Flow Complex

Joachim Giesen

Friedrich-Schiller-Universität Jena

Points

Surface reconstruction

Proteins: feature extraction

The Flow Complexjoint work with Matthias John

Distance function

Distance function

x

d(x)

x

d(x)

Distance function

Gradient flow

Critical points

maximasaddle points

Flow and critical points

Flow and critical points

Stable manifolds

Flow complex

Back to three dimensions

Stable manifolds

Surface Reconstruction (first attempt)joint work with Matthias John

Surface Reconstruction

Flow complex

Surface reconstruction

Pairing and cancellation

• Pairing of maxima and saddle points

Pairing and cancellation

• Pairing of maxima and saddle points

• Cancellation of pair with minimal difference between distance values

Pairing and cancellation

• Pairing of maxima and saddle points

• Cancellation of pair with minimal difference between distance values

Pairing and cancellation

• Pairing of maxima and saddle points

• Cancellation of pair with minimal difference between distance values

• Until “topologically” correct surface

Pairing and cancellation

• Pairing of maxima and saddle points

• Cancellation of pair with minimal difference between distance values

• Until “topologically” correct surface

Pairing and cancellation

• Pairing of maxima and saddle points

• Cancellation of pair with minimal difference between distance values

• Until “topologically” correct surface

Pairing and cancellation

• Pairing of maxima and saddle points

• Until “topologically” correct surface

• Cancellation of pair with minimal difference between distance values

Pairing and cancellation

Result is a (possibly pinched)closed surface

Experimental results

Buddha144,647 pts

Hip132,538 pts

Experimental results

Dragon100,250 pts Noise added

Pockets in Proteinsjoint work Matthias John

Pockets in proteins

Weighted flow complex

Pockets

in molecules

Power distance

Let (p,w) be a weighted point.

Power distance: |x-p|² - w

x

√w

p

Distance to weighted points

The weighted flow complex

The weighted flow complex is also defined as the collection of stable manifolds.

Pockets in proteins

Pockets in proteins

Growing balls model

Pockets in proteins

Topological eventscorrespond to critical points of the distance function

Pocket: connected component of union of stable manifolds of positive critical points

Visualization

Pocket visualization: stable manifolds of negative critical points in the boundary

Mouth: (connected component of) stable manifolds of positive critical points in the boundary of a pocket

Examples

Void(no mouth)

Ordinary pocket(one mouth)

Tunnel(two or more)

Examples

Alphatoxin

Surface Reconstruction joint work with Tamal Dey, Edgar Ramos

and Bardia Sadri

For a dense sample of a smooth surface the critical points are either close to the surface or close to the medial axis of the surface.

Theorem

Medial axis

Distance function is not differentiable

on medial axis.

Sampling condition

Theorem

For a dense -sample of a smooth surface the reconstruction is homeomorphic and geometrically close to the original surface.

Medial Axis Approximationjoint work with Edgar Ramos and Bardia Sadri

Gradient flow

Gradient flow

Unstable manifolds of medial axis critical points.

For a dense -sample of a smooth surface the union of the unstable manifolds of medial axis critical points is homotopy equivalent to the medial axis.

Theorem

The medial axis core

Shape Segmentation / Matchingjoint work with Tamal Dey and Samrat Goswami

Gradient flow and critical points

Anchor hulls and drivers of the flow.

Segmentation (2D)

Segmentation (3D)

Matching (2D)

Matching (3D)

Flow Shapes and Alpha Shapesjoint work with

Matthias John and Tamal Dey

Flow Shapes

Flow Shapes:inserting the stable manifolds in order of increasing valuesof the distance function at the critical points

Flow Shapes

Flow Shapes:inserting the stable manifolds in order of increasing valuesof the distance function at the critical points

Flow Shapes

Flow Shapes:inserting the stable manifolds in order of increasing valuesof the distance function at the critical points

Flow Shapes

Flow Shapes:inserting the stable manifolds in order of increasing valuesof the distance function at the critical points

Flow Shapes

Flow Shapes:inserting the stable manifolds in order of increasing valuesof the distance function at the critical points

Flow Shapes:inserting the stable manifolds in order of increasing valuesof the distance function at the critical points

Flow Shapes

Finite Sequence C¹…Cⁿ of cell complexes.

C¹ = P (point set)Cⁿ = Flow complex

Alpha Shapes

Alpha Shapes:Delaunay complexrestricted to a unionof balls centered atthe sample points

Alpha Shapes

Alpha Shapes:Delaunay complexrestricted to a unionof balls centered atthe sample points

Alpha Shapes

Alpha Shapes:Delaunay complexrestricted to a unionof balls centered atthe sample points

Alpha Shapes

Alpha Shapes:Delaunay complexrestricted to a unionof balls centered atthe sample points

Alpha Shapes

Alpha Shapes:Delaunay complexrestricted to a unionof balls centered atthe sample points

Finite Sequence C¹…Cⁿ´ of cell complexes, n´ ≥ n.

C¹ = P (point set)Cⁿ´ = Delaunay

triangulation

Theorem

For every α ≥ 0 the flow shape corresponding

to the distance value α and the alpha shape

corresponding to balls of radius α are

homotopy equivalent.

Comparison of the shapes

Flow shape Alpha shape

Comparison of the shapes

Flow shape Alpha shape

Comparison of the shapes

Flow shape Alpha shape

Comparison of the shapes

Flow shape Alpha shape

The End

Thank you!