An O(N) iterative solution to the Poisson equation in low-levelvision problems
Lai, S.H.
Vemuri, B.C.
Dept. of Electr. Eng., Florida Univ., Gainesville, FL;
Abstract
In this paper, we present a novel iterative numerical solution to
the Poisson equation whose solution is needed in a variety of low-level
vision problems. Our algorithm is an O(N) (N being the number of
discretization points) iterative technique and does not make any
assumptions on the shape of the input domain unlike the polyhedral
domain assumption in the proof of convergence of multigrid techniques.
We present two major results namely, a generalized version of the
capacitance matrix theorem and a theorem on O(N) convergence of the
alternating direction implicit method (ADI) used in our algorithm. Using
this generalized theorem, we express the linear system corresponding to
the discretized Poisson equation as a Lyapunov and a capacitance matrix
equation. The former is solved using the ADI method while the solution
to the later is obtained using a modified bi-conjugate gradient
algorithm. We demonstrate the algorithm performance on synthesized data
for the surface reconstruction and the SFS problems
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