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Computing geodesics and minimal surfaces via graph cuts Yuri Boykov, Siemens Research, Princeton, NJ joint work with

Computing geodesics and minimal surfaces via graph cuts Yuri Boykov, Siemens Research, Princeton, NJ joint work with Vladimir Kolmogorov, Cornell University, Ithaca, NY. Minimal geometric artifacts Solved via local variational technique (level sets). Possible metrication errors

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Computing geodesics and minimal surfaces via graph cuts Yuri Boykov, Siemens Research, Princeton, NJ joint work with

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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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