Abstract:
This invention presents a method to extract atomic parts of a graphics model using its skeleton. A skeleton is a fully collapsed body of the model, and is obtained through a novel way to contract edges of the model. From the skeleton, atomic parts or features each is a part of the model that is distinctively autonomous from its connected or neighboring body is formed through space sweeping. Next, atomic parts can be connected into a hierarchy depending on the eventual interactive visualization applications. The operation of the method includes the steps of interactively computing, displaying and recording skeleton, atomic parts, and object hierarchies in response to user commands to, for example, modifying skeleton, atomic parts or object hierarchies. Object hierarchies are useful to various applications such as object scene management, view-dependent simplification, mesh-mapping, morphing, and building bounding volume hierarchies.

Description:
This application is a Continuation-In-Part of PCT International Application No. PCT/SG00/00109 filed on Jul. 27, 2000, which designated the United States and on which priority is claimed under 35 U.S.C. §120, the entire contents of which are hereby incorporated by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to computer graphics and solid modeling, and more particularly, to methods and apparatuses for representing and displaying an object of its skeleton, atomic parts, and hierarchy. 
     BACKGROUND OF THE INVENTION 
     The quest for ease of modeling in three-dimensional computer graphics has led to the development of several well-known techniques, such as constructive solid geometry, free-form deformation, non-uniform rational B-splines, or more recently, implicit surfaces. Today&#39;s graphics hardware, however, is only capable of dealing with polygons efficiently. Hence, models are often tessellated before display for efficient rendering. 
     Related to 3D modeling is 3D model acquisition. The difficulty of obtaining a good 3D model has led to development in this area, using methods such as laser range scanners and turntable techniques. Such methods produce massive point sets representing points on the model&#39;s surface and require further processing such as 3D triangulation. The result is again a polygon mesh. 
     A tradeoff has to be established between having a high quality model with a high polygon count, and fast rendering with fewer polygons. The ever-increasing number crunching capability of today&#39;s processors tends to push the envelope of polygon count for efficient rendering, making it feasible to use more complex models. 
     Working on the premise of polygon mesh, however, leads to many difficulties. A polygon mesh is inherently unstructured, making it expensive to perform geometric operations such as intersection tests in collision detection and ray tracing. In the present invention, we are interested in object representation as in its object hierarchy. 
     The object hierarchy is most natural in terms of the human concept of shape. This has to do with the fact that cognition works best for hierarchically organized systems. In fact, 3D modelers often exhibit this phenomenon unknowingly when they organize an object in a top-down fashion. Although the object hierarchy is a natural representation of shape, it is often non-unique and designer-specific. Even so, the variations are usually minor and do not affect the conceptualization of the model on the whole. The present invention describes an algorithm for determining first a collection of atomic parts of an input polygon mesh and for determining a unique hierarchy from the atomic parts. 
     It is conceivably easy to obtain an object hierarchy from certain representations of models, for instance, constructive solid geometry models. However, the same cannot be said of geometric models, particularly B-rep models, which are by far the most prevalent. UCOLLIDE [TAN99] is a collision detection system that makes use of simplification to compute bounding volume hierarchies. The novelty of this work lies in the use of cluster-based simplification [LOW97] for extracting shape, and the use of this shape information for computing bounding volume hierarchies. Traditionally, bounding volume hierarchies are generated by top-down partitioning or bottom-up merging. Bottom-up methods only work well for organizing objects in a scene, and not polygons per se. Top-down methods perform poorly when the object to be partitioned consists of many sparsely arranged parts. Hence, it is difficult to achieve optimal or near optimal bounding volume hierarchies using either method. 
     [TAN99] uses simplification and shape analysis to extract the major components (or atomic parts) of an object. Further shape analysis on the simplified model yields the components of the model. Partitioning on each component can then be done using traditional top-down methods. 
     For complex models with numerous interconnected parts, it is insufficient to use only one simplified model or also called level-of-details (LOD) for shape analysis. This is because it is difficult to pinpoint one LOD that captures all the essential features of a model. By using a few LODs, the decomposition of a model can be guided along each node of the parent LOD and a hierarchy is naturally obtained. Although reasonably good results can be obtained using this method, there are some issues that remain to be addressed: 
     (i) the association of polygons in a lower LOD to a higher one may not be straightforward; 
     (ii) it is not clear what is the desired number of LOD and how to choose them; 
     (iii) different LODs produce different results; and 
     (iv) vertex clustering can cause topology change in the model. How this affects decomposition is unclear. 
     In addition, simplification using vertex clustering is not incremental. The algorithm needs to be invoked a number of times for LOD generation and this can cause a performance penalty. In light of all these issues, an alternative formulation of shape extraction is developed in the current invention. 
     SUMMARY AND OBJECTS OF THE INVENTION 
