Approximate inverse and Sobolev estimates for the attenuated Radon transform

Abstract

The ill-posedness of the attenuated Radon transform is a challenging issue in practice due to the Poisson noise and the high level of attenuation. The investigation of the smoothing properties of the underlying operator is essential for developing a stable inversion. In this paper, we consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm based on the method of the approximate inverse. The derived method inherits the efficiency and stability of the approximate inverse and supplies a method of extraction of contours. These algorithms appear to be efficient for an attenuation of human body type. However, for higher attenuations the ill-posedness increases exponentially what deteriorates accordingly the quality of reconstructions. Nevertheless, a high attenuation map affects less the contour extraction of a high contrast function and so can be neglected. This leads to simplifying the proposed method and circumvents in this case the artifacts due to the attenuation as attested by simulation results.

On inversion and Sobolev estimate results about the weighted Radon transform

“…Remark 3.6. Theorem 3.5 generalizes the conclusion of Theorem 4.2 in the study of Rigaud and Lakhal 16 to the n-dimensional space.…”

“…Remark Theorem generalizes the conclusion of Theorem 4.1 in the study of Rigaud and Lakhal to the dual operator of n ‐dimensional weighted Radon transform.…”

“…Although the reconstruction formulae in the absence of absorption and for constant absorption have been discussed for several decades, the inversion of the general attenuated Radon transform in the case of arbitrary absorption has been obtained recently; see the study of Novikov for the derivation of the formula and the study of Arbuzov, Bukhgeim, and Kazantsev for a different approach. Rigaud and Lakhal accomplished a reconstruction algorithm of the attenuated Radon transform, which was based on the method of the approximate inverse. Other researchers also have made lots of related work and obtained meaningful results.…”

3D Compton scattering imaging: study of the spectrum and contour reconstruction

“…They produce smooth artifacts, noted Ef in the reconstruction scheme, quite strong. By analogy with the attenuated Radon transform [32], the ill-conditioning of the reconstruction problem should increase exponentially with the intensity of the electron density which is observed in practice. The values of the electron density considered here as well as the variation of the physical factors are thus substantially amplified by the reconstruction strategy.…”

Superposition, reduction of multivariable problems, and approximation

“…Recall that the Radon transform (see e.g., [20]) is an integral transform taking a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. Below we need a higher dimensional variant (see Lemma 5.1) of this idea, and we shall refer to it also as a Radon transform.…”