An affine invariant tensor dissimilarity measure and its applications to tensor-valued image segmentation

Tensor fields specifically, matrix valued data sets, have recently attracted increased attention in the fields of image processing, computer vision, visualization and medical imaging. In this paper, we present a novel definition of tensor "distance" grounded in concepts from information theory and incorporate it in the segmentation of tensor-valued images. In some applications, a symmetric positive definite (SPD) tensor at each point of a tensor valued image can be interpreted as the covariance matrix of a local Gaussian distribution. Thus, a natural measure of dissimilarity between SPD tensors would be the KL divergence or its relative. We propose the square root of the J-divergence (symmetrized KL) between two Gaussian distributions corresponding to the tensors being compared that leads to a novel closed form expression. Unlike the traditional Frobenius norm-based tensor distance, our "distance" is affine invariant, a desirable property in many applications. We then incorporate this new tensor "distance" in a region based active contour model for bimodal tensor field segmentation and show its application to the segmentation of diffusion tensor magnetic resonance images (DT-MRI) as well as for the texture segmentation problem in computer vision. Synthetic and real data experiments are shown to depict the performance of the proposed model.