Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes

Abstract

A functional for joint variational object segmentation and shape matching is developed. The formulation is based on optimal transport w.r.t. geometric distance and local feature similarity. Geometric invariance and modelling of object-typical statistical variations is achieved by introducing degrees of freedom that describe transformations and deformations of the shape template. The shape model is mathematically equivalent to contour-based approaches but inference can be performed without conversion between the contour and region representations, allowing combination with other convex segmentation approaches and simplifying optimization. While the overall functional is non-convex, non-convexity is confined to a low-dimensional variable. We propose a locally optimal alternating optimization scheme and a globally optimal branch and bound scheme, based on adaptive convex relaxation. Combining both methods allows to eliminate the delicate initialization problem inherent to many contour based approaches while remaining computationally practical. The properties of the functional, its ability to adapt to a wide range of input data structures and the different optimization schemes are illustrated and compared by numerical experiments.

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Notes

  1. 1.

    Note that \(t_{\text {r}}\) is not a gradient field and thus \(\notin T_\mu {{\mathcal {W}}_{2}}({\mathbb {R}}^2)\). One could find a corresponding gradient version by lifting the rotation field from the contour to the interior, Sect. 2.3. However the functional is also meaningful with this non-gradient mode and its effect on the template is more intuitive.

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Acknowledgments

This work was supported by the DFG, Grant GRK 1653.

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Correspondence to Bernhard Schmitzer.

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Schmitzer, B., Schnörr, C. Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes. J Math Imaging Vis 52, 436–458 (2015). https://doi.org/10.1007/s10851-014-0546-8

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Keywords

  • Image segmentation
  • Object matching
  • Optimal transport
  • Shape analysis
  • Convex relaxation