Structural Similarity in Inverse Problems

Abstract

This thesis provides a theoretical basis for applying structural similarity to joint inverse problems with gradient-based variational regularization. This study develops an over-arching formulation for these types of problems which have been successfully applied in geophysics, image enhancement, and medical imaging in prior research. Via the Direct Method from the calculus of variations, the study identifies lower semi-continuity and coerciveness as essential properties for the well-posedness of the variational problem with regularizers that are integrals over an integrand specifying structural similarity. Informed by practice, well-posedness of the coupled inverse problem is proven for previously used specific integrands with solutions in W^(m,p), BV, SBV , and the space of finite Radon measures. Specifically, the use of gradient-difference, cross-gradient or Schatten norms as structural similarity quantifiers is theoretically justified. A generalized form of the cross-gradient that inherently works on N coupled problems is introduced and is proven to lead to a well-posed problem. Additionally, quasiconvex relaxation and compensated compactness are explored as alternative methods that provide insight when the Direct Method fails, in particular for the case of using a dot-product regularizer. This thesis also shows that both new and existing structural regularizers outperform traditional TV regularization in RGB image deblurring problems.