     The invention described herein satisfies the above-identified needs and overcomes the limitations of the prior invention by [TAN99]. The present invention describes a new system and method to effectively generate object hierarchies, for purposes of various interactive applications such as: 
     (i) Organizing an arbitrary mesh for scene management and view-dependent simplification; 
     (ii) Providing an alternative structure to one given by modeler; 
     (iii) Visualizing complex models; 
     (iv) Mesh-mapping; for morphing, establishing correspondence; and 
     (v) Building bounding volume hierarchy (BVH) for intersection tests in collision detection and ray tracing. 
     Specifically, disclosed herein is a method for execution by a data processor that generates effective object hierarchies. The method includes the following steps: 
     (1) Preprocessing. This is to prepare data structures for subsequent processes. 
     (2) Skeletonization. This is the process of deriving a skeleton of the input model, where skeleton is a fully collapsed body of the model. 
     (3) Generating Atomic Parts. This is the process of obtaining various features (which is a part of the model that is distinctively autonomous from its connected or neighboring body) from the skeleton. 
     (4) Postprocessing. This process connects various atomic parts obtained from previous step into a hierarchy. 
     Further scope of applicability of the present invention will become apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention will become more fully understood from the detailed description given hereinafter and the accompanying drawings which are given by way of illustration only, and thus are not limitative of the present invention, and wherein: 
     FIG. 1 is a block diagram of an exemplary raster graphics system; 
     FIG. 2 is a simplified diagram of a graphics processing system according to the present invention; 
     FIG. 3 is a flowchart that depicts the method steps according to the present invention; 
     FIG. 4 is an edge contraction example utilizing method as described in the present invention; 
     FIG. 5 is a flowchart that depicts the steps of computing atomic parts utilizing space sweeping; 
     FIG. 6 is an example on contracting edges to form a skeleton containing a virtual edge; 
     FIG. 7 is a diagram to illustrate the notion of below, on, and above the sweep plane; 
     FIG. 8 is a diagram to illustrate the cross-section, which is the intersection of the sweep plane with the polygon mesh; 
     FIG. 9 is a sample profiles of the geometric function F(t); and 
     FIG. 10 is a flowchart that depicts the operation of the method according to the present invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 1 illustrates an exemplary raster graphics system that includes a main (Host) processor unit  100  and a graphics subsystem  200 . The Host processor  100  executes an application program and dispatches graphics tasks to the graphics subsystem  200 . The graphics subsystem  200  outputs to a display/storage device  300  connected thereto. 
     The graphics subsystem  200  includes a pipeline of several components that perform operations necessary to prepare geometric entities for display on a raster display/storage device  300 . For the purposes of describing the invention, a model of the graphics subsystem is employed that contains the following functional units. It should be realized that this particular model is not to be construed in a limiting sense upon the practice of the present invention. 
     A Geometric Processor unit  210  performs geometric and perspective transformations, exact clipping on primitives against screen (window) boundaries, as well as lighting computations. The resulting graphics primitives, e.g. points, lines, triangles, etc., are described in screen space (integral) coordinates. 
     A Scan Conversion (Rasterization) unit  220  receives the graphics primitives from the geometric processor unit  210 . Scan converter unit  220  breaks down the graphics primitives into raster information, i.e. a description of display screen pixels that are covered by the graphics primitives. 
     A Graphics Buffer unit  230  receives, stores, and processes the pixels from the Scan Conversion unit  220 . The graphics buffer unit  230  may utilize conventional image buffers and a z-buffer to store this information. A Display Driver unit  240  receives pixels from the Graphics Buffer unit  230  and transforms these pixels into information displayed on the output display device  300 , typically a raster screen. 
     FIG. 2 is a simplified diagram of a graphics processing system according to the present invention. As shown in FIG. 2, an input device  10  inputs graphics data to be processed by the present invention. The CPU  100  processes the input data from input devices  10  by executing an application program. CPU  100  also dispatches graphics tasks to the graphics subsystem  200  connected thereto. The output results (skeleton, atomic parts, and object hierarchy) may then be stored and/or displayed by display/storage devices  300 . 
     Having described an exemplary graphics processing system that is suitable for use in practicing the present invention, a description is now provided of a method of extracting atomic parts from a polygon mesh and the arrangement of these parts into a hierarchy. 
     For the purposes of describing the present invention, it is assumed that objects are described in polygonal boundary representations (b-reps). Models with curved surfaces are tessellated, i.e. approximated with planar triangles. This is to facilitate display as well as edge contraction, as will be described in the following section. 
     FIG. 3 shows a flowchart of the algorithm. 
     Preprocessing Step 
     The purpose of preprocessing is to compute the necessary data structures required for subsequent processing. In particular, the following steps are performed: (P 1 ) making a copy of the original tessellated triangle list, (P 2 ) computing a vertex neighborhood graph, (P 3 ) computing an incident edge table, and (P 4 ) computing min heap for all the edges in the model. Notice that some of these steps, such as (P 2 ) and (P 3 ), are not necessary if the input object representation comes with such information. 
