Flow Complex

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Flow Complex

• Joachim Giesen
• Friedrich-Schiller-Universität Jena

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Points
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Surface reconstruction
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Proteins feature extraction
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The Flow Complexjoint work with Matthias John
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Distance function
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Distance function
x
d(x)
x
d(x)
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Distance function
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Gradient flow
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Critical points
maxima
saddle points
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Flow and critical points
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Flow and critical points
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Stable manifolds
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Flow complex
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Back to three dimensions
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Stable manifolds
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Surface Reconstruction (first attempt)joint work
with Matthias John
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Surface reconstruction
Surface Reconstruction
Flow complex
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Pairing and cancellation

• Pairing of
• maxima and
• saddle points

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Pairing and cancellation

• Pairing of
• maxima and
• saddle points

• Cancellation of
• pair with minimal
• difference between
• distance values

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Pairing and cancellation

• Pairing of
• maxima and
• saddle points

• Cancellation of
• pair with minimal
• difference between
• distance values

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Pairing and cancellation

• Pairing of
• maxima and
• saddle points

• Cancellation of
• pair with minimal
• difference between
• distance values

• Until topologically
• correct surface

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Pairing and cancellation

• Pairing of
• maxima and
• saddle points

• Cancellation of
• pair with minimal
• difference between
• distance values

• Until topologically
• correct surface

24
Pairing and cancellation

• Pairing of
• maxima and
• saddle points

• Cancellation of
• pair with minimal
• difference between
• distance values

• Until topologically
• correct surface

25
Pairing and cancellation

• Pairing of
• maxima and
• saddle points

• Cancellation of
• pair with minimal
• difference between
• distance values

• Until topologically
• correct surface

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Pairing and cancellation
Result is a (possibly pinched) closed surface
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Experimental results
Hip 132,538 pts
Buddha 144,647 pts
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Experimental results
Dragon 100,250 pts
Noise added
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Pockets in Proteinsjoint work Matthias John
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Pockets in proteins
Pockets in molecules
Weighted flow complex
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Power distance
Let (p,w) be a weighted point. Power distance
x-p² - w
vw
p
x
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Distance to weighted points
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The weighted flow complex
The weighted flow complex is also defined as the
collection of stable manifolds.
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Pockets in proteins
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Pockets in proteins
Growing balls model
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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
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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
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Examples
Void (no mouth)
Ordinary pocket (one mouth)
Tunnel (two or more)
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Examples
Alphatoxin
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Surface Reconstruction joint work with Tamal
Dey, Edgar Ramos and Bardia Sadri
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Theorem
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.
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Medial axis
Distance function is not differentiable on
medial axis.
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Sampling condition
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Theorem
For a dense ?-sample of a smooth surface the
reconstruction is homeomorphic and geometrically
close to the original surface.
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Medial Axis Approximationjoint work with Edgar
Ramos and Bardia Sadri
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Gradient flow
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Gradient flow
Unstable manifolds of medial axis critical
points.
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Theorem
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.
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The medial axis core
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Shape Segmentation / Matchingjoint work with
Tamal Dey and Samrat Goswami
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Gradient flow and critical points
Anchor hulls and drivers of the flow.
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Segmentation (2D)
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Segmentation (3D)
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Matching (2D)
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Matching (3D)
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Flow Shapes and Alpha Shapesjoint work with
Matthias John and Tamal Dey
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Flow Shapes
Flow Shapes inserting the stable manifolds in
order of increasing values of the distance
function at the critical points
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Flow Shapes
Flow Shapes inserting the stable manifolds in
order of increasing values of the distance
function at the critical points
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Flow Shapes
Flow Shapes inserting the stable manifolds in
order of increasing values of the distance
function at the critical points
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Flow Shapes
Flow Shapes inserting the stable manifolds in
order of increasing values of the distance
function at the critical points
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Flow Shapes
Flow Shapes inserting the stable manifolds in
order of increasing values of the distance
function at the critical points
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Flow Shapes
Flow Shapes inserting the stable manifolds in
order of increasing values of the distance
function at the critical points
Finite Sequence C¹Cn of cell complexes. C¹ P
(point set) Cn Flow complex
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Alpha Shapes
Alpha Shapes Delaunay complex restricted to a
union of balls centered at the sample points
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Alpha Shapes
Alpha Shapes Delaunay complex restricted to a
union of balls centered at the sample points
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Alpha Shapes
Alpha Shapes Delaunay complex restricted to a
union of balls centered at the sample points
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Alpha Shapes
Alpha Shapes Delaunay complex restricted to a
union of balls centered at the sample points
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Alpha Shapes
Alpha Shapes Delaunay complex restricted to a
union of balls centered at the sample points
Finite Sequence C¹Cn of cell complexes, n n.
C¹ P (point set) Cn Delaunay
triangulation
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Theorem

• For every a 0 the flow shape corresponding
• to the distance value a and the alpha shape
• corresponding to balls of radius a are
• homotopy equivalent.

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Comparison of the shapes
Flow shape
Alpha shape
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Comparison of the shapes
Flow shape
Alpha shape
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Comparison of the shapes
Flow shape
Alpha shape
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Comparison of the shapes
Flow shape
Alpha shape
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The End

• Thank you!