Laboratoire d'InfoRmatique en Image et Systèmes d'information FRE 2672 CNRS/INSA de Lyon/Université Claude Bernard Lyon 1/Université Lumière Lyon 2/Ecole Centrale de Lyon, Université Claude Bernard Lyon1 - Bâtiment Nautibus 43, boulevard du 11 Novembre 1918 – F Villeurbanne Cedex FRE 2672 – WorkShop Télégéo 6/11/03 Curvature Based Triangle Mesh Segmentation with Boundary Rectification Guillaume Lavoué, Florent Dupont, Atilla Baskurt

WorkShop Télégéo 6/11/03 2 Plan Context. Method overview. The region segmentation process.  Vertex classification.  Region growing.  Region merging. The boundary rectification process.  Why ?  The Boundary Score definition.  General algorithm.  Correct boundary marking.  Contour tracking. Perspectives.

WorkShop Télégéo 6/11/03 3 Context Semantic-3D RNRT national project.  Request and transmission of 3D CAD objects, multi- bandwidths, multi- platforms. Topics are Indexing, Watermarking and Compression.  Research of an adaptive, multi-resolution compression method. Chosen approach.  Subdivision surface fitting.  Arbitrary topology.  Intrinsically Multi-resolution.  Represented by a mesh.  Prior segmentation of the mesh into patches.  To simplify the fitting algorithm.  To permit adaptiveness. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

Complete process diagram Compressed model Manifold CAD mesh Segmentation Patch Subdivision Surface fitting Assembling Patch Subdivision Surface fitting Adaptive precision Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 5 Plan Context. Method overview. The region segmentation process.  Vertex classification.  Region growing.  Region merging. The boundary rectification process.  Why ?  The Boundary Score definition.  General algorithm.  Correct boundary marking.  Contour tracking. Perspectives. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 6 Method overview Boundary edges extraction 3D Triangle mesh Vertex curvature classification Classified vertices Region growing Region merging Segmented regions Constant curvature region segmentation Boundary edges Boundary Score processing Correct boundary edges Contour tracking Rectified boundaries boundary rectification We need known curvature surface patches with regular boundaries for the fitting process. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 7 Plan Context. Method overview. The region segmentation process.  Vertex classification.  Region growing.  Region merging. The boundary rectification process.  Why ?  The Boundary Score definition.  General algorithm.  Correct boundary marking.  Contour tracking. Perspectives.

WorkShop Télégéo 6/11/03 8 Vertex classification Extraction of discrete curvature. [ Meyer, Discrete Differential- Geometry Operators for Triangulated 2-Manifolds (VisMath 2002)]  Mean Curvature:  Gauss Curvature:  Principal Curvatures: avec Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 9 Vertex classification  Principal curvature directions dmax et dmin. Eigenvectors of the curvature tensor, found by least square fitting.. 0,25 2,5 -7,0 4,5 0,0 2,3 2,7 4,5 Mean Gauss abs(Min) Max dmin dmax Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 10 Vertex classification Curvature classification.  K-Means clustering applied to Kmin and Kmax.  Cluster regularization. kmax kmin Classification 5 clusters kmax kmin Classification 5 clusters Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 11 Region growing Region growing.  Triangle growing and labeling operation. triangle regions with holes between them.  Crack filling process. Not-labeled triangles are assigned to regions according to:  Their neighbor triangles labels.  Boundary criteria. Triangle growingCrack filling Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 12 Purpose:  Reduce the over-segmentation resulting from region growing.  Suppress results dependency to K-Means clusters number. General algorithm:  Region adjacency graph construction.  Nodes: connected regions.  Edges: adjacency between two regions.  Calculation of similarity distances between regions.  Valuation of the graph edges.  Graph reduction.  At each iteration the smallest adjacency edge is eliminated.  Stopping criteria.  A number of queried final regions.  A maximum distance threshold. Region merging Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 13 Region merging Region distance measurement.  The distance D ij between i th and j th regions is:  The curvature distance DC ij :  Two situations between two regions: RiRi RjRj RiRi RjRj No curvature difference but a significant boundary No significant boundary but a curvature difference Chosen curvature distance With:C i, C j, the curvature values of the i th and j th regions. C ij, the curvature value of their boundary. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 14 Region merging  The nesting coefficient N ij : Introduced in image processing by Schettini in Segmentation Algorithm For Color Images (Pattern Recognition Letters 1993).  The filtering coefficient S ij : The fusion of smallest regions is accelerated. With:P i, P j, the perimeters of the i th and j th regions. P ij, the common perimeter of the i th and j th regions. With:A i, A j, the areas of the i th and j th regions. A ij, a minimum fixed area., a positive number ~ 0. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 15 Samples Fandisk 6475 vertices Vertex classification Classified vertices 14 clusters Region growing Segmented Fandisk 68 regions Region merging Threshold = 50 Segmented Fandisk 16 regions Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 16 Samples Shark vertices Vertex classification Classified vertices 10 clusters Region growing Segmented Shark 343 regions Region merging Threshold = 50 Segmented Shark 16 regions Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 17 Curvature classification results dependency Tests conducted with different K-Means clusters number. Cluster Nb regularized Region Nb After growing Region Nb After merging Results Threshold =

