Circular Slit Maps of Multiply Connected Regions with Application to Brain Image Processing

Abstract

AbstractIn this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is $$O((M + 1)n)$$ O ( ( M + 1 ) n ) , where $$M+1$$ M + 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require $$O((M+1)^3 n^3)$$ O ( ( M + 1 ) 3 n 3 ) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented.