Regularization of Inverse Problems via Time Discrete Geodesics in Image Spaces
For imaging inverse problems with a reference image, this work provides a novel variational approach with theoretical guarantees and promising numerical results.
The paper combines a discrete geodesic path model for image metamorphosis with the L2-TV model to solve inverse problems in imaging given a reference image. Numerical results for sparse and limited angle CT and superresolution demonstrate the method's effectiveness.
This paper addresses the solution of inverse problems in imaging given an additional reference image. We combine a modification of the discrete geodesic path model for image metamorphosis with a variational model,actually the $L^2$-$TV$ model, for image reconstruction. We prove that the space continuous model has a minimizer which depends in a stable way from the input data. Two minimization procedures which alternate over the involved sequences of deformations and images in different ways are proposed. The updates with respect to the image sequence exploit recent algorithms from convex analysis to minimize the $L^2$-$TV$ functional. For the numerical computation we apply a finite difference approach on staggered grids together with a multilevel strategy. We present proof-of-the-concept numerical results for sparse and limited angle computerized tomography as well as for superresolution demonstrating the power of the method.