Bernadette Hahn-Rigaud

Abstract

Building on the well-known total variation, this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be adaptive (we speak of A-NID), i.e., the regularization varies during the iterates in order to incorporate prior information on the edges, deal with the evolution of the reconstruction and circumvent the limitations due to the nonconvexity of the proposed functionals. After a detailed analysis of the convergence and well-posedness of the method, the latter is validated by simulations performed on synthetic and real data on computerized tomography.

Abstract

The problem of dynamic 2D vector tomography is considered. Object motion is a combination of rotation and shifting. Properties of the dynamic ray transform operators are investigated. Singular value decomposition of the operators is constructed with usage of classic orthogonal polynomials.

Abstract

The classic regularization theory for solving inverse problems is built on the assumption that the forward operator perfectly represents the underlying physical model of the data acquisition. However, in many applications, for instance in microscopy or magnetic particle imaging, this is not the case. Another important example represent dynamic inverse problems, where changes of the searched-for quantity during data collection can be interpreted as model uncertainties. In this article, we propose a regularization strategy for linear inverse problems with inexact forward operator based on sequential subspace optimization methods (SESOP). In order to account for local modelling errors, we suggest to combine SESOP with the Kaczmarz’ method. We study convergence and regularization properties of the proposed method and discuss several practical realizations. Relevance and performance of our approach are evaluated at simulated data from dynamic computerized tomography with various dynamic scenarios.

Abstract

In this presentation we address the dynamic concentration reconstruction problem in magnetic particle imaging. The tradeoff between the signal-to-noise ratio in the processed data used for the reconstruction and the concentration dynamics of the tracer requires sophisticated reconstruction techniques to avoid motion artifacts. We incorporate motion priors in the concentration reconstruction and as an extra we can obtain an estimate for a flow field.

Abstract

This article deals with the problem of recovering an image from tomographic data corrupted by the motion of an object. Using generalized Radon transforms as modeling for the dynamic imaging operators, we present a novel approach for the reconstruction process. The proposed method, motivated by results on microlocal analysis, approximates the solution via a filtered backprojection-type algorithm which has the advantage of being quick to implement. The numerical results applied on dynamic photoacoustic tomography (PAT) data corroborate the efficiency and relevance of the proposed method.

Abstract

Compton scattering imaging is a nascent concept arising from the current development of high-sensitive energy detectors and is devoted to exploit the scattering radiation to image the electron density of the studied medium. Such detectors are able to collect incoming photons in terms of energy. This paper introduces potential 3D modalities in Compton scattering imaging (CSI). The associated measured data are modeled using a class of generalized Radon transforms. The study of this class of operators leads to build a filtered backprojection kind algorithm preserving the contours of the sought-for function and offering a fast approach to partially solve the associated inverse problems. Simulation results including Poisson noise demonstrate the potential of this new imaging concept as well as the proposed image reconstruction approach.

Abstract

A main challenge in computerized tomography consists in imaging moving objects. Temporal changes during the measuring process lead to inconsistent data sets, and applying standard reconstruction techniques causes motion artefacts which can severely impose a reliable diagnostics. Therefore, novel reconstruction techniques are required which compensate for the dynamic behavior. This article builds on recent results from a microlocal analysis of the dynamic setting, which enable us to formulate efficient analytic motion compensation algorithms for contour extraction. Since these methods require information about the dynamic behavior, we further introduce a motion estimation approach which determines parameters of affine and certain non-affine deformations directly from measured motion-corrupted Radon-data. Our methods are illustrated with numerical examples for both types of motion.

Abstract

Movements of the object during the data collection in computerized tomography can introduce motion artefacts in the reconstructed image. They can be reduced by employing information about the dynamic behaviour within the reconstruction step. However, inaccuracies concerning the movement are inevitable in practice. In this article, we give an explicit characterization of what is visible in an image obtained by a reconstruction algorithm with incorrect motion information. Then, we use this result to study in detail the situation of locally deforming objects, i.e. individual parts of the object have a different dynamic behaviour. In this context, we prove that additional artefacts arise due to the global nature of the Radon transform, even if the motion is exactly known. Based on our analysis, we propose a numerical scheme to reduce these artefacts in the reconstructed image. All our results are illustrated by numerical examples.

Abstract

In this paper, we use microlocal analysis to understand what X-ray tomographic data acquisition does to singularities of an object which changes during the measuring process. Depending on the motion model, we study which singularities are detected by the measured data. In particular, this analysis shows that, due to the dynamic behavior, not all singularities might be detected, even if the radiation source performs a complete turn around the object. Thus, they cannot be expected to be (stably) visible in any reconstruction. On the other hand, singularities could be added (or masked) as well. To understand this precisely, we provide a characterization of visible and added singularities by analyzing the microlocal properties of the forward and reconstruction operators. We illustrate the characterization using numerical examples.

