Bernadette N. Hahn

Efficient algorithms for linear dynamic inverse problems with known motion

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.

Reconstruction in the three-dimensional parallel scanning geometry with application in synchrotron-based x-ray tomography

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.

Detectable Singularities from Dynamic Radon Data

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.

Reconstruction of dynamic objects with affine deformations in computerized tomography

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.

Null space and resolution in dynamic computerized tomography

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 specimens dynamics results in a loss of resolution compared to the stationary setting. Our theoretical results are illustrated at a numerical example.

Combined reconstruction and edge detection in dimensioning

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.