Three-dimensional matching and digital repairing method for cultural relic fragments

Abstract

The invention discloses a three-dimensional matching and digital restoration method for cultural relic fragments, and relates to the technical field of cultural relic restoration, and the method comprises the steps: obtaining the point cloud data of each fragment belonging to the same cultural relic; constructing a triangular mesh model of each fragment by adopting a Poisson surface reconstruction method; repairing the corrosion edge in the triangular mesh model of each fragment; a multilevel matching algorithm is adopted to carry out registration on the triangular mesh model, subjected to edge restoration, of each fragment, and the optimal splicing position and attitude parameters of each fragment are obtained; according to the optimal splicing position and the attitude parameter of each fragment, splicing the triangular mesh model of each fragment after edge repair to obtain an initial cultural relic digital model; and repairing the missing area of the initial cultural relic digital model to obtain a repaired cultural relic digital model. According to the method, the precision and efficiency of cultural relic repair are improved, the risk of manual operation is reduced, and meanwhile, the traceability and reversibility of the repair steps are ensured.

Description

Technical Field

[0001] The present application relates to the technical field of cultural relic restoration, and in particular to a three-dimensional matching and digital restoration method for cultural relic fragments. Background Art

[0002] Throughout the history of human civilization, archaeological discoveries have always been an important way to explore history and understand ancient society. As one of the most eye-catching archaeological discoveries in the 20th century, the Sanxingdui site not only reveals the unique cultural features of ancient Shu, but also provides a large number of precious bronze artifacts. These artifacts not only have extremely high artistic value, but also carry rich historical information, which is of immeasurable value for the study of ancient civilization, cultural heritage and historical development.

[0003] However, after thousands of years of erosion, most of these cultural relics are broken and severely corroded, which not only affects the appreciation and research value of the cultural relics, but also brings great difficulties to the protection and restoration of the cultural relics.

[0004] In the field of cultural relics restoration, the traditional manual contrast bonding method is not only inefficient, but also has the risk of damaging cultural relics, and it is difficult to ensure the consistency of the restoration effect. At the same time, although the existing digital technology provides new ideas for cultural relics restoration, the high cost, non-portable equipment and insufficient precision have limited its wide application in the field of cultural relics restoration. In addition, the huge number of bronze fragments, irregular shapes, varying degrees of corrosion and blurred edges have further increased the difficulty of restoration work. Therefore, how to ensure the historical authenticity and integrity of cultural relics while ensuring the efficiency of restoration has become a technical problem that needs to be solved urgently in the current field of cultural relics restoration. Summary of the invention

[0005] The purpose of this application is to provide a three-dimensional matching and digital restoration method for cultural relic fragments, which can handle different types of corrosion conditions and fragments of various specifications, and achieve high-precision, high-efficiency, and damage-free restoration of cultural relics.

[0006] To achieve the above objectives, the present application provides a method for three-dimensional matching and digital restoration of cultural relic fragments, comprising:

[0007] Obtain point cloud data of each fragment belonging to the same cultural relic;

[0008] Based on the point cloud data of each fragment, a Poisson surface reconstruction method is used to construct a triangular mesh model of each fragment;

[0009] Repairing the corroded edges in the triangular mesh model of each fragment to obtain the triangular mesh model of each fragment with the edge repair completed;

[0010] A multi-level matching algorithm is used to align the triangular mesh models of each fragment that have completed edge repair, and the optimal splicing position and optimal posture parameters of each fragment are obtained;

[0011] According to the optimal splicing position and optimal posture parameters of each fragment, the triangular mesh model of each fragment with completed edge repair is spliced ​​to obtain the initial digital model of the cultural relic;

[0012] The missing areas of the initial digital model of the cultural relic are repaired to obtain a restored digital model of the cultural relic.

[0013] According to the specific embodiments provided in this application, this application has the following technical effects:

[0014] The present application provides a three-dimensional matching and digital restoration method for cultural relic fragments. By acquiring point cloud data of each fragment belonging to the same cultural relic, the problem of the difficulty in accurately acquiring three-dimensional information of cultural relic fragments in traditional restoration methods is solved, and the comprehensive digital capture of the morphological characteristics of cultural relic fragments is realized. By using the Poisson surface reconstruction method based on point cloud data to construct a triangular mesh model of each fragment and repairing the corroded edge in the triangular mesh model, the difficult problem of constructing a three-dimensional model of cultural relic fragments and restoring the integrity of the edge is solved, and the accurate construction and complete presentation of the three-dimensional model of cultural relic fragments is realized, providing more accurate boundary conditions for subsequent fragment matching and splicing. By using a multi-level matching algorithm to register the triangular mesh model that has completed edge repair, the problem of accurate matching and splicing of cultural relics fragments in three-dimensional space is solved, and high-precision alignment and posture adjustment between fragments are achieved; according to the registration results, the triangular mesh models of each fragment are spliced ​​to obtain the initial digital model of cultural relics. This step solves the problem of digital splicing of cultural relics fragments and preliminarily realizes the digital reconstruction of the entire cultural relics; finally, by repairing the missing areas of the initial digital model of cultural relics, the problem of restoring the integrity of the digital model of cultural relics is solved, and the digital model of cultural relics that has been repaired is obtained, realizing high-precision digital reproduction and protection of the original appearance of cultural relics. It can be seen that this application can not only achieve high-precision digital restoration of cultural relics, but also significantly improve the efficiency of restoration. More importantly, the entire restoration process is completely based on the digital model and will not cause any physical damage to the cultural relics fragments. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0016] Figure 1A schematic diagram of a process flow of a three-dimensional matching and digital restoration method for cultural relic fragments provided in one embodiment of the present application;

[0017] Figure 2 A detailed flowchart of a method for three-dimensional matching and digital restoration of cultural relic fragments provided in one embodiment of the present application. DETAILED DESCRIPTION

[0018] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0019] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0020] like Figure 1 and Figure 2 As shown, the present application provides a method for three-dimensional matching and digital restoration of cultural relic fragments, as shown in the following steps 101 to 106. Among them:

[0021] Step 101, obtaining point cloud data of each fragment belonging to the same cultural relic.

