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Computing geodesics and minimal surfaces via graph cuts

Explore geodesic active contours and minimal surfaces in image-based Riemannian spaces using graph cut techniques for accurate computation. Study how cut metrics approximate given Riemannian metrics presenting a theoretical connection between graph cuts and integral/differential geometry.

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Computing geodesics and minimal surfaces via graph cuts

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  1. Computing geodesics and minimal surfaces via graph cuts Yuri Boykov, Siemens Research, Princeton, NJ joint work with Vladimir Kolmogorov, Cornell University, Ithaca, NY

  2. Minimal geometric artifacts • Solved via local variational technique (level sets) • Possible metrication errors • Global minima Two standard object extraction methods Interactive Graph cuts [Boykov&Jolly ‘01] • Discrete formulation • Computes min-cuts on N-D grid-graphs Geodesic active contours [Caselles et.al. ‘97, Yezzi et.al ‘97] • Continuous formulation • Computes geodesics in image- based N-D Riemannian spaces Geo-cuts

  3. distance map distance map Geodesics and minimal surfaces • The shortest curve between two points is a geodesic B B A A Euclidian metric (constant) Riemannian metric (space varying, tensor D(p)) • Geodesic contours use image-based Riemannian metric • Generalizes to 3D (minimal surfaces)

  4. a cut hard constraint n-links hard constraint t s Graph cuts (simple example à la Boykov&Jolly, ICCV’01) Minimum cost cut can be computed in polynomial time (max-flow/min-cut algorithms)

  5. Minimum cost cut (standard 4-neighborhoods) Minimum length geodesic contour (image-based Riemannian metric) Continuous metric space (no geometric artifacts!) Metrication errors on graphs discrete metric ???

  6. C • Cost of a cut can be interpreted as a geometric “length” (in 2D) or “area” (in 3D) of the corresponding contour/surface. Cut Metrics :cuts impose metric properties on graphs • Cut metric is determined by the graph topology and by edge weights. • Can a cut metric approximate a given Riemannian metric?

  7. Our key technical result We show how to build a grid-graph such that its cut metric approximates any given Riemannian metric • The main technical problem is solved via Cauchy-Croftonformula from integral geometry.

  8. a subset of lines L intersecting contour C a set of all lines L Euclidean length of C : the number of times line L intersects C Integral Geometry andCauchy-Crofton formula C

  9. C Edges of any regular neighborhood system generate families of lines { , , , } Euclidean length graph cut cost for edge weights: the number of edges of family k intersecting C Cut Metric on gridscan approximate Euclidean Metric Graph nodes are imbedded in R2 in a grid-like fashion

  10. “Distance maps” (graph nodes “equidistant” from a given node) : 8-neighborhoods 256-neighborhoods “standard” 4-neighborhoods (Manhattan metric) Cut metric in Euclidean case • (Positive!) weights depend only on edge direction k.

  11. restoration with “standard” 4-neighborhoods original noisy image restoration with 8-neighborhoods using edge weights Reducing Metrication Artifacts Image restoration [BVZ 1999]

  12. (Positive!) weights depend on edge direction k and on location/pixel p. • Local “distance maps” assuming anisotropic D(p) = const 256-neighborhoods 4-neighborhoods 8-neighborhoods Cut Metric in Riemannian case • The same technique can used to compute edge weights that approximate arbitrary Riemannian metric defined by tensor D(p) • Idea: generalize Cauchy-Crofton formula

  13. C Convergence theorem Theorem: For edge weights set by tensor D(p)

  14. image-derived Riemannian metric D(p) regular grid edge weights Boundary conditions (hard/soft constraints) Global optimization Graph-cuts [Boykov&Jolly, ICCV’01] min-cut = geodesic “Geo-Cuts” algorithm

  15. Minimal surfaces in image inducedRiemannian metric spaces (3D) 3D bone segmentation (real time screen capture)

  16. variational optimization method for combinatorial optimization for fairly general continuous energies a restricted class of energies [e.g. KZ’02] • finds a local minimum finds a global minimum • near given initial solution for a given set of boundary conditions • numerical stability has to be carefully • addressed [Osher&Sethian’88]: • continuous formulation -> “finite differences” • numerical stability is not an issue • discrete formulation ->min-cut algorithms • anisotropic metrics are harder anisotropic Riemannian metrics • to deal with (e.g. slower) are as easy as isotropic ones Gradient descent method VS. Global minimization tool Our results reveal a relation between… Level Sets Graph Cuts [Osher&Sethian’88,…] [Greig et. al.’89, Ishikawa et. al.’98, BVZ’98,…] (restricted class of energies)

  17. Conclusions • “Geo-cuts” combines geodesic contours and graph cuts. • The method can be used as a “global” alternative to variational level-sets. • Reduction of metrication errors in existing graph cut methods • stereo [Roy&Cox’98, Ishikawa&Geiger’98, Boykov&Veksler&Zabih’98, ….] • image restoration/segmentation [Greig’86, Wu&Leahy’97,Shi&Malik’98,…] • texture synthesis [Kwatra/et.al’03] • Theoretical connection between discrete geometry of graph cuts and concepts of integral & differential geometry

  18. Geo-cuts (more examples) 3D segmentation (time-lapsed)

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