at each retrieval tier, reducing the total number of matching computations to 27% of exhaustive retrieval (36 versus 132 computations for 12 fragments) while preserving reassembly accuracy.

(4)

A pseudo-ground-truth accuracy evaluation method is introduced, providing a quantitative means to assess cumulative reassembly errors in scenarios where reliable reference models are unavailable.

Experimental results on public cultural relic datasets demonstrate that the proposed framework achieves a cumulative reassembly error of 1.58%, significantly lower than the 5.19%, 4.21% and 3.21% reported by the three most closely related baseline methods—while reducing retrieval computations by 73% relative to exhaustive matching. These results establish that the unified integration of refined contour extraction, complementary multi-feature description, hierarchical retrieval, and self-contained accuracy evaluation constitutes a meaningful advance over prior approaches that address each of these aspects in isolation.

2. Experimental Data and Preprocessing

2.1. Overall Technical Framework

The experimental data used in this study consist of the public Buddha head fragment dataset from the University of Vienna and the public simulation data from Stanford University. The Buddha head fragment data possess a certain thickness and fracture surface features that cannot be ignored, as well as rich surface texture features; these are movable, individual cultural artifacts of moderate volume. The Stanford public models are all historical artifact models.

Using fragment contour points as the basis for reassembly, the primary research content of this paper includes: point cloud preprocessing of fragments, extraction of fragment contour points, construction of feature descriptors, retrieval of adjacent fragments, and reassembly of cultural relic fragments. The specific technical roadmap is illustrated in Figure 1. The main innovations are as follows:

• Refined Extraction of Fragment Contour Points via Cylindrical Bounding Boxes: Initial extraction of fragment contour points is achieved based on roughness. Then, the endpoints of missing segments are determined according to the magnitude of the resultant force exerted by surrounding points on the sample points. Subsequently, a missing segment is selected under constraint conditions, and a cylinder with radius r is constructed using this segment as the axis. Using this cylinder as a bounding box, point clouds within the box are identified as contour points, while those outside are classified as non-contour points. During this process, to avoid erroneous extraction of feature points, the cylinder radius r should be kept as small as possible within a reasonable range.
• Proposed Spatial Cube Contour Shape Description Method: A cube is constructed centered on a contour point. This cube consists of n³ small cubes, where each small cube constitutes a voxel. Different values are assigned based on whether point clouds exist within the voxels to represent the contour shape in the neighborhood of the contour point. This shape can be represented as a [1 * ×n³] dimensional vector.
• Proposed Pseudo-Ground Truth Accuracy Verification Method: The reassembled cultural relics contain cumulative errors. This method utilizes the reassembly results of adjacent fragments to obtain the cumulative error, thereby calculating the pseudo-ground truth of the cultural relic fragments. Matching accuracy is obtained by comparing the reassembly error—before matching constraints are judged—with the pseudo-ground truth, thus verifying the effectiveness of the proposed algorithm.

2.2. Experimental Data

In the virtual restoration process of cultural relic fragments, 3D laser scanners are typically utilized to acquire and process point cloud data. Based on the principle of laser ranging, the 3D laser scanner emits a laser and receives the reflected signal to obtain information such as distance and angle of the scanned object, thereby deriving the spatial coordinates of the scanned points. This scanning method allows for the rapid and convenient acquisition of high-precision point cloud data, including X, Y, and Z coordinates and RGB color information. Compared to 2D images, point cloud data contain richer spatial information, enabling better extraction of the concave–convex features of the fracture surfaces and the texture features of surface carvings. Furthermore, 3D laser scanners can acquire data without contact, avoiding secondary damage to the fragments, and are widely used in the virtual reassembly and restoration of cultural relic fragments [24,25,26].

The experimental data in this study were obtained using a Minolta VIVID-900 3D laser scanner manufactured in Tokyo, Japan. The VIVID-900 3D laser scanner performs scanning operations under a laser beam and obtains the spatial relative position coordinates of the point cloud based on the principle of triangulation, which can be used to improve high-precision measurement of small objects. The equipment is shown in Figure 2. This scanning device has high precision, capturing 200,000 points per scan with a point accuracy within 0.05 mm and a point-to-point distance of 0.1 mm. It is capable of capturing fragment features with high precision and detailing fragment textures.

2.3. Data Preprocessing

Voxel-based point cloud simplification methods are divided into two types: voxel centroid downsampling and voxel center downsampling. First, the spatial points are voxelized—that is, all point clouds are partitioned into a grid where each small cell is called a voxel. After voxelization, the cells contain certain points; the centroid of the points in each non-empty voxel is taken to represent all points within that voxel [27].

In the voxel centroid simplification method, the formula for calculating the voxel centroid P_cen (x_cen, y_cen, z_cen) is as follows:

$$\left\{\begin{array}{c}{x}_{\mathit{cen}}={\displaystyle \frac{{\sum }_{i=1}^m{x}_i}{m}}\\ y_{\mathit{cen}}={\displaystyle \frac{{\sum }_{i=1}^m{y}_i}{m}}\\ z_{\mathit{cen}}={\displaystyle \frac{{\sum }_{i=1}^m{z}_i}{m}}\end{array}\right.$$

(1)

where m is the number of point clouds within the non-empty voxel, and (x_i, y_i, z_i) are the coordinates of the point clouds within the voxel.

In the voxel center simplification method, the formula for calculating the voxel center P_cen (x_cen, y_cen, z_cen) is as follows:

$$\left\{\begin{array}{c}{x}_{cen}={x\left(col-0.5\right)}_{min}\\ y_{cen}={y\left(row-0.5\right)}_{min}\\ z_{cen}={z\left(lay-0.5\right)}_{min}\end{array}\right.$$

(2)

where d is the edge length of the voxel, (col, row, lay) are the column, row, and layer numbers of the point cloud within the voxel, and (x_min, y_min, z_min) are the minimum coordinate values.

Consequently, the effect of point cloud simplification is directly related to the size of the voxel grid. A larger grid range results in fewer simplified point clouds and fewer preserved features, whereas a smaller grid range results in more simplified point clouds and more preserved features. Voxel-based point cloud simplification is efficient and produces a uniform distribution of point clouds, allowing the simplified point cloud spacing to be constrained by controlling the voxel grid size.

This study adopts the voxel centroid-based simplification method and tests four sampling distances, 0.004, 0.005, 0.006, and 0.007, shown in Figure 3. Through comparison, it was found that at a sampling distance of 0.004, the point cloud features are clear but the number of points is high. At sampling distances of 0.006 and 0.007, the number of points decreases, but features become blurred. Therefore, a sampling distance of 0.005 was selected for simplifying the point clouds of all cultural relic fragments.

3. Extraction of Cultural Relic Fragment Contour Points

After obtaining the 3D point cloud of the fragments, feature points can be extracted. Fracture surface feature points are located in the damaged regions of the fragments. Therefore, accurate extraction of these feature points is crucial for subsequent processing. This paper classifies the structure of cultural relic fragments with thickness into: fracture surfaces, original fragment surfaces, and fracture surface contours. For hollow artifacts, the classification is further refined: original surfaces are divided into inner and outer original surfaces, and fracture surface contours are divided into inner and outer contours, as shown in Figure 4.

In this study, fragment contour points are selected as feature points, and a contour point extraction method proceeding from initial extraction to refined extraction is proposed. First, initial extraction is performed based on the roughness of the point cloud neighborhood, followed by the exclusion of pseudo-contour points based on the number of neighboring points. Finally, the cylindrical bounding box method is proposed for refined extraction to obtain complete contour features. Experimental verification on fragment samples ensures the effectiveness and accuracy of the extraction method.

