Geometric Multigrid for High-Order Regularizations of Early Vision Problems Stephen L. Keeling 1 and Gundolf Haase 2 Abstract. The surface estimation problem is used as a model to demonstrate a framework for solving early vision problems by high-order regularization with natural boundary conditions. Because the ap- plication of algebraic multigrid is usually constrained by an M -matrix condition which does not hold for discretizations of high-order problems, a geometric multigrid framework is developed for the efficient solution of the associated optimality systems. It is shown that the convergence criteria of [5] are met, and in particular the general elliptic regularity required is proved. Further, the Galerkin formalism is used together with a multi-colored ordering of unknowns to permit vectorization of a symmetric Gauss-Seidel relaxation in image processing systems. The implementation is analyzed computationally and inaccu- racies are corrected by lumping and by proper floating point representations. Direct one-dimensional calculations are used to estimate the effect of regularization order, regularization strength, relaxation, and data support on the multigrid reduction factor. A finite difference formulation is ruled out in fa- vor of a finite element formulation. A representative problem from magnetic resonance coil sensitivity estimation is solved using increasingly higher orders of regularization, and the results are compared in terms of accuracy and multigrid convergence. 1 Introduction Early vision problems are those in which dense three-dimensional information is to be in- ferred from sparse and generally noisy two-dimensional imaging data [16]. The model problem used here to demonstrate the framework set forth in the present work is the surface estima- tion problem [11] [19] [20] [22]. However, this framework has also been used for other early vision problems, including optical flow and image registration, and those results will be re- ported separately. The variational approach to solving such problems involves to minimize the sum of a data residual term and a solution regularization term [16]. The optimality condi- tion determining the mimimizer is a partial differential equation whose discretization can be poorly conditioned depending upon the sparsity of data, the order of regularization, and the formulation of boundary conditions [3] [11] [19] [20] [22]. To reduce ill-conditioning, low-order regularization and low-order non-natural boundary conditions have been used [3] [11] [19] [20] [22]. Also creative conjugate gradient preconditioners [11], hierarchical [19] and wavelet bases [15] [22], and multigrid methods [3] [20] have been developed for such formulations. However, as shown below, low-order regularizations and non-natural boundary conditions have a corrupting effect on the solution to the early vision problem. For instance, in a surface estimation problem such as that reported in [10], the surface to be estimated typically has exponential growth at the boundary. Such boundary growth is naturally frustrated by low-order homogeneous boundary conditions. Thus, in the present work, high-order regularizations with natural boundary condi- tions are studied. Also, because the application of algebraic multigrid is usually constrained by an M -matrix condition [21] which does not hold for discretizations of high-order problems, a geometric multigrid framework is developed for the efficient solution of the associated optimality systems; see also [12] for related multigrid work. The present paper can be summarized as follows. In Section 2 the variational formulation is given and its analysis is summarized. The general optimality system is a high-order Neumann problem whose solution regularity is needed in numerical error estimates. Since in the present 1 Institut f¨ ur Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit¨at Graz, Heinrichstraße 36, 8010 Graz, Austria; email: stephen.keeling@uni-graz.at; tel: +43–316–380–5156; fax: +43–316–380–9815. 2 Institut f¨ ur Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit¨at Graz, Heinrichstraße 36, 8010 Graz, Austria; email: gundolf.haase@uni-graz.at; tel: +43–316–380–5178; fax: +43–316–380–9815. 1