# Variational Method for 3D Image Segmentation

Maïtine Bergounioux and David Vicente*

We deal with 3D MRI images of mouse vascular network. These images involve thin structures that can be modeled as very small section tubes. We propose a Mumford-Shah like variational model for segmentation. We introduce an approximate model and prove it $\Gamma$-converges to the original one. Then, we develop a strategy for tuning parameters with respect to the tube size: this allows us to detect structures which size (volume/area) above (or below) a fixed threshold. Finally, we present numerical simulations to illustrate the method.

Mathematics Subject Classification: 49K20

Keywords: Mumford-Shah functional ; variational method; $\Gamma$-converge; 3D image; tube

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