WO2000045237A2 - Compression progressive de maillages triangulaires - Google Patents

Compression progressive de maillages triangulaires Download PDF

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Publication number
WO2000045237A2
WO2000045237A2 PCT/IL2000/000053 IL0000053W WO0045237A2 WO 2000045237 A2 WO2000045237 A2 WO 2000045237A2 IL 0000053 W IL0000053 W IL 0000053W WO 0045237 A2 WO0045237 A2 WO 0045237A2
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WO
WIPO (PCT)
Prior art keywords
mesh
vertex
vertices
patch
hole
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Application number
PCT/IL2000/000053
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English (en)
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WO2000045237A3 (fr
Inventor
Daniel Cohen-Or
Offir Remez
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Enbaya Ltd.
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
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Publication date
Application filed by Enbaya Ltd. filed Critical Enbaya Ltd.
Priority to AU23165/00A priority Critical patent/AU2316500A/en
Priority to JP2000596429A priority patent/JP2002535791A/ja
Priority to EP00901872A priority patent/EP1194860A2/fr
Publication of WO2000045237A2 publication Critical patent/WO2000045237A2/fr
Publication of WO2000045237A3 publication Critical patent/WO2000045237A3/fr

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T9/00—Image coding
    • G06T9/001—Model-based coding, e.g. wire frame

