Structural Similarity in Joint Inverse Problems

Abstract Joint inverse problems occur in many practical situations, where different modalities are used to image the same object. Structural similarity is a way to regularize such joint inverse problems by imposing similarity between the images. While structural similarity has found widespread use in many practical settings, its theoretical foundations remain underexplored. This study develops an over-arching formulation for these types of problems and studies their well-posedness via the Direct Method from the calculus of variations. We focus in particular on lower semi-continuity and coerciveness as essential properties for the well-posedness of the variational problem in $W^{m,p}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> and $SBV$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> <mml:mi>B</mml:mi> <mml:mi>V</mml:mi> </mml:math> . Here quasiconvexity and growth properties of the structural similarity quantifier turns out to be essential. We find that 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 $N$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> coupled problems is introduced.