1/20 Using M-Reps to include a-priori Shape Knowledge into the Mumford-Shah Segmentation Functional FWF - Forschungsschwerpunkt S092 Subproject 7 „Pattern and 3D Shape Recognition“ Grossauer Harald

2/20 Outlook Mumford-Shah Mumford-Shah with a-priori knowledge Medial axis and m-reps Statistical analysis of shapes

3/20 Original Mumford-Shah functional: For minimizers (u,C) : –u … piecewise constant approximation of f –C … curve along which discontinuities of u are located Mumford-Shah

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5/20 Replace by d(S ap,S) S ap represents the expected shape (prior) d(S ap,∙) somehow measures the distance to the prior Mumford-Shah with a-priori knowledge How is „somehow“?

6/20 Curve representation Curves (resp. surfaces) are frequently represented as –Triangle mesh (easy to render) –Set of spline control points (smoother) –CSG, … Problems: –Local boundary description –No global shape properties

7/20 Blum‘s Medial Axis (in 2D) Medial axis for a given „shape“ S : Set of centers of all circles that can be inscribed into S, which touch S at two or more points Medial axis + radius function → Medial axis representation (m-reps)

8/20 Information derived from the m-rep (1) Connection graph: –Hierarchy of figure(s) –Main figure, protrusion, intrusion –Topology of surface –Connection and substance edges

9/20 Let be a parametrization of, then – is the „principal direction“ of S – describes the „bending“ of S – is the local „thinning“ or „thickening“ of S –Branchings of may indicate singular surface points (edges, corners) Information derived from the m-rep (2)

10/20 Problem of m-reps Stability: We never infer the medial axis from the boundary surface!

11/20 Discrete representation (in 3D) Approximate medial manifold by a mesh Store radius in each mesh node → Bad approximation of surface → Store more information per node: Medial Atoms

12/20 Medial Atoms (in 3D) Stored per node: Position and radius Local coordinate frame Opening angle Elongation (for „boundary atoms“ only)

13/20 Shape description by medial atoms One medial atom: Shape consisting of N medial Atoms: + connection graph

14/20 A distance between shapes? Current main problem: What is a suitable distance Or maybe even consider

15/20 Statistical analysis of shapes Goal: Principal Component Analysis (PCA) of a set of shapes Zero‘th principal component = mean value Problem: is not a vector space

16/20 Statistical analysis of shapes Variational formulation of mean value: No vector space structure needed, but not necessarily unique → All S i must be in a „small enough neighborhood“

17/20 How to carry over these concepts from the vector space to the manifold ? PCA in For data the k ‘th principal component is defined inductively by: – is orthogonal to – is orthogonal to the subspace, where: has codimension k the variance of the data projected onto is maximal

18/20 Principal Geodesic Analysis Vector spaceManifold Linear subspaceGeodesic submanifold Projection onto subspace Closest point on submanifold Problem again: not necessarily unique

19/20 Principal Geodesic Analysis Second main problem(s): –Under what conditions is PGA meaningful? –How to deal with the non-uniqueness? –Does PGA capture shape variability well enough? –How to compute PGA efficiently?

20/20 The End Comments? Ideas? Questions? Suggestions?