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Petter StrandmarkFredrik Kahl Centre for Mathematical Sciences, Lund University

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Segmentation Data term Length of boundary Segmentation by minimizing an energy:

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Data term Example from Schoenemann et al. 2009 Length of boundary Squared curvature

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Schoenemann, Kahl and Cremers, ICCV 2009 Schoenemann, Kahl, Masnou and Cremers, arXiv 2011 Schoenemann, Masnou and Cremers, arXiv 2011 Global optimization of curvature Schoenemann, Kuang and Kahl, EMMCVPR 2011 Improved multi-label formulation Kanizsa, Italian Journal of Psychology 1971 Dobbins, Zucker and Cynader, Nature 1987 Motivation from a psychological/biological standpoint This paper: Correct formulation,efficiency, 3D Goldluecke and Cremers, ICCV 2011 Continuous formulation

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Start with a mesh of all possible line segments variable for each region variables for each pair of edges Restricted to {0,1}

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Incorporate curvature: variable for each region; 1 means foreground, 0 background variables for each pair of edges

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Boundary constraints: Surface continuation constraints:

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Problem with the existing formulation: Nothing prevents a double boundary

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Simple fix? Require that Not optimal (fractional) Existing formulation Global solution! Not correct!

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Consistency:

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Existing formulation Global solution! Not correct! Not optimal (fractional) New constraints Global + correct!

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90 60 45 27 30 32 regions, 52 lines12 regions, 18 lines Too coarse!

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Always split the most important region; use a priority queue

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p. 69

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16-connectivity

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8-connectivity

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Approximate surface with a mesh of faces Want to measure how much the surface bends: Willmore energy

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One unit cell 8 unit cells (5 tetrahedrons)

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Area regularizationCurvature regularization Problem: Wrapping a surface around a cross

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Area regularizationCurvature regularization 491,000 variables637,000 variables 128 seconds Problem: Connecting two discs

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Curvature regularization is now more practical Adaptive meshes Hexagonal meshes New constraints give correct formulation Surface completion Source code available online (2D and 3D)

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