3.1. Initial Extraction of Contour Points

When feature points reach a certain degree of sharpness, the points within their spherical neighborhood are concentrated on one side, resulting in significant neighborhood roughness. In point cloud model $p$, a spherical neighborhood of radius $r$ is constructed for any point $p_i$, and a covariance matrix is built with the neighboring points within radius $r$ to obtain the eigenvalues of the matrix. The covariance matrix measures the correlation between multi-dimensional random variables. In a point cloud model, each point can be viewed as a random variable, and the covariance matrix describes the degree of correlation between these points. The eigenvalues and eigenvectors of the covariance matrix identify the primary feature directions of the data; the variance along these directions corresponds to the eigenvalues. Consequently, whether a point cloud is a feature point can be determined by these eigenvalues. The specific determination process is as follows:

Step 1: Construct a spherical neighborhood of radius r and obtain the points within this neighborhood for point cloud $p$, denoted as $N\left(p\right)$. The relationship is shown in Equation (3), where $p$ is the point cloud model, $N\left(p\right)$ is any point within the model, and $p_j$ are the neighboring points of $p$ within radius $r$:

$$N(p)=\left\{p_j\left|p_j\in P,‖p_j-p‖\right.\le r\right\}$$

(3)

Step 2: Calculate the centroid m of the neighboring points $p_j$ according to Equation (4), and construct the covariance matrix C according to Equation (5), where $k$ is the number of points within the neighborhood:

$$m={\displaystyle \frac{1}{k}}{\sum }_{j=o}^k{p}_j,$$

(4)

$$C={\displaystyle \frac{1}{k}}\sum _{p_j\in N\left(p\right)}\left(p_j-m\right){\left(p_j-m\right)}^T$$

(5)

Step 3: Decompose the covariance matrix to obtain its eigenvalues λ₁, λ₂, and λ₃, where λ₁ ≤ λ₂ ≤ λ₃.

Step 4: Compare the maximum eigenvalue λ₃ with a threshold a. If λ₃ > a, point p is an initial feature point; if λ₃ < a, point p is a non-feature point.

Step 5: Traverse all points in point cloud $p$ sequentially to obtain the initial feature points.

Step 6: Finally, eliminate pseudo-contour points based on the number of points in the neighborhood, as shown in Figure 5.

3.2. Refined Extraction of Contour Points

Initial extraction results show that while the extracted feature points belong to the contour, significant portions of both the inner and outer contour lines are missing, making it impossible to accurately describe the contour shape. Consequently, this paper proposes the Cylinder Box method to supplement the contour points based on the initial extraction, thereby perfecting the contour features and achieving refined extraction.

Analysis of the missing portions reveals that missing contour points are mostly clustered near the lines connecting existing points; these lines are termed “supplementary line segments”. The core idea of the Cylinder Box method is to use the supplementary line segment as the axis and select an appropriate radius r to construct a small cylindrical box. The contour points $Q\left(p\right)$ satisfy Equation (6):

$$Q\left(p\right)=\left\{p\in P,p\in O\left(L,r\right)\right\}$$

(6)

where $Q\left(p\right)$ is the set of feature points refined via the Cylinder Box method. The condition defines a cylinder with central axis $L$ and radius $r$.

Compared with the original fragment point cloud, points located within the cylindrical bounding box are identified as contour points. To ensure refined extraction, radius $r$ must be the minimum value within a reasonable range. For regions with large contour gaps, multiple iterations can be performed. Feature points extracted through this method belong to the original point cloud P and fall within the cylindrical bounding box defined by axis $L$ and radius $r$.

The specific steps of the Cylinder Box method are:

Step 1: Locate endpoints of supplementary line segments. The segment to be supplemented is denoted as L, with endpoints l₀ and l₁, as shown in Figure 6. As seen in the magnification of l₀ in Figure 7, the distribution range of neighboring points around l₀ is small, while the range around non-endpoint points is larger, occupying half or more of the circular neighborhood area. Thus, when neighboring points exert an outward “pulling force” on the sample point, the resultant force at the segment endpoints is larger than at non-endpoint points. To avoid errors caused by point cloud distance and non-uniform density, the resultant force is calculated as follows:

For any point P₀ in the initial contour set P, find the k nearest points P_i (i = 1, 2, …, k) and calculate the resultant force using Equation (7). The factor 1/k normalizes the number of surrounding points to avoid errors from non-uniform distribution. The latter part of the formula normalizes the tension from various directions to prevent inconsistent force magnitudes due to differing distances shown in Figure 8. Points with a resultant force greater than a threshold θ are selected as the set of line endpoints l_i. Since few endpoints are needed, θ should be set to a relatively large value.

