Laboratoire d'InfoRmatique en Image et Systèmes d'information FRE 2672 CNRS/INSA de...

WorkShop Télégéo 6/11/03

Laboratoire d'InfoRmatique en Image et Systèmes d'informationFRE 2672 CNRS/INSA de Lyon/Université Claude Bernard Lyon 1/Université Lumière Lyon 2/Ecole Centrale de Lyon,

Université Claude Bernard Lyon1 - Bâtiment Nautibus43, boulevard du 11 Novembre 1918 – F- 69622 Villeurbanne Cedex

http://liris.cnrs.fr

FRE 2672

[email protected] – http://www.semantic-3d.net/

Curvature Based Triangle Mesh Segmentation with Boundary Rectification

Guillaume Lavoué, Florent Dupont, Atilla Baskurt

WorkShop Télégéo 6/11/03 2

PlanContext.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.


ContextSemantic-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

Patch

Subdivision Surface

fitting

Subdivision Surface

fitting

Adaptive precision



PlanContext.

Method overview.The region segmentation process. Vertex classification. Region growing. Region merging.



Perspectives.



Method overview

Boundary edges extraction

3D Trianglemesh

Vertexcurvature

classification

Classifiedvertices

Region growingRegion merging

Segmentedregions

Constant curvature region segmentation

Boundaryedges

BoundaryScore

processing

Correctboundary

edges

Contourtracking

Rectifiedboundaries

boundary rectification

We need known curvature surface patches with regular boundaries for the fitting process.






Perspectives.


Vertex classificationExtraction of discrete curvature. [Meyer, Discrete Differential-Geometry Operators for Triangulated 2-Manifolds (VisMath 2002)]

Mean Curvature:

Gauss Curvature:

Principal Curvatures:

avec

iX

Xj

ijiij

))(cot(cot2

1)(

)(1

ijiNj

jji XXA

XK

f

jji AXKg

1

/)2()(

)()()( iii XXKhXKmin

)()()( iii XXKhXKmax

)()()(

)(2

1)(

2iii

ii

XKgXKhX

XKXKh



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



Vertex classificationCurvature classification.

K-Means clustering applied to Kmin and Kmax.Cluster regularization.

kmax

kmin

Classification 5 clusters

kmax

kmin

Classification 5 clusters



Region growingRegion 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 growing Crack filling



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



Region mergingRegion distance measurement.

The distance Dij between ith and jth regions is:

The curvature distance DCij : Two situations between two regions:

ijijijij SNDCD

ijjijiij CCCCDC

Ri

Rj

RiRj

No curvature differencebut a significant boundary

No significant boundary but a curvature difference

Chosen curvature distance

With: Ci , Cj , the curvature values of the ith and jth regions.

Cij , the curvature value of their boundary.



Region merging The nesting coefficient Nij :

Introduced in image processing by Schettini in Segmentation Algorithm For Color Images (Pattern Recognition Letters 1993).

The filtering coefficient Sij :

The fusion of smallest regions is accelerated.

ij

jiij P

PPN

),min(

With: Pi , Pj , the perimeters of the ith and jth regions.

Pij , the common perimeter of the ith and jth regions.

else

AAorAAifS jiij 1

)( minmin With: Ai , Aj , the areas of the ith and jth regions.

Aij , a minimum fixed area.

, a positive number ~ 0.



Samples

Fandisk 6475 vertices

Vertex classification

Classified vertices 14 clusters

Region growing

Segmented Fandisk 68 regions

Region merging Threshold = 50

Segmented Fandisk 16 regions



Samples

Shark 10234 vertices

Vertex classification

Classified vertices 10 clusters

Region growing Segmented Shark

343 regions

Region merging Threshold = 50

Segmented Shark 16 regions



Curvature classification results dependencyTests conducted with different K-Means clusters number.

Cluster NbCluster Nb

regularized

Region Nb

After growing

Region Nb

After merging

ResultsThreshold = 50

5 5 38 16

10 8 61 16

10 9 49 15

15 13 66 16

20 15 92 16



The boundary rectification process. Purpose. The Boundary Score definition. General algorithm.


Perspectives.


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.



The Boundary Score definition.

)()()( eSceSaeS c

dmin

dmax

22

2222

11

1111

KminKmax

KminmaxKmaxminKminKmax

KminmaxKmaxmin

eSa

)(


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.

),max(),max(

)()(

1212

1212

KmaxΚmaxΚminΚmin

KmaxΚmaxΚminΚmineSc

1min

2mine

),( 22 ΚminΚmax

),( 11 ΚminΚmax

1dmin

2dmin


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.

SeS )(

SeS )(



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 Pi depending of its score and its angle with its CBE.S(e)

),()()( CBEeeeSeP Sorted list


Potential edge

CBE

),( CBEee


Contour tracking

EndSorted list

construction

Lowest weight edge integration

List updateOnly closed boundaries?

Yes

No


Potential edgeCBEPotential edgeCBEPotential edgeCBE


Sample


3D-object

Vertex Classification

Region growing

Region merging

Correct boundary edges

marking

Contour tracking



The boundary rectification process. Purpose. The Boundary Score definition. General algorithm.


Perspectives.



Conclusion and perspectivesOur 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.


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