  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
$C^2$ singularity boundaries. In this paper, we demonstrate that the discrete
shearlet transform is suited as regularizer for the segmentation of curved
structures. Neither the shearlet nor the curvelet transform where used as
regularizer in a segmentation model so far. 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 comes into play. We demonstrate by
numerical examples 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 non-local means.
