  Intensity inhomogeneities in images constitute a considerable challenge in
image segmentation. In this paper we propose a novel biconvex variational model
to tackle this task. We combine a total variation approach for multi class
segmentation with a multiplicative model to handle the inhomogeneities. Our
method assumes that the image intensity is the product of a smoothly varying
part and a component which resembles important image structures such as edges.
Therefore, we penalize in addition to the total variation of the label
assignment matrix a quadratic difference term to cope with the smoothly varying
factor. A critical point of our biconvex functional is computed by a modified
proximal alternating linearized minimization method (PALM). We show that the
assumptions for the convergence of the algorithm are fulfilled by our model.
Various numerical examples demonstrate the very good performance of our method.
Particular attention is paid to the segmentation of 3D FIB tomographical images
which was indeed the motivation of our work.
