Abstract:
A method and apparatus for the simplification of a mesh surface is disclosed that preserves the original geometry of the shape of the surface and, at the same time, reduces undesirable triangle geometries. In one embodiment, a mesh simplification process first determines whether an edge swap operation should be performed as a function of a threshold criteria. Such a threshold may be a function of the span angles and cross angles associated with an edge or, alternatively, may be a predetermined span angle size threshold. In another embodiment, the decision as to whether to contract an edge is made by comparing the size of at least one span angle with a span angle threshold and by comparing the sizes of incident angles associated with the edge to an incident angle threshold.

Description:
This patent application claims the benefit of U.S. Provisional Application No. 60/742,503, filed Dec. 5, 2005, which is hereby incorporated by reference herein in its entirety. 
     CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is also related to U.S. Pat. application Ser. No. 11/466,194, titled Method and Apparatus for Non-Shrinking Mesh Smoothing Using Local Fitting; and U.S. patent application Ser. No. 11/466,211, now U.S. Pat. No. 7,623,127, titled Method and Apparatus for Discrete Mesh Filleting and Rounding Through Ball Pivoting, both of which are being filed simultaneously herewith and are hereby incorporated by reference herein in their entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates generally to two- and three-dimensional mesh shapes and, more particularly, to the simplification of the meshes on the surfaces of such shapes. 
     Many applications, such as medical and industrial design and manufacturing applications, involve manipulating and editing a digital model of an object. As one skilled in the art will understand, such a digital model may be created by scanning an object to create a point cloud representation of the object. The surface of such a model of a scanned object typically consists of a plurality of points, the number of which is a function of the resolution of the scanning process. Once such a point cloud representation has been obtained, the surface of the object may then be approximated by connecting the points of the point cloud to form a plurality of geometric shapes, such as triangles, on the surface of the model. This model may then, for example, be edited by using computer aided design (CAD) software programs or similar specialized image manipulation software programs. 
       FIG. 1  shows an illustrative model of the surface  101  of such an object. Referring to that figure, points  103  are, illustratively, the points in a point cloud that are obtained from the scanning of the object. Then, as discussed above, well-known methods are used to connect the points in a way such that the surface of the object is approximated by a plurality of triangles  102 , referred to herein collectively as a triangle mesh. One such method for constructing a surface by connecting points and forming triangles is described, for example, in F. Bernardini et al., The Ball-Pivoting Algorithm for Surface Reconstruction , IEEE Transactions on Visualization and Computer Graphics, 5(4), October-December, 1999, pp. 349-359, which is hereby incorporated by reference herein in its entirety. Many other methods for creating mesh surfaces have also been developed and are well-known. 
     As one skilled in the art will recognize, there is an inherent tradeoff between the accuracy of the approximation of a surface and the complexity of the triangle mesh used for the approximation. Specifically, a mesh consisting of a large number of relatively small triangles will typically produce a more accurate surface approximation. However, the larger the number of triangles, the greater the volume and complexity of the computations required to create and edit the surface. Therefore, as one skilled in the art will also recognize, it is thus advantageous to balance the complexity of the mesh surface with the computational cost to produce an accurate surface model while, at the same time, reducing the associated cost of creating and editing the model. To achieve such a balance, mesh simplification algorithms have been developed to simplify mesh models and, therefore, reduce the aforementioned computational costs while at the same time retaining satisfactory accuracy. 
     Prior attempts at simplifying a triangle mesh surface have involved either local or global methods, As one skilled in the art will recognize, local methods attempt to simplify a mesh by treating discrete portions of a mesh individually while, on the other hand, global methods attempt to simplify the mesh by treating the entire mesh as a whole. Global methods involve, for example, minimizing an error function that ranks all of the possible mesh simplification steps and then performs them, illustratively, in order according to the identified rank (i.e., starting with those steps that alter the geometry of the mesh the least). Given the larger number of triangles in meshes of complex objects, global methods require a relatively large number of computations to rank the simplification steps and then reconstruct a simplified mesh surface of a model. On the other hand, local methods require less computations since fewer points/triangles in a mesh are to be simplified at any one time. As one skilled in the art will recognize, local methods of mesh simplification are greedy algorithms in that they attempt to find a locally optimum choice (here, a mesh simplification choice) at each stage of the algorithm. The goal of such algorithms is to attempt to determine the global optimal solution to mesh simplification based on local results. Since local algorithms do not consider the entire set of triangles in the mesh, however, such a global optimal solution is typically not achieved. However, while such local algorithms typically do not result in the global optimal solution, they typically provide satisfactory approximations of such a solution. Therefore, since the computational cost of such local methods is typically much lower than global methods, and the results of such methods are adequate for most applications, many attempts at mesh simplification have relied on such local methods. 