     For the purpose of describing the preprocessing step, it is assumed that the original model is represented by a vertex table VT and a triangle list TL. The vertex table VT contains the supporting vertices used in the model. The triangle list TL contains the triangles used in the model and each triangle is represented as an ordered list of references (indices) to the vertex table VT. In addition, there may be other augmented structures such as a facet normal table and a vertex normal table, but these are normally not used. The contents of the vertex table VT and triangle list TL are volatile during the execution of the algorithm. In another embodiment, the original model can be represented as an edge-based data structure (instead of vertex-based as assumed in the description) such as the winged-edge data structure and quad-edge data structure [GUIB85]. It is straightforward for those skilled in the art to do the necessary adaptation from the following description to work on edge-based data structures. 
     (P1) Making a Copy of the Original Tessellated Triangle List 
     It is essential to make a copy of the original tessellated triangle list. This is because the edge contraction step performs in-place updating on the triangle list as new vertices are added. In one embodiment of this operation, the triangle list is used for rendering of intermediate results. When edge contraction is completed for all the edges in the model, the tessellated triangle list is restored so that the original model can be rendered. 
     (P2) Computing Vertex Neighborhood Graph VNB 
     The vertex neighborhood graph VNB is used to store connection information between vertices. More specifically, if vertex v 0  and v 1  are neighbors, e.g. (v 0 ,v 1 ) of an edge of a triangle, then v 1  ε VNB(v 0 ) and v 0  ε VNB(v 1 ). The vertex neighborhood graph VNB is used extensively during the edge contraction step. A simple way of implementing VNB is to use a two-dimensional array where each row represents the list of vertices related to the vertex index. A more efficient method is to use an adjacency list. 
     (P3) Computing Incident Edge Table INC 
     The incident edge table INC is another data structure used in edge contraction. The elements of the incident edge table INC are defined as follows: for any valid edge (v 0 , v 1 ) in the model, INC(v 0 , v 1 ) gives the list of triangles that are incident to this edge. The incident edge table INC can be implemented using a hash table. A simple hashing formula could be given as follows:        Key   =     {             v0   *   displacement_factor     +   v1             if                 v0     &lt;   v1                 v1   *   displacement_factor     +   v0         otherwise                                  
     displacement_factor is an arbitrary integer larger than the number of vertices in the model. 
     (P4) Computing Min Heap MH for all the Edges in the Model 
     The min heap is a data structure whereby the smallest element or the element with the smallest weight is always at the top of the heap. Thus, it is easy to retrieve the smallest weight item from the min heap. It is also efficient to remove the smallest weight item whilst maintaining the property of the min heap. After computing incident edge table INC, the list of all edges is easily known. The min heap is then obtained by adding all edges using the length of each edge as its weight. Thus, edges are retrieved in increasing order of length. In another embodiment of this method, the weight of each edge is computed based on some pre-calculated value of its vertices. These are variations that serve to improve the results and do not detract from the objective of this operation. The min heap can be implemented using a linear array. 
     Skeletonization Step 
     Simplification is the process of obtaining a simpler form of a model while preserving certain attributes, such as appearance, as much as possible. In the context of geometric models, a simpler model is one with fewer numbers of triangles and vertices. While the applications of simplification are mainly in the arena of level-of-detail modeling, there have been ingenious applications like Collision Detection [TAN99], Progressive Mesh [HOPP96] and Skeletonization [DEUS99]. The approach taken by the former is based on the vertex clustering principle while the other two use edge contraction. Hoppe [HOPP96] employs edge contraction for simplification and compression, with emphasis on appearance preservation. Deussen et. al. [DEUS99] in turn implements a simpler version of edge contraction for the purpose of defining intersection planes for non-photorealistic rendering. 
     The process of skeletonization in this invention is different in its methodology of edge contraction. Furthermore, the resulting skeleton is not the end product; rather the crux of the operation lies in the association of triangles into parts forming the skeleton. 
     A skeleton is defined as the result of a collapsed model. Edge contraction refers to the process of collapsing edges into vertex. For the purpose of describing the present invention, assume that the current edge to be collapsed is represented by (v 0 ,v 1 ) where v 0  and v 1  are the vertices of its endpoints. FIG. 4 illustrates this process. Unless otherwise stated, edge contraction is performed until the min heap MH of all edges is empty (S 0 ). An additional data structure is required in this method to store associated triangles and this is known as the associated triangle list ATL (which is initialized to empty set at the beginning of the skeletonization step). The detailed steps in edge contraction consist of: (S 1 ) creation of new vertex, (S 2 ) updating of associated triangle list ATL, (S 3 ) updating of incident edge table INC, (S 4 ) updating of vertex neighborhood graph VNB, (S 5 ) updating of triangle list TL, (S 6 ) updating min heap MH and skeleton S. FIG. 6 shows a complete example of the process. 
     (S1) Creating New Vertex 
