Patent Application
20050093862
This invention relates to the field of computer graphics. More specifically, the invention relates to converting arbitrary polygon meshes to multiresolution subdivision surfaces.
The following formally define fundamental mathematical concepts used throughout this description. See any standard textbook on fundamental concepts of mathematics and computational geometry.
An n-manifold is a topological space which is locally Euclidean, i.e., around every point, there is a neighborhood which is topologically the same as the open unit ball in n, n≧2.
A 2-manifold is an n-manifold with n=2.
A map φ: A→B is said to constitute a homeomorphism between A and B if it is continuous, bijective, and has a continuous inverse.
A coordinate chart is a way of expressing the points of a small neighborhood on a manifold M as coordinates in Euclidean space. Formally, a coordinate chart is a map: φ: U→V where U is an open set in M, V is an open set in n and n is the dimension of the manifold. The map φ must be a homeomorphism.
An atlas is a collection of consistent coordinate charts on a manifold. As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a manifold and looks like the flat two-dimensional Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas.
A Voronoi diagram (or Dirichlet tessellation) is a partitioning of an input data set around n points called seeds into n regions (also known as cells) such that each region contains exactly one point and every point in a given region is closer to its seed than to any other point. A centroidal Voronoi diagram (CVD) has the property that each seed lies at the center of its region (cell). A constrained centroidal Voronoi diagram (CCVD) has the property that the seed points are the constrained mass centroids of their associated regions (see also Du et al., Constrained Centroidal Voronoi Tessellations for Surfaces, SIAM Journal for Scientific Computing, 24(5), pp. 1488-1506, 2003).
A scalar field is a continuous function Φ: Ω→, where Ω is a connected domain
n, n≧1. The image of Φ embedded in
n+1 space, i.e.,
={(X,Φ(X))|XεΩ}⊂
n+1 is called a hypersurface. When n=2,
is called a height field.
The medial axis transform (MAT) of an object is defined as the set of maximal balls completely contained within the object. The medial axis (MA) consists of the centers of the balls and intuitively can be viewed as the skeleton of the object.
Subdivision defines a smooth surface recursively as the limit of a sequence of meshes. Each finer mesh is obtained from a coarser mesh by using a set of fixed refinement rules. Examples of suitable rules are the Catmull-Clark subdivision rules (see E. Catmull and J. Clark, Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes, CAD 10(6), pp. 350-355, 1978). In the preferred implementation Catmull-Clark rules are used, but this is not a limitation upon the practice of this invention. Any quadrilateral-based subdivision rules for 3D models can be used.
Multiresolution subdivision surfaces extend subdivision surfaces by introducing details at each level. The coarsest-level mesh is sometimes also called the base mesh. Each time a finer mesh is computed, it is obtained by adding detail offsets to the subdivided coarser mesh. If given a semi-regular mesh, i.e., a mesh with subdivision connectivity, one can easily convert it to a multiresolution surface by defining a smoothing operation to compute vertices on a coarser level from a finer level. The details are computed as differences between levels.
The valence of a vertex of a mesh is defined as the number of mesh faces adjacent to that vertex. An interior vertex of the base mesh is called an extraordinary vertex if it has valence different than four. All other interior vertices in a semi-regular representation have valence four and are called regular vertices. Vertices on the boundary of an open mesh are considered regular if they have valence two and extraordinary otherwise. A mesh with subdivision connectivity is also called a semi-regular mesh.
Depending on the application, mesh decompositions may have different requirements regarding the properties of the constituent elements. For example, for skeleton extraction, prior art methods produce decompositions With complex components, possibly with higher than zero-genus. [See S. Katz and A. Tal, ACM TOG. Special issue for SIGGRAPH conference, Proceedings of Siggraph 2003, 2003]. Such decompositions may be suitable for restricted problems such as approximate skeleton extraction, but would require further, non-trivial processing for most other applications. For example, texture mapping involves associating 2D color images with regions of a 3D mesh. To establish such an association, an atlas consisting of charts that can be easily mapped to the planar domains of the images is necessary. The same holds true for semi-regular remeshing problems, where the charts serve as faces of a coarse polyhedral base domain over which the remeshing is performed.