     Different local mesh simplification methods are known. One class of such methods, known as vertex decimation methods, function by removing vertices of triangles and then reconstructing the surface of the model to fill in the resulting gap in the surface left by the deleted vertex. However, such methods can be undesirable since such surface reconstruction requires a reconstruction of the entire mesh surface once one or more vertices of local triangles have been deleted. Thus, the computational cost of such methods is relatively high. 
     Another local mesh simplification method, referred to herein as edge contraction, also functions to reduce the number of vertices in a mesh. Unlike the vertex decimation technique, the edge contraction method does not delete the vertices and, therefore, create gaps in the mesh that require reconstruction of the surface. Instead, such edge contraction methods operate by identifying a target edge to be contracted and, then, by moving one or both of the two endpoints of the target edge to a single position. According to this method, and as discussed further herein below, all edges incident to the original vertices of the contracted edge are thus linked at this single position Any triangle faces that have degenerated into lines or points are then removed. 
       FIGS. 2A and 2B  show an example of how edge contraction can be used to simplify a 2D or 3D surface approximated by a triangle mesh. Specifically, referring to those figures, surface  201  is a surface of an object characterized by a mesh of triangles  202 . Each of the triangles  202  is characterized by having three vertices and three edges connecting those vertices. As discussed above, during edge contraction, one edge, such as edge  203  between vertices  206  and  207 , is contracted in a way such that vertex  206  and vertex  207  are moved to the same position. The result of such a contraction is shown in  FIG. 2B . Specifically, edge  203  is contracted so that vertex  206  and vertex  207  are positioned at position  204  in  FIG. 2B . As a result, triangle faces  212  and  213  are transformed into edges  218  and  219  in  FIG. 2B . Also, all incident edges to vertices  206  and  207 , such as edges  208 ,  209 ,  210  and  211 , now meet at new vertex  204 . Thus, the resulting mesh is simplified by the consolidation of two vertices into a single vertex and the removal of two triangle faces from the mesh surface. 
     A key determination in edge contraction operations is how to contract the edge into a single vertex and, more particularly, where vertex  204  should be positioned in the model after the contraction is completed. Typical prior efforts selected one of the vertices, such as vertex  206  in  FIG. 2A , and then contracted edge  203  in a way such that the position of vertex  204 , which combined vertices  206  and  207  together, was at the position of prior vertex  206 . Alternatively, an average of the position of vertices  206  and  207  was determined and the new vertex  204  was positioned at that average position. 
     SUMMARY OF THE INVENTION 
     The present inventors have recognized that, while prior methods for the simplification of a mesh surface are advantageous in many regards, they are also disadvantageous in certain respects. Specifically, while prior edge contraction methods would produce a simplified mesh surface (i.e., having fewer triangles and vertices), they could undesirably alter the original geometry of the model. In addition, such prior methods could often result in undesirable triangle geometries, such as overly-thin triangles, as a part of the mesh after simplification was performed. Such thin triangles having, for example, one very large vertex angle and one-or more very small vertex angles, are undesirable since they tend to introduce excessive errors in algorithms requiring robust numerical computations associated with the editing or manipulation of a mesh. 
     Accordingly, the present inventors have invented a method for simplification of a mesh surface that preserves the original geometry of the shape of the surface and, at the same time, reduces undesirable triangle geometries, such as the aforementioned thin triangles. More particularly, in accordance with an embodiment of the invention, a mesh simplification process first determines whether an operation, referred to herein as an edge swap operation and discussed further herein below, is desired. In a first embodiment such a determination is made as a function of a comparison of the size of span angles associated with an edge with the size of cross angles associated with that edge. In a second embodiment, this determination is made by the comparison of the size of at least one span angle with a span angle threshold. In accordance with another embodiment, a determination is next made whether to contract the edge. This determination may be made, illustratively, by referring to the comparison of the size of span angles associated with the edge with the size of cross angles associated with that edge. Alternatively, in yet another embodiment, the decision as to whether to contract an edge is made by comparing the size of at least one span angle with a span angle threshold and by comparing the sizes of incident angles associated with the edge to an incident angle threshold. Based on the foregoing comparisons, the edges on a triangle mesh surface are swapped and/or contracted so as to advantageously simplify that surface while maintaining a desired accuracy in the mesh representation of that surface. 
     These and other advantages of the invention will be apparent to those of ordinary skill in the art by reference to the following detailed description and the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a model of a 3D shape having a triangle mesh surface; 
         FIGS. 2A and 2B  show an illustrative mesh surface having an edge to be considered for mesh simplification; 
         FIGS. 3A and 3B  show a second illustrative mesh surface having an edge to be considered for mesh simplification; 
         FIG. 4  shows an expanded view of the mesh surface of  FIG. 2A ; 
         FIG. 5  is a flow chart showing the steps of a method in accordance with an embodiment of the present invention; and 
         FIG. 6  shows a computer adapted to perform the illustrative steps of the method of  FIG. 5  as well as other functions associated with the simplification of triangle mesh surfaces. 
     
