Geometry Based Connectivity Compression of Triangle Mesh

Transcription

1 Geometry ased onnectivity ompression of Triangle Mesh Huang Jin, Hua Wei, Wang Qing, ao Hujun. State Key Lab of D&G in Zhejiang University {hj, huawei, qwang, bstract Efficient encoding of triangle mesh has recently become a subject of intensive study and many methods have been proposed. Most of the triangle mesh compression algorithms up to now do not benefit from geometry information when compressing connectivity. In this paper, we present a novel method to encode the connectivity information by exploiting geometry information. In our scheme, a triangle mesh is incrementally reconstructed from local geometry information of input mesh. The reconstructed connectivity is similar to that of the original mesh. The difference between them is recorded into a stream of numbers to guide decompression. The vertex coordinates can be compressed using the method of Touma and Gotsman[] at the same time. The experimental results of our algorithm show that its compression rates are often below the connectivity entropy estimation [, 9] widely accepted. keywords: mesh compression, connectivity encoding, geometry encoding, surface reconstruction. Introduction The demands of efficient storing and transmission D computer models are increasing fast with the development of Internet and entertainment. mong the representations of three-dimensional surfaces, triangle mesh is a popular D geometry representation, because triangle is a basic rendering primitive for common graphics hardware, and it is easy to convert other representation to triangle mesh. triangle mesh is made up of two components: the positions (coordinates) of vertices and the connectivity information. Most of the current file formats for triangle meshes record the connectivity as triples of vertex indices. For a mesh containing n vertices, the connectivity information needs about 6n log n bits (from the Euler equation we know that the number of triangles is about twice of vertices) before any compression, and it is more expensive than vertex coordinates. Therefore, the connectivity-encoding scheme has drawn much attention. Lots of algorithms are proposed also to compress vertex coordinates. In the final compressed data, the bits to store vertex coordinates are much more than the ones to store connectivity information in general. In this paper, we propose a novel technique to compress triangle mesh based on geometry information, which is discarded in most of the previous connectivity compression algorithms. To guarantee effective geometry compression, many geometry compression schemes [,, 8] can be used in our framework. ompared with the theoretical analysis and result of [], our experiments show that our scheme often achieves better results.. Previous work Many published works have focused on developing optimal, economical encoding algorithm for triangle meshes. The pioneering work on mesh compression was done by Deering [6]. Most efforts has been put on connectivity compression, because the largest benefit can be achieved here, and the traverse of all vertices built up by the connectivity coder is helpful for geometry compression. The geometry information was discarded at the first stage of mesh compression in this procedure, and the mesh was dealt as the pure connectivity graph G{V, E}. Taubin and Rossignac proposed an algorithm [] encoding geometry and connectivity separately. vertex spanning tree is used to predict the position of each vertex from some of its ancestors in the tree, and the correction vectors are entropy encoded. The vertex spanning tree and a small set of jump edges are used to split the model into a simple polygon. triangle spanning tree and a sequence of marching bits are used to encode the triangulation of the polygon. ecause the geometry is compressed without connectivity support, the compression ratio of geometry is not very high. There are two traversal based methods reported: Edgereaker[7], and valence encoding[, 8](the typical one is TG coder []). The first algorithm starts from a seed triangle, and initiates the front to the boundary of it. Then the algorithm chooses an edge in front as the gate to encode/decode the opposite triangle. t the same time, it output/input a code corresponding to this operation. When the right vertex is unvisited, geometry information is encoded/decoded. fter the operation completing, the front is updated. Next step starts from choosing a gate again. The algorithm stops when the front become empty. This algorithm outputs a stream of five symbols {, L, E, R,