     Given v 0  and v 1  are vertices to be collapsed, a straightforward method of creating a new vertex vn would be (v 0  +v 1  )/2. In another embodiment, the new vertex is given the coordinate of v 0  or v 1  if either v 0  or v 1  belong to an existing (skeletal) edge in skeleton S. In either case, the new vertex vn is appended at the end of the vertex table VT. 
     (S2) Updating of Associated Triangle List ATL 
     Since v 0  and v 1  are vertices of at least one triangle, the collapsing of v 0  and v 1  would cause at least one triangle to be removed from the model. In addition, the change of vertex numbering, i.e. v 0  →vn and v 1  →vn, violates the validity of the associated triangle list ATL. The purpose of this step is to keep track of the removed triangles, as well as to update the vertex number in the associated triangle list ATL. The following algorithm exhaustively enumerates the types of updating involved: 
     (i) for each vertex v★ in VNB(v 0 )\{v 1  }, rename ATL(v 0 , v★) as ATL(vn, v★) 
     (ii) for each vertex v★in VNB(v 1 )\{v 0  }, rename ATL(v 1 , v★) as ATL(vn, v★) 
     (iii) for each triangle T in INC(v 0 , v 1 ) with v 2  be its third vertex, do 
     store T to ATL(v 2 , vn) 
     store ATL(v 0 , v 2 ) and ATL(v 1  v 2 ) to ATL(v 2 , vn) 
     store ATL(v 0 , v 1 ) to ATL(vn, vn) 
     store ATL(v 0 , v 0  ) and ATL(v 1 , v 1 ) to ATL(vn, vn) 
     (S3) Updating of Incident Edge Table INC 
     The collapsing operation (v 0 , v 1 )→vn requires that the incident edge table INC be updated. The algorithm for performing this update is given as follows: 
     (i) for each vertex v★in VNB(v 0 )\{v 1  }, rename INC(v 0 , v★) as INC(vn, v★) 
     (ii) for each vertex v★in VNB(v 1 )\{v 0  }, rename INC(v 1 , v★) as INC(vn, v★) 
     (iii) for each triangle T in INC(v 0 , v 1 ) with v 2  be its third vertex, do delete T from INC(v 2 , vn) 
     (iv) remove INC(v 0 , v 1 ) 
     (S4) Updating of Vertex Neighborhood Graph VNB 
     As in step (S3), it is mandatory to maintain the validity of vertex neighborhood graph VNB after the collapsing operation. The algorithm for this update is as follows: 
     (i) add VNB(vn)=VNB(v 0 )∪VNB(v 1 )\{v 0 , v 1 } 
     (ii) delete VNB(v 0 ) and VNB(v 1 ) 
     (iii) for each vertex v★in VNB(vn), update VNB(v★)=VNB(v★)\{v 0 , v 1 }∪{vn} 
     (S5-S6) Updating of Triangle List TL, Min Heap MH and Skeleton S 
     The triangle list TL is updated so that collapsed vertices are renamed to vn. Note that if a high fidelity rendering is required, the affected normals in facet normal table and vertex normal table are to be recomputed. For the sake of efficiency, the updating of min heap MH and skeleton S are also performed under this step. It is not efficient to update MH directly, as this would involve examining the entire heap. Instead, only affected edges are reinserted into the min heap MH. When an edge is retrieved, its validity can be ascertained by checking against the incident edge table INC. 
     The algorithm for performing this update is as follows: 
     for each vertex vi in VNB(vn), do 
     if INC(vn, vi) is not empty 
     for each triangle T in INC(vn, vi), do 
     find and rename vertex v 0  or v 1  in T to vn 
     let wt=f(vi, vn) where f could be the length function 
     heap_insert(MH, edge(vi, vn), wt) 
     otherwise 
     add edge (vn, vi), termed the skeletal edge (vn, vi), to S 
     Generating Atomic Parts 
     Having obtained the collection of skeletal edges forming the skeleton and the data structure ATL, the next step is to compute the list of atomic parts associated with the model (FIG. 3, step G 0 ). One simple embodiment is to define an atomic part as the collection of triangles that represent a connected part of the skeleton. More formally, it is obtained from the following derivation: 
     Let R be a relation on S such that aRb (i.e. a related to b) if there exists a path from a to b in S. 
     R is an equivalence relation under S. 
     Obtain the set of equivalence classes of R, called Q i  where i=1 to m. 
     For each Q i , form an atomic part P i =∪ATL(vj, vk) where (vj, vk)εS i  and vj, vkεQ i    
     In another embodiment, the space-sweep technique [HMMN84] is adopted to sweep the given polygon mesh with a plane, called the sweep plane, along some sweep path [t start , t end ]. FIG. 5 shows the detailed steps in extracting atomic part adopting the space sweeping. Skeletal edges provide a good basis for sweeping, but before that can be done the set of skeletal edges is organized (that are disjoint in general) into a linear order for sweeping. 
     To do this, a skeletal tree is obtained from the skeleton (Step G 1 ). Then a traversal order of the edges is defined of the skeletal tree by grouping them into so-called branches (Step G 2 ). The ordering of space sweeping is determined by the ordering the branches. The sweeping paths are determined by the positions of the branches. During the space sweeping in advancing the sweep plane (Step G 4 ) at some location called sweep location, the orientation of the sweep plane is adjusted to better intersect the polygon mesh (Step G 3 ). The intersection between the sweep plane and the polygon mesh defined the cross-section at the sweep location. Cross-section is used to compute a geometric function and a topological function (Step G 5 ). These functions define whether the sweep location is a critical point. The identification of critical points define the boundaries of some atomic parts (Step G 6 ). 
     (G1) Constructing the Skeletal Tree 
     During skeletonization, a triangle is either contracted to a skeletal edge or an endpoint on a skeletal edge. Consider triangles Δ 1  and Δ 2  that are incident to a same edge. Suppose Δ 1  is contracted to b and Δ 2  is contracted to d where (a, b) and (c, d) are two disjoint skeletal edges. To connect them, a virtual edge (b, d) is added to the collection of skeletal edges. For the example of FIG. 6, Δv 5 v 6 v 8  and Δv 5 v 2 v 6  sharing (v 5 , v 6 ) are contracted to v 12  and v 9  respectively; we thus have the virtual edge (V 12 , v 9 ). 