Attention is drawn to the problem of finding atlas decompositions which satisfy several important properties:
Prior methods for mesh decomposition can be classified into two main categories, according to whether they require some degree of user involvement in defining the component charts or not. Semi-automatic methods require manual outlining of one or more chart boundaries or some other defining properties of the charts. (See for example V. Krishnamurthy and M. Levoy, Fitting Smooth Surfaces to Dense Polygon Meshes, Proceedings of SIGGRAPH 96, pp. 313-324, 1996.) Fully automatic methods produce atlases without the need for interactive adjustment of charts. Our proposed method falls into the latter category and solves a significant number of problems which prior art fails to address.
Many atlas decomposition methods produce irregularly shaped charts which are not suitable for remeshing and may be used only with difficulty, loss in efficiency and/or additional processing for texture mapping. Most such methods take a greedy approach and generate charts by normal bucketing, accumulating triangles of the input mesh one-by-one while checking the orientation of the surface normal. Such approaches can result in a large number of patches for objects with many small features. In most cases, there is not enough control over the shape of the charts, which can become rather irregular. As a result, they cannot be used directly for mesh extraction. See M. Garland, A. Willmott, and P. Heckbert, Hierarchical Face Clustering on Polygonal Surfaces, Proceedings of ACM Symposium on Interactive 3D Graphics, 2001.
The majority of existing automatic methods for semi-regular remeshing target triangle-based subdivision schemes and are not applicable to quadrilateral remeshing. They are typically based on a mesh simplification scheme which takes a fine triangle mesh and decimates it into a coarse triangle mesh. The coarse triangle mesh is then used as a base mesh for Loop subdivision. (See, for example, A. W. F. Lee, W. Sweldens, P. Schroeder, L.
Cowsar, and D. Dobkin, MAPS: Multiresolution Adaptive Parameterization of Surfaces, Proceedings of SIGGRAPH 98, pp.95-104, 1998 and T. Kanai, “MeshToSS: Converting Subdivision Surfaces from Dense Meshes”, Proc. Modeling and Visualization 2001”, pp. 325-332, 2001). There is no straightforward process that would adapt this process to quadrilateral remeshing as there is no simplification scheme that decimates a polygon mesh into a coarse quadrilateral mesh.
The human eye is sensitive to the variations of light intensity reflected from an object.
The light intensity depends on the angle between the object surface normal and a light source. Shape variations that result in large variation variations in the surface normal—changes in curvature such as ridges, creases or ravines in the surface—are then particularly noticeable to a human observer, that is they are perceptually salient. Existing methods typically produce charts that are not specifically aligned with the salient features of the input model. A large number of charts is hence required to ensure a good approximation after remeshing with a reasonable number of subdivision levels.
Some methods have been proposed that produce a quadrilateral base mesh by first generating a triangle base mesh followed by a grouping of triangles into quadrilaterals. (See M. Eck and H. Hoppe, Automatic Reconstruction of B-Spline Surfaces of Arbitrary Topological type, Proceedings of Siggraph 1996, pp. 325-334, 1996). In addition to being inefficient, such an approach is not always guaranteed to work: the triangle mesh would have to have an even number of triangles and even then, there is no guarantee that a complete pair wise matching of triangles into quadrilaterals can be found. No alignment to salient model features is sought.
Finally, for texture mapping, if the charts are not height fields, they first have to be flattened before the texture can be applied. Hence, the texture images corresponding to the charts are distorted by flattening and they do not appear the same as they do on the model. This makes it difficult to do touch up edits of the textures in 2D, having to work instead with the texture applied to the 3D model.
All above cited references are herein incorporated by reference in their entirety.
An aspect of this invention is an improved system and method for converting arbitrary polygon meshes to multiresolution subdivision surfaces.
