KR20030016477A - Remeshing optimizing method and apparatus for polyhedral surface - Google Patents

Abstract

PURPOSE: A method and a system for optimizing remeshing of the surface of a three-dimensional polygon are provided to reduce distance error to preserve the original shape of mesh and improve regularity quality of the mesh. CONSTITUTION: The original mesh is simplified. The simplified mesh is subdivided to be refined. The subdivided mesh is optimized to deform the mesh into the original mesh shape. The subdivision step is repeated until the mesh deformed into the original shape satisfies a predetermined condition. The predetermined condition is Hausdorff distance. The simplification is carried out in such a manner that a procedure including a step of calculating the positions of all vertexes, a step of selecting the effective pair for all vertexes, a step of calculating the position of a new vertex for the selected effective pair, and a step of aligning the effective pair with heap at the calculated position to reduce edges is repeated.

Description

Remechanism Optimization Method and Apparatus for 3D Polygonal Surfaces {REMESHING OPTIMIZING METHOD AND APPARATUS FOR POLYHEDRAL SURFACE}

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to remeshing of polygonal surfaces, and more particularly, to methods and apparatus for remeshing 3D polygonal surfaces in the field of computer graphics and geometric modeling of geometric figures.

Currently, polygons are used a lot to express 3D objects. This is because any object can be expressed by approximating a polygonal surface.

In particular, triangular meshes have become the standard way of representing the shape of three-dimensional objects in computer graphics and figure modeling techniques, because it is fast and effective to represent objects in triangles.

However, in some cases, the object may be represented by millions of triangles. In this case, real-time processing (animation, rendering) is not possible with the current technology, and a lot of burden is placed on file storage or transmission.

To solve this problem, a simplification method and a multiresolution representation method of expressing the model with fewer triangles while maintaining the shape of the original model to the maximum have been proposed. In particular, wavelet transform was applied to the multi-resolution representation to obtain good results.

However, there is a disadvantage in that the wavelet transform is applied to a multi-resolution representation method to be limited to a mesh having subdivision connectivity.

In order to remedy this drawback, a remeshing method using parameterization that remaps an irregular mesh by mapping points of the original mesh to a simplified mesh has been proposed.

However, the remeshing method using parameterization has a problem in that when the mesh is not soft enough or complicated, one-to-one correspondence cannot be made at the time of mapping or mapped to the wrong point.

In order to solve this problem, a remeshing method using a local harmonic map, a conformal map, or shrink wrapping has been proposed.

However, the local harmonic map and the conformal map method are difficult to implement in the remeasurement method using parameterization or the remeshed mesh does not have the overall softness. In particular, the remeshing method using shrink wrapping applies only to a genus-zero triangle mesh where any vertex can have a value other than '6' and also has a limitation that the base domain mesh must be quite similar to the original mesh.

The present invention is applicable to non-manifold surfaces as well as to meshes where the remeshed mesh has overall softness and general connectivity and boundaries to improve conventional problems. An object of the present invention is to provide a method and apparatus for remeshing a 3D polygonal surface, which is designed to optimize remeshing to ensure a Hausdorff distance to a mesh within a standard.

1 is a block diagram of a remodeling optimization apparatus for an embodiment of the present invention.

FIG. 2 is a detailed block diagram of the simplified processing unit in FIG. 1; FIG.

3 is a view showing an edge reduction method.

4 illustrates edge reduction approximated in two dimensions.

5 illustrates an algorithm of surface streamlining.

Figure 6 is an exemplary view showing a remetering process in an embodiment of the present invention.

7 shows an example of B-to-B projection in a rabbit model.

8 is an enlarged view of an ellipse portion shown in FIG.

Figure 9 is an exemplary view showing an optimization remodeling process modeled on a mannequin.

10 is an exemplary view showing an optimization remodeling process using a horse model.

Figure 11 is an exemplary view showing the results of the remeet modeled on the rabbit in the present invention and the existing method.

12 is a chart comparing the mean squared distance error of the base domain to which the results of QEM and the globally optimized remeshing algorithm (GORA) method are applied.

Fig. 13 is a diagram showing distance errors in the MAPS and GORA (ICP) methods.

14 is a chart showing regularity quality.

15 is a chart showing mesh resolution.

16 shows an histogram of compactness.