     On the other hand, suppose Al is contracted to b and Δ 2  is contracted to (c, d), where (a, b) and (c, d) are two disjoint skeletal edges. An issue is as to whether (b, c) or (b, d) should be a virtual edge. This is resolved because as the computation of virtual edges is in fact an integral part of the skeletonization process. The skeletonization process creates a virtual edge whenever two disjoint parts are the resulted of an from edge contraction (rather than after the whole process). That is (b, c) should be a virtual edge if b was a result of contracting some edge incident to c, and likewise for (b, d). 
     For a connected mesh, the result of adding virtual edges is a connected graph. Such a graph is called a skeletal tree if it has no cycles. Otherwise, cycles are removed by applying the standard minimal spanning tree algorithm. The cost between each pair of nodes u and v is defined as the inverse of the size of ATL(u,v). 
     (G2) Locating a Branch for Sweeping 
     From a leaf vertex of the skeletal tree, a branch is defined—starting with the edge incident to the leaf vertex—as the maximal chain of edges whose vertices (excluding the first and the last) are incident to exactly two edges in the skeletal tree. Each branch corresponds to one continuous part of the whole sweep path [t start , t end ], with the leaf vertex as the starting point, and the last vertex of the chain as the ending point. In general, there are, as many branches as the number of leaves. 
     The union of the ATL over the edges of a branch, with first vertex u and last vertex v, corresponds to a part of the object. Its surface area can be approximated as:        A   =       (     length                 of                 branch     )     *         f        (   u   )       +     f        (   v   )         2                              
     where f(t) measures the perimeter of the cross-section at t. In one embodiment, we can assuming f(t) varies linearly over the branch (as the skeletonization results in fairly uniform branches), thus can simplify the computation to the following:        A   =       ∫   u   v            f        (   t   )                          t                                
     Branches are then ordered for sweeping based on surface area of their ATL. Those branches having a smaller area are favored for sweeping first. This ordering allows small but significant atomic parts to be extracted first so that they are not absorbed into larger part of some bigger atomic parts. During sweeping, edges of the skeletal tree that have been swept are removed. 
     (G3 and G4) Orienting and Advancing the Sweep Plane 
     To begin a space sweep, a branch is used and its corresponding set of triangles, which are obtained from the union of the ATL&#39;s of edges from the branch. The sweep plane, given by P(t)=P(v, n), is determined by a reference point v and a normal vector n as shown in FIG.  7 . For any given vertex u from the mesh, its direction from v may be computed. If (v−u)·n&lt;0, u is below the sweep plane. If it is equal to zero, u lies on the plane. Otherwise u is above the sweep plane. 
     The reference point v is either a vertex of the mesh or a point on an edge of the mesh. Specifically, v is on the boundary of the cross-section G(t). Initially, n is the normal vector along the first skeletal edge of a given branch B. v is then an arbitrary point along the intersection of the mesh and the sweep plane passing through the first endpoint of B. Since the polygon mesh is a discrete representation, only an approximate sweep path need be obtained, by advancing in discrete steps. To advance P(v, n) to a new v, one way is to move it to the next nearest vertex above the sweep plane. In this way, the total number of cross-sections will be proportional to the number of vertices of the mesh. For dense meshes, this is an expensive operation. For such cases, a more pragmatic embodiment is to advance in fixed steps. In order not to over- or underestimate the step size, the amount can also be computed as a function of the edge length distribution. 
     Each branch comprises a set of skeletal edges which provide a general direction for the sweep plane movement. However, the orientation of the skeletal edge is not necessarily good, as it is dependent on the result of edge contraction. To account for this, the orientation of the sweep plane is not entirely based on the skeletal edges but computed adaptively. 
     Hence, when the sweep plane advances from P(u, n) to P(v, n), its orientation is subjected to change. To compute n′, it is first set to n. It is then adjusted so that the sweep path will follow the natural orientation (n s ) of the volume. Specifically, it is advantageous to have n′ oriented based on the normal vectors of triangles Δ i  in Δ (which is the set of intersecting triangles at the cross-section—see step G 5  below). We define n s  as follows:            n   s     =       ∑     i   =   1     k            N        (     Δ   i     )       ×     N        (     Δ     i   +   1       )             ,                          
     where N(Δ i ) and N(Δ i+1 ) are normal vectors of consecutive triangles in Δ (with Δ k+1 =Δ 1 ), ordered in an anticlockwise manner about n′. Intuitively, the cross product of N(Δ i ) and N(Δ i+1 ) captures the essence of a local optimal direction for the sweep plane with respect to the shared edges of Δ i  and Δ i+1 . Thus, the sum over all such cross products points to an aggregated direction. 
     (G5) Computing Cross-Section, Geometric and Topological Functions 
     An edge in the polygon mesh is an intersecting edge of P(t) if one of its endpoints is either on or above P(t), and the other endpoint is on or below P(t). An intersecting triangle of P(t) is a triangle incident to one or more intersecting edges of P(t). The set of intersecting triangles of P(t) is denoted as Δ={Δ i |i=1, 2, . . . , k}. The intersection of P(t) with A is a collection of line segments forming the boundary of G(t). This boundary is either a single simple polygon (see t 1  of FIG. 8) or a collection of simple polygons (see t 2  of FIG.  8 ). From this boundary, two functions are defined: 
     (1) The Geometric function F(t) is given by 
     
