Numerical evaluation of Hankel transforms for oscillating functions

Investigation of the numerics of point spread function integration in single molecule localization

The computation of point spread functions, which are typically used to model the image profile of a single molecule, represents a central task in the analysis of single molecule microscopy data. To determine how the accuracy of the computation affects how well a single molecule can be localized, we investigate how the fineness with which the point spread function is integrated over an image pixel impacts the performance of the maximum likelihood location estimator. We consider both the Airy and the two-dimensional Gaussian point spread functions. Our results show that the point spread function needs to be adequately integrated over a pixel to ensure that the estimator closely recovers the true location of the single molecule with an accuracy that is comparable to the best possible accuracy as determined using the Fisher information formalism. Importantly, if integration with an insufficiently fine step size is carried out, the resulting estimates can be significantly different from the true location, particularly when the image data is acquired at relatively low magnifications. We also present a methodology for determining an adequate step size for integrating the point spread function.

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.

Zernike-Bessel representation and its application to Hankel transforms

The duality between the well-known Zernike polynomial basis set and the Fourier-Bessel expansion of suitable functions on the radial unit interval is exploited to calculate Hankel transforms. In particular, the Hankel transform of simple truncated radial functions is observed to be exact, whereas more complicated functions may be evaluated with high numerical accuracy. The formulation also provides some general insight into the limitations of the Fourier-Bessel representation, especially for infinite-range Hankel transform pairs.

Efficient deconvolution of noisy periodic interference signals

The interference signal formed by combining two coherent light beams carries information on the path difference between the beams. When the path difference is a periodic function of time, as, for example, when one beam is reflected from a vibrating surface and the other from a fixed surface, the interference signal is periodic with the same period as the vibrating surface. Bessel functions provide an elegant and efficient means for deconvoluting such periodic interference signals, thus making it possible to obtain the displacement of the moving surface with nanometer resolution. Here we describe the mathematical basis for the signal deconvolution and employ this technique to obtain the amplitude of miniature capillary waves on water as a test case.

Fast Hankel transform of nth order with improved performance

An improved algorithm for the numerical computation of the Hankel transform of nth order is presented and is tested with some well-known functions.

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.