CN110570524A - method for simplifying median plane of three-dimensional model for topology maintenance - Google Patents

Abstract

a simplified method of topologically preserving median planes in a three-dimensional model, comprising: (1) searching topological holes, namely defects, on a median plane of the three-dimensional model, and marking surface patches around the holes as topological key areas; (2) when the median plane is simplified by using an edge folding method, carrying out topology inspection on edges in a topology key area before simplification, and judging whether related holes disappear or not after the edges to be simplified are simplified, so that a topology structure is damaged; (3) dividing the bounding box of the median plane into even number of fragments, and carrying out parallel simplification processing in an odd/even interleaving mode on the fragments; the edges of the non-topological key area are directly folded and simplified; and (3) carrying out topology check in the step (2) on the edge in the topology critical area before simplification, if the topology is damaged, not simplifying the edge, if the topology is not damaged, simplifying the edge, and marking the newly generated patch as the topology critical area. A simplified median plane for topology preservation will be obtained. The method has the advantages of topology maintenance, high operation speed and small simplified error.

Description

Method for simplifying median plane of three-dimensional model for topology maintenance

Technical Field

the invention belongs to the field of computer graphics and computer vision, and particularly relates to a median plane simplification method for topology maintenance.

background

the brief representation method of geometric shapes plays an important role in computer graphics and geometric modeling technology, and is always a hot topic in the research field. The median Surface (media Surface) is used as a concise representation method of a three-dimensional model, and has wide application in the aspects of model compression and approximation, model animation and deformation, model retrieval and identification and the like.

The median plane of a three-dimensional model refers to the set of the centroids of all inscribed spheres inside the model. The inscribed sphere centered at these points is called the median sphere. The median plane and the function defined to describe the median spherical radius on the median plane together constitute the Medial Axis Transform (MAT). Any three-dimensional model can be represented by using the median plane, and the original model can be restored from the central axis transformation reconstruction. But the generation of the median plane is sensitive to noise and is easy to generate a plurality of burrs, which hinders the expression of the median plane from being concise. Therefore, the median plane is simplified, the expression simplicity is improved, and the method has important significance in many applications.

Existing median plane reduction methods can be broadly divided into two categories. One class of methods considers the median plane as a collection of median points, reduced by thresholding which defines some metric on the median points [1] [2] [3 ]. Such methods tend to sacrifice geometric accuracy while removing burrs, and destroy the topological representation of the median-facing model. Another class of methods generally represents the median plane as a grid model and reduces [4] [5] [6] [7] by shrinking or folding the edges or patches of the median plane. However, such methods either have difficulty in maintaining the topology or pay attention to maintaining the topology but the resulting geometry is highly inaccurate or computationally slow. For example, the QEM edge folding frame-based simplified method [5] [6], which generally can obtain better geometric accuracy, but relies on each simplified topology check to maintain the topology, and seriously affects the computational efficiency.

The topology of the median plane consistent with the original model is the root cause for its guarantee for many applications. The topology damage naturally affects the topology of the reconstructed model, and can cause wrong topology change in the application based on the median plane abstract expression. Therefore, in the process of simplifying the median plane, how to quickly and efficiently generate a simplified median plane with topology maintenance and high geometric precision has very important value.

[1]M.Foskey,M.C.Lin,and D.Manocha,“Efficient computation of a simplified medial axis,”Journal of Computing and Information Science inEngineering,vol.3,no.4,pp.274–284,2003.

[2]F.Chazal and A.Lieuer,“Theλ-medial axis,”Graphical Models,vol.67,no.4,pp.304–331,2005.

[3]B.Miklos,J.Giesen,and M.Pauly,“Discrete scale axis representations for 3d geometry,”ACM Transactions on Graphics(TOG),vol.29,no.4,p.101,2010.

[4]FARAJ N.,THIERY J.-M.,BOUBEKEUR T.:Progressive medial axis filtration.In SIGGRAPH Asia 2013Technical Briefs(2013),ACM,p.3

[5]SUN F.,CHOI Y.-K.,YU Y.,WANG W.:Medial meshes—a compact and accurate representation of medial axis transform.IEEE transactions onvisualization and computer graphics 22,3(2016),1278–1290.

[6]LI P.,WANG B.,SUN F.,GUO X.,ZHANG C.,WANG W.:Q-MAT:Computing medial axis transform by quadratic error minimization.ACM Transactions onGraphics(TOG)35,1(2015),8.

