This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden

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.

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Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set Theta, sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconvolutions with kernels having a Fourier transform which vanishes at high frequencies are examples of unstable inverse problems, where a thresholding in a wavelet basis is a suboptimal estimator. A new "mirror wavelet" basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals. This thresholding estimator is used to restore blurred satellite images.

The nonparametric Poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness. The methods and theory presented here share many of the desirable features associated with waveletbased estimators: computational speed, spatial adaptivity, and the capability of detecting discontinuities and singularities with high resolution. Unlike traditional wavelet-based approaches, which impose an upper bound on the degree of smoothness to which they can adapt, the estimators studied here guarantee non-negativity and do not require any a priori knowledge of the underlying signal’s smoothness to guarantee near-optimal performance. At the heart of these methods lie multiscale decompositions based on free-knot, free-degree piecewise-polynomial functions and penalized likelihood estimation. The degrees as well as the locations of the polynomial pieces can be adapted to the observed data, resulting in near minimax optimal convergence rates. For piecewise analytic signals, in particular, the error of this estimator converges at nearly the parametric rate. These methods can be further refined in two dimensions, and it is demonstrated that platelet-based estimators in two dimensions exhibit similar near-optimal error convergence rates for images consisting of smooth surfaces separated by smooth boundaries.

The inversion of the Radon transform is a classical ill-posed inverse problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A well-known consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affine-like system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides near-optimal rate of convergence in estimating a large class of images from noisy Radon data. This is achieved by introducing a shearlet-based decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level ɛ, the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentially optimal mean square error O(log(ɛ −1)ɛ 4/5), as ɛ → 0. Several numerical demonstrations show that its performance improves upon similar competitive strategies based on wavelets and curvelets. Key words: directional wavelets; inverse problems, Radon transform shearlets; wavelets

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

Abstract. Reconstructing low-dose computed tomography (CT) images is an unstable inverse problem, due to the presence of noise. To address this problem, we propose a new regularized reconstruction method that combines features from the Filtered Back-Projection (FBP) algorithm and regularization theory. The filtering part of FBP comprises Fourier-domain inversion followed by noise suppression based on thresholding procedure in complex wavelet domain. The proposed method exploits the properties of dual tree complex wavelet transform (DT-CWT) to remove blurring and noise without the need for assuming a specific noise model. Furthermore, it uses an adaptive shrinkage function based on median, mean and standard deviation of wavelet coefficients to suppress noise while preserving the sharpness of the reconstructed image. The efficacy of the proposed method was assessed with projections simulated from Shepp-Logan Phantom. Simulation results confirm that the proposed method produces consistently good reconstruction in terms of suppressing noise and preserving resolution in the reconstructed images.

iterative thresholding algorithm for linear inverse

iterative thresholding algorithm for linear inverse problems with a