An Analysis and Implementation of Multigrid Poisson Solvers With Verified Linear Complexity
Matías Di Martino, Gabriele Facciolo
⚠ This is a preprint. It may change before it is accepted for publication.

Abstract

The Poisson equation is the most studied partial differential equation, and it allows to formulate many useful image processing methods in an elegant and efficient mathematical framework. Using different variations of data terms and boundary conditions, Poisson-like problems can be developed, e.g., for local contrast enhancement, inpainting or image seamless cloning between many other applications. Multigrid solvers are among the most efficient numerical solvers for discrete Poisson-like equations. However, their correct implementation relies on: (i) the proper definition of the discrete problem, (ii) the right choice of interpolation and restriction operators, and (iii) the adequate formulation of the problem across different scales and layers. In the present work we address these aspects, and we provide a mathematical and practical description of multigrid methods. In addition, we present an alternative to the extended formulation of Poisson equation proposed in [M. Mainberger, A. Bruhn, J. Weickert, and S. Forchhammer, Edge-based compression of cartoon-like images with homogeneous diffusion, Pattern Recognition, 44 (2011), pp. 1859–1873]. The proposed formulation of the problem suits better multigrid methods, in particular, because it has important mathematical properties that can be exploited to define the problem at different scales in a intuitive and natural way. In addition, common iterative solvers and Poisson-like problems are empirically analyzed and compared. For example, the complexity of problems is compared when the topology of Dirichlet boundary conditions changes in the interior of the regular domain of the image. We also show that multigrid methods convergence can be achieved in linear time.

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