An aspect of this invention is an improved system and method for converting arbitrary polygon meshes to multiresolution subdivision surfaces accounting for model features.
An aspect of this invention is an improved system and method for converting arbitrary polygon meshes to quadrilateral-based multiresolution subdivision surfaces.
An aspect of this invention is an improved system and method for providing atlases that can be used for various applications (e.g., for low distortion texture mapping, for model parameterization)
The present invention is an improved computer system, method, and program product that has one or more input devices for receiving one or more input meshes representing a three dimensional model. The three dimensional model is capable of being represented as a 2-manifold polygon mesh. A conversion process automatically converts the input mesh to a multiresolution quadrilateral-based subdivision surface (MQSS) representation.
The foregoing and other objects, aspects, and advantages will be better understood from the following non limiting detailed description of preferred embodiments of the invention with reference to the drawings that include the following:
This invention relates to the remeshing of three dimensional models represented by 2-manifold polygon meshes. The remeshing method is automatic and efficient, allowing the processing of large meshes.
The data processor 101 is also coupled through the bus 102 to a user interface, preferably a graphical user interface (GUI) 105 that includes any one or more of the following well known user interface devices: a keyboard, a mouse, a trackball, a voice recognition interface, and any general purpose user display device, such as a high resolution graphical CRT display terminal, a LCD display terminal, or any suitable other display device.
The data processor 101 may also be coupled through the bus 102 to a network interface 106 that provides bidirectional access to a data communications network 107, such as an intranet and/or the internet. Coupled to the network 107 can be one or more sources and/or repositories of digital models, such as a remote digital model database 108 that is reachable through an associated server 109.
The data processor 101 is also preferably coupled through the bus 102 to at least one peripheral device 110, such as a scanner 110A (e.g., a 3D scanner) and/or a printer 110B and/or a 3D model making apparatus, such as a rapid prototyping system, and/or a computer controlled fabrication system.
In general, this invention may be implemented using one or more software programs running on a personal computer, a server, a microcomputer, a mainframe computer, a portable computer, and embedded computer, or by any suitable type of programmable data processor 101.
Given this disclosure, many other system architectures would become known to one skilled in the art.
The use of this invention substantially improves the remeshing of 3D model data for many applications (for example, model editing and compression).
The teachings of this invention can also be configured to provide high-quality atlas decompositions of 3D models for various applications (for example, texture mapping). The methods may be used to process the digital 3D model data stored in the 3D model database 104a and/or in the remotely stored 3D model database 108 over the network 107 and in cooperation with the server 109. As but one example, a 3D input model to be remeshed could be remotely stored in the 3D model database 108a, while the remeshed 3D model represented as a multiresolution subdivision surface could be stored in the local 3D model database 104.
In a preferred implementation, the atlas 104b, 108b may be implemented as a collection of clusters of faces of the input mesh, each cluster corresponding to a chart of the atlas and containing pointers to mesh faces that are part of it. Typically, this data would be stored in memory 103 as shown in
The invention includes a conversion process 300, shown in more detail in
In process 400, an atlas decomposition is found for the input mesh using a combination of clustering techniques as described below. The resulting altas is made of charts that have several properties:
The process 400 comprises the following steps (with more detail given below):
Reference can also be made to
To generate an atlas over the input mesh, a number N of initial seed points is generated over the mesh according to some distribution. For example, for a given N, the seed points could be generated using a random placement process that uniformly distributes points over the mesh (see R. Osada, T. Funkhouser, B. Chazelle, and D. Dobkin, Shape Distributions, ACM Transactions on Graphics, 21(4), October, 2002, for an example of such process). Alternatively, a suitable number N and initial locations of the N seeds on the mesh could be estimated using a seed generation process 410 as shown in