[7]YAN Y.,SYKES K.,CHAMBERS E.,LETSCHER D.,JU T.:Erosion thickness on medial axes of 3d shapes.ACM Trans.Graph.35,4(July 2016),38:1–38:12.

disclosure of Invention

The invention solves the problems: multiple application scenarios, such as model approximation and compression, animation and deformation, retrieval and identification, require a compact and accurate median plane. Therefore, the invention aims to solve the problems that the topology is damaged and the calculation efficiency is not high in the process of simplifying the median plane in the existing method.

the invention provides a simplifying method of a median plane of a three-dimensional model with maintained topology. The method simplifies the median plane through the edge folding operation which minimizes the geometric approximation error, and ensures that the simplified result has good geometric precision. Meanwhile, the method limits the topology checking step to a topology critical area which is possible to have topology change (namely, an area which is possible to cause the change of a topology structure by one-time simplifying processing, specifically, the folding simplifying processing of one edge in the invention can cause relevant topology holes to disappear), reduces a large amount of unnecessary operation, and further improves the simplifying efficiency through a parallel processing frame.

The technical scheme of the invention is as follows: a simplification method of a three-dimensional model median plane based on topology preservation comprises the following steps:

(1) searching holes, namely defects, on a median plane of the three-dimensional model, and marking surface patches around the holes on the median plane as topological key areas; the edges of the patches are key edges, namely edges which need topology check in simplified calculation;

(2) using an edge folding method to simplify the median plane of the three-dimensional model, carrying out topology inspection on the edges in the topological key area before simplification, if the edges to be simplified can cause the relevant holes to disappear after simplification, not simplifying the edges, otherwise, simplifying the edges;

(3) dividing a bounding box of a median plane of the three-dimensional model into an even number of fragments, and performing parallel simplified processing in a staggered mode on the fragments; and when the edges in the topology key area are simplified, the topology check in the step (2) is carried out to obtain a simplified median plane for maintaining the topology. The simplified median plane can be used as a brief expression of a three-dimensional model to be applied to a plurality of application scenes such as model compression and approximation, model animation and deformation, model retrieval and identification and the like.

In the step (1), finding the holes, namely the defects, on the median plane of the three-dimensional model, and marking the surface patches around the holes on the median plane as topological key areas, specifically comprising the following steps:

(11) Carrying out orthographic projection on the median plane in 6 orthogonal directions of +/-x, +/-y and +/-z of a coordinate system in which median plane data of the three-dimensional model are located to obtain 6 depth images and 6 corresponding patch serial number images, wherein the patch serial number images record the corresponding patch serial number of each pixel in the depth images on the median plane;

(12) Judging the depth difference between each pixel and the peripheral pixels of the depth image; if the difference is larger than a certain threshold value, marking the pixel as a key pixel;

(13) Finding out the sequence number of each key pixel, forming a topology key area by the patches, wherein the edges of the patches are key edges, namely the edges needing topology check in simplified calculation.

The certain threshold in the step (12) is the length of the longest side on the median plane. The threshold setting method can effectively screen out two adjacent pixels which are not adjacent to the median plane patch and belong to the depth image. Meaning that the position has undergone a depth transition, possibly at the boundary of the hole.

in the step (2), the specific steps of performing topology check on the edges in the topology critical area before simplification are as follows:

(21) Traversing the adjacent vertex sets of the two vertexes of the edge to find the common adjacent vertex;

(22) Traversing the adjacent patch sets of the two vertexes of the edge to find the common adjacent patches of the two vertexes;

(23) for each co-adjacent vertex, a determination is made as to whether there is a co-adjacent patch that contains the vertex. If all the common same-phase adjacent vertexes have the common adjacent surface patches, the topological structure cannot be damaged by folding the edge; otherwise, folding the edge destroys the topology.

in the step (3), the model bounding box is divided into an even number of fragments, and the specific steps of performing parallel simplified processing on the fragments in an interleaving manner are as follows:

(31) dividing a bounding box of the three-dimensional model into an even number of fragments; the width of the fragment is required to be more than four times of the length of the maximum side length of the median plane;

(32) Alternately selecting all the fragments with odd serial numbers and all the fragments with even serial numbers to carry out parallel simplification processing until a simplification target is reached;

(33) When the simplification processing is carried out on the median plane in one fragment in the step (32), the simplification processing is carried out in an edge folding mode, and the edge with the minimum simplification cost in the fragment is found each time; checking whether the edge is a key edge, if not, simplifying the edge, otherwise, carrying out topology check, and judging whether the simplification of the edge can cause the disappearance of the related holes; if the relevant holes disappear, the edge is not simplified, if the relevant holes disappear, the edge is simplified, the newly generated patch is marked as a topological key area, and the corresponding edge is marked as a key edge.