2 S} identifying the operation at each step of the traverse. The latter turns the connectivity information into a stream of valence of each vertex. oth traverse the mesh in an order determined by the connectivity. The vertex coordinates are compressed with connectivity information support by parallelogram, such as []. Hence, before decompressing the geometry information, the connectivity must be restored at first. Many of improvements have been proposed on them. The distribution of original output symbols in plain Edgereaker algorithm [7] is not so good. Thus, the encoding algorithm of LERS string is very important, and many works have been done to exploit the pattern in the stream. In [5,7], the symbol stream is analyzed, and pushed into a context-based entropy coding procedure to utilize the relationship among the symbols. While most costs of valence encoding algorithm is the split operation. This operation introduces additional offset value into the valence stream. Pierre lliez and Mathieu Desbrun designed a strategy [] to avoid it by choosing appropriate pivot vertex along conquest edge list, and used local index to reduce the offset range affiliating with the split code. Lee, lliez, and Desbrun choose the best gate in the edge list just like [] in paper [], but use geometry angle instead of valence as the criteria. esides the two traversal based methods, there s another algorithm, which use some triangulation rule to compress mesh []. SungSoo Kim presented a novel algorithm for Triangulated Irregular Network (TIN) by inversing the order of geometry decompression and topology decompression. The whole vertex position information must be known before restoring connectivity information. They encode the TIN topology with a rule, so TIN is decomposed into two parts: a rule (=Delaunay Triangulation) and elementary data (= vertex coordinates and some different edges). The geometry information is compressed isolated by IM s vertex coordinates compression method [], then the connectivity turns into the difference between original edge set and the Delaunay triangulation on the vertices. y saving much connectivity information in the rule, they get high compression ratio on terrain data. However, this method cannot avoid two problems. First, it cannot extend to generic mesh, because most of meshes cannot be mapped into a plan to apply Delaunay triangulation. Second, it is difficult to achieve high compression ratio when encoding the vertex coordinates independent of connectivity. Our new algorithm can overcome the two restrictions easily. We find a rule, which could predict the most of triangles in meshes, especially regular meshes, and it can give useful connectivity information to vertex coordinates compression.. Overview In the next section, we introduce some definitions. In section, we present our geometry based connectivity compression strategy. In section, and we provide some results and discuss the essential difference between our algorithm and the Edgereaker or TG coder. In the last section, we give some conclusions and show the future work. Definitions In this paper, we restrict mesh to -manifold triangle mesh. In such mesh, each edge is bounded by one or two triangles, and the star of each vertex (i.e., its incident triangles and edges) remains connected when it is removed. Edges that bound two triangles are called interior edges. Edges that bound exactly one triangle are called exterior edges and their union is called the boundary of mesh. Our method has no restriction on the boundary and genus of the target mesh. Just like Edgereaker algorithm or P(all-Pivoting lgorithm [5]), starting from a seed triangle, the algorithm outputs a number, which denotes the corresponding operation at each step while traveling the mesh. In a snapshot of the encoding procedure, we use the following definitions. D P Figure. Snapshot of encoding/decoding. The dark gray triangles is encoded/decoded, while the others is still unknown at this time. E F front: a set of edges, and it is initially composed of a single loop containing three edges defined by the first seed triangle. It locally separates the surface into two parts: an inner conquered region and an outer free region. The inner conquered region is the encoded submesh, while the outer free region is still unknown when decompression. The compression/decompression operation only occurs on the edges in the front. In Figure, it s the boundary of dark gray region. full vertex: a visited vertex, and all of the triangles adjacent to it have been encoded, such as in Figure. unknown vertex: a vertex, which have not been visited, such as F in Figure.