$$F(P_0)={\displaystyle \frac{1}{k}}\left|\sum _{i=1}^k{\displaystyle \frac{\overrightarrow{P_0{P}_i}}{\left|\overrightarrow{P_0{P}_i}\right|}}\right|$$

(7)

Step 2: Determine supplementary line segments. For any endpoint l₁ in L_i, the direction of the resultant force is determined as shown in Figure 9. For endpoint l₁ with force direction $\overrightarrow{F_{l1}}$ and another endpoint l₂ with force direction $\overrightarrow{F_{l2}}$, the supplementary segment is the vector $\overrightarrow{l_1{l}_2}$. Three constraints must be met:

The angle α between $\overrightarrow{F_{l1}}$ and $\overrightarrow{l_1{l}_2}$ approaches a large value.

The angle β between $\overrightarrow{l_1{l}_2}$ and the vector $\overrightarrow{F_{l2}}$ approaches a small value.

The point cloud roughness differs on either side of the vector projection.

Step 3: Define the cylindrical bounding box. Construct a cylindrical bounding box with the supplementary segment as the axis and radius r. Points within the box are determined to be contour points. If r is too large, it causes erroneous extraction; if too small, refined extraction cannot be achieved. Thus, r must take a small value within a reasonable range.

Step 4: Select refined extraction sequence. Apply the method starting with the shortest supplementary segments, then proceed based on length. For very long gaps, the algorithm should be cycled to fill the gap repeatedly.

Many thresholds and parameter settings are required throughout the experiment, as shown in

Table 1. Threshold setting parameters.

Parameter	Value
n	50
ω	0.01
a	17.8
θ	0.9
r	0.02

.

The specific technical route is shown in Figure 10.

4. Multi-Feature Description

Feature points alone cannot satisfy the requirements of fragment reassembly; matching requires considering feature similarity by constructing feature descriptors. Feature descriptors possess strong descriptiveness for feature points and exhibit rotation and scale invariance.

Given the significant differences between fracture surfaces and original surfaces, and the regular distribution of surface textures, this study calculates multiple features for contour points. This includes: Heat Kernel Signatures (HKSs) for the original surface in the neighborhood of feature points [28], geometric features for the fracture surface in the neighborhood, and contour descriptors constructed for the feature points themselves.

4.1. Geometric Feature Description

Geometric features effectively describe the surface distribution of the point cloud neighborhood, primarily through geometric relationships such as the direction, angle, and distance of the lines connecting feature points and their neighbors [29]. This paper utilizes Fast Point Feature Histograms (FPFHs) to calculate the geometric features of the fracture surface neighborhood. FPFH is an improvement over PFH [30]; it only calculates the geometric features of the feature point and its neighbors to obtain Simple Point Feature Histograms (SPFH). Then, calculations are performed for each neighbor and its k surrounding points, with the results weighted and averaged according to Equation (8).

$$FPFH\left(p_s\right)=SPFH\left(p_s\right)+{\displaystyle \frac{1}{k}}\sum _{i=1}^k{\displaystyle \frac{1}{{\omega }_k}}\cdot SPFH\left(p_t\right)$$

(8)

where ω_k is the weight, which is related to the distance from the feature point to the neighboring point.

4.2. Heat Kernel Feature Extraction

Manifold computations typically utilize mesh models. This study uses the Delaunay triangulation mesh reconstruction method on the simplified point clouds, as shown in Figure 11; the resulting model effectively avoids loss of detail and preserves surface texture shapes.

The Heat Kernel Signature (HKS) is a descriptor used for 3D models. It captures local features over short time durations and global features over long durations, effectively representing texture information and manifold shapes around feature points. To calculate HKS for the original fragment surface near feature points, the mesh Laplace operator is first used to calculate eigenvalues and eigenvectors for the contour points. Then, the HKS is restricted to the time domain according to Equation (9), and the remaining heat of the feature point is calculated at identical time intervals to form the descriptor.

$${\mathrm{h}}_{\mathrm{t}}\left(\mathrm{x},\mathrm{x}\right)=\sum _{\mathrm{i}=0}^{\mathrm{\infty }}\exp \left(-{\mathsf{\lambda }}_{\mathrm{i}}\mathrm{t}\right){\phi }_{\mathrm{i}}^2\left(\mathrm{x}\right)$$

(9)

In the formula, $\exp \left(-{\mathsf{\lambda }}_{\mathrm{i}}\mathrm{t}\right)$ is a low-pass filter whose cutoff frequency is determined by ${\lambda }_i$.