Compared with the prior art, the invention has the advantages that: the invention limits the topology check in the edge folding simplification process to the topology key area, avoids performing the topology check on all edge folding, greatly reduces the computation load and accelerates the speed of simplifying the median plane; meanwhile, the invention adopts a simpler and more direct topology checking method, further improves the simplification efficiency by utilizing a parallel framework, can quickly generate a simplified median plane with topology maintenance and high geometric precision, and can be effectively applied to the compression and approximation of the model, animation and deformation, retrieval and identification.

drawings

FIG. 1 is a flow chart of the present invention;

FIG. 2 is a flowchart of finding topological holes and marking topological critical areas; the method comprises the steps that (a) projection is carried out on a median plane from 6 orthogonal directions, (b) a depth image and a patch sequence number image obtained by projection are represented, (c) key pixels (pixels marked by black points in the image) with obvious depth change around the key pixels are detected in the depth image, and (d) a patch corresponding to the key pixels is found according to the patch sequence number image, and a key area and a key edge are marked;

FIG. 3 is a schematic view of projection direction selection;

FIG. 4 is a schematic diagram of parallelizing bounding box fragments;

FIG. 5 is a schematic diagram of slice width selection;

FIG. 6 is a graph of the results of a comparative experiment of the method of the present invention and the ET method; wherein (a), (b), (c) are simplified results of the ET process and (d), (e), (f) are simplified results of the present invention.

FIG. 7 is a graph showing the results of comparative experiments between the method of the present invention and the Q-MAT method; where (a) is the input model and initial median plane, (b) is the simplified result of Q-MAT, and (c) is the simplified result of the present invention;

FIG. 8 is a result of a median-plane reduction of a number of different topological models in accordance with the present invention; wherein (a) is the initial median plane of the input, (b), (c) are the simplified results of simplifying to 1000 and 500 vertices, (d) is the reconstructed model from (c), (e) is the error visualization result of reconstructing the model compared with the original model; # v denotes the number of vertices and e denotes the reconstruction approximation geometry error.

Detailed Description

The invention is described in detail below with reference to the figures and examples.

the invention mainly relates to 3 steps:

(1) finding topological holes and marking topological key areas: in the existing median plane simplification algorithm based on edge folding, the topology check is carried out on each edge when the edge is folded, which is a main reason of low efficiency of the algorithm. In fact, it is only when the edge around the hole is folded that it may cause the hole to disappear, thereby changing the topology. Therefore, the step aims to mark key areas around the holes, and only the key edges in the key areas are subjected to topology check in the subsequent simplification process, so that a large amount of unnecessary operations are avoided.

(2) checking the edge folding topology: by elaborating on the reason for the disappearance of the hole, it can be seen that if one edge is not one of the three edges of the triangular hole, folding it does not destroy the topology. Therefore, the main purpose of this module is to determine whether a triangle hole is formed by local connection information of one edge. If so, folding the edge destroys the topology; if not, folding the edge does not change the topology.

(3) and (3) parallel edge folding simplification: the invention further improves the algorithm efficiency by parallelizing the edge folding simplification process. The folding of one edge affects the connection between adjacent regions of the edge. Therefore, parallelization needs to ensure that the edge folding of the parallel processing does not affect each other. The invention divides the model bounding box into even number of fragments, and only simultaneously carries out edge folding operation in the odd number or even number fragments which are spaced each time. As long as the interval length meets a certain condition, the parallel folding can be guaranteed not to influence each other.

The steps of the present invention will be described separately below.