    
    
     DETAILED DESCRIPTION 
     The present inventors have recognized that it may be desirable in some instances to determine whether an edge swap is desirable either prior to or in place of an edge contraction operation.  FIGS. 3A and 3B  show an example of an illustrative edge swap operation that is useful for reducing undesirable geometry features that are present in a triangle mesh, such as thin triangles, while, at the same time, preserving an accurate approximation of the surface of the object represented by the triangle mesh. Specifically, referring to those figures, illustrative surface  301  is a mesh surface comprising a mesh of triangles  302 . These triangles  302  are, as discussed above, each characterized by three edges connecting three vertices, for example edge  303  connecting vertices  305  and  306 . Edge  303  has span angle S 1   309  associated with vertex  307  and span angle S 2   310  associated with vertex  308 . As shown in  FIG. 3A , triangles  311  and  312  having common edge  303  are characterized in that they are thin, i.e., they have relatively large span angles S 1   309  and S 2   310  and small incident angles I 1   315  and I 2   316  (in triangle  311 ) and I 3   319  and I 4   320  (in triangle  312 ). As discussed above, thin triangles are typically undesired since the existence of such triangles tends to lead to a total increased number of triangles in the mesh which, for example, increases the computation cost of mesh simplification operations. Alternatively, such thin triangles may lead to inaccuracy in the numerical computations necessary to perform various operations, such as editing operations, on the respective mesh surface. Therefore, in accordance with an embodiment of the present invention and as shown in  FIG. 3B , an edge swap may be performed to eliminate these undesirable thin triangles Specifically, referring to  FIG. 3B , such an edge swap is accomplished by removing edge  303  and replacing that edge with edge  304  in  FIG. 3B  between vertices  307  and  308 . As can be seen by reference to  FIG. 3B , such a swap eliminates thin triangles  311  and  312  and, instead, creates more desirable triangles  313  and  314 , i.e., which have vertex angles closer to being equal than do triangles  310  and  311  that existed before the swap. 
     As discussed above, the present inventors have discovered that it is desirable to consider whether one or both an edge swap or an edge contraction should be performed on an edge. In order to make this determination, various criteria may be used. Illustratively, in accordance with a first embodiment of the present invention, threshold criteria are defined for both span angles S 1   309  and S 2   310 , and incident angles I 1   315 , I 2   316 , I 3   319  and I 4   320 . More particularly, according to one embodiment, let a first cross angle C 1  between triangles  311  and  312  on either side of common edge  303  be defined as:
 
 C 1= I 1+ I 3  (Equation 1)
 
and a second cross angle C 2  between those triangles be defined as:
 
 C 2= I 2+ I 4  (Equation 2)
 
Then, the determination whether to perform an edge swap or an edge contraction may be performed according to the expressions:
 
 S 1+ S 2&gt; C 1+ C 2=&gt;swap edge  (Equation 3)
 
and
 
 S 1+ S 2&lt; C 1+ C 2=&gt;contract edge  (Equation 4)
 
where, once again S 1  and S 2  are span angles of an edge and Cl and C 2  are the cross angles associated with that same edge.
 