4.3. Construction of Contour Shape Descriptors

In manual restoration, contour shape is the primary basis for reassembly. Therefore, this paper further describes the contour shape based on refined extraction.

A cube of size n × n × n is established centered on any contour point, consisting of n³ small cubic voxels with edge length d. If a voxel contains point clouds, its value is set to 1; otherwise, it is set to 0. Increasing n expands the description range, while decreasing d makes the description more refined. The specific method is:

Step 1: For two fragments P and Q, establish n × n × n cubes centered on their contour points to obtain contour shape descriptors Con(pi) and Con(qi). As shown in the 2D plane in Figure 12 (using a 6 × 6 grid), the descriptor for a feature point can be represented as a vector.

Step 2: When matching using the Contour Shape Descriptor, the difference between Con(pi) and Con(qi) is calculated using Equation (10), and the norm of the result is taken to obtain the final matching result φ:

$$\phi ={‖Con\left(p_i\right)-Con\left(q_i\right)‖}_1$$

(10)

Step 3: Set a threshold ω. If φ < ω, points p_i and q_i are considered a feature point pair; if φ > ω, they are considered a non-feature pair.

5. Cultural Relic Fragment Reassembly

This study proposes a tree-based fragment retrieval method that utilizes a strategy ranging from coarse-to-fine registration to progressively reassemble cultural relic fragments. Furthermore, a pseudo-ground truth accuracy verification method is proposed, which leverages matching errors to derive the pseudo-ground truth of the fragments and subsequently calculate the reassembly precision.

5.1. Tree-Based Fragment Retrieval Method

In the process of identifying adjacent fragments, the feature matching degree is calculated for two sets of features. If the matching degree exceeds a certain threshold, the two fragments containing these feature sets are determined to be adjacent. To ensure the accuracy of fragment reassembly and avoid omissions or mismatches, a one-by-one traversal of all fragments is typically performed.

Consider a set of unknown fragments numbered as field A, containing n fragments: A(a₁, a₂, a₃, a₄, …, a_n). To improve matching efficiency, the strategy of excluding already-retrieved objects is considered. For the fragment field A, using fragment a₁ as the first object to be reassembled, the adjacent fragment a₂ is found after n − 1 loops. Then, using a₂ as the second object, the remaining fragments (a₃, a₄,…,a_n)in the field are retrieved, completing the second retrieval after (n − 2) loops. After traversing all fragments, the total computational load is A(n − 2).

Building upon this, this paper proposes a tree-based fragment retrieval method that dynamically changes the retrieval field and constraint rules during the process to enhance efficiency. As shown in Figure 13, for the fragment field A, the fragment with the largest number of points is selected as the first sub-fragment. Matching degree calculations are performed sequentially for the remaining (n − 1) fragments to identify the top L adjacent fragments with the highest matching degrees. Subsequently, these L fragments serve as the second-tier sub-fragments, and matching degree calculations are performed against the remaining (n − 1 − L) fragments to find the top M adjacent fragments with the highest matching degrees. After the retrieval is complete, any remaining fragments in the field—which may have low matching degrees due to small fracture surface areas—return to traverse all fragments except themselves to select the fragment with the highest matching degree as the neighbor.

In this method, l and m are matching parameters, representing either the number of fragments or the matching degree. The fragment matching degree is defined as the ratio of matching point pairs between two fragments to the number of feature points in the sub-fragment.

5.2. Bidirectional Hausdorff Distance Determination Method

While Euclidean distance is commonly used to determine feature matching degrees, the varying number of feature points across fragments necessitates a more robust approach. This paper adopts the bidirectional Hausdorff distance method. In point cloud computation, the Hausdorff distance essentially represents the maximum distance from one set of point clouds to the nearest point in another set. The calculation steps are as follows:

Step 1: Given two point cloud sets, A(a₁, a₂, a₃, …, a_i) and B(b₁, b₂, b₃, …, b_j), where i ≠ j, select any point a_i from set A and calculate the distance to points in B to find the nearest point. As shown in Figure 14, a_i is compared with every point in set B to find the nearest point b₁.

