Dimensionality reduction of curvelet sparse regularizations in limited angle tomography.

We investigate the reconstruction problem for limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well in such situations. To stabilize the inversion we propose the use of a sparse regularization technique in combination with curvelets. We argue that this technique has the ability to preserve edges. As our main result, we present a characterization of the kernel of the limited angle Radon transform in terms of curvelets. Moreover, we characterize reconstructions which are obtained via curvelet sparse regularizations at a limited angular range. As a result, we show that the dimension of the limited angle problem can be significantly reduced in the curvelet domain

Reconstructions in limited angle X-ray tomography: Characterization of classical reconstructions and adapted curvelet sparse regularization.

This thesis is devoted to the problem of tomographic reconstruction at limited angular range. In the first part, we prove a characterization of filtered backprojection reconstructions from limited angle data. Moreover, we develop a strategy for artifact reduction and stabilization. In the second part, we introduce a new edge-preserving reconstruction algorithm for limited angle tomography and analyze this algorithm mathematically. Some numerical experiments are also presented

How to characterize and decrease artifacts in limited angle tomography using microlocal analysis.

The filtered backprojection algorithm (FBP) in limited angle tomography reliably reconstructs only specific features of the original object and creates additional artifacts in the reconstruction. While the former is well understood mathematically, the added artifacts have not been studied very much in the literature. In our paper Inverse Problems 29125007 we mathematically explain why additional artifacts are created by the FBP and Lambda-CT algorithms for a limited angular range, and we derive an artifact reduction strategy using microlocal analysis

Characterization and reduction of artifacts in limited angle tomography.

We consider the reconstruction problem for limited angle tomography using filtered backprojection (FBP) and lambda tomography. We use microlocal analysis to explain why the well-known streak artifacts are present at the end of the limited angular range. We explain how to mitigate the streaks and prove that our modified FBP and lambda operators are standard pseudodifferential operators, and so they do not add artifacts. We provide reconstructions to illustrate our mathematical results

Blind recursive estimation of SIMO channels

International audienceWe present a blind recursive algorithm for the estimation of nonstationary SIMO systems. An adaptive algorithm is developed to track the noise projector, to evaluate the degree of intersymbol interference (ISI), and the channel parameters. This method is also capable of handling abruptly changing channels during the observation interval. We define a new filtering operator that matches this projector with the subspace method. Finally, we relate this work with Capon's approach and the beamforming method. Simulation results concerning the performance of the developed method and its comparison with the classical subspace method are presented

Blind noise and channel estimation

International audienceIn the classical methods for blind channel identification (subspace method, TXK, XBM) (Moulines et al, 1995; Tong et al, 1994; Xavier et al, 1997), the additive noise is assumed to be spatially white or known to within a multiplicative scalar. When the noise is non-white (colored or correlated) but has a known covariance matrix, we can still handle the problem through prewhitening. However, there are no techniques presently available to deal with completely unknown noise fields. It is well known that when the noise covariance matrix is unknown, the channel parameters may be grossly inaccurate. In this paper, we assume the noise is spatially correlated, and we apply this assumption for blind channel identification. We estimate the noise covariance matrix without any assumption except its structure which is assumed to be a band-Toeplitz matrix. The performance evaluation of the developed method and its comparison to the modified subspace approach (MSS) (Abed-Meraim et al, 1997) are presented

X-ray computed tomography using curvelet sparse regularization.

Purpose: Reconstruction of x-ray computed tomography (CT) data remains a mathematically challenging problem in medical imaging. Complementing the standard analytical reconstruction methods, sparse regularization is growing in importance, as it allows inclusion of prior knowledge. The paper presents a method for sparse regularization based on the curvelet frame for the application to iterative reconstruction in x-ray computed tomography. Methods: In this work, the authors present an iterative reconstruction approach based on the alternating direction method of multipliers using curvelet sparse regularization. Results: Evaluation of the method is performed on a specifically crafted numerical phantom dataset to highlight the method’s strengths. Additional evaluation is performed on two real datasets from commercial scanners with different noise characteristics, a clinical bone sample acquired in a micro-CT and a human abdomen scanned in a diagnostic CT. The results clearly illustrate that curvelet sparse regularization has characteristic strengths. In particular, it improves the restoration and resolution of highly directional, high contrast features with smooth contrast variations. The authors also compare this approach to the popular technique of total variation and to traditional filtered backprojection. Conclusions: The authors conclude that curvelet sparse regularization is able to improve reconstruction quality by reducing noise while preserving highly directional features. &nbsp