Interferometric optical Fourier-transform processor for calculation of selected spatial frequencies

A novel interferometric optical Fourier-transform processor is presented th at calculates the complex-valued Fourier transform of an image at preselect ed points on the spatial-frequency plane. The Fourier spectrum of an arbitr ary input image is interfered with that of a reference image in a common-pa th interferometer. Both the real and the imaginary parts of the complex-val ued spectrum are determined. The source and the reference images are easily matched to guarantee good fringe visibility. At least six interferograms a re postprocessed to extract the real and the imaginary parts of the Fourier spectrum at preselected points. The proposed hybrid optical-digital techni que is computationally appropriate when the number of desired spatial frequ encies is small compared with the number of pixels in the image. When the n umber of desired points is comparable with the number of image pixels,a con ventional or pruned two-dimensional fast Fourier transform is more appropri ate. The number of digital operations required by the hybrid optical-digita l Fourier processor is proportional to the number of desired spatial freque ncies rather than the number of pixels in the image. The points may be regu larly distributed over the spatial-frequency plane or concentrated in one o r several irregularly shaped regions of interest. The interferometric optic al Fourier processor is demonstrated in a moving-object trajectory estimati on system. The system successfully estimates the trajectory of multiple obj ects moving over both stationary and white-noise backgrounds. A comparison of performance was made with all-digital computation. With everything else equal, our hybrid optical-digital calculation was more than 3 orders of mag nitude faster. (C) 2000 Optical Society of America OCIS code: 070.0070.