Multiscale Likelihood Analysis and Image Reconstruction
Willett, Rebecca; Nowak, Robert David
The nonparametric multiscale polynomial and platelet methods presented here are powerful new tools for signal and image denoising and reconstruction. Unlike traditional wavelet-based multiscale methods, these methods are both well suited to processing Poisson or multinomial data and capable of preserving image edges. At the heart of these new methods lie multiscale signal decompositions based on polynomials in one dimension and multiscale image decompositions based on what the authors call platelets in two dimensions. Platelets are localized functions at various positions, scales and orientations that can produce highly accurate, piecewise linear approximations to images consisting of smooth regions separated by smooth boundaries. Polynomial and platelet-based maximum penalized likelihood methods for signal and image analysis are both tractable and computationally efficient. Polynomial methods offer near minimax convergence rates for broad classes of functions including Besov spaces. Upper bounds on the estimation error are derived using an information-theoretic risk bound based on squared Hellinger loss. Simulations establish the practical effectiveness of these methods in applications such as density estimation, medical imaging, and astronomy.