Statistical methods for image and signal processing
Philip Andrew Sallee, Bruno A. Olshausen · 2004
Statistical methods provide a principled means for solving many types of problems which require the estimation of missing or uncertain information. This dissertation discusses methods for adapting statistical models to images, sounds and other types of signals for applications in image and signal processing. Wavelets provide a multi-scale representation which has been shown to be well suited for describing many naturally occurring signals. These are typically designed by hand based on certain mathematical properties and may not achieve the best match to the data. We describe an approach for using an overcomplete wavelet framework as part of a generative statistical model with a sparse prior placed on the wavelet coefficients. The wavelet functions are adapted to a given dataset by maximizing the average log likelihood of the model. This is demonstrated for natural images, sounds, and EEG data. The learned representations are shown to have a higher degree of sparsity than other wavelet bases. This statistical framework also provides a principled approach for performing certain types of signal estimation, such as denoising, in terms of a statistical inference process. We explore two inference methods for the overcomplete wavelet models presented: A Gibbs sampling method, and a greedy optimization procedure known as matching pursuit. We also demonstrate how a statistical model may be applied to a form of secure information hiding, known as steganography, in which the objective is to hide information in an image or some other media so that it cannot be detected without a cryptographic “key”. This model-based approach provides a means for maximizing the capacity of stored information while obtaining provably secure steganography insofar as the model is accurate. Using this methodology, a steganography method is proposed for JPEG images which achieves higher embedding efficiency and message capacity than previous methods, while remaining secure against first order statistical attacks. Methods for applying statistical models for steganalysis, the art of detecting steganographic messages, are also presented.