Figure 17 is an exemplary diagram showing an edge length histogram.

Explanation of symbols on the main parts of the drawings

100: simplified processing unit 110 to 130: simplified circuit

200: segmentation processing unit 210: intermediate point segmentation circuit

220: planarization filter 300: optimization processing unit

400: reference mesh input unit

The present invention is to simplify the original mesh to achieve the above object, subdivision and refine the refined mesh (simplified) in the above, and to transform the subdivided mesh in the shape of the original mesh (deformation) optimization process, and repeating the above-mentioned subdivision process until the mesh deformed in the form of the original mesh satisfies a predetermined condition so as to obtain an optimized new mesh having subdivision connectivity. It features.

The simplification process includes calculating positions of all vertices, selecting valid pairs for all vertices, and calculating new positions for the selected valid pairs to align the valid pairs on the heap. Repeating to reduce the edges.

The segmentation process is characterized in that it comprises the step of subdividing the base mesh with respect to the midpoint, and the step of flattening the subdivided mesh by the surface streamline method.

The surface streamlining method includes the steps of calculating an error from neighboring vertices for all vertices, adding a value by any multiple of the error for each vertex, multiplying the error value by a random constant value, and It is characterized by consisting of steps that add to the vertex.

The optimization process includes defining a method of projecting a point uniformly distributed from an original mesh to a mesh to be deformed, and minimizing an energy function by using the projection information defined above. It is characterized by using a GORA (ICP) algorithm to minimize the distance between the two meshes by performing a step of deforming the mesh by repeatedly obtaining the position of the vertex (vertex).

The optimization process is characterized by first applying the Iterated Closed Point (ICP) method to the base region in order to optimize the maintenance and simplification of the shape and boundary of the original mesh.

Hereinafter, the present invention will be described based on the drawings.

1 is a block diagram of a remeshing apparatus for an embodiment of the present invention, as shown in FIG. 200 and the subdivided mesh that is the output from the subdividing processing unit 200 from the base mesh in the simplified processing unit 100 ( Optimization processing unit 300 for minimizing the distance between the two meshes by adjusting the vertex position of the subdivided mesh by performing projection onto the; and the base mesh sequentially simplified by the original mesh and the simplified processing unit 100. To the base mesh ( It is configured to include a reference mesh input unit 400 to be input to the optimization processing unit 300 as).

As shown in the block diagram of FIG. 2, the simplification processing unit 100 includes a simplification circuit 110 for firstly inputting the original mesh to the reference mesh input unit 400 and the mesh simplification circuit 110. Second simplification process of the mesh in the input to the reference mesh input unit 400, the simplified circuit 120 and the mesh in this mesh simplification circuit 120 to the third process to simplify the input to the optimization processor 300 A simplified circuit 130 is provided.

The refinement processing unit 200 is an intermediate point refinement circuit 210 for subdividing the intermediate point with a coarse base mesh output from the optimization processing unit 300, and the optimization by flattening the fine mesh. The flattening filter 220 which inputs to the process part 300 is comprised.

The flattening filter 220 is configured as a mesh streamlined filter.

When described in detail the operation and effect of the embodiment of the present invention configured as described above.

According to an embodiment of the present invention, a process of obtaining a new mesh having subdivision connectivity by simplifying the original mesh by the simplification processing unit 100, and by using the simplification processing unit 100, the simplification processing unit 100 Subdivision and refinement, and the optimization processing unit 300 includes a process of deforming the subdivided mesh in the shape of the original mesh by the refinement processing unit 200 to form the original mesh. The process is repeated until the deformed mesh satisfies a preset condition.

If the operation in the embodiment of the present invention will be described for each process as follows.

1. A process of simplification of a polygonal surface representing an arbitrary object will be described with reference to FIGS. 3 and 5.

The simplification process in the embodiment of the present invention performs iterative edge contraction on the original model to represent the model with fewer polygons while maintaining the shape of the original model as much as possible. This is executed in the simplified processing unit 100.

The simplification processor 100 sequentially simplifies the original mesh representing the target object through a plurality of simplification circuits 110 to 130, and finally inputs the simplified mesh to the optimization processor 300.

And, the reference mesh input unit 400 is a reference mesh (a mesh input to the simplified circuits 110 to 130) ) To be input to the optimization processing unit 300.