       
           F(t)= measure( G ( t )), 
       
     
     where measure is the measurement, such as the area or perimeter of cross-section G(t). 
     (2) The Topological function T(t) is given by          T        (   t   )       =     {           0   ,           topology                 of                   G        (     t   -   ɛ     )                     is                 different                 from                   G        (   t   )                   1   ,         otherwise                                  
     where ε is an arbitrarily small positive number. 
     With intersecting triangles of P(t) in Δ, it is easy to compute the geometric function F(t). To compute the derivatives of F(t) (as needed in the computation of critical point in the following discussion), we use previous values (up to five) of F(t) to find delta differences in the computation. Also, standard filters, such as Gaussian or median, can be used to smooth out the values. 
     To compute T(t), we first define H, a function of the cross-section G(t), as the number of simple polygons in G(t). Note that G(t−ε) and G(t) are homotopic if G(t−ε) can be deformed continuously into G(t) and vice versa. This property implies that their boundaries are also homotopic, that is, H( G(t))=H(G(t−ε)). With that, we arrive at the following alternative definition of T(t):          T        (   t   )       =     {           1   ,             H        (     G        (     t   -   ɛ     )       )       =     H        (     G        (   t   )       )                   0   ,             H        (     G        (     t   -   ɛ     )       )       ≠       H        (     G        (   t   )       )                     o                 r                   G        (   t   )                     degenerates                 to                 a                 point                                      
     Since we only need to compare between G(t−ε) and G(t), only two cross-sections need to be stored throughout the computation. 
     (G6) Decomposing Mesh into Atomic Part 
     A critical point in the sweep path is a boundary point between atomic parts that captures change in either geometry, topology or both. FIG. 9 shows some sample profiles of F(t). Other possible profiles such as symmetric ones or combination of the given examples are considered in a similar way. At t c  as indicated by dashed lines, each of the profiles is divided into two segments. Naturally, it is desirable to grant the interval [t 1 , t c ] and [t c , t 2 ] to two different atomic parts. By computing a few derivatives of the profiles, it is found that the common characteristic of all profiles at t c  is that F (n) (t c )=0, where F (n)  is the first vanishing derivative of F(t) for some n, and F (n)  crosses zero at t c . In order to restrict atomic parts to simple shapes, there is an additional constraint that topology must be constant, i.e. T(t) must be non-zero in each atomic part. From these conditions, a critical point at t is defined as: 
     
       
         ( F   (n) ( t )=0 and  F   (n) ( t−ε )· F   (n) ( t+ε )&lt;0) or  T ( t ) =0 
       
     
     In the course of sweeping, the geometric and topological functions are computed and analyzed (Step G 5 ). When a consecutive pair of critical points is found, the part of the polygon mesh that is swept is extracted as an atomic part. 
     From T(t) and the derivatives of F(t), it is determined whether t is a critical point as defined above. When a critical point is reached, triangles swept so far that are not part of other atomic parts are extracted as a new atomic part. For intersecting triangles in Δ, they can be included in the new atomic part, or split up along their intersection with the sweep plane, depending on the application. 
     Postprocessing Step 
     The objective of post-processing is to derive the object hierarchy representing the original model M by organizing the atomic parts in a meaningful way. The object hierarchy is defined as a rooted tree where the root node is the original model M and each level below it is a breakdown of the node into mutually exclusive parts. The leaf nodes are atomic parts obtained in previous step. Hierarchy construction makes use of the relationship between connected atomic parts and this relationship is defined by the connection graph G as follows: 
     
       
           G={ ( P   i   , P   j )|∃ t   p   εP   i   , t   q   εP   j  and  t   p   , t   q  shares an edge}, 
       