Abstract

One major challenge in computerized tomography is to image objects which change during the data acquisition and hence lead to inconsistent data sets. Motion artefacts in the reconstructions can be reduced by applying specially adapted algorithms which take the dynamic behaviour into account. Within this article, we analyse the achievable resolution in the dynamic setting in case of two-dimensional affine deformations. To this end, we characterize the null space of the operator describing the dynamic case, using its singular value decomposition and a necessary dynamic consistency condition. This shows that independent of any reconstruction method, the specimen's dynamics results in a loss of resolution compared to the stationary setting. Our theoretical results are illustrated at a numerical example.

Abstract

One challenge for modern imaging methods is the investigation of objects which change during the data acquisition. This occurs in non-destructive testing as well as in medical applications, e.g. on account of patient or organ movements. Due to the object's deformations, the respective imaging modality is described by a dynamic inverse problem. In this paper, a classification scheme for linear dynamic problems depending on the object's motion is provided. Based on this scheme, we study the class in detail where the dynamic problem is still related to the operator in the static case, and where we call the deformations moderate. We proof important properties of the dynamic operator, derive a singular value decomposition and develop suitable regularization methods. The application of these methods to specific problems is illustrated at two examples including dynamic computerized tomography.

Abstract

In this paper, we propose a new approach for the atmospheric tomography based on the method of the approximate inverse. The image quality of Earth-bound telescopes is severely degraded by turbulences of the atmosphere of the Earth. In order to receive sharp images, the incoming light is corrected for these distortions using deformable mirrors. In atmospheric tomography, the turbulence profile is reconstructed so that the shape of the mirrors can be adjusted in an optimal way. We perform this reconstruction step by applying the method of the approximate inverse, a non-iterative regularization method which requires the solution of an adjoint problem, to Multi-Conjugate Adaptive Optics. We show that the approximate inverse leads to efficient algorithms and give numerical examples. The adjoint problem, which can be precomputed, is solved with an iterative Kaczmarz-type algorithm. In this first study of the new algorithm, tip/tilt effects and spot elongation are not included.

Abstract

The data acquisition in computerized tomography takes a certain amount of time since the x-ray source has to be rotated around the specimen. An object that changes during the scanning causes inconsistent data sets. To avoid the motion artefacts in reconstructions, the algorithm has to take the dynamic behavior of the specimen into account. In this context, some a priori information about the movement is required. A reconstruction method is proposed that compensates for the motion with a special focus on affine deformations. It also permits the combination of reconstruction and image analysis tools to extract features of the object without motion artefacts. The algorithm is validated with a numerical example from medical imaging.

Abstract

An inverse problem is called dynamic if the object changes during the data acquisition process. This occurs e.g. in medical applications when fast moving organs like the lungs or the heart are imaged. Most regularization methods are based on the assumption that the object is static during the measuring procedure. Hence, their application in the dynamic case often leads to serious motion artefacts in the reconstruction. Therefore, an algorithm has to take into account the temporal changes of the investigated object. In this paper, a reconstruction method that compensates for the motion of the object is derived for dynamic linear inverse problems. The algorithm is validated at numerical examples from computerized tomography.

Abstract

The result of tomographic examination is a series of two-dimensional (2D) or three-dimensional (3D) images, on which a diagnosis is based. Automatic evaluations of these images are rather common in nondestructive testing, in medical analysis this may partially be the case in the future. Typically the two tasks, image reconstruction and image evaluation, are treated separately. This paper describes an approach where the two steps are combined in just one method. By joint optimization of the two steps, the results are much superior to the separate treatment of the two tasks, as the comparisons in Louis (2008 SIAM J. Imaging Sci. 1 188–208) show. By constructing corresponding reconstruction filters, the algorithms are of filtered backprojection type, hence the computing time essentially remains the same as for the reconstruction of the density itself. We consider the reconstruction problem in x-ray tomography for the fan-beam geometry with flat detectors and edge detection. This method is then extended to filtered backprojection with Feldkamp-type kernels for the 3D cone-beam case. We calculate special reconstruction kernels which work also in the case of very noisy real data, and we present numerical examples from the measured data.

Abstract

Common reconstruction approaches in the three-dimensional parallel scanning geometry, that occurs for example in synchrotron-based x-ray tomography, use layerwise 2D procedures. A finite number of layers of the investigated object are reconstructed by a 2D method, before a 3D visualization is obtained by interpolation between these layers. In this paper, a fully three-dimensional reconstruction algorithm for the parallel scanning geometry is presented. It allows us to calculate arbitrary slices or even the complete 3D density function directly without causing any interpolation errors. Furthermore, the 3D method enables the fast and stable combination of reconstruction and image analysis step to extract features of the investigated object, for example, derivatives used in canny edge detectors. The application to real data illustrates the stability and advantageous regularization properties of the new approach.