When the mesh is simplified in the simplification processing unit 100, an edge is removed. When an arbitrary edge is removed, both endpoints of the edge are replaced with a new vertex. Usually one face is removed for the borderline edge and two faces are removed for the borderline edge.

The simplification process can be subdivided in several ways, depending on the selection criteria for which edges are removed first and the conditions for determining the position of the new vertices in order to maintain the shape of the original model as much as possible.

In an embodiment of the present invention, iterative edge collapse using QEM (Quadric Error Metrics) proposed by Garland, which has less constraints on geometry or topology, is relatively fast in execution, and obtains good results. The method will be described as an example.

That is, the simplification processing unit 100 describes a process of achieving surface simplification by executing iterative edge contraction using quadrature error metrics (QEM) with reference to FIG. 3.

(1) Compute Q matrices for all initial vertices.

(2) Select a valid pair (v ₁ , v ₂ ) of all edges or vertices to be reduced.

(3) a new vertex position for each valid pair (v ₁ , v ₂ ) Calculate At this time, the new vertex error ( Is the cost of this pair.

(4) Sort all pairs on the heap according to the values obtained above. At this time, the pair with the smallest value is placed on top of the heap.

(5) Iteratively calculate the pair with the smallest cost (v ₁ , v ₂ ) in the heap. At this time, the valid pairs v ₁ and v ₂ are contracted and the values of all other pairs including the vertex v ₁ are updated.

3 shows one pair of vertices (v ₁ , v ₂ ) The figure shows an edge reduction method that is reduced to), where the value of the effective pair (v ₁ + v ₂ ) and the new vertex ( Use QEM to calculate the position of.

By the way, iterative edge reduction method using quadrature error matrix (QEM) has less constraints on mesh geometry or topology and is relatively fast to execute and can get good quality. Sometimes spikes occur and the vertex's angle grows around its boundaries.

That is, quadratic error (E _Q ) may occur.

Below, the process of defining and deriving quadratic error (E _Q ) is described.

Several vertices are adjacent to each vertex on the mesh. For example, in a regular mesh, six vertices are adjacent to each vertex.

In this case, an error approximated at each vertex may be defined as a sum of squares of distances to adjacent surfaces.

The adjacent face belongs to a plane that satisfies n ^T v + d = 0, as shown in FIG. Where n = [abc] ^T is the unit normal vector of the plane and d is a constant defined in the plane.

4 illustrates edge reduction approximated in two dimensions.

The square of the distance from any vertex (v = [x, y, z] ^T ) to the plane can be defined as shown in Equation 1 below.

If the quadratic surface function Q is defined as (A, b, c) = (nn ^T , dn, d ² ), then the quadratic surface function Q (v) at the vertex v is Q (v) = v ^T Av + 2b ^T v + c.

Further, the quadratic surface error E _Q at any vertex v is defined as the sum of the quadratic surface function Q _i as shown in Equation 2 below.

Therefore, the new vertex (Equation 2) We can calculate the position of) as

Function (Q ( )) Is quadratic, so the vertex with the minimum value of distance error ( ) Is the point that satisfies Equation 3 below.

At this time, Become, Solve for optimal vertex position = -A ^-1 b can be found.

In summary, the value of the effective pair (v ₁ + v ₂ ) is obtained by Equation 2 above to reduce the edge using a second order error matrix. From this the new vertex position is = -A ^-1 b, and the new vertex ( The approximate distance error of) is defined as the sum of Q (v ₁ ) and Q (v ₂ ).

Therefore, limiting the position of the new vertex to the point that minimizes the QEM on the boundary edge prevents spikes and limits the degree that the vertex can have.

2. A subdivision process for the base mesh will be described with reference to FIG.

The segmentation process is a process of obtaining a mesh having subdivision connectivity by applying successive segmentation to the base domain when the base mesh is generated for the original mesh in the simplification process, which is performed by the segmentation processor 200.

The refinement processing unit 200 divides the intermediate point refinement circuit 210 into a coarse base mesh output from the optimization processing unit 300 with respect to the intermediate point, and the flattening filter 320 refines the refined mesh. Is flattened and input to the optimization processor 300.

The flattening filter 320 flattens the subdivided mesh without reducing the mesh volume.

In other words, the segmentation process uses a loop subdivision method that applies iterative segmentation and smooths the segmented mesh at that time.