     
     For models containing disjointed parts, the connection graph G is also disjointed. In that case, each connected subgraph is treated as an independent connection graph and is dealt with separately. Spatial analysis (see [TAN99]) could later be applied to join them up into a hierarchy. For the rest of the discussion on hierarchy building, G is assumed connected. 
     G is undirected and could possibly contain cycles. The presence of cycles could create problems for some hierarchy construction methods. Hence it is necessary to first transform the graph into a tree. One way of doing this, in step T 1 , is to compute the minimal spanning tree (MST) G′ of G. The cost of each edge in G could be some function, such as cosine, of the angle formed by orientations of the endpoints. The cost function is chosen as such because a sharp angle formed by the parts indicate that they probably should not be joined in the first place. 
     One way of computing an orientation would be to use the axis of least inertia, which is fairly expensive to compute in three-dimensions. Alternatively, the orientation is approximated by the largest eigenvector in the principal component analysis (PCA) of the object. 
     Having obtained G′ for each connected subgraph of G, hierarchy construction can begin (step T 2 ). Three types of hierarchical construction method are disclosed in the following paragraphs: (1) unrooted folding method (2) rooted folding method (3) common edge method. The use of a particular method may depend on the eventual applications of the constructed hierarchy. 
     (1) Unrooted Folding Method 
     This method applies to the default configuration, in which G′ is unrooted. Nodes of degree larger than one with at least one or more leaf neighbors are tagged. At each step of the algorithm, tagged nodes are collapsed to form a new node in the object hierarchy H. The algorithm for building H is as follows: 
     
       
         add  P   1   , P   2   , . . . ,P   m  to  H   
       
     
     
       
         while |G′|&gt;1 
       
     
     add nodes each is of degree larger than one and has at least one or more leaf neighbors to a queue, Q 
     while P=dequeue(Q) is not empty 
     create a new node M′ in H to be parent of leaf neighbors of P 
     create a new node M″ in H to be parent of M′ and P 
     delete leaf neighbors of P from G′ 
     rename P to M″ in G′ 
     (2) Rooted Folding Method 
     The unrooted folding method gives a simple way of constructing the hierarchy, without a priori knowledge of the model. However, the hierarchy constructed this way may be lopsided and not really reflect the natural partitioning of the model. A better way would be to pick a representative node in G′ to be the root. A representative node could be the node with the largest bounding volume (representing a dominant part), or one with the highest degree (representing a center of focus), or could be selected manually by the user. Hierarchy construction would be similar as before, except that it is now done bottom-up. The rooted folding method for building H is as follows: 
     
       
         add  P   1   , P   2 , . . . , P m  to H 
       
     
     
       
         while | G′|&gt; 1 
       
     
     let h be the height of G′ 
     add nodes at level (h−1) to Q 
     while P=dequeue(Q) is not empty 
     create a new node M′ in H to be parent of leafs of P 
     create a new node M″ in H to be parent of M′ and P 
     delete leafs of P from G′ 
     rename P to M″ in G′ 
     (3) Common Edge Method 
     For geometric objects with many interconnected parts it may be difficult to choose a representative node to be the root. In that case, it may be useful to have a metric for evaluating the “strength” of connection between parts. This metric should measure the relative importance of an atomic part and its neighbors. In one embodiment, the metric that is used in this method is the relative length of the common edges between two parts and is defined as follows: 
     
       
         common( P   i   , P   j )=Σ||( v   a   , v   b )|| s.t.  ( v   a   , v   b ) in  t   k  where  t   k   εP   i  and ( v   a   , v   b ) in  t   l   
       
     
     where t l εP j   
     
       
         boundary( P   i )=Σ||( v   a   , v   b )|| s.t.  ( v   a   , v   b ) in  t   k  where  t   k   εP   i  and not  t   l   εP   i    s.t.  ( v   a   , v   b ) in  t   l   
       
     
     
       
         relative( P   i   , P   j )=common( P   i   , P   j )/min(boundary( P   i ), boundary( P   j )) 
       
     
     The method of construction is to keep removing edges based on a function of its metric. Such function may either be taking, in one embodiment, the largest, or, in another embodiment, the smallest metric. The algorithm for building H is as follows: 
     
       
         add  P   1   , P   2   , . . . , P   m  to  H   
       
     
     
       
         while | G′|&gt; 1 
       
     
     get (P i , P j ) from G′ with f (relative(P i , P j )) where f could be max or min function 
     create a new node M′ in H to be the parent of P i  and P j    
     collapse P i  and P j  to new node M′ (corresponding to M′ in H) in G′ 
     The above completes the description of the hierarchy building process. As mentioned in the summary and objects of the invention, there are various interactive applications that can benefit from the computed object hierarchy. In the following, the use of the object hierarchy for one such application is described explicitly; other applications within the spirit and scope of the invention will become apparent to those skilled in the art from such a discussion. 
     Application to Collision Detection 
     One of the most time consuming operation in collision detection is the object intersection test. This is also true for the ray intersection test in ray-tracing. It is generally accepted that a tight bounding volume is good for such tests since it will ensure that pruning is done as soon as possible. Recent developments in this area lead to efforts such as BOXTREE [BARE96], OBBTree [GOTT96] and k-DOP [KLOS98], to name a few. Most of them either take the top-down approach of partitioning or the bottom-up approach of merging, with no inherent interest in the shape of the model as a whole. Top-down partitioning is ineffective for models that are sparse, as the level of representation is too coarse at the top levels. Bottom-up methods suffer from efficiency problems since obtaining a globally optimal solution is prohibitively expensive. 
     As described earlier, UCOLLIDE [TAN99] uses a series of simplified models to extract parts of models, hence taking advantage of their shape. However, it is not clear how many simplified models to use or how to select them to get the best result. 
     The present invention presents an automated way of obtaining the object hierarchy that is easily convertible into a bounding volume hierarchy suitable for collision detection and ray tracing. The atomic parts P 1 , P 2 , . . . , P m  which form the leaf nodes of H are in general simple shapes. Hence top-down partitioning methods can be applied effectively to reduce P i  to leaf nodes containing a single or small number of triangles. 
     For a disjoint connection graph G, the MST of each connected subgraph G i  forms a hierarchy H i  representing the object hierarchy of congregate part R i , the collection of polygons of the nodes in G i . Let bounding_box(R i ) be the size of the bounding volume of R i . The choice of bounding volumes is application dependent and transparent to the algorithm. The algorithm for merging the various H i  makes use of pair-wise merging of smallest enclosing bounding volume to obtain a locally optimal solution. 
     
