LAP: A Linearize and Project Method for Solving Inverse Problems with Coupled Variables

摘要

Many inverse problems involve two or more sets of variables thatrepresent different physical quantities but are tightly coupled with each other.For example, image super-resolution requires joint estimation of the imageand motion parameters from noisy measurements. Exploiting this structureis key for efficiently solving these large-scale optimization problems, whichare often ill-conditioned.In this paper, we present a new method called Linearize And Project(LAP) that offers a flexible framework for solving inverse problems withcoupled variables. LAP is most promising for cases when the subproblemcorresponding to one of the variables is considerably easier to solve than theother. LAP is based on a Gauss-Newton method, and thus after linearizingthe residual, it eliminates one block of variables through projection. Dueto the linearization, this block can be chosen freely. Further, LAP supportsdirect, iterative, and hybrid regularization as well as constraints. ThereforeLAP is attractive, e.g., for ill-posed imaging problems. These traits differentiateLAP from common alternatives for this type of problem such as variableprojection (VarPro) and block coordinate descent (BCD). Our numerical experimentscompare the performance of LAP to BCD and VarPro using threecoupled problems whose forward operators are linear with respect to oneblock and nonlinear for the other set of variables.