However, when applying the loop segmentation method, the remeasurement result may have insufficient quality at the temporary vertices.

That is, the loop segmentation method reduces the volume of the sparse mesh and such volume reduction can make the optimization process difficult.

Therefore, in the embodiment of the present invention, the mesh is segmented with the intermediate point and then surface fairing filtering is performed to improve the flattening and miniaturization in the temporary vertices as much as the normal vertices.

In other words, the planarization filter 220 performs a surface fairing process to increase the smoothness of the mesh subdivided in the midpoint segmentation circuit 210.

An example of the surface fairing method is the method proposed by Taubin.

Taubin's method attempts a signal-processing approach to wireline triangular meshes with a common topology, using Laplacian's discrete approximation to define the common frequencies in the mesh. And its eigen vector was defined as frequency.

Defining discrete Laplacian of discrete surface signal is expressed by Equation 4 below.

Here, the weight w _ij is a non-negative value, = 1.

In the embodiment of the present invention, the surface streamlining process consists of three steps.

Step 1: For all vertices, an approximation DELTA x is calculated from the neighboring vertices as shown in Fig. 5 (a).

Step 2: Add λ times of the approximation Δx to each vertex to prevent shrinkage during smoothing of the mesh. Here, lambda has a value between 0 and 1.

Step 3: For each vertex, add μ multiplied by λ instead of λ. Where μ is a negative number and has a value less than -λ.

The process is repeated until the mesh is smooth enough.

This surface wired method is effective because the memory of the implemented circuit is proportional to the number of vertices and the execution time is very fast.

3. The optimization process to optimize the refined mesh close to the shape of the reference mesh will be described.

The refined mesh in the refinement processing unit 200 may have a certain distance error and may not be sufficiently flattened.

The optimization process is a process of optimizing the subdivided mesh close to the shape of the reference mesh in the refinement processing unit 200, and is performed by the optimization processing unit 300.

The optimization processor 300 initially sets the base mesh in the simplified circuit 130 of the simplified processor 100 in the reference mesh input unit 400. ) Is projected to the shape of the reference mesh and deformed to the refinement processing unit 200. Then, the base mesh in the refinement processing unit 200 is converted into a reference mesh () in the reference mesh input unit 400. ) And repeatedly deform to approximate the shape of the reference mesh.

The reference mesh input unit 400 sequentially converts the base mesh of the simplified circuits 120 and 110 of the simplified processor 100 and the original mesh input to the simplified circuit 100 into a reference mesh ( ) To be input to the optimization processor 300.

Therefore, the optimization processor 300 restores the mesh deformed close to the original mesh by repeating the above optimization process.

This optimization process is shown in the example of FIG.

This optimization process improves mesh shape and planarization by minimizing the global energy function.

In other words, minimization of the energy function is a refined mesh to minimize the equations. Compute the position of all vertices in the c-squared, which solves the least square problem using the conjugate gradient method.

That is, the linear least square problem is solved using the Conjugate Gradient method when minimizing the energy function E (K, V) for the position (V) of the subdivided mesh.

It uses the projection information from the original mesh to another mesh to repeatedly find the positions of the vertices that minimize the energy function. This is called the Iterated Closest Points method.

At this time, the reference mesh ( Meshes on points Iterative projection to the nearest points on the) Calculate the gradient of the energy function for the vertices of) and adjust the positions of all vertices from it.

That is, a mesh subdivided within each resolution level ( Shape of the reference mesh ( In order to calculate the distance between two meshes as close as possible to the shape of the The set of points on) is a refined mesh ( ) Onto the image.

If arbitrary points are subdivided into a mesh ( Subdivided mesh when not projected to any point on 4 points on the face of the ) Is projected to the nearest points on

In this case, assuming that the nearest points are coincident points of four points, the reference mesh ( The set of points in the neighboring area with the four nearest points on Any point is a refined mesh ( ) Is reprojected to the neighboring area of 4 points.

And, a fine mesh Granular mesh () when coarse Since we cannot describe the input mesh in detail from Apply multilevel projection, used with.

In other words, since the coarse base mesh is generated by simplifying the input mesh, the shape of the base mesh is closer to that of the simplified input mesh than the fine input mesh.

Thus, the projection from the simplified input mesh to the base mesh is more reliable than the projection from the fine input mesh to the base mesh.

