Inversion of noisy Radon transform by SVD based needlet

A linear method for inverting noisy observations of the Radon transform is developed based on decomposition systems (needlets) with rapidly decaying elements induced by the Radon transform SVD basis. Upper bounds of the risk of the estimator are established in $L^p$ ($1\le p\le \infty$) norms for functions with Besov space smoothness. A practical implementation of the method is given and several examples are discussed

Quantum Tomography

This is the draft version of a review paper which is going to appear in "Advances in Imaging and Electron Physics"Comment: To appear in "Advances in Imaging and Electron Physics". Some figs with low resolutio

Radon needlet thresholding

We provide a new algorithm for the treatment of the noisy inversion of the Radon transform using an appropriate thresholding technique adapted to a well-chosen new localized basis. We establish minimax results and prove their optimality. In particular, we prove that the procedures provided here are able to attain minimax bounds for any $\mathbb {L}_p$ loss. It s important to notice that most of the minimax bounds obtained here are new to our knowledge. It is also important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover and to inhomogeneous smoothness. We perform a numerical study that is of importance since we especially have to discuss the cubature problems and propose an averaging procedure that is mostly in the spirit of the cycle spinning performed for periodic signals

Image Reconstruction from Truncated Data in Single-Photon Emission Computed Tomomgraphy with Uniform Attenuation

International audienceWe present a mathematical analysis of the problem of image reconstruction from truncated data in two-dimensional (2D) single-photon emission computed tomography (SPECT). Recent results in classical tomography have shown that accurate reconstruction of some parts of the object is possible in the presence of truncation. We have investigated how these results extend to 2D parallel-beam SPECT, assuming that the attenuation map is known and constant in a convex region $\Omega$ that includes all activity sources. Our main result is a proof that, just like in classical tomography accurate SPECT reconstruction at a given location x ∈ $\Omega$,does not require the data on all lines passing through $\Omega$; some amount of truncation can be tolerated. Experimental reconstruction results based on computer-simulated data are given in support of the theory

Single-Scattering Optical Tomography

We consider the problem of optical tomographic imaging in the mesoscopic regime where the photon mean free path is of order of the system size. Within the accuracy of the single-scattering approximation to the radiative transport equation, we show that it is possible to recover the extinction coefficient of an inhomogeneous medium from angularly-resolved measurements. Applications to biomedical imaging are described and illustrated with numerical simulations.Comment: Finalized and submitted to PR