1. Finding topological holes and marking topological key areas

the topological properties remain unchanged under continuous transformations of tension, compression, bending, etc., and only the tearing and adhesion caused changes the topological properties. The edge folding operation is always a continuous transformation that preserves the topology except for the case where the hole disappears. Thus, for simplicity of edge folding, maintaining the topology can be attributed to ensuring that edge folding does not result in the disappearance of holes. Edge folding is an operation that only affects the connection relationships in the vicinity of the folded edge, and only when the edge around the hole is folded, may a topology change result. The check of whether the edge fold would disrupt the topology is only necessary around the hole. The invention firstly finds the topological key area around the hole and limits the topological check in the area, thereby obviously improving the calculation efficiency.

the initial median plane tends to be a non-manifold mesh, and extracting the area around the hole on a non-manifold mesh model is a difficult problem. However, it is much easier to extract pixels around the hole in the two-dimensional image obtained by projection of the mesh model. While the projection of the depth of the median plane will necessarily have a significant variation in depth values around the visible aperture. Therefore, the invention firstly carries out hole detection on the two-dimensional depth projection of the median plane, namely, finds out the pixel with great difference with the depth value of the surrounding pixels as the key pixel around the hole. And then finding the corresponding patch serial numbers of the key pixels on the median plane grid model, and marking the edges of the patches as key edges so as to mark a topological key area on the median plane.

The new edge created after the critical edge is folded remains around the hole and is therefore also marked as the critical edge. This transitivity means that as long as the partial area around a hole is marked as a critical area, a part of the edge near the hole always needs to be subjected to topology inspection in the subsequent simplification process, and the structure of the hole is not damaged. And at least a part of the peripheral area of one hole is marked as a key area as long as the hole is visible in one projection image. Therefore, the projection direction may be selected so long as it is ensured that each hole is partially visible in the projected image. As shown in the left image of fig. 3, a hole, when unobstructed, is not visible in the projection associated with the projection direction y if and only if its plane is parallel to the projection direction y. Thus, if two other orthogonal projection directions x, z are added, it is ensured that each unobstructed aperture is found. The right image of fig. 3 shows the occlusion common to the median plane, with the holes visible on one side only. Thus, by adding the opposite-x, -y, -z directions on the basis of the three orthogonal directions, the occlusion problem can be substantially solved.

The specific implementation steps are as follows:

(1) as shown in fig. 2(a), forward projecting the median plane from 6 orthogonal directions of ± x, ± y, ± z to obtain 6 depth images and 6 corresponding patch number images shown in fig. 2(b), where the patch number images record the corresponding patch number of each pixel in the depth images on the median plane;

(2) Judging the depth difference between each pixel of the depth image and the peripheral pixels; if the difference is greater than a certain threshold, marking the pixel as a key pixel (e.g., the pixel marked by the black dot in (c) of fig. 2); the threshold is set to the length of the longest edge on the median plane, because once the depth difference between a pixel and the surrounding neighborhood pixels is greater than the threshold, it means that the two pixels correspond to two non-adjacent patches. This indicates that a transition with a larger depth value occurs in the neighborhood of the pixel, which means that the pixel may be located in the area around the hole.

(3) as shown in fig. 2(d), the patch number image can be used to find the patch number corresponding to each key pixel, the patches form a topology key region, the edges of the patches are the key edges, and topology inspection is required in simplified calculation;

2. Edge folding topology inspection

when an aperture is made up of more than three sides, folding any one side does not result in the disappearance of the aperture. Only when the constituting sides of the triangular hole are folded will the hole disappear. If an edge is a side of a triangular hole, it is only necessary to determine whether a vertex is adjacent to both vertices of the edge, and the three vertices do not form a patch, thereby forming the triangular hole. The specific implementation steps are as follows:

(1) traversing the adjacent vertex sets of the two vertexes of the edge to find the common adjacent vertex;

(2) Traversing the adjacent patch sets of the two vertexes of the edge to find the common adjacent patches of the two vertexes;

(3) For each co-adjacent vertex, a determination is made as to whether there is a co-adjacent patch that contains the vertex. If all the common same-phase adjacent vertexes have the common adjacent surface patches, the topological structure cannot be damaged by folding the edge; otherwise, folding the edge destroys the topology.

