Infinite-Dimensional Measure Spaces and Frame Analysis

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ℋ, we study three cases of measures. We first show that, for ℋ infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ℋ. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures

CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

This paper proposes a spatial-Radon domain CT image reconstruction model based on data-driven tight frames (SRD-DDTF). The proposed SRD-DDTF model combines the idea of joint image and Radon domain inpainting model of \cite{Dong2013X} and that of the data-driven tight frames for image denoising \cite{cai2014data}. It is different from existing models in that both CT image and its corresponding high quality projection image are reconstructed simultaneously using sparsity priors by tight frames that are adaptively learned from the data to provide optimal sparse approximations. An alternative minimization algorithm is designed to solve the proposed model which is nonsmooth and nonconvex. Convergence analysis of the algorithm is provided. Numerical experiments showed that the SRD-DDTF model is superior to the model by \cite{Dong2013X} especially in recovering some subtle structures in the images

A new sparse representation framework for compressed sensing MRI

Abstract(#br)Compressed sensing based Magnetic Resonance imaging (MRI) via sparse representation (or transform) has recently attracted broad interest. The tight frame (TF)-based sparse representation is a promising approach in compressed sensing MRI. However, the conventional TF-based sparse representation is difficult to utilize the sparsity of the whole image. Since the whole image usually has different structure textures and a kind of tight frame can only represent a particular kind of ground object, how to reconstruct high-quality of magnetic resonance (MR) image is a challenge. In this work, we propose a new sparse representation framework, which fuses the double tight frame (DTF) into the mixed-norm regularization for MR image reconstruction from undersampled k -space data. In this framework, MR image is decomposed into smooth and nonsmooth regions. For the smooth regions, the wavelet TF-based weighted L 1 -norm regularization is developed to reconstruct piecewise-smooth information of image. For nonsmooth regions, we introduce the curvelet TF-based robust L 1 , a -norm regularization with the parameter to preserve the edge structural details and texture. To estimate the reasonable parameter, an adaptive parameter selection scheme is designed in robust L 1 , a -norm regularization. Experimental results demonstrate that the proposed method can achieve the best image reconstruction results when compared with other existing methods in terms of quantitative metrics and visual effect

Data-Driven Multi-scale Non-local Wavelet Frame Construction and Image Recovery

10.1007/s10915-014-9893-2Journal of Scientific Computing632307-32