19 pagesInternational audienceIn this article, we focus on tomographic reconstruction. The problem is to determine the shape of the interior interface using a tomographic approach while very few X-ray radiographs are performed. We use a multi-marginal optimal transport approach. Preliminary numerical results are presented

We deal with a severe ill posed problem, namely the reconstruction process of an image during tomography acquisition with (very) few views. We present different methods that we have been investigated during the past decade. They are based on variational analysis. This is a survey paper and we refer to the quoted papers for more details.Mathematics Subject Classification (2010). 49K40, 45Q05,65M32.

Abstract. This paper deals with a method of tomographic reconstruction of radially symmetric objects from a single radiograph, in order to study the behavior of shocked material. The usual tomographic reconstruction algorithms such as generalized inverse or filtered back-projection cannot be applied here because data are very noisy and the inverse problem associated to single view tomographic reconstruction is highly unstable. In order to improve the reconstruction, we propose here to add some a priori assumptions on the looked after object. One of these assumptions is that the object is binary and consequently, the object may be described by the curves that separate the two materials. We present a model that lives in BV space and leads to a non local Hamilton-Jacobi equation, via a level set strategy. Numerical experiments are performed (using level sets methods) on synthetic objects.1991 Mathematics Subject Classification. 68U10, 44A12, 49N45.The dates will be set by the publisher.

In this paper, we propose an edge detection technique based on some local smoothing of the image followed by a statistical hypothesis testing on the gradient. An edge point being defined as a zero-crossing of the Laplacian, it is said to be a significant edge point if the gradient at this point is larger than a threshold s(ε) defined by: if the image I is pure noise, then P( ∇I ≥ s(ε) ∆I = 0) ≤ ε. In other words, a significant edge is an edge which has a very low probability to be there because of noise. We will show that the threshold s(ε) can be explicitly computed in the case of a stationary Gaussian noise. In images we are interested in, which are obtained by tomographic reconstruction from a radiograph, this method fails since the Gaussian noise is not stationary anymore. But in this case again, we will be able to give the law of the gradient conditionally on the zero-crossing of the Laplacian, and thus compute the threshold s(ε). We will end this paper with some experiments and compare the results with the ones obtained with some other methods of edge detection.