1 / 20

Using M-Reps to include a-priori Shape Knowledge into the Mumford-Shah Segmentation Functional

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

vincent-warner
Download Presentation

Using M-Reps to include a-priori Shape Knowledge into the Mumford-Shah Segmentation Functional

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

  4. Mumford-Shah with a-priori knowledge • Replace by d(Sap,S) • Sap represents the expected shape (prior) • d(Sap,∙)somehow measures the distance to the prior How is „somehow“?

  5. 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

  6. 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)

  7. Information derived from the m-rep (1) • Connection graph: • Hierarchy of figure(s) • Main figure, protrusion, intrusion • Topology of surface • Connection and substance edges

  8. Information derived from the m-rep (2) • 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)

  9. Problem of m-reps • Stability: • We never infer the medial axis from the boundary surface!

  10. 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

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

  12. Shape description bymedial atoms • One medial atom: • Shape consisting of N medial Atoms:+ connection graph

  13. A distance between shapes? • Current main problem:What is a suitable distanceOr maybe even consider

  14. 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

  15. 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“

  16. 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 ontois maximal • How to carry over these concepts from the vector space to the manifold ?

  17. Principal Geodesic Analysis Problem again: not necessarily unique

  18. 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?

  19. The End Comments? Ideas? Questions? Suggestions?

More Related