Step 2: Traverse all points in set A to obtain the set of shortest distances d.

Step 3: Select the maximum distance h from set d. This is the unidirectional Hausdorff distance from A to B, denoted as h(A, B)24. The formula is expressed as:

$$h\left(A,B\right)=\underset{a\in A}{max}\left\{\underset{b\in B}{min}‖a-b‖\right\}$$

(11)

Step 4: Swap the roles of sets A and B to calculate the unidirectional Hausdorff distance from B to A, denoted as h(B, A).

$$h\left(B,A\right)=\underset{b\in B}{max}\left\{\underset{a\in A}{min}‖b-a‖\right\}$$

(12)

Utilizing the bidirectional Hausdorff distance effectively avoids the impact of differing feature point counts between two fragments, thereby improving reassembly precision.

5.3. Cultural Relic Fragment Registration

Since a single-step reassembly often fails to meet precision requirements, this paper adopts a coarse-to-fine registration strategy. The SAC-IA algorithm is used for coarse registration, followed by the ICP algorithm for fine registration [31].

The SAC-IA algorithm first downsamples the point cloud, extracts normal vectors and FPFH features, and determines matching point pairs based on the feature distance of the points to solve for the rigid transformation matrix. The process iterates until the stop condition is met. After coarse registration, the two fragments possess transformed coordinate poses but still retain some reassembly error; therefore, the ICP algorithm is employed for precise fine registration.

5.4. Pseudo-Ground Truth Accuracy Verification Method

Traditional accuracy verification relies on comparing experimental errors with true values. However, due to the unique nature of cultural relic fragments, true values are often missing during experiments. This paper proposes a pseudo-ground truth accuracy verification method. As illustrated in Figure 15, if Head05 and Head02 are determined to be adjacent, and Head02 and Head04 are also adjacent, the matching relationship between Head05 and Head04 remains unknown. Accuracy verification is therefore conducted using these adjacent fragments with unknown matching relationships. The matching error is derived based on the distance difference in matching point pairs in the reassembled artifact.

The total length of the reassembled Buddha head is measured, which includes the reassembly errors. By calculating the matching error from the matching point pairs of adjacent fragments, the pseudo-ground truth of the artifact is obtained. The calculation steps are as follows:

For two adjacent fragments P and Q with feature points P(p₁, p₂, p₃, …, p_k) and (q₁, q₂, q₃, …, q_k), where (p_k, q_k) represents a matching pair, the matching errors along the x, y, and z axes are calculated:

$${\theta }_x={\displaystyle \frac{{\sum }_{i=1}^k\left(x_{pi}-x_{qi}\right)}{k}}$$

(13)

$${\theta }_y={\displaystyle \frac{{\sum }_{i=1}^k\left(y_{pi}-y_{qi}\right)}{k}}$$

(14)

$${\theta }_z={\displaystyle \frac{{\sum }_{i=1}^k\left(z_{pi}-z_{qi}\right)}{k}}$$

(15)

The average error θ for the two fragments is then calculated:

$$\theta =\sqrt{{\theta }_x^2+{\theta }_y^2+{\theta }_z^2}$$

(16)

Using the obtained matching error, the total length of the reassembled Buddha head is adjusted to find the pseudo-ground truth $h$:

$$h=\sqrt{{\left(x-{\theta }_x\right)}^2+{\left(y-{\theta }_y\right)}^2+{\left(z-{\theta }_z\right)}^2}$$

(17)

where x, y, and z are the projected lengths of the reassembled head along each axis. Finally, the fragment reassembly error $\omega$ is calculated as:

$$\omega ={\displaystyle \frac{\theta }{h}}\times 100\%$$

(18)

6. Experimental Results and Analysis

6.1. Contour Point Extraction Results

Fragments Head02 and Head03 were selected for experimental testing and compared with methods based on normal vector angles and point cloud roughness shown in Figure 16 and Figure 17. It was clearly observed that contour points obtained via those traditional methods suffered from significant loss and could not effectively describe the fragment shapes. After supplementation using the Cylinder Box method, small gaps were completely filled in a single pass, while larger gaps were resolved through iterative supplementation. The resulting contour points accurately describe the fragment morphology.