[0022] In an exemplary embodiment, the fragment point cloud data is obtained by a three-dimensional scanning device (3D CaMegaPCP-400). The 3D CaMegaPCP-400 three-dimensional scanner is used to quickly scan the fragments of the same cultural relic, with a scanning frequency of 0.25 seconds. The fragment surface data is obtained by irradiating the laser beam emitted by the device onto the surface of the fragment. There are 640 laser points distributed on each laser line, and 480 laser lines are emitted in one scan. More than 300,000 points of point cloud data are collected, including XYZ (three-dimensional coordinates) and RGB (color) information. The scanning accuracy is set to 0.1 mm. The modeling accuracy is 0.1 mm.

[0023] In another exemplary embodiment, it is also possible to adopt:

[0024] Structured light scanning technology, which obtains three-dimensional information by projecting structured light patterns;

[0025] Photogrammetry technology uses photos from multiple angles to reconstruct three-dimensional models.

[0026] Step 102 : Based on the point cloud data of each fragment, a Poisson surface reconstruction method is used to construct a triangular mesh model of each fragment.

[0027] In an exemplary embodiment, after acquiring the point cloud data of each fragment in step 101, the point cloud data is preprocessed, specifically including:

[0028] Noise point removal. Statistical Outlier Removal (SOR) is used to remove noise points. For each vertex in the point cloud data, the average distance between it and its neighboring points is calculated:

[0029]

[0030] Among them, μ d represents the average distance between the current vertex and its neighboring points, N' represents the number of neighboring points, and d n' Indicates the distance from the current vertex to the n'th neighboring point.

[0031] Then calculate the standard deviation σ according to the following formula d :

[0032]

[0033] If the average distance from the current vertex to its neighboring points exceeds the threshold range, that is, μ d< μ d -α'·σ d or μ d> μ d -α'·σ d , then the current vertex is marked as a noise point and deleted. Where α' is the standard deviation multiple threshold, and in this embodiment, α'=2.

[0034] Coordinate system 1. Convert the point cloud data in the local coordinate system to the global coordinate system through the rigid body transformation matrix T1:

[0035]

[0036] Where R is a 3×3 rotation matrix, t is a 3×1 translation vector, and x1, y1, and z1 represent the horizontal, vertical, and vertical coordinates of the point cloud data, respectively. The optimal transformation matrix is ​​solved using the Iterated Closest Points (ICP) algorithm:

[0037]

[0038] Among them, T * Represents the optimal transformation matrix of point cloud data in the local coordinate system, p b' represents the point cloud in the local coordinate system, q b' represents p in the global coordinate system b' The corresponding point cloud, B' represents the total number of valid corresponding point pairs involved in the registration.

[0039] Merge the overlapping point clouds. For the overlapping point cloud data, the voxelization method is used to downsample and merge. The space is divided into a cubic grid with a side length of r, and the average value of the pixels falling in the same grid is taken:

[0040]

[0041] Where M' is the number of points in the grid. In this embodiment, the voxel size is r = 0.5 mm, p m' Represents the point cloud data that falls within the same voxel (cube grid), including: the average of the XYZ coordinate values ​​(spatial position) and the average of the RGB color values ​​(color information), with p merged To represent the point cloud data value in each voxel.

[0042] Mesh model conversion. Since point cloud data is a discrete set of points, in order to obtain a continuous surface description and facilitate subsequent operations such as texture mapping, it is converted into a triangular mesh form.

[0043] In this implementation, the point cloud data of each fragment is converted into a triangular mesh model using the Poisson surface reconstruction method. First, the normal vector of each vertex is estimated:

[0044]

[0045] Among them, n p Represents the normal vector of the current vertex, N p is the neighborhood point set of the current vertex, |N p | is the number of neighborhood points, p is the current vertex, q is a point in the neighborhood of point p, q next Indicates the next point of q in the neighborhood point set of p. The value of each vertex is p merged , get the normal vector n of each vertex p Constructing a vector field The exponential function x2 can be calculated based on Poisson's equation (the goal of the solution is to define the surface of the object, the isosurface with a value of 0, that is, the surface composed of all points where x2=0, where x2>0 inside the object, x2<0 outside the object, and x2=0 on the surface of the object).

[0046] Then solve the Poisson equation:

[0047]

[0048] in, is the Laplace operator, Represents the normal vector n p The vector field The divergence of . That is, we can get an isosurface with a value of 0. Then triangulate the zero isosurface using the Marching Cubes algorithm: the first step is to divide the space into a regular cube grid; the second step is to check the scalar values ​​(indicator function values) of the 8 vertices of each cube, and determine whether it intersects with the isosurface based on the positive or negative vertex values; the third step is to interpolate and calculate the intersection points on the intersecting edges; the fourth step is to connect the intersection points to form triangles, thus obtaining a triangular mesh model.

[0049] Specifically, the first step is the division of space and the calculation of scalar values. First, the spatial region containing the point cloud needs to be divided into a regular grid of cubes, each with 8 vertices. It should be noted that these vertices are not the original point cloud data points, but the vertices of the regular grid. At each grid vertex, we need to calculate the value of the indicator function x2, which is obtained by solving the Poisson equation. When the vertex is inside the object, the value of x2 is greater than 0; when the vertex is outside the object, the value of x2 is less than 0.

[0050] The second step is to determine the intersection of isosurfaces. Check each edge of the cube. If the scalar values ​​of the two endpoints of an edge are one positive and one negative, it means that the isosurface (that is, the surface with x2=0) passes through this edge. This judgment process determines the position of the object surface in space.

[0051] The third step is intersection calculation. On the edges that are determined to intersect with the isosurface, the specific intersection positions of the isosurface are calculated by linear interpolation. Assuming that the two endpoints of the edge are v1 and v2, and their corresponding scalar values ​​are s1 and s2, the intersection position can be calculated by the formula v1+t(v2-v1), where t=s1 / (s1-s2). These calculated intersection points are newly generated points, and they are not original point cloud data points.

[0052] The fourth step is to generate a triangular mesh. According to the distribution of intersection points in the cube, a predefined triangle connection pattern (usually a lookup table with 256 cases) is used to connect the intersection points into triangles to form local surface fragments. All the triangle fragments of the cube together constitute the complete object surface.