Definitions

  • the present invention relates to compact representation of 3D geometric models and, more particularly, to a progressive mesh compression and reconstruction method.
  • the geometry is represented by the set of coordinates of the mesh's vertices. To enable effective compression of the geometry, the coordinate values are first quantized to a fixed number of bits.
  • the connectivity data is the vertex/triangle adjacency list, sometimes also referred to as the topology. In a naive representation, the connectivity data is about twice as large as the geometry data.
  • Mesh compression algorithms are normally required to use a lossless compression of the connectivity data.
  • Current mesh compression methods are based on the triangle-strips technique, in which the triangular mesh is traversed along sequences of triangles, which look like peeled strips.
  • the progressiveness property is important to compensate for low network bandwidth and transmission latency.
  • the compression method of the present invention is based on a multiresolution decomposition, which inherently has a progressive property. However, unlike the progressive meshes, here the size of the data that is required to faithfully recover the original mesh is comparable to any known technique of mesh compression. Recently, Renato Pajarola and Jarek Rossignac ("Compressed progressive meshes", Technical Report GIT-GNU-99-05, Georgia Institute of Technology Washington, 1999) have developed a new progressive meshes technique, where a batch of vertex-split operations are encoded efficiently to yield a compressed progressive mesh representation. Multiresolution analysis and wavelets have matured as a versatile tools for representing functions and analyzing features at multiple levels of detail.
  • Multiresolution analysis methods for the compression of 3D meshes have been applied only in terms of the number of triangles representing the mesh at various levels of detail.
  • lossless compression methods of 3D meshes which compress in terms of the total number of bits required to represent the mesh, have not been reported.
  • the present invention includes a multiresolution analysis for which the representation of a given mesh is compact at every level of detail, and in particular for the original mesh.
  • Traditional wavelets are constructed over regular structures. The construction of wavelets over arbitrary meshes is currently an interesting challenge [10].
  • Pioneering work by Michael Lounsbery, Tony D. DeRose and Joe Warren Multiresolution analysis for surfaces of arbitrary topological type", ACM Transactions on Graphics, Vol. 16 No. 1 pp.
  • this two-step technique does not provide the means to fully restore the original mesh. It should be emphasized that mesh compression techniques are required to restore the original connectivity. Indeed, most of the efforts of the mesh compression methods have been invested in a compact encoding of the unstructured connectivity of the triangulation.
  • a method for compressing and reconstructing a mesh representation of an object including a plurality of mesh vertices connected by edges
  • the method including the steps of:(a) selecting at least one mesh vertex to be removed from the mesh; (b) for each selected vertex, computing an approximation of the selected vertex and a difference between the approximation and the selected vertex; (c) removing, from the mesh representation, the at least one selected vertex and the edges whereto the at least one selected vertex is connected, thereby creating, for each selected vertex, a hole in the mesh representation, the hole including a plurality of hole vertices; and (d) triangulating each at least one hole, thereby replacing each at least one hole with a corresponding patch, and thereby producing a compressed mesh.
  • a method of compressing a mesh representation, of an object, that is to be transmitted from a server to a client, the mesh representation including a plurality of mesh vertices connected by edges the method including the steps of: (a) selecting at least one mesh vertex to be removed from the mesh; (b) for each selected vertex, computing an approximation of the selected vertex and a difference between the approximation and the selected vertex; (c) removing, from the mesh representation, the at least one selected vertex and the edges whereto the at least one selected vertex is connected, thereby creating, for each selected vertex, a hole in the mesh representation, the hole including a plurality of hole vertices; and (d) triangulating each at least one hole, thereby replacing each at least one hole with a corresponding patch, and thereby producing a compressed mesh.
  • the present invention is a lossless compression method based on a multiresolution decomposition where the detail coefficients have a compact representation and thus smaller entropy than the original mesh.
  • the present invention uses a hierarchical simplification scheme, which generates a multiresolution model of the given triangular mesh. By reversing the process a hierarchical progressive refinement process is defined, where a simple prediction plus a correction is used for reconstructing vertices to form a finer level.
  • the connectivity of triangulation is encoded efficiently and recovered incrementally during the progressive reconstruction of the original mesh.
  • the original mesh Mn can be decomposed into M n _j and W n by applying a decimation algorithm over its vertices, where W n consists of the set of removed vertices, and M n _, is the simplified mesh. Then by interpolating over the triangles of M n _, we create a set of points P(c), which serves as a prediction for the set W n .
  • the displacement vectors between the removed vertices and the interpolated points are the shorter coefficients watty
  • the key idea is to construct a multiresolution of an arbitrary mesh with irregular connectivity. Unlike traditional wavelets, here the domain is unstructured, and therefore the refinement is not applied uniformly during the reconstruction stage.
  • An interpolation scheme predicts a point to which we add a displacement vector to recover a vertex ?. This new vertex is inserted into the triangulation while restoring its connectivity in M l+ _. Note that recovering the original connectivity is necessary to correctly decode the data encoded over the representation of M l+ _. This is made clearer below where we show how to encode and decode a given mesh.
  • FIG. 1 illustrates the 4-color encoding scheme of the present invention
  • FIGs. 2 and 3 illustrate the 2-color encoding technique of the present invention
  • FIG. 4 illustrates the prediction of the vertex corresponding to a patch
  • FIG. 5 illustrates the first four steps of progressive compression of a 3D mesh
  • FIG. 6 illustrates four more initial 3D meshes
  • FIG. 7 illustrates a system to which the present invention may be applied.
  • the present invention is of a progressive compression and reconstruction method which can be used to compress and reconstruct 3D triangular meshes with no loss of information.
  • the principles and operation of mesh compression and reconstruction according to the present invention may be better understood with reference to the drawings and the accompanying description.
  • patches are interpolated to predict a set of points, which serves as a base for the displacement vector to the removed vertices.
  • the predicted points are quantized so the displacement vectors can be represented by a small number of bits, with smaller entropy than the original vertices.
  • one displacement vector is stored.