3 gate: current active edge in the front (Figure: ). There must be one visited triangle, called active triangle, for a gate, e.g. in Figure. We perform the parallelogram rule on the triangle along the gate to get prediction vertex. prediction vertex: a dummy vertex (Figure : P ) defined by the parallelogram rule P = +, where is current active triangle, and is the gate. right vertex: the actual vertex, which connects with the gate (Figure : D). Obviously, it must be in front, when there re two triangles bound the gate. When the gate is on the mesh boundary, there is no right vertex. We just record a skip operation to identify this situation. Geometry ased onnectivity ompression In this section, we replace the framework in [] by Edgereaker to compress vertex coordinates and connectivity interweaved, and gains more geometry compression. t first, parallelogram rule is adapted as prediction rule. Then, to achieve higher compression ratio, simplified P[5] is used.. Parallelogram as prediction rule The algorithm [] mentioned above can be viewed as a guided reconstruction. We can generalize this scheme to the following. Given a triangle mesh {V, E}, we try to reconstruct E by rule R. Then we get the difference of the reconstruction result E and E, and save the difference as compression output data. While the cost of the reconstruction algorithms [5, 6,, 5] is too high to be suitable for compression. Fortunately, the parallelogram rule can be adapted to fit this demand. The most common technique for geometry compression is the parallelogram rule. During compression, the traversal-based algorithms generate a triangle strip tree. For the active triangle, we predict the coordinates of vertex D, which composes triangle D, is closed to P = +. This rule is called the parallelogram, introduced by Touma and Gotsman[]. pplying the parallelogram rule to these strips, recording the corrective vector DP of the prediction position and the actual coordinates. Then the vertex coordinates can be mapped into a clamped region to achieve geometry compression. The parallelogram rule works well when the mesh is smooth and sampled regularly. DP can be rather small compared with the distance of the nearest vertices pair, i.e. the prediction position is close to the right vertex D very much. The right vertex cannot be too far away from the prediction vertex. Thus if we sort the vertices into a list by the distance to prediction vertex ascent, then we can expect the right vertex is in the front of the list. The rank of right vertex(the position of the right vertex in the list) can be viewed as a guide for reconstruction during decompression (Table ). In most of the cases, the right vertex rank. When more than one vertex have the same value for rank, We simply add another vertex property to sort them, such as the index of vertex in the mesh. Rank = {V i V i front, and V i P < DP } () t each step of Edgereaker, there s an identified active triangle, and a specified gate. pplying the parallelogram rule, we get the prediction dummy vertex P and find the nearest vertex V (which ranks ) in front to P. If V P is too large, we predict inserting an unknown vertex into current submesh, otherwise we predict connecting the gate to V. For simple implementation, the threshold of inserting new vertex can be set to the mean length of current triangle edges. The corresponding operation is recorded into connectivity stream as table. In the stream, is corresponding to making a right prediction, and corresponding to an edge in mesh boundary, while the others denotes that we make a wrong prediction. When a new vertex is inserted into current encoded submesh, we output the corrective vector DP the right vertex into geometry stream. These two streams can be encoded by entropy encoder to get final compressed data. Table. This table represents the relationship of code and operation. During compressing, we know the actual condition (the first column of the table) and the prediction operation (the first row of the table), and output the code. On the other hand, we get the original mesh from the code and the prediction operation when decompressing. Predict Predict inserting connecting vertex new vertex which rank Right vertex is unvisited Right vertex If RN == is in conquest RN + edge list, Else and rank RN RN + There s no right vertex (hole). Simplified P as prediction rule The rank of right vertex and the rule to determine inserting new vertex affect the effectiveness of the al-

