The nonparametric multiscale platelet algorithms presented in this paper, unlike traditional wavelet-based methods, are both well suited to photon-limited medical imaging applications involving Poisson data and capable of better approximating edge contours. This paper introduces platelets, localized functions at various scales, locations, and orientations that produce piecewise linear image approximations, and a new multiscale image decomposition based on these functions. Platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries. For smoothness measured in certain H older classes, it is shown that the error of m-term platelet approximations can decay significantly faster than that of m-term approximations in terms of sinusoids, wavelets, or wedgelets. This suggests that platelets may outperform existing techniques for image denoising and reconstruction. Fast, platelet-based, maximum penalized likelihood methods for photon-limited image denoising, deblurring and tomographic reconstruction problems are developed. Because platelet decompositions of Poisson distributed images are tractable and computationally efficient, existing image reconstruction methods based on expectation-maximization type algorithms can be easily enhanced with platelet techniques. Experimental results suggest that platelet-based methods can outperform standard reconstruction methods currently in use in confocal microscopy, image restoration, and emission tomography.

This paper introduces a new multiscale method for nonparametric piecewise polynomial intensity and density estimation of point processes. Fast, piecewise polynomial, maximum penalized likelihood methods for intensity and density estimation are developed. The recursive partitioning scheme underlying these methods is based on multiscale likelihood factorizations which, unlike conventional wavelet decompositions, are very well suited to applications with point process data. Experimental results demonstrate that multiscale methods can outperform wavelet and kernel based density estimation methods. 1 Point Process Estimation In a variety of applications, data is acquired by observing the times or locations at which events occur. Frequently, either by choice or due to limitations of the measuring device, events are only observed on discrete intervals, resulting in a discrete signal representing the number of events occurring within each interval. Examples include the arrival of photons at a detector or observations of a random variable; we can estimate the intensity of photon emission or the probability density function by modeling these processes as Poisson and multinomial processes, respectively. Wavelet-based methods are powerful tools for nonparametric signal denoising and have been used in ap-

The nonparametric multiscale polynomial and platelet algorithms presented in this thesis are powerful new tools for signal and image denoising and reconstruction. Unlike traditional waveletbased multiscale methods, these algorithms are both well suited to processing Poisson and multinomial data and capable of preserving image edges. At the heart of these new algorithms lie multiscale signal decompositions based on polynomials in one dimension and multiscale image decompositions based on platelets in two dimensions. This thesis introduces platelets, localized atoms at various locations, scales and orientations that can produce highly accurate, piecewise linear approximations to images consisting of smooth regions separated by smooth boundaries. Polynomialand platelet-based maximum penalized likelihood methods for signal and image analysis are both tractable and computationally efficient. Simulations establish the practical effectiveness of these algorithms in applications such as medical and astronomical, density estimation, and networking

Given observations of a one-dimensional piecewise linear, length-M Poisson intensity function, our goal is to estimate both the partition points and the parameters of each segment. In order to determine where the breaks lie, we develop a maximum penalized likelihood estimator (MPLE) based on information-theoretic complexity penalization. We construct a probabilistic model of the observations within a multiscale framework, and use this framework to devise a computationally ecient optimization algorithm, based on a tree-pruning approach, to compute the MPLE.

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.