×

Inversion of the Radon transform, based on the theory of \(A\)-analytic functions, with application to 3D inverse kinematic problem with local data. (English) Zbl 1127.44002

The authors consider the following two-dimensional stationary transport equation: \[ Pu=\langle \omega, \nabla _{x }u(x,\alpha)\rangle + \mu (x)u(x,\alpha)=a(x)\quad x\in \Omega,\; \alpha\in \mathbb{R} \] where \(\omega=(\cos \alpha, \sin \alpha)\), \(u(x,\alpha)\) is a \(2\pi\)-periodic function, describing the density of particles at a point \(x\), moving in the direction \(\omega\), \(\mu (x)\) is a function, defining the attenuation at a point \(x\), \(a(x)\) is the radiation source, and \(\Omega \) is a strictly convex domain with smooth boundary. The inverse problem consists in finding the function \(a(x)\) from the flow \(u(x,\alpha)\) on \(\partial \Omega\times[0,2\pi]\) and attenuation function \(\mu \).
They show that this inverse problem reduces to Cauchy problem for the so-called A-analytic functions. By studing the properties of such A-analytic functions, they obtain the inversion of the generalized Radon transform in terms of the A-analog of the Cauchy transform. This leads to stable algorithms which are applied to a three-dimensional inverse kinematic problem.

MSC:

44A12 Radon transform
65R10 Numerical methods for integral transforms
65R32 Numerical methods for inverse problems for integral equations
PDFBibTeX XMLCite
Full Text: DOI