3D Thinning on Cell Complexes for Computing Curve and Surface Skeletons Lu Liu Advisor: Tao Ju Master Thesis Defense Dec 18 th, 2008

Outline Motivation Goal and Rationale Cell Complex Our thinning algorithm Conclusion & Future work

Thin geometric structure lying in the center Skeleton as a Shape Descriptor Curve (1D) 1D Skeleton2D Object Elongated Part Tube Plate Curve (1D) Surface (2D) 1D/2D Skeleton 3D Object Dimension Reduction

Skeleton as a Shape Descriptor Thin geometric structure lying in the center Applications – Handwritten character recognition – Shape matching and retrieval – Shape segmentation – Shape deformation – Medical image visualization homepages.inf.ed.ac.uk/rbf/HIPR2/thin.htm

Skeleton as a Shape Descriptor Thin geometric structure lying in the center Applications – Handwritten character recognition – Shape matching and retrieval – Shape segmentation – Shape deformation – Medical image visualization based_image_retrieval.html

Thin geometric structure lying in the center Applications – Handwritten character recognition – Shape matching and retrieval – Shape segmentation – Shape deformation – Medical image visualization Skeleton as a Shape Descriptor ublication

Skeleton as a Shape Descriptor Thin geometric structure lying in the center Applications – Handwritten character recognition – Shape matching and retrieval – Shape segmentation – Shape deformation – Medical image visualization Item.do?contentType=Article&hdAction=lnkhtml&c ontentId=

Skeleton as a Shape Descriptor Thin geometric structure lying in the center Applications – Handwritten character recognition – Shape matching and retrieval – Shape segmentation – Shape deformation – Medical image visualization Bone MatrixProtein VesselsNerve cells

Goal Thinness – Definition of skeleton (1-dimension reduction) – Easy to detect curve and surface components Topology preservation – Genus, connectivity – Handwritten character recognition, shape matching Shape preservation – Curve skeleton for tube-like shape components – Surface skeleton for plate-like shape components – Shape segmentation, shape deformation

Computing Skeletons On continuous models – As simplified Medial Axes [Sud et. al., 2005] On digital models – As a subset of lattice points 1. In many applications, such as medical imaging, data come as a collection of digital points 2. Computing skeleton on digital model is simple to implement and stable to perform

Computing Skeletons Digital model is represented as a set of points on a spatial grid Geometry and topology – Adjacency relation 2D 4-connectivity 2D 8-connectivity 3D 6-connectivity 3D 26-connectivity

Computing Skeletons Thinning on point based representation – Peeling off boundary points – Topology preservation: simple points – Shape preservation: curve/surface enpoints – Local operations: simple

Computing Skeletons Obstacles – Topology preservation under parallel thinning – Thinness 4 points joints – Shape preservation Endpoints detection is sensitive to noise A fundamental different representation

Cell Complexes In N-D, a set of k-cells (k<=N) – A closed set: the facets of each k-cell (e.g., edges of a square) also belong to the same set Point (0-cell)Edge (1-cell)Square (2-cell)Cube (3-cell)

Cell Complexes Construction: – All those k-cells whose boundary points are in the “points on a spatial grid” representation – Result in a closed set – Any grid, any dimension

Cell Complexes – Simplicial Collapse Removal of k-simple pair – Dimension, – is only on the boundary of Topology preserving Local operation δσδσ A k-cell σ and a (k-1)-cell δ, so that δ is not contained in another k-cell than σ.

Proposition 1 (Topology-preservation): Simultaneous removal of multiple simple pairs preserves the homotopy of a cell complex. Proposition 2 (Thinness): Removal of all simple N-simple pairs deletes all N-cells in a for a N dimensional cell complex. Cell Complexes – Simplicial Collapse

Our Thinning Algorithm Simplicial collapse – Topology preservation – Thinness Significance measures – Shape preservation Our thinning algorithm

Significance Measures Shape elongation(dimension awareness): – k-D skeleton is elongated in k directions: curve-skeleton: 1; surface skeleton: 2 D, d measures, significance measures S1, S2 – cells with large significance measures are preserved (1)S1 = d – D (2) S2 = 1 – D/d 1-D skeleton 2-D shape

Significance measures computation – Approximation of D,d Significance Measures – Approximation

Significance measures – approximation D of a cell is the index of the iteration in which the cell becomes isolated d of a cell is the index of iteration in which the cell becomes simple

Significance Measures – Approximation D measure d measure S1 measure S2 measure

T1 = 5; T2 = 0.5 S1 measure S2 measure Significance Measure – Approximation

Our Thinning Algorithm Parallel thinning Approximate D Approximate d Significance measures

Our Thinning Algorithm Algorithm is simple to implement Skeleton is thin, topology preserving, and shape preserving

Results - T Shape Model t1 = 5, t2 = 0.5

Results – Rocker Arm Model t1 = 5, t2 = 0.5

Results – Hip Bone Model t1 = 5, t2 = 0.5

Results – Hip Bone Model t1 = 9, t2 = 0.5

Results – Fertility Model t1 = 5, t2 = 0.5

Results – Dragon Model t1 = 5, t2 = 0.5

Results – Protein tim Model t1 = 5, t2 = 0.5

Performance Model Before ThinningAfter ThinningTime(s) pointedgequadcubepointedgequad Tshape Rocker arm Hip Hip(t2=9) Dragon Fertility Protein Tim X 3 X * 128 * 128 uniform grid

Performance

Future Work Queue structure for outmost layer in thinning – To overcome time consuming Adaptive thinning algorithm on octree grid – To overcome memory consuming Other topology preserving operators – ? Growing operator: skinning – Growing operator + simplicial collapse: topology preserving and volume preserving deformation

Conclusion Present and prove two properties of simplicial collapse on cell complexes – Thinness – Topology preservation under parallel thinning Propose two significant measures – Shape preservation Develop a simple thinning algorithm

Acknowledgement Great thanks goes to – My advisor: Professor Tao Ju – My committee members: Professor Cindy Grimm Professor Robert Pless

Q & A