  • the key idea is to encode the triangles of a patch by means of coloring them, such that the decoder can detect the patches during the reconstruction stage based on the triangle colors.
  • adjacent patches cannot be assigned the same color, where two patches are said to be adjacent if they share an edge.
  • the triangles of M_ are recursively traversed and each patch is assigned a color that is different from the colors assigned to its adjacent patches. Because the patches do not tessellate the entire mesh, we use a null color for the triangles that are not included in any patch. The rest of the triangles are colored in only three colors.
  • Figure 1A shows a mesh 10 of vertices 12 connected by edges 14.
  • Three colors are not always enough, but in practice such cases are rare, and can be avoided by giving up the removal of some vertices.
  • the coloring technique illustrated in Figure 1 requires 2 bits per triangle.
  • the cost of encoding a vertex is the cost of coloring the triangles of the patch created by its removal. Assuming the degree of a vertex is 6, then its removal requires coloring four triangles, that is, 8 bits per vertex removal. Note that there is some overhead because some triangles are not included in any patch. To reduce this overhead, when selecting the vertices to be removed we strive to create a maximal independent set.
  • the above coloring technique can be improved by triangulating the patches by a dependent triangulation that can be encoded with only one bit per triangle.
  • a hexagonal patch 20 (the most popular patches in common triangulation) is triangulated by the three edges 22 of the shape of the letter Z.
  • the two middle triangles 24 of patch 20 are encoded with a '1' bit (shaded) and the two external triangles 26 with a '0' bit (unshaded).
  • Pentagons are triangulated with three triangles where the middle one is encoded with a T and the others two with a '0'.
  • the middle triangle T m must be selected such that the two other triangles share a vertex with the smallest angle in T m
  • the sequences of "alternating' triangles 32 are colored with 'l's and the two externals 34 with 'O's. While encoding adjacent patches we need to avoid edge- adjacent '1 '-encoded triangles. Recovery of the patches is guaranteed since the sequence of adjacent 'l's has a known shape, from which the two external 'O'-encoded triangles are uniquely recovered and, as a consequence, the boundary of the patch is also uniquely recovered. Note that with this technique, a quadrilateral cannot be encoded.
  • Figure 3 illustrates a 2-coloring (light gray and dark gray) of mesh 10.
  • a sequential order of the triangles of is defined by a breadth- first traversal of the triangles.
  • One bit is associated with each triangle and stored in a binary vector, which represents the colors of the triangles.
  • the length of the vector is [M_ ] the number of triangles of the mesh, M_.
  • the vector is then compressed by some lossless compression technique.
  • the preferred lossless compression technique is an LZ encoder.
  • the patches can be colored by using either four colors or two.
  • the 4-color technique requires 2 bits per triangle and the 2-color technique only 1 bit per triangle. Denoting by m the number of bits used to color the triangles, the cost of encoding a d- vertex is m(d-2) bits, because the patch created by removing a d- degree vertex consists of only d-2 triangles.
  • the removal of a 6-degree vertex requires 4 bits, and a 5-degree vertex only 3 bits.
  • the average of the degrees is always close to six.
  • the distribution of the vertex's degrees can vary. If the mesh of a given level of detail consists mainly of vertices of degree 5 and higher, the 2-color technique is very effective. However, if the mesh consists mainly of vertices of degree 4 and 3, the 4-color technique is more effective since the 2-color technique cannot be applied to low degree vertices.
  • the patches created by the removal of low degree vertices consists of one or two triangles only, the cost of 2 bits per triangle means that the cost of encoding the insertion is only 2-4 bits.
  • the encoding of the connectivity requires no more than 4 bits per vertex. Because the independence set is not optimal, there are many triangles that are not included in any of the patches. Thus, in practice the cost is higher than 4 bits per vertex. In any case the stream of bits that colors the mesh is further compressed by an LZ encoder.
  • the coloring technique is selected according to the distribution of degrees in the given level of detail.
  • Figure 5 shows the coloring of the first four intermediate levels of the progressive compression of a mesh that defines a typical 3D object. The first two levels use a 4-coloring, because most of the vertices are of degree three or four. The second two levels use a 2-coloring.
  • the 2-coloring encoding technique of the present invention requires only one bit per triangle. However, in terms of the number of bits per vertex the cost is at least 4 bits per vertex, or 2 bits per triangle. Note that the number of triangles in the entire hierarchy is about three times the number of triangles in the original mesh. This cost is about that reported by Gabriel Taubin et al., "Progressive forest split compression", SIGGRAPH 98 Conference Proceedings, pp. 123-132, ASM SIGGRAPH, July 1998, almost twice more than reported by C. Touma and C. Gotsman, "Triangle mesh compression", Graphics Interface '98, pp. 26-34, June 1998, and eight times better than the cost of a vertex split of the progressive mesh of Hugues Hoppe, "Progressive meshes", SIGGRAPH 96 Conference Proceedings, pp. 99-108, August 1996.
  • the stream of the displacement preferably is encoded using Huffman encoding.
  • Huffman encoding We have tested the results with 12-bit precision per coordinate. The following tables compare our results with those of Touma and Gotsman's technique, which are the best published so far, for the 3D meshes illustrated in Figures 5 and 6.
  • Table 1 connectivity compression results
  • FIG. 7 illustrates a system in which the present invention is particularly useful.
  • a server 50 stores representations of 3D objects as triangular meshes and transmits these representations to a client 52 over a communications line 54 that suffers from low bandwidth and/or high transmission latency.
  • the 3D objects whose representations are stored in server 50, may be sculptures in a virtual museum.
  • the user of client 52 wishes to perform an interactive walkthrough of the museum. This is done by changing the viewpoint of client 52.
  • server 50 compresses the triangular mesh representation of the new sculpture, following the principles of the present invention.
  • Server 50 transmits the compressed representation to client 52, and client 52 reconstructs the full triangular mesh representation from the compressed representation, also following the principles of the present invention.
  • Server 50 transmits the compressed representation of the sculpture in the order ⁇ M 0 ,w_,w 2 ,... ⁇ , i.e., in order of increasing resolution.
  • Client 52 first receives the lowest resolution of the sculpture, set 0 , and displays a low resolution representation of the sculpture based on 0 .
  • client 52 adds these difference sets to its representation of the sculpture and displays the sculpture at successively higher resolutions.
  • the lowest resolution is adequate for a realistic rendition of the sculpture when the sculpture first comes into view.
  • sufficient difference sets w_ have arrived at client 52 to enable client 52 to render the sculpture at the necessary level of resolution.