[0053] When applied to fragments, the original point cloud data is mainly used in three aspects: calculating the normal vector field, guiding the solution of Poisson's equation, and providing approximate position information of the surface. The vertices of the final mesh model come from the intersection points calculated by the Marching Cubes (voxel-level reconstruction method, MC) algorithm. These points are optimally placed to better represent the surface of the object, rather than directly using the original point cloud data points as mesh vertices.

[0054] The above method has several advantages: it can handle noise and incomplete data well; it can generate continuous and smooth surfaces; it can naturally handle complex topological structures; and it can fill small data holes. In general, the final triangular mesh model is a continuous surface reconstructed from point cloud data, rather than directly using point cloud data points as mesh vertices, which can achieve better surface reconstruction results.

[0055] Among them, by implementing this implementation method, the triangular mesh can show the continuous surface of the fragment. Since corroded bronze may have small hole corrosion, for holes with an area smaller than the threshold S in the triangular mesh model, the moving least squares method is used to perform surface fitting to fill the holes:

[0056]

[0057] Among them, w m” (x') is the weight function:

[0058] w m” (x') = exp(-||x'-x' m” || 2 / h 2 ).

[0059] Where m'' is the number of holes whose area is less than the threshold S, h is the smoothing parameter, x' is the spatial position point in the area to be repaired, f(x') is the surface function value fitted at position x', and x' m” is a known point on the hole boundary. This process uses the information of the surrounding known points to estimate the surface shape of the hole area through weighted averaging. The final output is a completed continuous surface segment that seamlessly integrates into the original mesh model.

[0060] In this implementation, the quadratic error metric (QEM) method is used to simplify the mesh. For each edge, the cost is calculated:

[0061]

[0062] Among them, F e is the set of faces adjacent to edge e, K f is the quadratic error matrix of the patch f(x'). The edges are gradually shrunk from small to large according to the cost until the target number of patches is reached.

[0063] Through the above steps, noise points are removed, coordinate system 1 is performed, the local coordinate system is converted to a unified coordinate system, and the point cloud merging in the overlapping area is processed, the mesh model is converted, mesh holes are repaired, and mesh model simplification is performed. High-precision three-dimensional data is obtained to provide basic data support for subsequent processing.

[0064] Step 103: Repair the corroded edges in the triangular mesh model of each fragment to obtain the triangular mesh model of each fragment with the edge repaired.

[0065] In some embodiments, since the edges of the corroded bronze fragments may be blurred, it is necessary to repair the edges. For better fragment matching later, step 103 repairs the corroded edges in the triangular mesh model of each fragment to obtain the triangular mesh model of each fragment with the edge repaired, which specifically includes steps 201 to 203:

[0066] Step 201 , performing edge curvature analysis on the triangular mesh model of the fragment to obtain the curvature analysis result of each vertex in the triangular mesh model.

[0067] Step 202, based on the edge score of each vertex in the triangular mesh model and the adaptive threshold, determine the type of each vertex in the triangular mesh model, the type being a complete edge point, a corroded edge point, and a transition region point.

[0068] Step 203: repair the vertices of the eroded edge points in the triangular mesh model by using a surface extrapolation method, so as to obtain a triangular mesh model of the fragment with the edge repaired.

[0069] Wherein, in implementing this embodiment, the curvature analysis result in step 201 at least includes: average curvature and maximum principal curvature.

[0070] For each triangular patch T with p as a vertex, calculate the area of ​​the Thiessen polygon (Voronoi) region belonging to p within the patch: if the triangle is an obtuse triangle: the area is half of the triangle area; if the triangle is an acute triangle: the Voronoi region is surrounded by the perpendicular bisector perpendicular to the edge.

[0071] The calculation formula of the average curvature is:

[0072]

[0073] Among them, A Mixed (p) represents the mixed Voronoi area of ​​the current vertex, p represents the current vertex, p1 and p2 represent the first adjacent vertex and the second adjacent vertex of the current vertex respectively, T represents the triangle with p, p1 and p2 as vertices, S T represents the area of ​​the triangle, α and α' respectively represent the angle with the first adjacent vertex as the vertex and the angle with the second adjacent vertex as the vertex, H(p) represents the average curvature of the current vertex, N(p) represents the vertex set in the preset neighborhood of the current vertex p, α j and β jThey represent the angle with the current vertex as the vertex and the angle with the second adjacent vertex as the vertex, respectively. j Represents the three-dimensional coordinates of the j-th vertex in the neighborhood.

[0074] The calculation formula of the maximum principal curvature is:

[0075]

[0076] Among them, K(p) represents the Gaussian curvature of the current vertex, θ i It represents the inner angle of the adjacent triangle at the current vertex, n is the number of triangles adjacent to the current vertex, and k1(p) represents the maximum principal curvature of the current vertex.

[0077] In this implementation, the edge score calculation formula in step 202 is:

[0078] EdgeScore(p)=w1·ΔH(p)+w2·||k1(p)||+w3·D(p).

[0079] Wherein, EdgeScore(p) represents the edge score of the current vertex, w1, w2 and w3 are the first weight coefficient, the second weight coefficient and the third weight coefficient respectively, ΔH(p) represents the average curvature change rate, and D(p) represents the distance from the current vertex to the nearest non-edge point.

[0080] The calculation formula of the adaptive threshold is:

[0081] Thresh local (p) = OTSU (EdgeScore (N k (p)));

[0082] Among them, Thresh local (p) represents the adaptive threshold of the current vertex, N k (p) represents the neighborhood area formed by the k nearest neighbor vertices with the current vertex as the center, k=30, and OTSU represents the maximum between-class variance method.

[0083] The type of each vertex in the triangular mesh model is determined according to the following conditions:

[0084] If EdgeScore(p)<0.8×Thresh local (p), then the current vertex is a complete edge point.

[0085] If EdgeScore(p)>1.2×Thresh local (p), the current vertex is a corrosion edge point.

[0086] If 0.8×Thresh local(p)≤EdgeScore(p)≤1.2×Thresh local (p), the current vertex is a transition region point.

