using m-reps to include a-priori shape knowledge into the mumford-shah segmentation functional
DESCRIPTION
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. Outlook. Mumford-Shah Mumford-Shah with a-priori knowledge Medial axis and m-reps - PowerPoint PPT PresentationTRANSCRIPT
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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
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