Fast and Robust Biomedical Image Reconstruction from Nonuniform Samples

Abstract

We consider the problem of reconstructing images from non-uniformly under-sampled spatial point measurements with emphasis on robustness to noise. The computational methods that deals with this problem are known as scattered data approximation (SDA) methods. Among these, well-performing methods achieve the reconstruction by minimizing a cost that is a weighted sum of data fidelity term measuring the accuracy of the fit at the measurement locations, and a regularization term. The latter term incorporates certain smoothness, and is constructed by summing the squared derivative values of a chosen order. The relative weight between these two terms is known as the smoothing parameter. Prominent methods in this category are known as thin-plate spline (TPS) and radial basis function (RBF) methods, and they require solving large numerically ill-conditioned and/or dense linear system of equations. Subspace variational method alleviates the numerical instability and the computational complexity associated with the TPS and RBF methods. However, this approach involves solving large and sparse linear system of equation requiring specialized numerical methods. In the first part of the thesis, we propose a novel method for SDA that eliminates the need for solving dense linear system of equations, and even the need for storing matrix representing linear system. This is achieved by handling the reconstruction problem in two stages. In the first stage, the given non-uniform data are transformed into a pair of regular grid images, where, one image represents the measured samples and the other represents the sample density map. In the second stage, the required image is computed as the minimizer of a cost that is completely expressed in terms of regular grid discrete operations. It is expressed as a sum of weighted quadratic data fitting term involving the transformed image pair, and and discrete quadratic roughness functional...