First, the mesh is split (412) into relatively flat regions based on information provided by the mesh normals. Mesh normals may be computed per vertex or per face. Without restricting the generality of our approach, in what follows we use the term normal to mean face normal. We encode each normal vector n as a color by assigning each of the three components of the vector n=(nx, ny, nz)T to each of the red, green, and blue components of a color. If we want to visualize the normal variation over the input mesh, we can render the model with colors, as illustrated in
Having a set of N seeds on the input mesh, we compute a constrained centroidal Voronoi decomposition (CCVD) (420) of the mesh around the given seeds S1, . . . , SN (see
For each region that exhibits a normal variation greater than a certain threshold (432), we further split it into height fields using a color quantization process (434) similar to the one used to produce the initial quantization, but this time we restrict it within the cell. Otherwise the region remains unchanged (436). The set of all resulting regions forms the partition P (438). See also
After a partition of the input mesh into height fields is obtained, a step of further ensuring that the resulting regions are nicely shaped consists of checking the shape of each region against a shape measure that quantifies whether or not the region is hoemomorphic to a disk (i.e., region boundary has exactly one connected component) and whether or not it is approximately convex (440). Regions that do not satisfy a shape compactness criterion (442) are further split into compact regions homeomorphic to disks (444) using a CCVD decomposition around medial axis samples. The CCVD decomposition process 420 is applied restrictively to each region that has to be split. The seeds used in this case can be selected randomly or using samples from the approximate medial axis of the region, similar to process 419 in
An optional cleanup of the charts may optionally follow (450), see
The atlas thus obtained can be used directly for applications like texture mapping and mesh parameterization. In the following we describe the necessary processing to use the atlas decomposition for remeshing with semi-regular connectivity.
It is observed that each chart of the atlas has, on average, at most six neighboring charts. This follows from the process of constructing the atlas using Voronoi decompositions and from the property of Voronoi diagrams that the average number of Voronoi edges per Voronoi polygon does not exceed six (see A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, pp. 65, Wiley & Sons, Ltd., 2000).
A set of representative colors (415a) is defined to correspond to a small subset of all possible normal directions. For example, by choosing the subset of normal directions to be the 14 normal vectors corresponding to the faces and vertices of the unit cube and by mapping these normals to colors as previously described, we obtain a set of 14 colors that can be viewed as a seed set for the quantization process. All faces of the input mesh can be classified based on their color using MacQueen's approach (415b) to belong to one of the classes defined by the seeds (the one with the closest seed in color space). Note that the choice of 14 representative colors is just an example, any other number and choice of subset of normal directions may be used. The seeds themselves are adjusted during the process to the centroids of their regions (this is, in effect, a CCVD in color space).
Having found an atlas decomposition over the input mesh with charts corresponding to approximately flat regions of the models and homeomorphic to disks, a quadrilateral base mesh corresponding to this decomposition is extracted using a quadrilateral mesh extraction process 500 as shown in
Having established a coarse quadrilateral mesh MQ, a multiresolution quadrilateral subdivision surface MQSS can be extracted by resampling the original geometry M over the faces of the coarse polygonal mesh at dyadic positions corresponding to the finest subdivision level desired, followed by multiresolution analysis to propagate the data from the finest level of the subdivision hierarchy to coarser levels. In a preferred implementation this could be done by first applying L-1 steps of subdivision to the quadrilateral mesh extracted using process 500; where L is the desired number of levels in the final multiresolution hierarchy, thus generating a fine mesh D which constitutes a resampling domain for the final representation (610). A resampling of the original geometry at positions corresponding to the vertices of the subdivided domain mesh D yields a resampled version R of the original mesh with subdivision connectivity (620). Multiresolution analysis can then be used to decompose R over the faces of the base mesh, thus generating a multiresolution quadrilateral-based subdivision surface (MQSS) representation for the input mesh (630). For an example of multiresolution analysis of meshes see H. Biermann et al., Cut-and-Paste Editing of Multiresolution Surfaces, ACM Transactions on Graphics, vol 21(3), pp. 312-321, Proceedings of ACM Siggraph 2002, San Antonio, July 2002. An optional optimization of the resulting MQSS may also follow (640).
All references cited above are herein incorporated by reference in their entirety.