3. Parallel edge folding simplification

Parallelization can further accelerate the speed of algorithm simplification, but because edge folding will affect the connection relationship in the neighborhood of the folded edge, the connection relationship changed by folding the edges needs to be ensured not to generate conflict when the edges are folded at the same time. As shown in FIG. 4, the present invention divides the model bounding box into an even number of slices, each time folding an edge within a spaced slice simultaneously with a minimum fold reduction cost. Ls in the figure is the length of the longest side of the bounding box, O, E denote odd numbered slices and even numbered slices, respectively. When the folding is carried out in parallel, the folding operation can be ensured not to be influenced by each other and can be carried out simultaneously as long as the distance between the folding operations is sufficiently large. In the process, an edge is counted as a member of a fragment as long as one vertex is located in the fragment. Fig. 5 shows the case where no collision occurs when two edges are folded in parallel, and the dotted circle in the figure represents the maximum possible influence range of the edge folding operation. When the slice width W is set to be greater than four times the maximum side length r of the median plane, the folding simplification of the sides in the spaced slices must not affect each other, and also simplification can be performed simultaneously.

The specific implementation steps are as follows:

(1) Dividing a bounding box of the model into an even number of fragments; the width of the fragment is required to be more than four times of the maximum side length of the median plane;

(2) Alternately selecting all the fragments with odd serial numbers and all the fragments with even serial numbers to carry out parallel simplification processing until a simplification target is reached;

(3) When the simplification processing is performed on the patches of the median plane in one patch, the simplification processing is performed in an edge folding mode. Each time finding the edge with the minimum simplification cost; checking whether the edge is a key edge, if not, simplifying the edge, otherwise, carrying out topology check, and judging whether the simplification can cause the disappearance of the related holes; if so, the edge is not simplified, otherwise, the edge is simplified, the newly generated patch is marked as a topological key area, and the corresponding edge is marked as a key edge;

the following are some experimental data of the present invention:

the experiment was performed on a microcomputer equipped with an Intel i7-8700(3.2GHz) CPU, 16G RAM and an NVIDIA GeForce GTX 1080 TiGPU. The models used in the experiments were all from the pool of Princeton Shape Benchmark and Aim @ Shape models commonly used in the academia.

First, the method is simplified with the currently advanced median plane in the world, namely the etching Thickness method (ET, Erosis Thick [7]]) Comparative tests were performed. FIG. 6 is a comparative experiment of the inventive method and ET method with simplified Cylinder model initial median plane, where # v represents the number of vertices of the median plane. As can be seen from fig. 6 (a), the median plane simplified by the ET method can maintain the topology, but has many burrs and a huge number of peaks; and the adjustment parameter theta can be seen from (b) and (c) in FIG. 6₁,θ₂with the simplification degree improved, the simplification result of the ET method eliminates the burrs, but much geometric information is lost, and the redundancy improvement of the number of vertexes is small. Fig. 6 (d), (e) and (f) are the simplification results of the invention at different degrees of simplification, and it can be seen that the results of the invention also maintain the topology, but the number of required vertices is less, and the geometric and topological information of the model can be more accurately expressed. In the aging aspect of the algorithm, the ET method needs 65.05 seconds to calculate two ET measurement values, and then a simplified median plane is obtained through parameter threshold adjustment. The present invention only requires 23.38 seconds, 24.51 seconds, and 24.84 seconds to generate the results (d), (e), and (f) in fig. 6.

FIG. 7 shows the results of comparative experiments with Q-MAT according to the present invention, in which # v indicates the number of peaks and # genus indicates the number of model defects, i.e., the number of wells, compared to another representative median plane-by-edge folding method, the Q-MAT [6 ]. To avoid the efficiency penalty of topology checking, the Q-MAT method starts topology checking only after a certain number of vertices is reduced. This results in that the method may have some topology results corrupted before starting the topology check. As shown in fig. 7 (a), the initial model and median plane deficit number are 8, but fig. 7(b) shows that the Q-MAT method causes a change in the deficit number, holes disappear, and topology is destroyed; while (c) in fig. 7 shows that the present invention can still maintain the topology unchanged with the same degree of simplification.

The table below counts the geometric errors and the required time for the median plane of the multiple models reduced to 2000 vertices using the Q-MAT method and the present invention. Experiments show that the invention not only can always maintain topology, but also has geometric precision similar to Q-MAT and faster calculation speed.

Q-MAT (median plane simplification method based on quadratic error minimization)

further examples of the simplification of the median plane in the three-dimensional model can be seen in fig. 8. Where (a) is the initial median plane of the input,

(b) the simplified result simplified to 1000 and 500 vertexes, (d) the model reconstructed from (c), (e) the error visualization result of the reconstructed model compared with the original model; # v denotes the number of vertices and e denotes the reconstruction approximation geometry error. It can be seen that the invention can always generate a simple median plane for maintaining topology from complex and redundant initial median planes for models with different topological structures, and has great significance for numerous applications (such as three-dimensional model animation technology and the like) expressed based on the median plane.