6.2. Fragment Reassembly Experimental Results

The fragment retrieval results for 12 fragments are shown in Figure 18. With Head05 as the starting point for the first retrieval, the three most compatible fragments (Head02, Head06, Head07) were identified. These were then used as starting points for the second retrieval against the remaining 8 fragments. In the third retrieval, with a 1% matching degree threshold, Head01 was identified as the neighbor for Head09. Finally, the remaining Head11 and Head12 traversed all fragments to complete the fourth retrieval.

The results indicate that except for Head11 and Head12, which required global calculations, most fragments involved only partial calculations, significantly reducing processing time and improving efficiency.

Table 2. Accuracy test results.

Serial Number	Matching Relationship Unknown Point Pairs	Average Distance Difference θ	Splicing Error ω/% (The Value of a Genuine Buddha Head h = 5.8296)
1	head01–head02	0.0011	0.0189
2	head01–head03	0.0087	0.1492
3	head01–head06	0.0026	0.0446
4	head02–head12	0.0117	0.2007
5	head03–head04	0.0150	0.2573
6	head04–head05	0.0154	0.2642
7	head05–head11	0.0123	0.2110
8	head06–head07	0.0061	0.1046
9	head07–head11	0.0023	0.0395
10	head08–head09	0.0075	0.1287
11	head09–head10	0.0073	0.1252
Cumulative error		0.09	1.54

presents the average error and reassembly error for adjacent fragments with unknown matching relationships. The pseudo-ground truth for the total length of the Buddha head is h = 5.8296. There are 11 pairs of adjacent fragments without explicit matching constraints. The maximum distance difference between feature point pairs is 0.0154, with a maximum precision error of 0.26%. The cumulative distance difference is 0.9, resulting in a cumulative reassembly error of 1.54%.

6.3. Comparative Experimental Results

The proposed tree-based fragment retrieval method was compared with one-by-one retrieval and the AFSF method. For reassembling 12 Buddha head fragments, one-by-one retrieval required 132 calculations, AFSF required 66, and the proposed tree-based method required only 36. This represents only 27% of the computational load of one-by-one retrieval and 55% of the AFSF method, demonstrating that the tree-based method effectively reduces complexity and enhances matching efficiency as

Table 3. Comparison of retrieval times.

Search Methods	Step by Step	AFSF	Tree-Structured Search
count	132	66	36
Percentage of searches for this article	27%	55%

.

A novel multi-repeated experimental procedure was designed, in which the randomization component is a coarse registration step employing random sampling of point correspondences. With all other parameters held constant, five independent runs of SAC-IA initialization were performed using different random seeds. To ensure statistical fairness, the same five-run protocol was applied to four methods to examine the stability of each method, The results are shown in Figure 19. The standard deviations of all methods are relatively low.

We evaluate all 7 possible non-empty subsets of the three feature types: three single-feature configurations, three two-feature combinations, and the full three-feature configuration. All other pipeline components are kept identical. Results are averaged over 5 runs on the Buddha head dataset. As can be seen from Figure 20, individual feature performance: FPFH alone (3.82%) outperforms HKS alone (4.21%) and Contour alone (5.63%), confirming that local geometric description is most informative for fracture surface matching. HKS provides useful topological context, while the contour descriptor alone is insufficient but complementary. Pairwise complementarity: Every two-feature combination outperforms all single-feature configurations. FPFH + Contour (2.27%) is the strongest pair, as FPFH handles interior surface geometry while the contour descriptor handles boundary shape—two non-overlapping aspects of the fragment. Marginal gain of full fusion: Adding HKS to the best two-feature combination (FPFH + Contour) further reduces error by 0.73 percentage points (2.27% → 1.54%). This demonstrates that HKS provides non-redundant global topology information that neither FPFH nor the contour descriptor captures.

The proposed method was also compared with a contour-weight-based method. Because fragments have rich surface textures but blurred contour information, traditional weight allocation often leads to errors and “offset penetration” phenomena. Furthermore, when compared with a feature area-based method, the proposed method avoids the matching offsets that occur in the former due to the similarity of descriptors in flat fracture regions shown in Figure 21.

