Radial Walsh functions form a closed set of orthogonal functions over a given finite interval, each function taking merely one constant value (either +1 or −1) in each of a finite number of subintervals into which the entire interval is divided. This set provides a remarkable set of orthogonal radial filters for the pupil of an imaging system. We report analytical expressions for members of the set and their Hankel transforms of order zero. Far-field diffraction characteristics, namely, the far-field amplitude distribution and the optical transfer functions, are presented for the first eight members of the set.

Far-field diffraction properties of radial Walsh filters

The calculation of image assessment critiera, eg, the Strehl ratio, the point spread function, or the optical transfer function, involves the evaluation of an integral where the integrand is highly oscillatory over a large range of integration Prefaced with a brief description of the well-known numerical quadrature methods adopted for the purpose, this paper presents a new quadrature technique that obviates the need for knowledge of derivatives of the argument of the exponential integrand Some illustrative numerical results are presented

P. K. Purkait

Papers

Walsh Functions in Lens Optimization: II. Evaluation of the Diffraction-based OTF for On-axis Imagery