The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinite-dimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via so-called generalized sampling (GS). Second, the extension of generalized sampling to inverse and ill-posed probl...

Data saving capability of “Compressed sensing (sampling)” in signal discretization is disputed and found to be far below the theoretical upper bound defined by the signal sparsity. On a simple and intuitive example, it is demonstrated that, in a realistic scenario for signals that are believed to be sparse, one can achieve a substantially larger saving than compressing sensing can. It is also s...

Compressed sensing technique is a recent framework for signal sampling and recovery. It allows signal acquisition with less sampling than required by Nyquist-Shannon theorem and reduces data acquisition time in MRI. When the sampling rate is low, prior knowledge is essential to reconstruct the missing features. In this paper, a different reconstruction method is proposed by using the principal ...

We show that a discrete version of Tchakaloff’s theorem on the existence of positive algebraic cubature formulas, entails that the information required for multivariate polynomial approximation can be suitably compressed. 2000 AMS subject classification: 41A10, 65D32.

Recently, a segmented AIC (S-AIC) structure that measures the analog signal by K parallel branches of mixers and integrators (BMIs) was proposed by Taheri and Vorobyov (2011). Each branch is characterized by a random sampling waveform and implements integration in several continuous and non-overlapping time segments. By permuting the subsamples collected by each segment at different BMIs, more ...

The recent theory of Compressed Sensing (Candès, Tao & Romberg, 2006, and Donoho, 2006) states that a signal, e.g. a sound record or an astronomical image, can be sampled at a rate much smaller than what is commonly prescribed by Shannon-Nyquist. The sampling of a signal can indeed be performed as a function of its “intrinsic dimension” rather than according to its cutoff frequency. This chapte...

The sampling patterns, cost functions, and reconstruction algorithms play important roles in optimizing compressed sensing magnetic resonance imaging (CS-MRI). Simple random sampling patterns did not take into account the energy distribution in k-space and resulted in suboptimal reconstruction of MR images. Therefore, a variety of variable density (VD) based samplings patterns had been develope...

There are two main approaches in compressed sensing: the geometric approach and the combinatorial approach. In this paper we introduce an information theoretic approach and use results from the theory of Huffman codes to construct a sequence of binary sampling vectors to determine a sparse signal. Unlike other approaches, our approach is adaptive in the sense that each sampling vector depends o...

We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finite-dimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finite-dimensional techniques are ill-suited for solving a number of import...