flow complex joachim giesen friedrich-schiller-universität jena
TRANSCRIPT
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
The weighted flow complex
The weighted flow complex is also defined as the collection of stable manifolds.
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
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
Theorem
For a dense -sample of a smooth surface the reconstruction is homeomorphic and geometrically close to the original surface.
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
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.