The reconstruction of the images obtained via the Cone Beam Computerized Tomography (CBCT) and Positron
Emission Tomography (PET) Scanners are ill-posed inverse problems. One needs to adress carefully the problem
of inversion of the mathematical operators involved. Recent advances in optimization have yielded efficient
algorithms to solve very general classes of inverse problems via the minimization of non-differentiable convex
functions. We show that such models are well suited to solve the CBCT and PET reconstruction problems. On
the one hand, they can incorporate directly the physics of new acquisition devices, free of dark noise; on the
other hand, they can take into account the specificity of the pure Poisson noise.
We propose various fast numerical schemes to recover the original data and compare them to state-of-the-art
algorithms on simulated data. We study more specifically how different contrasts and resolutions may be resolved
as the level of noise and/or the number of projections of the acquired sinograms decrease. We conclude that the
proposed algorithms compare favorably with respect to well-established methods in tomography.
In this paper, a general framework for the inversion of a linear operator in the case where one seeks several components
from several observations is presented. The estimation is done by minimizing a functional balancing discrepancy terms by
regularization terms. The regularization terms are adapted norms that enforce the desired properties of each component.
The main focus of this paper is the definition of the discrepancy terms. Classically, these are quadratic. We present
novel discrepancy terms adapt to the observations. They rely on adaptive projections that emphasize important information
in the observations. Iterative algorithms to minimize the functionals with adaptive discrepancy terms are derived and their
convergence and stability is studied.
The methods obtained are compared for the problem of reconstruction of astrophysical maps from multifrequency
observations of the Cosmic Microwave Background. We show the added flexibility provided by the adaptive discrepancy
terms.
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