[0087] Wherein, in implementing this embodiment, step 203 specifically includes:

[0088] For each eroded edge point, fit a quadratic surface in its k nearest neighbor region (k is 20 by default):

[0089] f(x,y)=ax 2 +by 2 +cxy+dx+ey+f.

[0090] Use weighted least squares method to solve the coefficients:

[0091] w r' =exp(-||p r' -p|| 2 / h 2 ).

[0092] Among them, p r' represents the coordinates of the neighboring points in the k-nearest region of the current eroded edge point, h is the smoothing parameter, which is 1 / 3 of the radius of the neighboring region, and w r' It represents the weight coefficient, which is calculated in advance based on the distance. It determines the importance of each data point in solving the surface coefficient. It is used to determine the influence of each neighboring point on the fitting, so as to use the weighted least squares method to help solve the surface coefficients a, b, c, d, e, f to be solved.

[0093] Specifically, we need to minimize the weighted sum of squared errors to solve for the coefficients. The error function can be expressed as:

[0094]

[0095] Among them, x r' ,y r' and z r' Respectively represent p r' In order to solve the surface coefficients a, b, c, d, e, f, it is necessary to calculate the error function F error This can be done by minimizing F error This is done by taking partial derivatives with respect to each surface coefficient and setting them equal to zero. This leads to a system of linear equations that can be solved to find the optimal values ​​of the surface coefficients. Specifically, we need to error Find the partial derivatives about a, b, c, d, e, and f respectively, and set them to zero:

[0096]

[0097] We get a linear system of six equations. By solving this system, we can find the values ​​of the surface coefficients a, b, c, d, e, and f.

[0098] The formula for repairing the vertices of the eroded edge point type in the triangular mesh model is:

[0099]

[0100] Among them, p smooth Indicates the reconstructed corroded edge point after repair, p new represents the reconstructed erosion edge point, p new =f(x+δ x ,y+δ y ), x and y represent the horizontal and vertical coordinates of the corrosion edge point, respectively, δ x and δ y represents the extrapolation step size of the horizontal coordinate of the corrosion edge point and the extrapolation step size of the vertical coordinate of the corrosion edge point N(p new ) represents the set of neighboring points in the preset neighborhood of the reconstructed corrosion edge point, f() represents the fitting quadratic surface of the current reconstructed corrosion edge point, p i' N(p new ), Gs represents the space domain Gaussian kernel, Gr represents the value domain Gaussian kernel, G s (||p new -p i' ||) represents the spatial distance weight between the reconstructed erosion edge point and its neighboring points, G r (|H(p new )-H(p i' )|) represents the average curvature weight of the reconstructed eroded edge point and its neighboring points.

[0101] Gaussian kernel G in spatial domain s It is used to measure the spatial distance relationship between the reconstructed corrosion edge point and its neighboring points. The closer the distance, the greater the weight. The spatial position relationship of the reconstructed corrosion edge points is considered through the spatial domain Gaussian kernel, and the local geometric shape is maintained by fitting the quadratic surface f(), thereby repairing the geometric features of the reconstructed corrosion edge points.

[0102] Range Gaussian kernel G r It is used to measure the average curvature difference between the reconstructed corrosion edge point and its neighboring points. The smaller the curvature difference, the greater the weight. The local texture changes are maintained by the difference in average curvature, thereby repairing the texture features of the reconstructed corrosion edge point.

[0103] Specifically, for each corrosion edge point, the surface is extrapolated first: with the current corrosion edge point as the center, k'=20 nearest neighbor points are selected, and a quadratic surface f() is fitted with these points. δx and δy are extrapolated in the x and y directions respectively to obtain the new point p new .

[0104] Then, weighted smoothing based on the idea of ​​bilateral filtering is performed:

[0105] In p new Collect the vertex set N(p new ), apply the spatial domain weight Gs and the range weight Gr to each neighborhood point, and obtain the final repair point Psmooth by weighted average.

[0106] Where Gs(x)=exp(-x 2 / 2σs 2 ), Gr(x)=exp(-x 2 / 2σr 2 ), σs is the spatial domain standard deviation, and σr is the range standard deviation.

[0107] Step 104 , using a multi-level matching algorithm, aligns the triangular mesh models of each fragment that have completed edge repair, and obtains the optimal splicing position and optimal posture parameters of each fragment.

[0108] In some embodiments, accurate automatic matching of fragments is achieved through multi-feature fusion and multi-level matching strategies, and the consistency of overall splicing is ensured through global optimization. Step 103 uses a multi-level matching algorithm to align the triangular mesh models of each fragment that have completed edge repair, and obtain the optimal splicing position and optimal posture parameters of each fragment, which specifically includes steps 301-304:

[0109] Step 301 : extracting features based on the triangular mesh model of each fragment after edge repair.

[0110] Step 302: Calculate the weighted similarity between any two fragments based on the features extracted from each fragment to obtain a first matching screening result.

[0111] Step 303: construct a nonlinear optimization objective function based on the first matching screening result, and iteratively optimize the nonlinear optimization objective function until a preset matching accuracy is reached to obtain a second matching screening result.

[0112] Step 304: Use a global consistency optimization algorithm to adjust the splicing position and posture parameters of each fragment in the second matching screening result to obtain the optimal splicing position and optimal posture parameters of each fragment.

[0113] In this implementation, the feature extraction in step 301 at least includes: geometric feature extraction and texture feature extraction.

[0114] The geometric feature extraction is: constructing a curvature distribution descriptor of each vertex according to a curvature analysis structure, and extracting the broken edge curve features of each fragment based on a triangular mesh model of each fragment that has completed edge repair.

[0115] The expression of the curvature distribution descriptor is:

[0116] C dist (p)={k1(q),k2(q)|q∈N r (p)}.

[0117]

[0118] Among them, C dist (p) represents the curvature distribution descriptor of the current vertex, Nr(p) represents the neighborhood point set with the current vertex as the center and r1 as the radius, r1 is equal to 2 times the average curvature, k2() represents the second principal curvature value, and q represents the vertex in the neighborhood point set.

[0119] The expression of the fracture edge curve characteristic is:

[0120] E feat (s)=[κ(s),τ(s),n(s),θ(s)].

[0121] Among them, E feat (s) represents the fracture edge curve feature of the current fragment, s is the fracture edge parameter of the current fragment, κ(s) is the fracture edge curvature of the current fragment, τ(s) is the fracture edge torsion of the current fragment, n(s) is the fracture edge surface normal vector of the current fragment, and θ(s) is the fracture surface direction angle of the current fragment.

[0122] The texture feature extraction is as follows: based on the triangular mesh model of each fragment with completed edge repair, the fracture surface normal map feature of each fragment is calculated, and a local height field descriptor is constructed according to the preset shape field of each vertex.

[0123] The calculation formula of the fracture surface normal map feature is:

[0124]

[0125] Among them, N v map represents the normal map feature of the fracture surface of the current fragment, (u, v) represents the parameterized plane coordinates of the current fragment, W(u, v) represents the vertex area of ​​the current fragment, n q Represents the vertex normal vector within the vertex region of the current fragment.

[0126] The expression of the local height field descriptor is:

[0127] H desc (p)={h(q)|q∈N d (p)}.

[0128] Among them, H desc (p) represents the local height field descriptor of the current vertex, Nd(v) represents the circular neighborhood with the current vertex as the center and a diameter of d, where d is 5 times the average side length of the grid, q represents the vertex in the circular neighborhood, and h(q) represents the relative height value of the qth vertex in the circular neighborhood relative to the current vertex.

[0129] Wherein, in implementing this embodiment, step 302 specifically includes:

[0130] The weighted similarity of any two fragments f and f' is calculated according to the following formula:

[0131]

[0132] Among them, S ff' represents the weighted similarity between any two fragments, F k” represents the kth type of feature, k”=1, 2, 3, 4 correspond to the curvature distribution descriptor, fracture edge curve feature, fracture surface normal map feature and local height field descriptor respectively, w k” represents the weight of the k-th class feature, and sim() is the similarity measurement function between any two fragments.

[0133] If the weighted similarity between any two current fragments is greater than the preset similarity, the current any two fragments are added to the feature matching point pair set.

[0134] The following formula is used to perform the first matching screening on the fragments of the feature matching point pair set:

[0135]

[0136] Among them, T init represents the first matching screening result, R represents the rotation matrix, t represents the translation vector, M represents the feature matching point pair set, R×f represents the rotation coordinates of fragment f, and (f,f') represents the fragment pair in the feature matching point pair set.

[0137] In this implementation, the formula for constructing the nonlinear optimization objective function in step 303 is:

[0138] E(T init )=E geo (T init )+λ1E norm(T init )+λ2E curv (T init ).

[0139] Among them, E(T init ) represents the nonlinear optimization objective function value constructed based on the first matching screening result, that is, the second matching screening result E, Egeo represents the geometric distance term, Enorm represents the normal consistency term, Ecurv represents the curvature continuity term, λ1 and λ2 represent the first weight coefficient and the second weight coefficient respectively.

[0140] Wherein, in implementing this implementation mode, step 304 specifically includes:

[0141] A matching graph G(V, E) is constructed according to the second matching screening result, wherein V is a set of fragment nodes and E is a set of matching edges.

[0142] Use the following formula to find a minimum set of edges that connect all shards:

[0143] w(e)=-log(S ff' ).

[0144] MST=argmin∑ e∈E w(e).

[0145] Among them, w(e) represents the weight of the matching edge, and MST represents the set of minimum weight edges connecting all fragments.

[0146] An initial global transformation matrix is ​​constructed for each fragment based on the minimum weight edge set, and the splicing position and posture parameters of each fragment in the second matching screening result are adjusted using the following formula until the preset number of iterations is reached to obtain the optimal splicing position and optimal posture parameters of each fragment:

[0147]

[0148] Among them, E global represents the global error, (j, j') represents the fragment pair in the second matching screening result, T j and T j' denote the global transformation matrices of fragments j and j', respectively, T jj' Represents the relative transformation matrix of the fragment pair (j, j').

[0149] Step 105 , stitching the triangular mesh models of the fragments with the edges repaired according to the optimal stitching positions and optimal posture parameters of the fragments to obtain an initial digital model of the cultural relic.

[0150] Step 106: repair the missing area of ​​the initial digital model of the cultural relic to obtain a repaired digital model of the cultural relic.

[0151] In some embodiments, the feature database provides a reconstruction reference to achieve accurate reconstruction of geometric shapes, ensure the natural transition and continuity of textures, and generate a complete digital model that conforms to historical features. Step 106 repairs the missing areas of the initial cultural relic digital model to obtain a repaired cultural relic digital model, which specifically includes steps 401 to 403:

[0152] Step 401: determining the missing area in the initial digital model of the cultural relic according to the missing boundary curve.

[0153] Step 402: Use the B-spline method to perform surface reconstruction on the missing area to obtain a reconstructed area.

[0154] Step 403, synthesizing the texture of the reconstructed area based on the texture feature library to obtain a digital model of the restored cultural relic.

[0155] Wherein, in implementing this implementation mode, step 401 further includes:

[0156] Extract the boundary points of the missing region:

[0157] Among them, C boundary represents the set of boundary points of the missing region, represents the missing region boundary, p b” are the boundary points of the missing area.

[0158] The constraint condition of the missing region boundary is calculated by the following formula:

[0159] B constraint ={n(s),κ(s),θ(s)}.

[0160] Among them, B constraint represents the set of boundary constraints, s represents the current boundary point in the set of boundary points of the missing region, n(s) represents the normal vector of the current boundary point, κ(s) represents the boundary curvature of the current boundary point, and θ(s) represents the tangent angle of the current boundary point.

[0161] Wherein, in implementing this implementation mode, step 402 specifically includes:

[0162] Step 501, based on the matched fragments, extract the following features for building a geometric feature database:

[0163] Global shape features. Compute the overall shape descriptor:

[0164] F global ={V',S',R',P'}.

[0165] Among them, V' is the volume ratio characteristic, S' is the surface area distribution, R' is the radial distribution function, and P' is the principal axis direction characteristic.

[0166] Local feature template. Extract local feature template from the complete area:

[0167] F local (p) = {H(p), K(p), D(p)}.

[0168] Among them, H(p) is the mean curvature field, K(p) is the Gaussian curvature field, and D(p) is the shape diameter function.

