Sobolev gradient preconditioning for image-processing PDEs

摘要

The article explores the relationship between Sobolev gradients and H~(-1) mixed methods for a variety of partial differential equations (PDEs) from image processing. A first-order system least-squares problem is used to introduce the method and compare the Euclidean with the Sobolev gradient. The standard two-term decomposition of an image as f = u + v with u ∈ H~1 and v ∈ L~2 = H~0 yields a second-order linear PDE, while minimizing other Lp norms give nonlinear PDEs. Finally, a three-term decomposition f = u + v + w with ue H~1 ,v ∈ H~(-1), w ∈ H~0 requires the solution of a fourth-order system with the Inharmonic operator.