Image edge detection is one of the most important techniques in digital image processing. It is used, among other things, as the first step of image segmentation. Therefore, it remains an area of interest for researchers trying to develop ever-better detection approaches. The main objective of this Thesis is to find a suitable method for image edge detection using convex optimisation. The proposed method is based on sparse modelling, and its main part is formulated as a convex optimisation problem solved by proximal algorithms. For defining the optimisation problem, it is assumed that the signal can be modelled as an over-parametrised, piecewise-polynomial signal that consists of disjoint segments. The number of these segments is significantly smaller than the number of signal samples, which encourages the use of sparsity. The formulation of a suitable optimisation problem is first performed on one-dimensional signals since the implementation and comparison of the different algorithms is significantly easier and less time-consuming for one-dimensional signals than two-dimensional ones.

The first part of the Thesis introduces the basic theory in signal processing, sparsity, convex optimisation and proximal algorithms. It also presents a cross-section of the methods used for image edge detection. The second part of the Thesis focuses on the formulation and the subsequent evaluation of individual optimisation problems for the segmentation of one-dimensional synthetic signals corrupted by noise. The evaluation is conducted in terms of both denoising and breakpoint detection accuracy. The last part of the Thesis is dedicated to expanding the best-performing approach for breakpoint detection in one-dimensional signals for the application to image edge detection. The proposed approach is tested on a standardised dataset of images containing manually labelled edges of several subjects. The results of the proposed method are evaluated using precision-recall curves and their maximum F-measure score, and then compared with other edge detection methods.