Shape-adaptive DCT for denoising of 3D scalar and tensor valued images.

During the last ten years or so, diffusion tensor imaging has been used in both research and clinical medical applications. To construct the diffusion tensor images, a large set of direction sensitive magnetic resonance image (MRI) acquisitions are required. These acquisitions in general have a lower signal-to-noise ratio than conventional MRI acquisitions. In this paper, we discuss computationally effective algorithms for noise removal for diffusion tensor magnetic resonance imaging (DTI) using the framework of 3-dimensional shape-adaptive discrete cosine transform. We use local polynomial approximations for the selection of homogeneous regions in the DTI data. These regions are transformed to the frequency domain by a modified discrete cosine transform. In the frequency domain, the noise is removed by thresholding. We perform numerical experiments on 3D synthetical MRI and DTI data and real 3D DTI brain data from a healthy volunteer. The experiments indicate good performance compared to current state-of-the-art methods. The proposed method is well suited for parallelization and could thus dramatically improve the computation speed of denoising schemes for large scale 3D MRI and DTI.

Total variation regularization of matrix-valued images.

We generalize the total variation restoration model, introduced by Rudin, Osher, and Fatemi in 1992, to matrix-valued data, in particular, to diffusion tensor images (DTIs). Our model is a natural extension of the color total variation model proposed by Blomgren and Chan in 1998. We treat the diffusion matrix D implicitly as the product D = LL(T), and work with the elements of L as variables, instead of working directly on the elements of D. This ensures positive definiteness of the tensor during the regularization flow, which is essential when regularizing DTI. We perform numerical experiments on both synthetical data and 3D human brain DTI, and measure the quantitative behavior of the proposed model.

A binary level set model and some applications to Mumford-Shah image segmentation.

In this paper, we propose a PDE-based level set method. Traditionally, interfaces are represented by the zero level set of continuous level set functions. Instead, we let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e., it can only be 1 or -1; thus, our method is related to phase-field methods. Some of the properties of standard level set methods are preserved in the proposed method, while others are not. Using this new method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth. We show numerical results using the method for segmentation of digital images.