However, when the input mesh is very complicated, the closest point on the subdivided mesh with respect to the point on the input mesh may not be coincident.

In this case, even if the energy function includes a spring energy representation, minimizing the overall energy function can result in insufficient performance, namely face flip, insufficient planarization, and large distance error. In other words, the wrong projection can cause the optimized mesh to have bad geometric figure properties, namely face flip and bad planarization.

Therefore, we will use a new projection method to prevent face flipping and consequently the compactness of the surface.

In general, a mesh may have a boundary line.

Preserving these boundaries is very important to preserve the overall shape of the mesh.

To preserve the overall shape of the mesh, we define F-to-F (Face to Face) projections and B-to-B (Boundary to Boundary) projections, respectively.

First, F-to-F projection is to select points that are uniformly distributed according to area on the surface on the reference mesh and project the selected points onto the surface of the subdivided mesh.

In this case, the closer the projected point is to a certain vertex, the more it affects the gradient of the vertex in proportion to the corresponding gravity center coordinate vector.

That is, the gradient of the energy function at a vertex is governed by the projection point when a point projected into one face approaches one of the three vertices of the face.

If an arbitrary point is not projected onto one face, the gradient does not contain information about the face, so that the face may not be simplified, and the area around the face may also have insufficient planarization. Select the points of and project them onto the reference mesh.

Since F-to-F projection is a time-consuming task, the projection time is reduced by using the spatial division method.

Coherence of coupling at each step can be used for further speedup. That is, the use of the fact that projection at any level is performed near the projection position at the previous level can greatly shorten the execution time.

B-to-B projection is a defined way to maintain the boundary shape, making all the points on the boundary edges project only to the boundary edges of the other mesh, making the shape of one mesh as close as possible to the shape of the other mesh. Way.

An example of B-to-B projection is shown in FIG.

FIG. 7 is an exemplary B-to-B projection of a bunny model 920 faces, with red dots representing boundaries on the original mesh and black dots representing projected points. FIG. 8 is an enlarged view of an ellipse portion in FIG. 7.

In other words, F-to-F projection can be defined as projection from plane to plane and B-to-B projection from edge to edge projection.

However, the shape of the optimized mesh depends largely on which points are projected. If the boundary is a complicated shape, the deformation using only the face-to-face projection is difficult to maintain the complex shape of the boundary.

Therefore, in the embodiment of the present invention, the F-to-F projection and the B-to-B projection are implemented together, whereby the B-to-B projection has a greater weight than the F-to-F projection.

Further, in the embodiment of the present invention, the F-to-F projection and the B-to-B projection may be implemented to be performed separately.

In terms of the overall energy function, the B-to-B projections have a greater weight than the F-to-F projections.

Meanwhile, the above-described optimization process will be described using mathematical equations.

The overall energy function is expressed in two ways: the distance from each subdivision mesh to the input mesh, and the spring energy function.

And a similar function is used for the generation of optimized meshes from unorganized pints.

The optimization processor 300 may be configured to refine the refined mesh to be as close as possible to its original shape. By adjusting the position of the vertices, the distance between the two meshes is minimized.

To do this, define an energy function as shown in Equation 5 below, and find a deformed mesh that minimizes this energy function.

In order to measure the distance energy between the two meshes, a projection which is uniformly distributed according to the area from one mesh to another mesh is used. In this case, a spatial segmentation data structure is used to improve the projection speed. use.

The sum of the squared distances between the two meshes may be expressed as distance energy as shown in Equation 6 below.

P ₁ is a point on the original mesh and p _c is the point projected onto another mesh.

At this time, if p _c is a point on the face, the reference mesh ( ) Points on Mesh refined to Nearest point on ) When you are on the ) Can be expressed as Equation 7 below.

V _i , v _j , and v _k represent three vertices of a face, respectively, , , Denotes a barycentric coordinate vector.

here, , , Is a value between '0' and '1' to be.

However, when the input mesh is very complicated, the closest point on the subdivided mesh with respect to the point on the input mesh may not be coincident.

In this case, minimization of the overall energy function can lead to insufficient performance, namely face flip, insufficient planarization and large distance error. In other words, the wrong projection can cause the optimized mesh to have bad geometric figure properties, namely face flip and bad planarization.

Thus, the spring energy function is used to prevent face flipping and consequently to the compactness of the surface.

