## Sobolev gradient preconditioning for image-processing PDEs.(English)Zbl 1168.68596

Summary: 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 $$u\in H^1$$ and $$v\in L^{2} = H^{0}$$ yields a second-order linear PDE, while minimizing other $$L^p$$ norms give nonlinear PDEs. Finally, a three-term decomposition $$f = u + v + w$$ with $$u\in H^1, v \in H^{-1}, w \in H^{0}$$ requires the solution of a fourth-order system with the biharmonic operator.

### MSC:

 68U10 Computing methodologies for image processing 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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