[0169] Step 502, establishing a texture feature library based on the known region:

[0170] T pattern ={T color ,T normal ,T rough}.

[0171] Among them, T color is the color distribution mode, T normal is the normal distribution mode, T rough is the roughness distribution pattern.

[0172] Step 503, construct an initial control point grid by the following formula to obtain a reconstruction area:

[0173]

[0174] Among them, S(u',v') represents the reconstruction area, (u',v') represents the parameterized plane coordinates of the control points in the reconstruction area, and P i”,j” is the control point in the reconstruction area, N i”,k' (u') and N j”,l (v') represents the first B-spline basis function and the second B-spline basis function, respectively. k' and l represent the spline order of the first B-spline basis function and the spline order of the second B-spline basis function, respectively.

[0175] Step 504, optimize the boundary of the reconstruction area using the following formula:

[0176] E surface =E smooth +λ3E boundary +λ4E feature .

[0177] Wherein, Esmooth represents the smoothing term of the reconstructed area, Eboundary represents the boundary constraint term of the reconstructed area, Efeature represents the feature preservation term of the reconstructed area, and λ3 and λ4 represent the third weight coefficient and the fourth weight coefficient, respectively.

[0178] In another exemplary embodiment, step 402 may also utilize the symmetry characteristics of the cultural relic to obtain the reconstruction area; or perform reconstruction based on a similar cultural relic template.

[0179] Wherein, in implementing this implementation mode, step 403 specifically includes:

[0180] The reconstruction area is parameterized using the following formula:

[0181]

[0182] Where E' represents the vertex set of the reconstruction area, i' and j' represent the horizontal and vertical coordinates of the current vertex in the reconstruction area, respectively, and u i” and u j” Respectively represent the parameter domain horizontal coordinate and parameter domain vertical coordinate of the current vertex in the reconstruction area, d i”j” represents the geodesic distance of the current vertex in the reconstruction area, w i”j” Represents the edge weight of the current vertex in the reconstruction region.

[0183] The texture elements are matched to the parameterized reconstructed region according to the following formula:

[0184]

[0185] Where p' represents the current vertex in the reconstruction area, T new Indicates the texture in the texture feature library that is similar to the boundary texture of the reconstructed area, T pattern represents the texture feature library, T boundary represents the boundary texture of the reconstructed area, T represents the texture in the texture feature library, and D() represents the texture similarity measurement function; T new It is obtained by matching the similarity between the texture feature library and the boundary texture, and is used as the ideal texture state that the vertices inside the reconstructed area should achieve.

[0186] The texture transition error is minimized by the following formula to obtain the texture of the reconstructed area and the digital model of the restored cultural relic:

[0187]

[0188] Among them, E texture represents the texture transition error, Ω represents the reconstruction area, p' represents the current vertex in the boundary area of ​​the reconstruction area, represents the boundary area of ​​the reconstruction area, T known represents the texture of the non-missing area, and μ represents the boundary weight coefficient of the reconstructed area.

[0189] The undamaged original area on the artifact fragment contains the intact original texture information T known, T known Serves as a reference for texture synthesis in the reconstructed area.

[0190] The whole process can be understood as follows: first, parameterize the reconstructed area and establish a unified texture coordinate system; then select the texture sample that best matches the boundary texture of the reconstructed area from the texture feature library; finally, minimize the texture transition error to achieve: the continuity of the texture inside the reconstructed area and the smooth transition of the boundary of the known area. This ensures that the repaired texture not only maintains the original characteristics but also achieves a natural transition.

[0191] In summary, the present application method adopts modular design, each functional module is independently encapsulated and equipped with a standardized data interface. Not only does it enable the various modules to work together efficiently, but it also greatly facilitates the maintenance and upgrade of the system. When a module needs to be updated or optimized, it can be done separately without making large-scale changes to the entire system, thereby reducing maintenance costs and improving the flexibility and scalability of the system. The data pipeline architecture is adopted to make the processing flow clear, the data transmission efficient and fast, and the processing process traceable. Under this architecture, every link from input to output of data is clearly defined and transmitted through a standardized data interface. This not only ensures the accuracy and integrity of the data, but also enables the processing process to be effectively monitored and recorded, providing strong support for subsequent data analysis and troubleshooting. Integrating intelligent processing technology, with the support of deep learning algorithms, adaptive parameter adjustment and multi-feature fusion matching are realized. The system can automatically adjust the processing parameters according to different situations, thereby optimizing the processing effect. At the same time, the application of multi-feature fusion matching technology also improves the accuracy and robustness of matching, providing a more reliable technical guarantee for the three-dimensional matching and digital restoration of cultural relics fragments.

[0192] In another exemplary embodiment, in addition to the modular design, a cloud computing architecture may be used to place computing tasks in the cloud or to use a distributed system and multi-node parallel processing.

[0193] The present application also provides an application scenario, which applies the above-mentioned three-dimensional matching and digital restoration method of cultural relic fragments. Specifically: The three-dimensional matching and digital restoration method of cultural relic fragments provided in this embodiment can be applied in the comprehensive scenario of cultural relic protection and digital restoration. The comprehensive scenario includes the preliminary processing link of cultural relic fragments, the three-dimensional digital restoration link, and the application and display link of restoration results. The cultural relic fragments first enter the preliminary processing link, and after cleaning, classification and preliminary recording, enter the three-dimensional digital restoration link. In this link, the three-dimensional matching and digital restoration method of cultural relic fragments provided in this embodiment uses technical means such as high-precision scanning, three-dimensional modeling, edge restoration and multi-level matching algorithms to accurately reconstruct and digitally restore the cultural relic fragments in three dimensions. After the restoration is completed, the digital model of the cultural relic will enter the application and display link of the restoration results, which is used in the protection, research, education and public display of cultural heritage. The three-dimensional matching and digital restoration method of cultural relic fragments provided in this embodiment belongs to the key technology in the three-dimensional digital restoration link. Specifically in this link, efficient and precise restoration of cultural relics fragments can be achieved, which not only improves the efficiency and quality of cultural relic restoration, but also provides strong technical support for subsequent cultural relic restoration, display and dissemination.