     Referring to  FIG. 3A , assume that the following angles of that figure have the following values: 
                                   TABLE 1                   Values for angles for consideration of edge swap                Angle   Value                       S1 309   135 degrees            S2 310   120 degrees            I1 315   20 degrees           I2 316   20 degrees           I3 319   30 degrees           I4 320   35 degrees                        
Thus, according to Table 1 and the foregoing equations, S 1 +S 2 =255 degrees total. C 1 =I 1 +I 3 =20 degrees+30 degrees=50 degrees and C 2 =I 2 +I 4 =20 degrees+35 degrees=55 degrees. Thus, C 1 +C 2 =50 degrees+55 degrees=105 degrees total. Therefore, S 1 +S 2 &gt;C 1 +C 2  and, as a result, edge  303  should be swapped as per Equation 3.
 
     The same analysis may be applied, for example, to determine if an edge should be contracted. Specifically,  FIG. 4  shows the surface of  FIG. 2A  for which such a determination is desired. Referring to  FIG. 4 , edge  203  has span angles S 1 ′ 401  and S 2 ′ 402  as well as incident angles I 1 ′ 403 -I 4 ′ 406 . In this case, the angles of  FIG. 4  have the following values: 
                                   TABLE 2                   Values for angles for consideration of edge contraction                Angle   Value                       S1′ 401   50 degrees           S2′ 402   55 degrees           I1′ 403   80 degrees           I2′ 404   50 degrees           I3′ 405   55 degrees           I4′ 406   70 degrees                        
Therefore, S 1 ′+S 2 ′=105 degrees, and C 1 ′+C 2 ′=(I 1 ′+I 3 ′)+(I 2 ′+I 4 ′)=135 degrees+120 degrees=255 degrees. Thus, S 1 ′+S 2 ′&lt;C 1 ′+C 2 ′and, as a result, edge  203  should be contracted according to Equation 4. The results of such an illustrative edge swap may, once again, be seen by referring to  FIG. 2B .
 
     One skilled in the art will recognize that variations on the above criteria for determining whether to swap an edge or contract an edge may be used in place of a direct comparison of the sum of the span angles and cross angles of triangles having an adjacent edge. For example, one illustrative criteria could be defined such that, if both span angles for an edge are greater than a specific angle threshold, such as each angle being greater than 90 degrees, then an edge swap is performed. Alternatively, another threshold test could be illustratively defined such that, if at least one span angle (e.g., span angle S 1 ′ 401  in  FIG. 4 ) is less then a particular degree threshold while, at the same time, at least one of the two incident angles (e.g., here I 1 ′+I 2 ′) associated with that span angle are larger than a particular incident angle threshold, then the edge is contracted. 
     One skilled in the art will also recognize that, in some cases, an edge swap may be followed by an edge contraction. Specifically, with reference once again to  FIGS. 3A and 3B , once an edge swap has been performed, the resulting edge  304  of  FIG. 3B  may then advantageously be contracted. As discussed herein above, therefore, the determination as to whether such a contraction should be performed would, illustratively, be made according to Equation 4 and according to the discussion herein above. 
     Other such variations on the threshold for performing an edge swap or an edge contraction in accordance with the principles described herein are also possible. For example, prior to either swapping or contracting an edge a determination as to whether various conditions exist. Specifically, in one illustrative embodiment, a determination is made whether a triangle mesh is a manifold. As one skilled in the art will recognize, a triangular manifold mesh is a mesh that satisfies four general criteria:
         1) There is at most one edge linking two points;   2) Two end points of an edge must be different points;   3) No two triangles intersect each other (a common edge between triangles, of course, is allowed); and   4) An edge can be a common edge of, at most, only two adjacent triangles.
 
