Variational Reconstruction of Vector and Scalar Images from Non-Uniform Samples

We address the problem of reconstructing scalar and vector functions from non-uniform data. The reconstruction problem is formulated as a minimization problem where the cost is a weighted sum of two terms. The first data term is the quadratic measure of goodness of fit, whereas the second regularization term is a smoothness functional. We concentrate on the case where the later is a semi-norm involving differential operators. We are interested in a solution that is invariant with respect to scaling and rotation of the input data. We first show that this is achieved whenever the smoothness functional is both scaleand rotation-invariant. In the first part of the thesis, we address the scalar problem. An elegant solution having the above mentioned invariant properties is provided by Duchon’s method of thin-plate splines. Unfortunately, the solution involves radial basis functions that are poorly conditioned and becomes impractical when the number of samples is large. We propose a computationally efficient alternative where the minimization is carried out within the space of uniform B-splines. We show how the B-spline coefficients of the solution can be obtained by solving a well-conditioned, sparse linear system of equations. By taking advantage of the refinable nature of B-splines, we devise a fast multiresolution-multigrid algorithm. We demonstrate the effectiveness of this method in the context of image processing. Next, we consider the reconstruction of vector functions from projected samples, meaning that the input data do not contain the full vector values, but only some directional components. We first define the rotational invariance and the scale invariance of a vector smoothness functional, and then characterize the complete family of such functionals. We show that such a functional is composed of a weighted sum of two sub-functionals: (i) Duchon’s