4 gorithm greatly. Ranking the vertex in front by distances to prediction vertex will make some wrong predictions. The figure shows this condition. The triangle () is the current active triangle, and edge is the gate. Vertices V i and V j are both visited vertices. If we use the distance as rank criteria, V j rank, but it is more reasonable to predict V i rank. We use a more subtle method to gain higher compression ratio, which can give the right prediction for this case. P [5] is a classical point cloud reconstruction algorithm. Starting from a seed triangle, it incrementally builds a -manifold mesh on given point set. t each step, the algorithm makes a ball touching the three vertices of current active triangle, and pivots the ball along one of the edges. The first vertex touched by the ball is just the vertex connect to the edge. ompare with our algorithm, the pivoting edge in P is corresponding to the gate. t the same time, the sequence of vertex touched by this ball during pivoting can be viewed as the order of the vertex. The right vertex of a gate ranks in most of the conditions in this criterion. Therefore, we can adapt the P to our algorithm. For the case mentioned above, we place a ball on triangle, and pivot it along edge, vertex V i rank, and V j rank as we expected. Most of triangle pairs bound an edge are almost coplanar. ased on this observation, we simplify P to following rule: at gate, for each vertex V i in front, calculate the angle V i, and sort V i by these angle descendant. We can use the above order to approximate the sequence of vertex touched by ball pivoting(see equation ). onsequently, the threshold of inserting new vertex must be amended. If there s no vertex in front will be touched by this ball during the whole pivoting, we predict to insert a new vertex. Rank = () {V i V i front, and cos V i < cos D} Vi Vj Vi Vj () Figure. Simplified P. The left figure is the side view and the right figure is the top view. The radius of the ball is chosen carefully by the regular property of the mesh. onsidering the most regular case, the mesh is composed all of regular hexagons. If there s no vertex found in the blue circle passing vertex, (see figure ), inserting a new vertex at the center of hexagon will make the mesh more regular. Otherwise, we connect the gate to a vertex fallen into the circle. We find.6 is the best choice for cos V. For more general case, the mesh is not regular, to estimate the best threshold of inserting vertex, we rescale local region to make it regular. The rescaling is based on the active triangle, for we don t know the actual mesh before decoding the whole mesh. It consists of three steps. Figure. Inserting vertex rule. From the left to right, the figure shows three cases:most regular mesh, predict connect vertex V and prediction inserting vertex P. First, we map the all vertices in front and the active triangle to the plane defined by the active triangle. Then an affine transform is performed on this plane to make the active triangle to V P ( (,, )(r, )(.5r, r) ), and keep the area of the triangle. Eventually, we lift all of these vertices to the original distance to the plane. The simplified P method can be used on these vertices rescaled to get the rank. If vertex V ranks, and cos V >.6, we predict inserting a new vertex. We check many models and find the rescaling is very useful to improve the compression ratio by reducing the inserting vertex prediction error. Results and Discussion We apply our algorithm on some typical meshes, such as bunny, mannequin, tree bulk, feline, fandisk, foot and dinosaur. Table is the result of our algorithm on these models. The vertex coordinates is quantified into 6 bits before compression. The entropy denotes the minimal number of bits required per symbol for lossless encoding of a given code sequence. N entropy = p i log p i i= When calculating the valence entropy (denotes as E V ), p i denotes the proportion of vertex whose valence is i in all vertices, and N is just the number of vertices. For the entropy of our output code stream E, p i denotes the proportion of code i in all codes. code denotes encoding a triangle, or finding an edge in the boundary. Therefore, the number of codes is the sum of triangle number and number of boundary edges. We must normalize the entropy of our code stream E into unit bit per vertex to match the valence entropy E V. The conversion is: number of codes E G = E number of vertices V

5 The E G represent the entropy of connectivity information from the aspect of geometry regularity. E V is the limitation of valence based algorithm, when using simple encoder such as zeros order entropy encoder. Typically, the final rate of [] must be higher than E V. Our algorithm first introduces a way to predict connectivity based on geometry information, so this limitation can be surpassed. In the output code stream of our algorithm, the probability of code is very high, i.e. the probability of making right prediction is rather high. When mesh is nearly uniformly sampled, E G < E V. While when non-uniformed sampled, E G surpasses E V. It shows that compressing connectivity information by geometry information is reasonable. The code of is the major component of the entropy. It comes from the wrong prediction for inserting vertex, although we have reduced it by local scaling. If the geometry information is compressed/decompressed before the connectivity, just as the [], it will be removed, and the ratio of connectivity compression will improve greatly. The figure shows the valence distribution and our code distribution of feline model Entropy: Entropy: Figure. Valence distribution and our code distribution of feline model(the unit of entropy is bit/vertex). We can generalize our algorithm to following rule by modifying the rank equation as: Rank = {V i V i front, and Φ(V i ) < Φ(D)} and