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
Slide 2
Background Thinning: a widely used approach in discrete domain
to compute skeleton
Slide 3
Background Applications of skeletons Shape matching and
retrieval Hand writing recognition Shape segmentation
Animation
Slide 4
Motivation Problems Thinning: sensitive to perturbation Goals
Robust
Slide 5
Motivation Problems Thinning: sensitive to perturbation
Pruning: complex Goals Robust Simple [Sud 05] [Shaked 98] The angle
constraint (local) The area constraint (global)
Slide 6
Motivation Problems Thinning: sensitive to perturbation
Pruning: complex Hard to control Goals Robust Simple Controllable
AnimationShape descriptor Surface skeleton Curve skeleton
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
A Nave Thinning Process Peel off layer by layer by removing
simple pairs 11
Slide 12
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
Slide 13
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
Slide 14
Medial Persistence Measure (MP) 14 LowHigh
Slide 15
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
Slide 16
Our Thinning Algorithm 2D measure 1 st round thinning 2 nd
round thinning OutputInput Preserving the medial edges with
measures larger than thresholds
Slide 17
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
Slide 18
Output Input Our Thinning Algorithm 3D Thresholding 2 nd round
thinning for color, for Size Play Video 1 st round thinning
Slide 19
MP of faces Input Mixed dimensional skeletons MP of edges Curve
skeletons only (infinity thresholds for faces)
Slide 20
MP of faces Input Mixed dimensional skeletons MP of edges Curve
skeletons only (infinity thresholds for faces)
Slide 21
MP of faces Input Mixed dimensional skeletons MP of edges Curve
skeletons only (infinity thresholds for faces)
Slide 22
Strength of Our Algorithm Robust to noise and cell shapes Noisy
Tetrahedral Cubic
Slide 23
Strength of Our Algorithm Robust to noise and cell shapes Noisy
Tetrahedral Cubic
Slide 24
Strength of Our Algorithm Robust to different resolutions
Slide 25
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
Slide 26
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
Slide 27
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
Slide 28
Slide 29
Alpha helix Beta sheets Protein (Cryo-EM volume) Secondary
structure
Slide 30
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
Slide 31
Discussion & Future work Artifacts Measure is anisotropic
on isotropic shapes Rely on regular grid Future: distance guided
thinning, octree
Slide 32
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
Slide 33
Discussion & Future work Artifacts Measure is anisotropic
on isotropic shapes Different representatin: octree Remedy:
distance based thinning Observations: Different resolutionsL
Continuous definition
Slide 34
Our thinning algorithm 2D 34 2D model in cell complex
representation Intermediate measure The stable part thinning Low
High