Multiscale Methods for Statistical Inference on Regular Lattice Data

Abstract

This thesis presents methods for multiscale statistical inference on random fields on a regular two-dimensional lattice. There are two distinct concepts of scale that are used in the thesis. The first one is connected to the computer vision community's understanding of scale-space as a family of smooths of a digital image, with fine structure being revealed at low levels of smoothing and the coarser structures standing out at high levels. The second way scale enters is through use of the two-dimensional wavelet transform, and the different scales can be though of as providing information on the energy content in different frequency bands. All the methods that are developed herein are in the form of statistical hypothesis tests. Paper I uses a Bayesian framework, and statistically significant gradient and curvature is determined through thresholding their posterior probabilities. The last two papers use standard frequentist methods, in Paper II for determining if a random field is isotropic, meaning that the second order statistical properties are independent of direction. Paper III addresses the problem of determining if two samples from a random field are realizations from the same underlying random field. The methods are demonstrated in practical examples, for Paper I on dermascopic images of skin lesions where it is used to detect hairs and dots/globules, potentially important components of a system for evaluating the risk of melanoma. In Paper II the test for isotropy is applied to paper density images of handsheets, useful for determining the quality of the paper. Finally, the methodology from Paper III is applied on small projections of the cosmic microwave background (CMB) temperature map from the Planck mission, and offers an alternative test for the overall isotropy of the CMB.