1. Circular Slit Maps of Multiply Connected Regions with Application to Brain Image Processing
- Author
-
Ali W. K. Sangawi, Ali Hassan Mohamed Murid, and Khiy Wei Lee
- Subjects
General Mathematics ,Fast multipole method ,Mathematical analysis ,Image processing ,Conformal map ,010103 numerical & computational mathematics ,01 natural sciences ,Integral equation ,Generalized minimal residual method ,010101 applied mathematics ,Complex geometry ,Bounded function ,Nyström method ,0101 mathematics ,Mathematics - Abstract
In 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.
- Published
- 2020