A novel interferometric optical Fourier-transform processor is presented that calculates the complex-valued Fourier transform of an image at preselected points on the spatial-frequency plane. The Fourier spectrum of an arbitrary input image is interfered with that of a reference image in a common-path interferometer. Both the real and the imaginary parts of the complex-valued spectrum are determined. The source and the reference images are easily matched to guarantee good fringe visibility. At least six interferograms are postprocessed to extract the real and the imaginary parts of the Fourier spectrum at preselected points. The proposed hybrid optical–digital technique is computationally appropriate when the number of desired spatial frequencies is small compared with the number of pixels in the image. When the number of desired points is comparable with the number of image pixels, a conventional or pruned two-dimensional fast Fourier transform is more appropriate. The number of digital operations required by the hybrid optical–digital Fourier processor is proportional to the number of desired spatial frequencies rather than the number of pixels in the image. The points may be regularly distributed over the spatial-frequency plane or concentrated in one or several irregularly shaped regions of interest. The interferometric optical Fourier processor is demonstrated in a moving-object trajectory estimation system. The system successfully estimates the trajectory of multiple objects 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 magnitude faster.
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