Sobolev gradient preconditioning for image‐processing PDEs

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 ∈ H1 and v ∈ L2 = H0 yields a second‐order linear PDE, while minimizing other Lp norms give nonlinear PDEs. Finally, a three‐term decomposition f = u + v + w with u ∈ H1, v ∈ H−1, w ∈ H0 requires the solution of a fourth‐order system with the biharmonic operator.