The invention has not been described in detail and is within the skill of the art.

The above description is only a part of the embodiments of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention.

Claims (5)

1. a method for simplifying the median plane of a three-dimensional model with topology preservation is characterized by comprising the following steps:

(1) Searching holes, namely defects, on a median plane of the three-dimensional model, and marking surface patches around the holes on the median plane as topological key areas; the edges of the patches are key edges, namely edges which need topology check in simplified calculation;

(2) when the median plane of the three-dimensional model is simplified by using an edge folding method, carrying out topology check on edges in a topology key area before simplification, and judging whether related holes disappear or not after the edges to be simplified are simplified, so that the topology is damaged;

(3) Dividing a bounding box of a median plane of the three-dimensional model into even slices, and performing parallel simplification processing in an odd/even interleaving mode on the slices; the edges of the non-topological key area are directly folded and simplified; when the edge in the topological key area is processed, the topological check in the step (2) is needed, if the topology is damaged, the edge is not simplified, if the topology is not damaged, the edge is simplified, and a newly generated patch is marked as the topological key area; finally, a simplified median plane of topology maintenance is obtained; the simplified median plane can be used as a brief expression of a three-dimensional model to be applied to a plurality of application scenes such as model compression and approximation, model animation and deformation, model retrieval and identification and the like.

2. a method of simplifying the median plane of a topologically preserved three-dimensional model according to claim 1, characterized in that: in the step (1), finding the holes, namely the defects, on the median plane of the three-dimensional model, and marking the surface patches around the holes on the median plane as topological key areas, specifically comprising the following steps:

(11) Carrying out orthographic projection on the median plane in 6 orthogonal directions of +/-x, +/-y and +/-z of a coordinate system in which median plane data of the three-dimensional model are located to obtain 6 depth images and 6 corresponding patch serial number images, wherein the patch serial number images record the corresponding patch serial number of each pixel in the depth images on the median plane;

(12) judging the depth difference between each pixel and the peripheral pixels of the depth image; if the difference is larger than a certain threshold value, marking the pixel as a key pixel;

(13) Finding out the sequence number of each key pixel, forming a topology key area by the patches, wherein the edges of the patches are key edges, namely the edges needing topology check in simplified calculation.

3. the method of claim 2 for simplifying the median plane of a topology preserving three-dimensional model, characterized in that: in the step (12), the certain threshold in the step (12) is a length of a longest side on a median plane.

4. A method of simplifying the median plane of a topologically preserved three-dimensional model according to claim 1, characterized in that: in the step (2), the specific steps of performing topology check on the edges in the topology critical area before simplification are as follows:

(21) traversing the adjacent vertex sets of the two vertexes of the edge to find the common adjacent vertex;

(22) Traversing the adjacent patch sets of the two vertexes of the edge to find the common adjacent patches of the two vertexes;

(23) for each co-adjacent vertex, a determination is made as to whether there is a co-adjacent patch that contains the vertex. If all the common same-phase adjacent vertexes have the common adjacent surface patches, the topological structure cannot be damaged by folding the edge; otherwise, folding the edge destroys the topology.

5. a method of simplifying the median plane of a topologically preserved three-dimensional model according to claim 1, characterized in that: in the step (3), the model bounding box is divided into even slices, and the specific steps of performing parallelization simplification processing in an odd/even interleaving mode on the slices are as follows:

(31) Dividing a bounding box of the three-dimensional model into an even number of fragments; the width of the fragment is required to be more than four times of the length of the maximum side length of the median plane;

(32) Alternately selecting all the fragments with odd serial numbers and all the fragments with even serial numbers to carry out parallel simplification processing until a simplification target is reached;

(33) When the simplification processing is carried out on the median plane in one fragment in the step (32), the simplification processing is carried out in an edge folding mode, and the edge with the minimum simplification cost in the fragment is found each time; checking whether the edge is a key edge, if not, simplifying the edge, otherwise, carrying out topology check, and judging whether the simplification of the edge can cause the disappearance of the related holes; if the relevant holes disappear, the edge is not simplified, if the relevant holes disappear, the edge is simplified, the newly generated patch is marked as a topological key area, and the corresponding edge is marked as a key edge.