Coherent optical implementation of 1-D Mellin transforms

Abstract

Peter Kellman and Joseph W. Goodman Stanford University, Department of Electrical Engineering, Stanford, California 94305. Received 20 May 1977. Recent interest in Mellin power spectra, complex filtering, and correlation is due to the scale invariance of the Mellin transform. Mellin correlation has been cited for application to 2-D target detection and 1-D signal processing where parameters of scale and Doppler are unknown. In such applications, Fourier domain matched filtering requires either sequential hypothesis testing or multichannel processing. Loss of correlation in the Fourier implementation due to scale difference between received signal and reference is exchanged, however, for an equally severe loss in the Mellin correlation due to time or positional displacement. A coherent optical technique for Mellin transforms is described here in which signals may be processed in real time to provide scale invariant correlations. The parameterization on time uncertainty is provided for automatically by utilizing an acoustooptic spatial light modulator. Methods for performing general linear integral transforms using parallel processing in both spatial dimensions allow real time operation. Several of these methods have been outlined recently; one configuration, well suited for implementing Mellin spectrum analysis, is reviewed here. The general linear superposition integral may be written as