choosing an appropriate inserting vertex rule. The Φ in our algorithm comes from P. We can deduce other Φ from many of reconstruction algorithms, such as [6, ]. The algorithms derived from this structure share the same time complexity. t each step, there re a number of vertices in front needed to be ranked, thus the complexity is higher than the most of previous methods. For a mesh contains N vertices, the number of vertices in front will be O(N ) on average. In the compression procedure, to find the rank of right D vertex is a linear operation, for we just need to compare each cos V i and cos D. In the decompression procedure, given the rank of right D vertex, the average complexity of finding D by nth element(a function in ++ Standard Template Library) is linear also. So the complexity of compression or decompression is O(N ). In our implementation, 6 minutes are taken to compression the model bunny, and about triangles per second on a PIII 66MHz PU with 56M memory. We expect to speedup the algorithm by optimizing the code. t the same time, we can ignore many of vertices in front, which is distant to the gate, for the rank of those vertices is unmeaning. Table. ompression results. #V and #F denotes the number of vertices and triangles respectively. Model #V #F E G E V sphere 6 8. bunny mannequin tree bulk feline fandisk dinosaur foot Summary and future work In this paper, we described a novel lossless, singleresolution connectivity compression algorithm, and shows that the entropy of connectivity can be measured from the aspect of geometry regularity. This method can achieve high compression ratio, and the output bitrate is lower than the entropy of valence distribution on some models. We just tried simplest zero order entropy encoder. If the high order adaptive context based entropy encoder is used, the size of final compressed data will be smaller. This needs additional work to exploit the pattern in our code stream, just as the improvement of [7] on original Edgereaker algorithm. cknowledgments: This paper is partly sponsored by National Natural Science Foundation of hina (Grant No. 6, 6) and key research project of Ministry of Education of hina (Grant No.9) References []. Touma and. Gotsman. Triangle mesh compression. In Graphics Interface 98 onference Proceedings, pages 6-, 998. [] P. lliez and M. Desbrun. Valence-driven connectivity encoding for D meshes. In Eurographics onference Proceedings, pages 8-89,. [] G. Taubin and J. Rossignac. Geometric compression through topological surgery. M Transactions on Graphics, 7():8-5, 998. [] Martin Isenburg and Pierre lliez, ompressing Polygon Mesh Geometry with Parallelogram Prediction, Visualization, pages -6,. 5

6 [5] ernardini, F., J. Mittleman, H. Rushmeier, and. Silva. The all-pivoting lgorithm for Surface Reconstruction. IEEE Transactions on Visualization and omputer Graphics, October-December 999, Vol. 5, No., pp [6] M. Gopi, S. Krishnan, and.t. Silva. Surface reconstruction based on lower dimensional localized delaunay triangulation. In Proceedings EUROGRPHIS, pages 67-78,. [7] J. Rossignac. Edgebreaker: onnectivity compression for triangle meshes. IEEE Transactions on Visualization [] Pierre lliez and Mathieu Desbrun. Progressive ompression for Lossless Transmission of Triangle Meshes. In SIGGRPH onference Proceedings, pages 98 5,. [] SungSoo Kim, YangSoo Kim, MiGyung ho and HwanGue ho, Geometric ompression lgorithm for Massive Terrain Data Using Delaunay Triangulation, In Proceedings of WSG 99, pp. -, Feb. [] Nina menta, Marshall ern and Manolis Kamvysselis, new Voronoi-based surface reconstruction algorithm, Siggraph 98, pages 5- (998). and omputer Graphics, 5():7-6, 999. [5].Szymczak,D.King,and J.Rossignac. n [8] Martin Isenburg: ompressing Polygon Mesh onnectivity Edgebreaker-based efficient compression scheme with Degree Duality Prediction. Graphics Inter- for connectivity of regular meshes.in Proceedings face : 6-7 of th anadian onference on omputational [9]. Kronrod and. Gotsman. Optimized compression of Geometry, pages 57-6,. triangle mesh geometry using prediction trees. In Proceedings [6] M. Deering. Geometry compression. In SIG- of st International Symposium on D Data Processing, Visualization and Transmission, pages 6-68,. [] H. Lee, P. lliez, and M. Desbrun. ngle-analyzer: triangle-quad mesh codec. In EUROGRPHIS onferernce Proceedings,. [] Martin Isenburg, Jack Snoeyink: Spirale Reversi: Reverse decoding of the Edgebreaker encoding. omputational Geometry (-): 9-5 (). GRPH 95, pages -, 995. [7]. Szymczak, Optimized Edgebreaker Encoding for Large and Regular Meshes, Data ompression onference, Poster, page 7,. [8] Eung-Seok Lee and Hyeong-Seok Ko, Vertex Data ompression for Triangular Meshes, Pacific Graphics, pages 5-,. [9]. Khodakovsky, P. lliez, M. Desbrun, and P. Schroeder. Near-optimal connectivity encoding of - manifold polygon meshes. to appear in GMOD,. Model Zoom in ode Ratio(%) Model Zoom in ode Ratio(%) Table. Models used for our bitrate measurements. We zoom in part of the mesh (column ), and display it in the second column. The last column shows the percentages of corresponding codes (column ) in the output stream. 6