# A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images

### J. F. Garamendi, E. Schiavi

#### Abstract

We present a general model for multi-class segmentation of multi-channel digital images. It is based on the minimization of an anisotropic version of the Mumford-Shah energy functional in the class of piecewise constant functions. In the framework of geometric measure theory we use the concept of common interphases between regions (classes) and the value of the jump discontinuities of the (weak) solution between adjacent regions in order to define a minimal partition energy functional. The resulting problem is non-smooth and non-convex. Non-smoothness is dealt with highlighting the relationship of the proposed model with the well known Rudin, Osher and Fatemi model for image denoising when piecewise constant solutions (i.e partitions) are considered. Non-convexity is tackled with an optimal threshold of the ROF solution which we which generalize to multi-channel images through a probabilistic clustering. The optimal solution is then computed with a fixed point iteration. The resulting algorithm is described and results are presented showing the successful application of the method to Light Field (LF) images.

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#### Paper Citation

#### in Harvard Style

Garamendi J. and Schiavi E. (2017). **A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images** . In *Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 4: VISAPP, (VISIGRAPP 2017)* ISBN 978-989-758-225-7, pages 468-475. DOI: 10.5220/0006127804680475

#### in Bibtex Style

@conference{visapp17,

author={J. F. Garamendi and E. Schiavi},

title={A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images},

booktitle={Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 4: VISAPP, (VISIGRAPP 2017)},

year={2017},

pages={468-475},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006127804680475},

isbn={978-989-758-225-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 4: VISAPP, (VISIGRAPP 2017)

TI - A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images

SN - 978-989-758-225-7

AU - Garamendi J.

AU - Schiavi E.

PY - 2017

SP - 468

EP - 475

DO - 10.5220/0006127804680475