WorkShop Télégéo 6/11/03 18 Plan Context. Method overview. The region segmentation process.  Vertex classification.  Region growing.  Region merging. The boundary rectification process.  Purpose.  The Boundary Score definition.  General algorithm.  Correct boundary marking.  Contour tracking. Perspectives.

WorkShop Télégéo 6/11/03 19 Purpose ? The subdivision surface fitting needs clean boundaries without discontinuities.  Method:  Extraction of the boundary edges from the segmented object.  Marking of the “correct” boundary edges with a boundary score calculation.  Contour tracking to complete boundaries according to the correct ones. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 20 The Boundary Score definition. dmin dmax Context - Method Overview - Region segmentation - boundary rectification -Perspectives Principal curvature directions are used.  The curvature varies according to dmax and tend to be constant following dmin. boundaries are parallel to dmin. Boundary Score processing.

WorkShop Télégéo 6/11/03 21 General algorithm.  Boundary edges extraction.  Correct boundary edges marking. A threshold S is fixed to determine Correct Boundary Edges (CBE), among the boundary edges which come from the region segmentation step. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 22 General algorithm.  Contour tracking.  Correct boundary edges form pieces of boundary contours.  For each not closed boundary, we extract the edges potentially being able to complete it (edges adjacent to one extremity CBE).  Each potential edge e is associated with a weight function P i depending of its score and its angle with its CBE. Sorted list Context - Method Overview - Region segmentation - boundary rectification -Perspectives Potential edge CBE

WorkShop Télégéo 6/11/03 23 Contour tracking End Sorted list construction Lowest weight edge integration List update Only closed boundaries? Yes No Context - Method Overview - Region segmentation - boundary rectification -Perspectives Potential edge CBE Potential edge CBE Potential edge CBE

WorkShop Télégéo 6/11/03 24 Sample Context - Method Overview - Region segmentation - boundary rectification -Perspectives 3D-object Vertex Classification Region growing Region merging Correct boundary edges marking Contour tracking

WorkShop Télégéo 6/11/03 25 Plan Context. Method overview. The region segmentation process.  Vertex classification.  Region growing.  Region merging. The boundary rectification process.  Purpose.  The Boundary Score definition.  General algorithm.  Correct boundary marking.  Contour tracking. Perspectives. Context - Method Overview - Region segmentation - boundary rectification -Perspectives

WorkShop Télégéo 6/11/03 26 Conclusion and perspectives Our method:  The segmentation algorithm.  Curvature classification. Curvature transition detection, not only hard edges cutting. Constant curvature region extraction.  Mixed Vertex-Triangle approach. Regions boundaries are clearly distinguishable edges.  The boundary rectification method:  Original method based on principal curvature directions. Perspectives:  Considering curvature variance or distribution histogram analysis. Improve classification. Automatically process the merging threshold.  Subdivision surface fitting in our adaptive multi-resolution compression objective. Context - Method Overview - Region segmentation - boundary rectification -Perspectives