id: 05152363
dt: j
an: 05152363
au: Keeling, Stephen L.; Haase, Gundolf
ti: Geometric multigrid for high-order regularizations of early vision
    problems.
so: Appl. Math. Comput. 184, No. 2, 536-556 (2007).
py: 2007
pu: Elsevier Science Publishing Co. (North-Holland), New York
la: EN
cc: 
ut: early vision; surface estimation; geometric multigrid; elliptic regularity;
    finite elements; lumping; multicolored ordering; magnetic resonance
    coil sensitivity; Galerkin method; regularization; convergence;
    Gauss-Seidel relaxation; image processing; finite difference
ci: Zbl 0729.65018
li: doi:10.1016/j.amc.2006.05.209
ab: 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
    application 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 {\it W. Hackbusch} [Iterative
    solution of large sparse systems of equations, Springer (1991; Zbl
    0729.65018)] are met, and in particular the general elliptic regularity
    required is proved. Further, the Galerkin formalism is used together
    with a multicolored ordering of unknowns to permit vectorization of a
    symmetric Gauss-Seidel relaxation in image processing systems.  The
    implementation is analyzed computationally and inaccuracies 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 favor 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.
rv: Yves Cherruault (Paris)