Convex Multiclass Segmentation with Shearlet Regularization

TL;DR

This paper introduces a novel shearlet-based regularization method for convex multi-label image segmentation, effectively capturing curved textures and structures, outperforming traditional total variation approaches.

Contribution

The paper pioneers the use of shearlet transforms as regularizers in segmentation models, providing a new approach for better curved structure segmentation.

Findings

Shearlet regularization outperforms TV in segmenting curved textures.

The proposed method effectively incorporates shearlet transforms via FFT and ADMM.

Numerical examples demonstrate improved segmentation of curved structures.

Abstract

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.