Convex Multilabel Segmentation with Shearlet Regularization

Segmentation plays an important role in many preprocessing stages in image processing. Recently, convex relaxation methods for image multi-labeling were proposed in the literature. Often these models involve the total variation (TV) semi-norm as regularizing term. However, it is well-known that the TV functional is not optimal for the segmentation of textured regions. In recent years directional representation systems were proposed to cope with curved singularities in images. In particular, curvelets and shearlets provide an optimally sparse approximation in the class of piecewise smooth functions with C singularity boundaries. In this paper, we show that the discrete shearlet transform is suited as regularizer for the segmentation of curved structures. To this end, we have implemented a translation invariant finite discrete shearlet transform based on the FFT. We describe how the shearlet transform can be incorporated within the multi-label segmentation model and show how to find a minimizer of the corresponding functional by applying an alternating direction method of multipliers. Here the Parseval frame property of our shearlets come into the play. We demonstrate by numerical example that the shearlet regularized model can better segment curved textures than the TV regularized one and that the method can also cope with regularizers obtained from nonlocal means.