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A Simple and Robust Thinning Algorithm on Cell Complexes Lu Liu +, Erin Wolf Chambers*, David Letscher*, Tao Ju + + Washington University in St. Louis * St. Louis University

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Background Thinning: a widely used approach in discrete domain to compute skeleton

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Background Applications of skeletons Shape matching and retrieval Hand writing recognition Shape segmentation Animation

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Motivation Problems Thinning: sensitive to perturbation Goals Robust

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Motivation Problems Thinning: sensitive to perturbation Pruning: complex Goals Robust Simple [Sud 05] [Shaked 98] The angle constraint (local) The area constraint (global)

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Motivation Problems Thinning: sensitive to perturbation Pruning: complex Hard to control Goals Robust Simple Controllable AnimationShape descriptor Surface skeleton Curve skeleton

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Our Thinning Algorithm 2D measure 1 st round thinning 2 nd round thinning OutputInput

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Our Thinning Algorithm 2D measure 1 st round thinning 2 nd round thinning OutputInput

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Cell Complexes A closed set of cells at various dimensions 0-cell (point), 1-cell (edge), 2-cell (face), 3-cell (cube), etc. Why cell complexes: Has explicit geometry Easy to maintain topology during thinning Removing simple pairs Simple pair: (, ) where is the only higher- dimensional cell adjacent to

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Our Thinning Algorithm 2D measure 1 st round thinning 2 nd round thinning OutputInput

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A Nave Thinning Process Peel off layer by layer by removing simple pairs 11

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Our Observation 12 15 6 11 16 20 10 15 Removed in iteration 20 I = 6, R = 20, R >> I Isolated in iteration 6 Highlighted medial edge Neighboring faces

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Our Observation 13 15 6 11 16 20 10 15 I = 2, R = 4, R I Isolated in iteration 2 Highlighted medial edge Removed in iteration 4 Neighboring faces

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Medial Persistence Measure (MP) 14 LowHigh

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Geometric Explanation I and R approximate different shape measures I : Radius of largest inscribing disc Thickness R : Half-length of longest inscribing tube Length MP captures tubular-ness: R-I : Scale 1-I/R : Sharpness I R

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Our Thinning Algorithm 2D measure 1 st round thinning 2 nd round thinning OutputInput Preserving the medial edges with measures larger than thresholds

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Medial Persistence (3D) Same computation Get isolation (I) and removal (R) iterations for each edge and face Compute absolute (R-I) and relative (1-I/R) medial persistence Simple computation Higher MP means: Edges: more significant tubular-ness Faces: more significant plate-likeness Absolute/Relative MP measures the scale/sharpness of feature Robust to boundary perturbation

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Output Input Our Thinning Algorithm 3D Thresholding 2 nd round thinning for color, for Size Play Video 1 st round thinning

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MP of faces Input Mixed dimensional skeletons MP of edges Curve skeletons only (infinity thresholds for faces)

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MP of faces Input Mixed dimensional skeletons MP of edges Curve skeletons only (infinity thresholds for faces)

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MP of faces Input Mixed dimensional skeletons MP of edges Curve skeletons only (infinity thresholds for faces)

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Strength of Our Algorithm Robust to noise and cell shapes Noisy Tetrahedral Cubic

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Strength of Our Algorithm Robust to noise and cell shapes Noisy Tetrahedral Cubic

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Strength of Our Algorithm Robust to different resolutions

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Summary Proposed a thinning algorithm on cell complexes Simple: 2 rounds of thinning, multiple dimensions Robust: stable medial persistence measure (MP) Noise Different cell shapes Different resolutions Controllable: different thresholds for medial geometry in different dimensions

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Limitations and Future Work Limitations Skeletons vary with the structure of the cell complex Medial persistence can be biased by grid directions Future work Continuous formulation of thinning and skeleton measures diagonal bias Smoother skeleton with resolution increase cubic tetrahedral

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Check out our project page (program, data, video, and more) Project page: http://www.cse.wustl.edu/~ll10/paper/pgcc/pgcc.html Google (Keywords) Cell complex, skeleton, project

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Alpha helix Beta sheets Protein (Cryo-EM volume) Secondary structure

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Scale dependent Scale independent I R T(Mabs)= 0.05L, T(Mrel) = 0.5 for both k = 1,2 (faces, edge) L is the width of the bounding box

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Discussion & Future work Artifacts Measure is anisotropic on isotropic shapes Rely on regular grid Future: distance guided thinning, octree

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Discussion & Future work Artifacts Measure is anisotropic on isotropic shapes Rely on regular grid Future: distance guided thinning, octree Observations Smoother skeleton with the increase of resolution Future: continuous definition

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Discussion & Future work Artifacts Measure is anisotropic on isotropic shapes Different representatin: octree Remedy: distance based thinning Observations: Different resolutionsL Continuous definition

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Our thinning algorithm 2D 34 2D model in cell complex representation Intermediate measure The stable part thinning Low High