The spring energy function that makes the energy function have a minimum value is defined as in Equation 8 below.

remind , Silver granular mesh ( On edge () Are the two end vertices of

At this time, edge length, surface area, etc. can be considered as k value, but simply interpreting as '1' can reduce the amount of calculation and can obtain good results.

Therefore, minimizing Equation 8 causes each vertex to be drawn with neighboring vertices, thereby making the mesh flat and miniaturized as a whole.

That is, a refined mesh ( ) Can be expressed as shown in Equation 9 below.

remind Is a point on the original mesh, Is the point corresponding to the other mesh, and Is the point on each subdivided mesh ( ) Is a weighting factor for projecting), and λ is a spring constant that can be arbitrarily determined by the user.

In addition, the operation of the embodiment of the present invention will be described with reference to the drawings.

First, the overall process of the present invention will be described with reference to the exemplary diagram of FIG. 6. The meshes displayed in the clockwise direction from the bottom of the left column to the top of the right column are meshes that are sequentially created in the simplified processing unit 100 and the right column. The mesh displayed from the upper end to the lower end of the right column is a mesh made by the segmentation processing unit 200 and the projection process between the left and right columns of each row is performed by the optimization processing unit 300.

And, in order to prove the effectiveness in the embodiment of the present invention will be described with reference to Figure 9 and Figure 10 for the remodeling process and results for various models.

FIG. 9 illustrates the remeshing process and results using a mannequin as a model. FIG. 9 (a) shows an original mesh M (25327 faces), FIG. 9 (b) shows a base mesh (4faces), and FIG. 9 (c) is Remesh M ^J (J = 1; 16 faces), FIG. 9 (d) is Remesh M ^J (J = 2; 64 faces), FIG. 9 (e) is Remesh M ^J (J = 3; 256 faces), Fig. 9 (f) shows Remesh M ^J (J = 4; 1024 faces), and Fig. 9 (g) shows Remesh M ^J (J = 8; 262144faces).

First, the base domain of FIG. 9 (b) is made by simplifying the original mesh of FIG.

Subsequently, the base domain mesh of FIG. 9 (b) is subdivided and then transformed as close as possible to the original mesh as shown in FIG. 9 (c) using an ICP algorithm.

Subsequently, the segmentation and deformation are repeated until the given condition (Hausdorff distance) is satisfied, and the strain is sequentially transformed into the base mesh as shown in FIGS. 9 (d) to 9 (f).

Thus, by repeating the above process, it is possible to finally obtain a remeasurement result as shown in FIG. 9 (g). 9 (g) shows the result of applying the GORA (ICP) method eight times when the Hausdorff distance is set to 5 × 10 ⁻⁴ .

And, Figure 10 shows a remodeling process based on the horse (horse) and the results, Figure 10 (a) is an orginal mesh M (96966 faces), Figure 10 (b) is a base mesh (200 faces), Fig. 10 (c) shows Remesh M ^J (J = 2; 3200 faces), Fig. 10 (d) shows Remesh M ^J (J = 3; 12800 faces), and Fig. 10 (e) shows Remesh M ^J (J = 2; 3200 faces). to be.

In addition, the results of the present invention were compared with existing methods.

FIG. 11 is a diagram illustrating remeasurement results using each method (MAPS, shrink wrapping, GORA) using a rabbit as a model. FIG. 11 (a) is an original bunny model and FIG. 11 (b) is a using MAPS. 11 (c) is using shrink wrapping, and FIG. 11 (d) is using GORA.

Therefore, when comparing the results of the embodiment of the present invention with the results of remechanization (MAPS) using the previously proposed parameters, it can be confirmed that the smoothing (smooth) over the entire portion of the mesh.

In general, it is very difficult to measure the quality of a remeshed mesh. This is not because there is an absolute standard, but because the criteria may vary depending on the given situation.

In general, the following are the criteria for measuring the quality of a commonly used mesh.

First, the most important factor is to maintain the shape of the original object as much as possible. In other words, the remeshed mesh should be as close as possible to the shape of the original mesh.

This includes maintaining a boundary shape.

For this purpose, Hausdorff distance or average squared distance error is usually used.

Second, if the mesh looks similar, fewer vertices are better. This means that other processing time, such as data storage and rendering, is reduced.