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  • Engineering & Computer Science (AREA)
  • Multimedia (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Image Generation (AREA)
  • Processing Or Creating Images (AREA)
  • Image Processing (AREA)

Abstract

L'invention se rapporte à un procédé de compression et de reconstruction d'une représentation maillée d'un objet. Selon ce procédé, on supprime du maillage des sommets sélectionnés et les arêtes associées, puis on remplace les trous résultants par des triangles. Pour chaque remplacement, on calcule une approximation du sommet supprimé et l'on sauvegarde la différence entre le sommet supprimé et son approximation. En outre, on associe aux triangles de remplacement un code couleur. On répète l'algorithme de compression autant de fois que nécessaire. Afin de reconstruire la représentation maillée originale, pour chaque remplacement, on calcule l'approximation du sommet supprimé, on calcule la différence entre l'approximation et le sommet supprimé pour reconstruire le sommet, et on relie le sommet reconstruit aux sommets du triangle de remplacement de manière à reconstruire les arêtes supprimées.
PCT/IL2000/000053 1999-01-27 2000-01-27 Compression progressive de maillages triangulaires WO2000045237A2 (fr)

Priority Applications (3)

Application Number Priority Date Filing Date Title
AU23165/00A AU2316500A (en) 1999-01-27 2000-01-27 Progressive compression of triangular meshes
JP2000596429A JP2002535791A (ja) 1999-01-27 2000-01-27 三角形網目のプログレッシブ圧縮
EP00901872A EP1194860A2 (fr) 1999-01-27 2000-01-27 Compression progressive de maillages triangulaires

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US11742699P 1999-01-27 1999-01-27
US60/117,426 1999-01-27

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Cited By (16)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2002039380A2 (fr) * 2000-11-13 2002-05-16 Siemens Aktiengesellschaft Procede et systeme de reconstruction d'une surface
US6901310B2 (en) 2000-11-06 2005-05-31 Siemens Aktiengesellschaft Method and system for approximately reproducing the surface of a workpiece
US7157422B2 (en) 2000-01-03 2007-01-02 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
US7378432B2 (en) 2001-09-14 2008-05-27 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
US7446092B2 (en) 2002-12-12 2008-11-04 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
EP2533202A3 (fr) * 2011-06-09 2015-04-15 Visual Technology Services Ltd. Compression de maillage de couleur
US9589318B2 (en) 2014-08-25 2017-03-07 Ge Aviation Systems Llc Method and system for generating airport surface map graphics in aircraft cockpit displays
US9787321B1 (en) 2016-11-17 2017-10-10 Google Inc. Point cloud data compression using a space-filling curve
US10313673B2 (en) 2016-10-19 2019-06-04 Google Llc Methods and apparatus to encode and/or decode normals of geometric representations of surfaces
US10430975B2 (en) 2016-11-17 2019-10-01 Google Llc Advanced k-D tree encoding for point clouds by most significant axis selection
US10496336B2 (en) 2016-11-17 2019-12-03 Google Llc K-D tree encoding for point clouds using deviations
US10553035B2 (en) 2017-06-02 2020-02-04 Google Llc Valence based implicit traversal for improved compression of triangular meshes
US10733766B2 (en) 2016-10-19 2020-08-04 Google, Llc Methods and apparatus to encode and/or decode normals of geometric representations of surfaces
US10891758B2 (en) 2018-07-23 2021-01-12 Google Llc Geometry encoder
CN112419178A (zh) * 2020-11-18 2021-02-26 芯勍(上海)智能化科技股份有限公司 破洞修复方法、终端设备及计算机可读存储介质
US10950042B2 (en) 2017-06-02 2021-03-16 Google Llc Guided traversal in compression of triangular meshes

