UC San Diego

Template matching, as an important topic in image processing, is the process of matching a clean and noiseless template to an observed, typically noisy signal. This topic is closely related to the problem of matching two or more noisy signals, sometimes referred to as 'aligning' or 'registering' the signals, and to methodology in spatial statistics falling under the umbrella name of 'scan statistic'. When the template has a point of discontinuity, matching a template to the signal can be interpreted as detecting the location of the discontinuity, a more specialized task more broadly referred to as 'change-point detection' in statistics. In Chapter 2, we study a standard mathematical model for matching a template to a noisy signal by M-estimation. While the most popular method may still be based on maximizing the (Pearson) correlation, the estimators we study can be made much more robust to heavy-tailed noise or the presence of outlying observations. We draw on standard empirical process theory and decision theory, to derive limit distributions and minimax convergence rates of M-estimators in a wide array of situations. In Chapter 3, we suggest a rank-based method for matching a template to a noisy signal, and study its asymptotic properties using some well-established techniques in empirical process theory combined with Hajek's projection method. The resulting estimator of the shift is shown to achieve a parametric rate of convergence and to be asymptotically normal.

Texture segmentation, as another crucial role in image processing, is the process of partitioning the image into differently textured regions. In Chapter 4, we present our related work on texture segmentation. We provide some theoretical guarantees for the prototypical approach which consists in extracting local features in the neighborhood of a pixel and then applying a clustering algorithm for grouping the pixel according to these features. The proposed algorithms work on both stationary and non-stationary random fields.