       
         while | R|&gt; 1 
       
     
     get R i , R j  from R with the smallest bounding_box(R i ∪R j ) 
     create a new node M′ in H to be the parent of H i  and H j    
     collapse R i , R j  to new node M′ (corresponding to M′ in H) in R 
     This manner of hierarchy building is similar in nature to existing bottom-up methods described in [BARE96] and [TAN99]. 
     The resulting tree H may be reorganized by picking a new root to achieve a more balanced result. To control the branching factor, H may also be converted into a binary tree. Note that this is done on a per node basis, and does not involve alteration to the original model M. 
     Operation of the Method of the Invention 
     FIG. 10 is a flowchart that depicts the operation of the method of the present invention in obtaining object hierarchies. First, the system loads or generates a database of objects appearing in a scene (M 1 ). These objects may be elements of, for example, a CAD database or a virtual environment. For each of these objects, in step M 2 , skeleton, atomic parts, and object hierarchies are generated as detailed in FIG.  3 . The input models and the skeletons, atomic parts, and object hierarchies are stored in the memory as shown in FIG.  2 . At this point, the system is ready to begin an interactive display session with a user. 
     A user can issue commands to, for example, display skeleton, atomic parts, or object hierarchies. Also, the user can interactively do adjustment to modify skeletons, atomic parts or object hierarchies, and the system accesses the memory to obtain the suitable data structures for modification and to pass them to the graphics subsystem for display of the data structures on the display screen. Additionally, in response to the user requests M 4 , the system can use skeletons, atomic parts, and object hierarchies to compute other data structures such as the bounding volume hierarchies, as described earlier, for use in ray-tracing or virtual environment navigation to determine collisions M 5 . 
     The invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as departure from the spirit and scope of the invention, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims. 
     References 
     [BARE96] G. Barequet, B. Chazelle, L. Guibas, J. Mitchell and A. Tal, “ BOXTREE: A Hierarchical Representation for Surfaces in  3 D ”, Proceedings Eurographics &#39;96, Vol. 15(3), August 1996, pp. C-387-396, C-484. 
     [DEUS99] Deussen O., Hamsel J., Raab, A., Schlechtweg, S., Strothotte T., “ An Illustration Technique Using Hardware-based Intersections and Skeletons”,  to appear in Proceedings Graphics Interface, June 1999. 
     [GOTT96] S. Gottschalk, M. C. Lin and D. Manocha, “OBBTree: A Hierarchical Structure for Rapid Interference Detection”, Computer Graphics (SIGGRAPH &#39;96 Proceedings), 1996, pp. 171-179. 
     [GUIB85] Guibas and Stolf, “ Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams”,  ACM Transaction of Graphics, vol. 4, (1985), 74-123. 
     [HMMN84] S. Hertel, K. Mehlhorn, M. Mantyla, and J. Nievergelt, “ Space Sweep Solves Intersection of Two Convex Poyhedra”,  ACTA INFORMATICA 21, pp. 501-519, 1984. 
     [HOPP96] H. Hoppe, “ Progressive Meshes”,  SIGGRAPH 96 Conference Proceedings, Annual Conference Series, Addison-Wesley, August 1996, pp. 99-108. See also European Patent Documents EP0789330A2, entitled  Selective Refinement of Meshes.    
     [KLOS98] J. Klosowski, M. Held, J. Mitchell, H. Sowizral and K. Zikan, “ Efficient Collision Detection Using Bounding Volume Hierarchies of k-DOPs”,  IEEE Transactions on Visualization and Computer Graphics, vol. 4 (1), 1998, pp. 21-36. 
     [LOW97] Kok-Lim Low and Tiow-Seng Tan, “ Model Simplification Usin Vertex-Custering”,  Proceedings on Symposium On Interactive 3D Graphics, 1997, pp. 75-81. 
     [TAN99] Tiow-Seng Tan, Ket-Fah Chong and Kok-Lim Low, “ Computing Bounding Volume Hierarcies Using Model Simplification”,  Proceedins on Symposium On Interactive 3D Graphics, 1999, pp. 63-69.