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Using M-Reps to includea-priori Shape Knowledge

into the Mumford-Shah Segmentation Functional

FWF - Forschungsschwerpunkt S092Subproject 7

„Pattern and 3D Shape Recognition“Grossauer Harald

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Outlook

• Mumford-Shah

• Mumford-Shah with a-priori knowledge

• Medial axis and m-reps

• Statistical analysis of shapes

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• 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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• Replace by d(Sap,S)

• Sap represents the expected shape (prior)

• d(Sap,∙) somehow measures the distance to the prior

Mumford-Shah with a-priori knowledge

How is „somehow“?

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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

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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)

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Information derived from the m-rep (1)

• Connection graph:– Hierarchy of figure(s)– Main figure, protrusion, intrusion– Topology of surface– Connection and substance edges

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• 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)

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Problem of m-reps

• Stability:

• We never infer the medial axis from the boundary surface!

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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

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Medial Atoms (in 3D)

Stored per node:• Position and radius

• Local coordinate frame

• Opening angle

• Elongation (for „boundary atoms“ only)

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Shape description bymedial atoms

• One medial atom:

• Shape consisting of N medial Atoms:

+ connection graph

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A distance between shapes?

• Current main problem:

What is a suitable distance

Or maybe even consider

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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

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Statistical analysis of shapes

• Variational formulation of mean value:

• No vector space structure needed, but not necessarily unique → All Si must be in a „small enough neighborhood“

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• 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

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Principal Geodesic Analysis

Vector space Manifold

Linear subspace Geodesic submanifold

Projection onto subspace

Closest point on submanifold

Problem again: not necessarily unique

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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?

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The End

Comments?

Ideas?

Questions?

Suggestions?