[0194] This application has the following technical effects:

[0195] By using high-precision 3D scanning equipment to obtain raw data, each scan collects more than 300,000 point cloud data, and directly performs edge detection and repair in 3D space, avoiding information loss caused by 2D-3D conversion. The edge feature extraction method based on curvature analysis can accurately capture local geometric details, and adopt a multi-feature fusion matching strategy, comprehensively consider geometric and texture features, and improve matching accuracy. The repair accuracy of this application reaches more than 95%, and the scanning accuracy can reach 0.1 mm.

[0196] Through the automated 3D scanning system, a single scan takes only 0.25 seconds to obtain complete data; the improved feature extraction algorithm is used to achieve rapid edge detection and repair; based on the multi-level matching strategy, the convergence speed is accelerated through the progressive optimization from coarse registration to fine registration; and the intelligent reconstruction method based on the feature database reduces manual intervention and significantly improves the repair speed. This application shortens the average processing time of a single fragment to 30 minutes, which is 80% more efficient than the traditional manual method.

[0197] This application uses laser 3D scanning technology, without touching the surface of the cultural relics, to perform fragment matching and restoration verification in a virtual environment, establish a digital cultural relic feature database, reduce repeated operations on the physical objects, and use a completely non-contact digital processing method to avoid secondary damage to the cultural relics.

[0198] This application uses a complete three-dimensional data acquisition and storage mechanism to retain the original feature information during the edge repair process and record all posture transformation parameters during the matching process. It also realizes digital recording of the entire repair process through scientific management and storage of the feature database, ensuring that the repair process is traceable and reversible.

[0199] This application adapts to various edge forms through direct feature analysis methods in three-dimensional space, improves the robustness of matching based on a matching strategy based on multi-feature fusion, and can handle various missing situations based on an intelligent reconstruction method based on a database, enabling this application to handle different types of corrosion situations and fragments of various specifications.

[0200] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0201] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A three-dimensional matching and digital restoration method for cultural relic fragments, characterized in that: The three-dimensional matching and digital restoration method of cultural relic fragments includes: Obtain point cloud data of each fragment belonging to the same cultural relic; Based on the point cloud data of each fragment, a Poisson surface reconstruction method is used to construct a triangular mesh model of each fragment; Repairing the corroded edges in the triangular mesh model of each fragment to obtain the triangular mesh model of each fragment with the edge repair completed; A multi-level matching algorithm is used to align the triangular mesh models of each fragment that have completed edge repair, and the optimal splicing position and optimal posture parameters of each fragment are obtained; According to the optimal splicing position and optimal posture parameters of each fragment, the triangular mesh model of each fragment with completed edge repair is spliced ​​to obtain the initial digital model of the cultural relic; The missing areas of the initial digital model of the cultural relic are repaired to obtain a restored digital model of the cultural relic.

2. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 1 is characterized in that: Repairing the eroded edges in the triangular mesh model of each fragment to obtain the triangular mesh model of each fragment with the edge repaired, specifically including: Perform edge curvature analysis on the triangular mesh model of the fragment to obtain the curvature analysis results of each vertex in the triangular mesh model; Based on the edge score and the adaptive threshold of each vertex in the triangular mesh model, determining the type of each vertex in the triangular mesh model, wherein the type is a complete edge point, a corroded edge point, and a transition area point; The vertices of the eroded edge points in the triangular mesh model are repaired by using a surface extrapolation method to obtain a triangular mesh model of the fragment with the edge repaired.

3. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 2 is characterized in that: The curvature analysis result includes at least: average curvature and maximum principal curvature; The calculation formula of the average curvature is: Among them, A Mixed (p) represents the mixed Thiessen polygon area of ​​the current vertex, p represents the current vertex, p1 and p2 represent the first adjacent vertex and the second adjacent vertex of the current vertex respectively, T represents the triangle with p, p1 and p2 as vertices, S T represents the area of ​​the triangle, α and α' represent the angle with the first adjacent vertex as the vertex and the angle with the second adjacent vertex as the vertex, respectively, H(p) represents the average curvature of the current vertex, N(p) represents the vertex set in the preset neighborhood of the current vertex p, α j and β j They represent the angle with the current vertex as the vertex and the angle with the second adjacent vertex as the vertex, respectively. j represents the three-dimensional coordinates of the jth vertex in the neighborhood; The calculation formula of the maximum principal curvature is: Among them, K(p) represents the Gaussian curvature of the current vertex, θ i It represents the inner angle of the adjacent triangle at the current vertex, n is the number of triangles adjacent to the current vertex, and k1(p) represents the maximum principal curvature of the current vertex.

4. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 2 is characterized in that: The calculation formula of the edge score is: EdgeScore(p)=w1·ΔH(p)+w2·||k1(p)||+w3·D(p); Wherein, EdgeScore(p) represents the edge score of the current vertex, w1, w2 and w3 are the first weight coefficient, the second weight coefficient and the third weight coefficient respectively, ΔH(p) represents the average curvature change rate, and D(p) represents the distance from the current vertex to the nearest non-edge point; The calculation formula of the adaptive threshold is: Thresh local (p)=OTSU(EdgeScore(N k (p))); Among them, Thresh local (p) represents the adaptive threshold of the current vertex, N k (p) represents the neighboring area formed by the k nearest neighboring vertices with the current vertex as the center, k = 30, and OTSU represents the maximum inter-class variance method; The type of each vertex in the triangular mesh model is determined according to the following conditions: If EdgeScore(p)<0.8×Thresh local (p), then the current vertex is a complete edge point; If EdgeScore(p)>1.2×Thresh local (p), then the current vertex is a corrosion edge point; 0.8×Thresh local (p)≤EdgeScore(p)≤1.2×Thresh local (p), the current vertex is a transition region point.

5. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 2 is characterized in that: The formula for repairing the vertices of the eroded edge point type in the triangular mesh model is: Among them, p smooth Indicates the reconstructed corroded edge point after repair, p new represents the reconstructed erosion edge point, p new =f(x+δ x ,y+δ y ), x and y represent the horizontal and vertical coordinates of the corrosion edge point, respectively, δ x and δ y represents the extrapolation step size of the horizontal coordinate of the corrosion edge point and the extrapolation step size of the vertical coordinate of the corrosion edge point N(p new ) represents the set of neighboring points in the preset neighborhood of the reconstructed corrosion edge point, f() represents the fitting quadratic surface of the current reconstructed corrosion edge point, p i' N(p new ), Gs represents the space domain Gaussian kernel, Gr represents the value domain Gaussian kernel, G s (||p new -p i' ||) represents the spatial distance weight between the reconstructed erosion edge point and its neighboring points, G r (|H(p new )-H(p i' )|) represents the average curvature weight of the reconstructed eroded edge point and its neighboring points.

6. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 1 is characterized in that: A multi-level matching algorithm is used to align the triangular mesh models of each fragment after edge repair to obtain the optimal splicing position and optimal posture parameters of each fragment, including: Feature extraction is performed based on the triangular mesh model of each fragment after edge repair; According to the features extracted from each fragment, the weighted similarity between any two fragments is calculated to obtain a first matching screening result; Constructing a nonlinear optimization objective function based on the first matching screening result, iteratively optimizing the nonlinear optimization objective function until a preset matching accuracy is reached, and obtaining a second matching screening result; A global consistency optimization algorithm is used to adjust the splicing position and posture parameters of each fragment in the second matching screening result to obtain the optimal splicing position and optimal posture parameters of each fragment.

7. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 6 is characterized in that: The feature extraction at least includes: geometric feature extraction and texture feature extraction; The geometric feature extraction is as follows: constructing a curvature distribution descriptor of each vertex according to the curvature analysis result, and extracting the broken edge curve features of each fragment based on the triangular mesh model of each fragment after edge repair; the expression of the curvature distribution descriptor is: C dist (p)={k1(q),k2(q)|q∈N r (p)}; Among them, C dist (p) represents the curvature distribution descriptor of the current vertex, Nr(p) represents the neighborhood point set with the current vertex as the center and r as the radius, where r is equal to 2 times the average curvature, k2() represents the second principal curvature value, and q represents the vertex in the neighborhood point set; The expression of the fracture edge curve characteristic is: E feat (s)=[κ(s),τ(s),n(s),θ(s)]; Among them, E feat (s) is the fracture edge curve feature of the current fragment, s is the fracture edge parameter of the current fragment, κ(s) is the fracture edge curvature of the current fragment, τ(s) is the fracture edge torsion of the current fragment, n(s) is the fracture edge surface normal vector of the current fragment, and θ(s) is the fracture surface direction angle of the current fragment; The texture feature extraction is as follows: based on the triangular mesh model of each fragment with completed edge repair, the fracture surface normal map feature of each fragment is calculated, and a local height field descriptor is constructed according to the preset shape field of each vertex. The calculation formula of the fracture surface normal map feature is: in, represents the normal map feature of the fracture surface of the current fragment, (u, v) represents the parameterized plane coordinates of the current fragment, W(u, v) represents the vertex area of ​​the current fragment, n q Represents the vertex normal vector within the vertex area of ​​the current fragment; The expression of the local height field descriptor is: H desc (p)={h(q)|q∈N d (p)}; Among them, H desc (p) represents the local height field descriptor of the current vertex, Nd(v) represents the circular neighborhood with the current vertex as the center and a diameter of d, where d is 5 times the average side length of the grid, q represents the vertex in the circular neighborhood, and h(q) represents the relative height value of the qth vertex in the circular neighborhood relative to the current vertex.

8. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 6 is characterized in that: According to the features extracted from each fragment, the weighted similarity between any two fragments is calculated to obtain the first matching screening result, which specifically includes: The weighted similarity of any two fragments f and f' is calculated according to the following formula: Among them, S ff' represents the weighted similarity between any two fragments, F k” represents the kth type of feature, k”=1, 2, 3, 4 correspond to the curvature distribution descriptor, fracture edge curve feature, fracture surface normal map feature and local height field descriptor respectively, w k” represents the weight of the kth class feature, sim() is the similarity measurement function between any two fragments; If the weighted similarity between any two fragments is greater than the preset similarity, the two fragments are added to the feature matching point pair set; The following formula is used to perform the first matching screening on the fragments of the feature matching point pair set: Among them, T init represents the first matching screening result, R represents the rotation matrix, t represents the translation vector, M represents the feature matching point pair set, R×f represents the rotation coordinates of fragment f, and (f,f') represents the fragment pair in the feature matching point pair set.

9. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 1 is characterized in that: Repair the missing areas of the initial digital model of the cultural relic to obtain a restored digital model of the cultural relic, including: Determine the missing area in the initial digital model of the cultural relic based on the missing boundary curve; Use the B-spline method to perform surface reconstruction on the missing area to obtain the reconstructed area; The texture of the reconstructed area is synthesized based on the texture feature library to obtain the digital model of the restored cultural relic.

10. The three-dimensional matching and digital restoration method of cultural relic fragments according to claim 9, characterized in that: The texture of the reconstructed area is synthesized based on the texture feature library to obtain the digital model of the restored cultural relic, including: The reconstruction area is parameterized using the following formula: Where E' represents the vertex set of the reconstruction area, i' and j' represent the horizontal and vertical coordinates of the current vertex in the reconstruction area, respectively, and u i” and u j” Respectively represent the parameter domain horizontal coordinate and parameter domain vertical coordinate of the current vertex in the reconstruction area, d i”j” represents the geodesic distance of the current vertex in the reconstruction area, w i”j” Represents the edge weight of the current vertex in the reconstruction area; The texture elements are matched to the parameterized reconstructed region according to the following formula: Where p' represents the current vertex in the reconstruction area, T new Indicates the texture in the texture feature library that is similar to the boundary texture of the reconstructed area, T pattern represents the texture feature library, T boundary represents the boundary texture of the reconstructed area, T represents the texture in the texture feature library, and D() represents the texture similarity measurement function; The texture transition error is minimized by the following formula to obtain the texture of the reconstructed area and the digital model of the restored cultural relic: Among them, E texture represents the texture transition error, Ω represents the reconstruction area, p' represents the current vertex in the boundary area of ​​the reconstruction area, represents the boundary area of ​​the reconstruction area, T known represents the texture of the non-missing area, and μ represents the boundary weight coefficient of the reconstructed area.

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