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Gestabiliseerde deconvolutiemethodes voor inverse problemen - met toepassing in lineaire (Magnetic Resonance beelvorming) en niet-lineaire (Ground Penetrating Radar beeldvorming) beeldreconstructie -: Final report project "Stabilized deconvolution methods for inverse problems – With application to linear (Magnetic Resonance imaging) and nonlinear (Ground Penetrating Radar imaging) image reconstruction –" Authors: B. Truyen Publication Date: Nov. 2004
Abstract: Inverse problems certainly are one of the most dominant research themes in contemporary applied mathematics, proof of this, the enormous increase in publications in recent years. So-called, discrete ill-posed problems typically arise in connection with the numerical treatment of inverse problems where one wants to compute information about some continuously distributed parameter using discrete measurements.
The numerical treatment of discrete ill-posed problems is such a huge field that it clearly exceeds the competence domain of a single research group. In this project, however, we will deal with one clearly confined subproblem that is to a significant degree unexplored, namely ill-posed linear and nonlinear deconvolution such as it frequently occurs in image reconstruction. This type of discrete ill-posed problems essentially boils down to that of repetitively solving systems of equations with a Toeplitz or Hankel structure, or some perturbation of it. Not only does this topic seem to be dealt with in a rather restrictive way in the classical literature, it also appears that there exists a clear gap in the exchange of concepts between the diverse application domains.
We will study this subject by means of two applications, namely Magnetic Resonance (MR) image reconstruction from signals sampled on a non-uniform grid in k-space - essentially a linear problem - and subsurface imaging using Ground Penetrating Radar (GPR) measurements, requiring the solution of a nonlinear inverse problem.Although both applications are drawn from completely different scientific domains, the two problem classes share many similar characteristics, such that it is only natural to deal with them in a combined study.
The setting of this research is that of numerical linear algebra rather than that of abstract functional analysis. In particular, emphasis will be on the analysis of some new and existing numerical algorithms, tailored specifically to the particularities of real-world image reconstruction. It is in this aspect that this research differs from previous work on stable inversion of linear and nonlinear operators. In particular, real-world image reconstruction problems are dominated by their highly ill-posed nature in combination with an intrinsic high dimensionality, and the non-regular spreading of the available discrete data.
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