A variational model is introduced for the segmentation problem of thin structures, like tubes or thin plates, in an image. The energy is based on the Mumford-Shah model with a surfacic term perturbed by a Finsler metric. The formulation in the special space of functions with bounded variations is given and, in order to get an energy more adapted for numerics, a result of Γconvergence is proved.
We present a variational model to perform the segmentation of thin structures in MRI images (namely codimension 1 objects). It is based on the classical Mumford-Shah functional and we have added geometrical priors as constraints. We precisely describe the structure model (that we call tubes) and write the problem as a bilevel problem. We focus on the lower level optimization problem and give existence, uniqueness and regularity results for the solution. The keypoint is the fact that 2D/3D problems are equivalent to 1D ones. This gives hints to perform an automatic parameter tuning for numerical purpose.
This work is a contribution to the problem of detection of thin structures, namely tubes, in a 2D or 3D image. We introduce a variational bimodal model for the case where the histogram of the image has two main modes. This model involves an energy functional that depends on a function and a Riemannian metric. One of the terms of this energy is the anisotropic perimeter associated to the dual metric. We perform an approximation of this functional and prove that it Γ-converges to the original one.
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