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JP3967626B2 (ja) * 2002-04-30 2007-08-29 独立行政法人科学技術振興機構 画像データ圧縮処理方法および画像処理装置

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US5905507A (en) * 1996-01-16 1999-05-18 International Business Machines Corporation Compression of geometric models using spanning trees

Cited By (22)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US7157422B2 (en) 2000-01-03 2007-01-02 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
US7348308B2 (en) 2000-01-03 2008-03-25 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
US6901310B2 (en) 2000-11-06 2005-05-31 Siemens Aktiengesellschaft Method and system for approximately reproducing the surface of a workpiece
WO2002039380A2 (fr) * 2000-11-13 2002-05-16 Siemens Aktiengesellschaft Procede et systeme de reconstruction d'une surface
WO2002039380A3 (fr) * 2000-11-13 2003-06-12 Siemens Ag Procede et systeme de reconstruction d'une surface
US7062353B2 (en) 2000-11-13 2006-06-13 Siemens Aktiengesellschaft Method and system for reconstructing a surface
CN1318929C (zh) * 2000-11-13 2007-05-30 西门子公司 重构表面的方法和系统
US8088941B2 (en) 2001-09-14 2012-01-03 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
US7378432B2 (en) 2001-09-14 2008-05-27 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
US7446092B2 (en) 2002-12-12 2008-11-04 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
US7833974B2 (en) 2002-12-12 2010-11-16 Tel Aviv University Future Technology Development L.P. Glycogen synthase kinase-3 inhibitors
EP2533202A3 (fr) * 2011-06-09 2015-04-15 Visual Technology Services Ltd. Compression de maillage de couleur
US9589318B2 (en) 2014-08-25 2017-03-07 Ge Aviation Systems Llc Method and system for generating airport surface map graphics in aircraft cockpit displays
US10313673B2 (en) 2016-10-19 2019-06-04 Google Llc Methods and apparatus to encode and/or decode normals of geometric representations of surfaces
US10733766B2 (en) 2016-10-19 2020-08-04 Google, Llc Methods and apparatus to encode and/or decode normals of geometric representations of surfaces
US9787321B1 (en) 2016-11-17 2017-10-10 Google Inc. Point cloud data compression using a space-filling curve
US10430975B2 (en) 2016-11-17 2019-10-01 Google Llc Advanced k-D tree encoding for point clouds by most significant axis selection
US10496336B2 (en) 2016-11-17 2019-12-03 Google Llc K-D tree encoding for point clouds using deviations
US10553035B2 (en) 2017-06-02 2020-02-04 Google Llc Valence based implicit traversal for improved compression of triangular meshes
US10950042B2 (en) 2017-06-02 2021-03-16 Google Llc Guided traversal in compression of triangular meshes
US10891758B2 (en) 2018-07-23 2021-01-12 Google Llc Geometry encoder
CN112419178A (zh) * 2020-11-18 2021-02-26 芯勍(上海)智能化科技股份有限公司 破洞修复方法、终端设备及计算机可读存储介质

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AU2316500A (en) 2000-08-18
WO2000045237A3 (fr) 2000-11-02
EP1194860A2 (fr) 2002-04-10
JP2002535791A (ja) 2002-10-22

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