Third, the more regular the distance between vertices in a general mesh, the better, which means that the overall resolution of the mesh increases.

Therefore, in order to determine the quality of the mesh remeshed in the present invention, Hausdorff distance, mean square distance error, triangle compactness, mesh resolution, edge length spread were measured and compared.

The comparison result is shown in FIGS. 12-17.

The results of the remeeting of the present invention are compared with those of the method proposed by Lee and Kobelt.

First, FIG. 12 is a table comparing the QEM result and the mean square distance error of the base domain to which the GORA method is applied. It can be seen that the error is small when the GORA method is applied. At this time, all meshes were converted into unit cubes for uniform processing regardless of size, and then simplified and remechanized were applied.

And, in order to compare the accuracy of the remeshing, the distance error between the original mesh and the remeshed mesh was measured, and the result is shown in the diagram of FIG.

As shown in the diagram of FIG. 13, it can be seen that the results according to the method GORA of the present invention are both small at the Hausdorff distance and the average squared distance. In other words, it can be said that the remeshing method proposed in the present invention maintains the original shape better than the method of MAPS.

Here, Kobelt's remeasurement results are excluded from the distance error comparison because the topology is different from the original model.

Next, the mean and variance of the compactness were compared for all faces in order to measure the regularity quality of the remodeled rabbit model. This result is as shown in the table of FIG. 14 and the exemplary view of FIG.

Figure 16 shows a histogram of compactness in accordance with the present invention and the existing method.

Therefore, it can be seen that the overall compactness of the GORA method proposed in the present invention is the highest. This means that the regular quality of each face is better than the other methods.

Finally, the mesh resolution and edge length spread of the remeshed rabbit model were measured. This result is as shown in FIG.

Here, the resultant mesh of the Shrink Wrapping method cannot be compared with each other due to different face numbers, but compared with the MAPS method, the result of the proposed method has high resolution as shown in the table of FIG. 15 and the histogram of FIG. It can be seen that the edge length width is narrow.

17 shows an edge length histogram.

As described in detail above, the present invention has the effect of reducing the projection error than the conventional method by simplifying the original mesh to obtain the base domain, and then subdividing the base domain to make it as close as possible to the original mesh. .

That is, the present invention preserves the shape of the original mesh well because the distance error is small and the mesh resolution is higher than that of the conventional method, thereby improving the regular quality of the overall mesh.

The present invention can be effectively applied to mesh compression using wavelets or other multiresolution representations.

Claims (25)

