Analysis for Inverse Problem
Inverse problems are ill-posed when measurable data are either insufficient for uniqueness or insensitive to perturbations of parameters to be imaged. To solve an inverse problem in a robust way, we should adopt a reasonably well-posed modified model at the expense of a reduced spatial resolution and/or add additional a priori information. Finding a well-posed model subject to practical constraints of measurable quantities requires deep knowledge about various mathematical theories in partial differential equations (PDEs) and functional analysis, including uniqueness, regularity, stability, layer potential techniques, micro-local analysis, regularization, spectral theory and others. In this chapter, we present various mathematical techniques that are frequently used for rigorous analysis and investigation of quantitative properties in forward and inverse problems.
4.1 Examples of Inverse Problems in Medical Imaging
Most inverse problems in imaging are to reconstruct cross-sectional images of a material property P from knowledge of input data X and output data Y. We express its forward problem in an abstract form as
where F is a nonlinear or linear function of P and X. To treat the problem in a computationally manageable way, we need to figure out its sensitivity, explaining how a perturbation P + ΔP influences the output data Y + ΔY. In ...