Theory and operational rules for the discrete Hankel transform

Previous definitions of a discrete Hankel transform (DHT) have focused on methods to approximate the continuous Hankel integral transform. In this paper, we propose and evaluate the theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The proposed transform also possesses requisite orthogonality properties which lead to invertibility of the transform. The standard set of shift, modulation, multiplication, and convolution rules are derived. In addition to the theory of the actual manipulated quantities which stand in their own right, this DHT can be used to approximate the continuous forward and inverse Hankel transform in the same manner that the discrete Fourier transform is known to be able to approximate the continuous Fourier transform.

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields

The method originally proposed by Yu et al. [Opt. Lett. 23, 409 (1998)] for evaluating the zero-order Hankel transform is generalized to high-order Hankel transforms. Since the method preserves the discrete form of the Parseval theorem, it is particularly suitable for field propagation. A general algorithm for propagating an input field through axially symmetric systems using the generalized method is given. The advantages and the disadvantages of the method with respect to other typical methods are discussed.

Numerical evaluation of Hankel transforms for oscillating functions

Six methods for the numerical calculation of zero-order Hankel transforms of oscillating functions were evaluated. One method based on Filon quadrature philosophy, two published projection-slice methods, and a third projection-slice method based on a new approach to computation of the Abel transform were implemented; alternative versions of two of the projection-slice methods were derived for more accurate approximations in the projection step. These six algorithms were tested with an oscillating sweep signal and with the calculation of a three-dimensional diffraction-limited point-spread function of a fluorescence microscope. We found that the Filon quadrature method is highly accurate but also computationally demanding. The projection-slice methods, in particular the new one that we derived, offer an excellent compromise between accuracy and computational efficiency.

A Kalman filtering approach to stochastic global and region-of-interest tomography

We define two forms of stochastic tomography. In global tomography, the goal is to reconstruct an object from noisy observations of all of its projections. In region-of-interest (ROI) tomography, the goal is to reconstruct a small portion of an object (an ROI) from noisy observations of its projections densely sampled in and near the ROI and sparsely sampled away from the ROI. We solve both problems by expanding the object and its projections in a circular harmonic (Fourier) series in the angular variable so that the Radon transform becomes Abel transforms of integer orders applied to the harmonics. The algorithm has three major components. First, we fit state-space models to each order of Abel transform and thus represent the Radon transform operation as a parallel bank of systems, each of which computes the appropriate Abel transform of a circular harmonic. A variable transformation here allows either the global or ROI problem to be solved. Second, the object harmonics are modeled as a Brownian branch. This is a two-point boundary value system, which is Markovianized into a form suitable for the Kalman filter. Finally, a parallel bank of Kalman smoothing filters independently estimates each circular harmonic from the noisy projection data. Numerical examples illustrate the proposed procedure.