A process of simplifying the original mesh, a process of subdivisioning and refining the simplified mesh in the process of simplification, and an optimization of optimizing the subdivided mesh to the shape of the original mesh And repeating the subdividing process until the mesh deformed in the form of the original mesh satisfies a predetermined condition. The method of claim 1, wherein the preset condition is a Hausdorff distance. The method of claim 1, wherein the simplification process includes calculating positions of all vertices, selecting valid pairs for all vertices, calculating positions of new vertices for the selected valid pairs, and 3D polygon, characterized in that iterative edge reduction process using a QEM (Quadric Error Matrix) that repeats the step of reducing the edge by aligning the effective pair in the heap at the calculated position How to optimize remeshing of surfaces. The method of claim 1, wherein the process of deforming as close as possible to the shape of the original mesh defines a projection that is uniformly distributed from the original mesh to the mesh to be deformed to minimize the distances between the two meshes and defines a defined projection. Iteratively using the information to find the position of the vertices (vertex) to minimize the function, characterized in that using the ICP algorithm including the procedure of finding the deformation of the mesh to minimize the defined projection distance How to optimize remeshing three-dimensional polygonal surfaces. 2. The method of claim 1, wherein the segmentation process comprises subdividing the base mesh with respect to the midpoint to have subdivision connectivity and flattening the subdivision mesh. Mesh optimization method. 6. The method of claim 5, wherein the planarization step applies a surface streamlining method. 7. The method of claim 5 or 6, wherein the planarization step comprises a first step of calculating an error from neighboring vertices for all vertices, a second step of summing up any multiple of the error for each vertex, and And a third step of multiplying the error value by an arbitrary constant value and adding it to each vertex. The method of claim 1, wherein the optimization process comprises a first step of defining a projection from the original mesh to a point that is uniformly distributed from the mesh to be deformed, and using the projection information above to produce an energy function. A second step of recursively obtaining positions of vertices to minimize the number of vertices; and a third step of minimizing the distance between two meshes by referring to the positions of the vertices obtained above. Optimization method. 10. The method of claim 8, wherein the first step of projection is a face-to-face projection that selects points distributed equally on the reference mesh and projects the points on the subdivided mesh. Way. 10. The method of claim 8, wherein the first step of projection comprises a Face-to-Face projection that selects points distributed equally on the reference mesh and projects the points on the subdivided mesh, and a boundary on all other points on the boundary edge. A method of remeshing 3D polygonal surfaces characterized by using a B-to-B projection projected to an edge. 11. The method of claim 10, wherein the weight of the B-to-B projection is set higher than the weight of the F-to-F projection. 11. The method of claim 9 or 10, wherein the spatial segmentation data structure is used to improve the speed of the F-to-F projection. 9. The method of claim 8, wherein the second step first applies an Iterated Closest Point (ICP) method to the base area to optimize the shape, boundary, and simplified mesh of the original mesh. Mesh optimization method. 9. The method of claim 8, wherein the third step minimizes an energy function expressed as a sum of distance energy and spring energy function as in the following equation. 15. The method of claim 14, wherein the distance energy is expressed as the sum of the squares of the distances between the two meshes, as shown below. P ₁ is a point on the original mesh and p _c is the point projected onto another mesh. 15. The method of claim 14, wherein the spring energy function is defined as in the following equation so that the energy function has a minimum value. remind , Silver granular mesh ( On edge () Are the two end vertices of 17. The method of claim 16, wherein the value of k is set to '1'. The method of claim 14 wherein the energy function is a reference mesh ( ) Points on Mesh refined to Nearest point on ) A method of remeshing a 3D polygonal surface, characterized in that it is deformed as follows. remind Is a point on the original mesh, Is the point corresponding to the other mesh, and Is the point on each subdivided mesh ( ) Is a weighting factor for projecting), and λ is a spring constant that can be arbitrarily determined by the user. here, Where v _i , v _j , v _k represent the three vertices of the face, respectively, , , Denotes a barycentric coordinate vector. 19. The method of claim 18 wherein the center of gravity coordinate vector ( , , ) Is a value between '0' and '1' Remethod optimization method of a three-dimensional polygonal surface, characterized in that satisfying the. Simplified processing means for simplifying the shape of the input mesh, subdividing processing means for subdividing and flattening the simplified base mesh, and a subdivided mesh in the subdividing processing means from the base mesh in the simplified processing means ( Optimization processing means for minimizing the distance between the two meshes by performing projection onto the refined mesh and adjusting the vertex positions of the subdivided mesh, and a base mesh sequentially processed by the original mesh and the simplified processing means. And a reference mesh input means for inputting the input to the optimization processing means. 21. The method of claim 20, wherein the simplification processing means comprises: a first simplification circuit for firstly simplifying the original mesh and inputting the reference mesh to the reference mesh input means, and secondly simplifying the mesh in the first simplification circuit to input the reference mesh. A second simplification circuit for inputting by means, and a third simplification circuit for inputting the mesh in the second simplification circuit to the optimization processing means by third order simplification. Optimization device. 21. The method according to claim 20, wherein the refinement processing means comprises: a midpoint refinement circuit for subdividing a coarse base mesh output from the optimization processing means with respect to an intermediate point, and the optimized meshing means by flattening the fine mesh. And a flattening filter to be inputted to the three-dimensional polygonal surface remodeling optimization device. 23. The apparatus of claim 22, wherein the flattening filter comprises a mesh streamlined filter. The method of claim 20, wherein the optimization processing means comprises a reference mesh ( ) Points on Mesh refined to Nearest point on ) And a minimizing energy function as shown in the following equation. remind Is a point on the original mesh, Is the point corresponding to the other mesh, and Is the point on each subdivided mesh ( ) Is a weighting factor for projecting), and λ is a spring constant that can be arbitrarily determined by the user. here, Where v _i , v _j , v _k represent the three vertices of the face, respectively, , , Denotes a barycentric coordinate vector. The method of claim 24, wherein the center of gravity coordinate vector ( , , ) Is a value between '0' and '1' Remeshing optimization device for a three-dimensional polygonal surface characterized in that.