To assess the accuracy of the proposed method, a pseudo-ground-truth-based precision evaluation was conducted to compare two representative approaches: the contour weight-based method reported in Reference [32] and the feature region-based assembly method presented in Reference [33], as summarized in

Table 4. Accumulated error comparison.( The ↓ figure shows how much the error has decreased compared to this method.)

Methods	Reference [32]	Reference [33]	Reference [34]	Ours
False true value	6.1754	5.4893	5.7012	5.8296
Cumulative difference	0.3209	0.2310	0.1830	0.0921
Cumulative error	5.19%	4.21%	3.21%	1.58%
Compare	↓ 3.61%	↓ 2.63%	↓ 1.63%	—

. The cumulative error of the contour weight-based method in Reference [32] is 5.19%, which represents an increase of 3.61% compared with the proposed method. Meanwhile, the feature region-based assembly method in Reference [33] yields a cumulative error of 4.21%, exceeding that of the proposed method by 2.63%. The Fragment Measurement Iterative Closest Point Feature Point Matching method in Reference [34] yields a cumulative error of 3.21%, exceeding that of the proposed method by 1.58%.

6.4. Limitations and Future Work

This study has been validated on public datasets (the University of Vienna Buddha Head Fragment Dataset and the Stanford University Simulation Dataset), demonstrating the effectiveness of the overall method and achieving good results, but some limitations still exist. Artifact categories such as thin-walled ceramics, flat stone tablets, or irregular bone fragments present additional challenges in contour completeness and fracture surface regularity that remain to be evaluated. Future work will extend validation to a broader range of artifact types, ideally through collaboration with cultural heritage institutions to obtain more varied scan datasets. The proposed framework relies on handcrafted multi-feature descriptors (FPFH, HKS, and the spatial cube contour shape descriptor) and does not include a systematic quantitative comparison with deep learning-based methods. Although CNN-based and point cloud network-based approaches have demonstrated competitive performance in related reassembly tasks, their direct applicability in the cultural relic domain is constrained by limited annotated training data. A comprehensive comparative study against representative learning-based baselines remains an important direction for future research.

The pseudo-ground-truth accuracy verification method depends on the availability of a sufficient number of adjacent fragment pairs with unknown relative transformations. In the present experiments, 11 such pairs were available, which provided a stable basis for cumulative error estimation. When the number of fragments is small or adjacent surface contacts are limited, the reliability of this evaluation strategy may decrease. Future work will characterize the stability boundary of this approach using synthetic datasets with a controlled number of adjacent surfaces, and will explore anchor-based complementary verification strategies for low-adjacency scenarios.

7. Conclusions

The automated virtual reassembly framework proposed in this study—integrating multi-dimensional feature extraction and heuristic tree-based retrieval—effectively overcomes the precision limitations caused by incomplete contours. First, the combination of initial extraction based on point cloud roughness and the innovative Cylinder Box method achieves precise and complete contour point extraction, significantly outperforming traditional methods based on normal vector angles or roughness. Furthermore, by fusing geometric features, Heat Kernel Signatures, and the newly designed spatial cube contour shape descriptor, a highly discriminative multi-feature description system was constructed to comprehensively represent both local and global fragment geometry. Finally, the tree-based fragment retrieval method optimizes the matching path, reducing computational complexity to 27% of that of exhaustive retrieval. Combined with the coarse-to-fine registration strategy the reassembly process remains robust and efficient. Experimental results demonstrate that the framework produces excellent visual restoration without mismatching phenomena such as penetration or translation. The cumulative reassembly error is as low as 1.58%, a significant reduction compared to mainstream methods based on contour weights and feature area partitioning. Additionally, the proposed pseudo-ground truth accuracy verification method provides a quantitative basis for evaluating reassembly reliability in the absence of true values. This research not only confirms the effectiveness of multi-feature fusion and intelligent retrieval in complex fragment reassembly but also provides a high-precision, high-efficiency technical solution for the digital restoration of cultural heritage, possessing significant theoretical value and broad application prospects.