Accordingly, prior to edge swap or contraction operations, a determination can illustratively be made whether these 4 criteria are satisfied. Illustratively, if any of the foregoing criteria are not satisfied, then an edge is not contracted and/or is not swapped.
       

       FIG. 5  is a flow chart showing the illustrative steps of a method for mesh simplification in accordance with one embodiment of the present invention. Referring to that figure, at step  501  an edge to be considered for mesh simplification is identified. Then, at step  502 , the cross angles are calculated as a sum of each pair of adjacent incident angles. Then, at step  503 , the span and cross angles associated with the edge are compared. At step  504 , a determination is made whether the sum of the span angles is greater than the sum of the cross angles. If so, and if any edge swapping criteria are met, such as the foregoing requirements for a manifold surface, then at step  505  the edge is swapped. If not, or after the edge swap has been performed at step  505 , then at step  506 , a determination is made whether the edge is shorter than a threshold and whether the sum of the span angles is less than the sum of the cross angles. If so, and if any edge contracting criteria are met, such as, once again, the foregoing criteria for a manifold surface, then at step  507  the edge is contracted. If not, or after the edge contraction is performed at step  507 , then at step  508 , a determination is made whether any additional edges require consideration and, if so, at step  509 , another edge is identified to be considered for mesh geometry improvement or simplification and the process returns to step  502 . Otherwise, at step  510  the determination is made that all edges have been considered and the process ends. 
     The foregoing embodiments are generally described in terms of manipulating objects, such as edges and triangles, in order to simplify a triangle mesh model of a 2D or 3D shape. One skilled in the art will recognize that such manipulations may be, in various embodiments, virtual manipulations accomplished in the memory or other circuitry/hardware of an illustrative computer aided design (CAD) system. Such a CAD system may be adapted to perform these manipulations, as well as to perform various methods in accordance with the above-described embodiments, using a programmable computer running software adapted to perform such virtual manipulations and methods. An illustrative programmable computer useful for these purposes is shown in  FIG. 6 . Referring to that figure, a CAD system  607  is implemented on a suitable computer adapted to receive, store and transmit data such as the aforementioned positional information associated with the edges and triangles of a triangle mesh model. Specifically, illustrative CAD system  607  may have, for example, a processor  602  (or multiple processors) which controls the overall operation of the CAD system  607 . Such operation is defined by computer program instructions stored in a memory  603  and executed by processor  602 . The memory  603  may be any type of computer readable medium, including without limitation electronic, magnetic, or optical media. Further, while one memory unit  603  is shown in  FIG. 6 , it is to be understood that memory unit  603  could comprise multiple memory units, with such memory units comprising any type of memory. CAD system  607  also comprises illustrative modem  601  and network interface  604 . CAD system  607  also illustratively comprises a storage medium, such as a computer hard disk drive  605  for storing, for example, data and computer programs adapted for use in accordance with the principles of the present invention as described hereinabove. Finally, CAD system  607  also illustratively comprises one or more input/output devices, represented in  FIG. 6  as terminal  606 , for allowing interaction with, for example, a technician or database administrator. One skilled in the art will recognize that CAD system  607  is merely illustrative in nature and that various hardware and software components may be adapted for equally advantageous use in a computer in accordance with the principles of the present invention. 
     One skilled in the art will also recognize that the software stored in the computer system of  FIG. 6  may be adapted to perform various tasks in accordance with the principles of the present invention. In particular, such software may be graphical software adapted to import surface models of shapes, for example those models generated from three-dimensional laser scanning of objects. In addition, such software may allow for selective editing of those models in a way that simplifies those models, as described above, or that permits a user to remove various portions of those models as described above. The software of a computer-based system such as CAD system  607  may also be adapted to perform other functions which will be obvious in light of the teachings herein. All such functions are intended to be contemplated by these teachings. 
     The foregoing Detailed Description is to be understood as being in every respect illustrative and exemplary, but not restrictive, and the scope of the invention disclosed herein is not to be determined from the Detailed Description, but rather from the claims as interpreted according to the full breadth permitted by the patent laws. It is to be understood that the embodiments shown and described herein are only illustrative of the principles of the present invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention. Those skilled in the art could implement various other feature combinations without departing from the scope and spirit of the invention.