Application Of The Radon Transform To Optical Production Of The Wigner Distribution Function

Abstract

The Wigner distribution function (WDF), a simultaneous coordinate and frequency representation of a signal, has properties useful in pattern recognition. Because the WDF is computationally demanding, its use is not usually appropriate in digital processing. Optical schemes have been developed to compute the WDF for one-dimensional (1 -D) signals, often using acousto-optic signal transducers. Some recent work has demonstrated the computation of two-dimensional (2-D) slices of the four-dimensional (4-D) WDF of a 2-D input transparency. In this latter case, the required 2-D Fourier transformation is performed by coherent optics. We demonstrate that computation of the WDF of real 2-D signals is susceptible to Radon transform solution. The 2-D operation is reduced to a series of 1 -D operations on the line-integral projections. The required projection data are produced optically, and the Fourier transformation is performed by efficient 1 -D processors (surface acoustic wave filters) by means of the chirp-transform algorithm. The resultant output gives 1 -D slices through the 4-D WDF nearly in real time, and the computation is not restricted to coherently illuminated transparencies. This approach may be useful in distinguishing patterns with known texture direction. The optical setup is easily modified to produce the cross-Wigner distribution function, a special case of the complex, or windowed, spectrogram.