The complex-valued optical Fourier transform and its application to moving-object trajectory estimation

This dissertation investigates interferometric optical Fourier processors for the purpose of computation. Both processors contain a conventional optical Fourier processor and employ a common-path type of interferometer to indirectly measure the complex-valued optical distribution produced in the back focal-plane of the Fourier transform lens. The first employs a point-diffraction interferometer while the second uses a novel interferometric technique. Both interferometric optical Fourier processors are demonstrated in a moving-object trajectory-estimation application.;A novel interferometric technique, referred to as joint-transforrn interference (JTI), is proposed to recover the complex-valued Fourier transform of an image at selected points on the spatial frequency plane. The Fourier spectrum of a source image is interfered with the Fourier spectrum of a reference image in a common-path interferometer. Both the real and imaginary parts of the complex-valued spectrum are determined, and in addition, the source and reference images are easily matched in order to guarantee good fringe visibility. Six interferograms are post-processed to extract the real and imaginary parts of the Fourier spectrum of the image at a selected number of points on the spatial frequency plane. When the number of desired points is comparable to the number of pixels in the image, a digital two-dimensional (2-D) fast Fourier transform (FFT) is appropriate, however, when a relatively small number of frequency-domain points is desired, the proposed hybrid optical-digital technique can offer substantial computational savings. The number of operations required by the hybrid optical-digital Fourier processor is proportional to the number of desired points. If only a small number of points are required, the optical Fourier processor can be much more computationally efficient than a pruned 2-D FFT, especially when the input images are very large.;The interferometric optical Fourier processor was employed to demonstrate the mixed-domain method of trajectory estimation. The mixed-domain technique is intended for the trajectory estimation of small, barely discernable, moving objects of unknown position and velocity. The interferometric optical Fourier processor is employed to determine the frequency-domain representation of moving objects embedded in an image sequence. The trajectory estimation system, based on the novel joint-transform interferometric optical Fourier processor, estimated the trajectory of multiple objects moving over both stationary and white noise backgrounds. The trajectories estimated using the optical Fourier processor compared favorably with those calculated using a digital FFT. The hybrid optical-digital calculation of the complex-valued Fourier transform at the selected frequency domain points required the measurement of six interferograms and approximately 200 floating-point operations. Calculation of the Fourier transform at the same points via the FFT required more than three orders of magnitude more floating-point operations. (Abstract shortened by UMI.).