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), where 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) 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.
Highlights
Conformal mapping has been applied to the study of the visual cortex
This paper illustrates a new integral equation method using the adjoint generalized Neumann and Neumann-type kernels for conformal mapping and its inverse of a bounded multiply connected region onto a disk and annulus with circular slits, extending the work presented in Sangawi et al [16,17]
Conformal mapping can be used to map an irregular surface onto a disk and annulus while preserving the angle, which is useful for visualization of magnetic resonance imaging (MRI)
Summary
Conformal mapping has been applied to the study of the visual cortex. Frederick and Schwartz [1] presented the conformal mapping of retina as a case of identification of a single point (the representation of the blind spot, or optic disk) in the eye. Explicit formulae for conformal mappings of multiply circular regions onto canonical slit regions in terms of Schottky–Klein prime function and their applications are described in [4,5,6]. This paper illustrates a new integral equation method using the adjoint generalized Neumann and Neumann-type kernels for conformal mapping and its inverse of a bounded multiply connected region onto a disk and annulus with circular slits, extending the work presented in Sangawi et al [16,17].
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