The Abel Inversion of Noisy Data Using Discrete Integral Transforms

The determination of a radially symmetric two-dimensional function from its one dimensional projection is known as Abel inversion. This finds applications in several fields of engineering and science including astronomy, image processing, plasma diagnostics and optics. One radial slice of the function, which completely specifies the two-dimensional function, is the inverse Hankel transform of the Fourier Transform of the projection. With the projection data available as a discrete signal, the Abel inversion can be performed using the discrete Fourier transform and the inverse Hankel transform. The existing techniques to compute inverse Hankel transforms are discussed and a modification to one of those methods is introduced and shown to yield reasonably accurate results with significantly fewer calculations. The efficient fast Fourier transform is used in the evaluation of both the transforms.

Experimentally obtained projection data, which is usually noisy and off-center, is dealt with in the Fourier domain using a frequency domain filter and a maximum likelihood estimator derived from the assumed noise characteristics. The maximum likelihood estimator is used to estimate the Fourier transform of the properly centered data. Results of numerical experiments are given to verify the method. The proposed method is computationally more efficient than the existing curve-fitting methods for performing the Abel inversion of noisy data.