Flow Complex Joachim Giesen Friedrich-Schiller-Universität Jena

Points

Surface reconstruction

Proteins: feature extraction

The Flow Complex joint work with Matthias John

Distance function

x d(x) x

Distance function

Gradient flow

Critical points maxima saddle 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 Buddha 144,647 pts Hip 132,538 pts

Experimental results Dragon 100,250 pts Noise added

Pockets in Proteins joint 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

Growing balls model

Pockets in proteins Topological events correspond 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 Approximation joint work with Edgar Ramos and Bardia Sadri

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 / Matching joint 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 Shapes joint work with Matthias John and Tamal Dey

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

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

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

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

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

Flow Shapes: inserting the stable manifolds in order of increasing values of 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 complex restricted to a union of balls centered at the sample points

Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points

Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points

Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points

Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the 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 shapeAlpha shape

Comparison of the shapes Flow shapeAlpha shape

Comparison of the shapes Flow shapeAlpha shape

Comparison of the shapes Flow shapeAlpha shape

The End Thank you!