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A modification of a previously published two-dimensional phase-retrieval algorithm based on the sampling theorem [ Opt. Commun.47, 380 ( 1983)] is introduced. With an appropriate choice of the intermediate sampling grid, a significant reduction in the number of computations is achieved, because an independent system of equations may be solved for each line of the Fourier transform of the object. The results of computer simulations on real objects are given. The algorithm converges rapidly.

The properties of the image of a nonuniform transparency degraded by speckle noise are studied. Attention is focused on the spatial details of an object that cannot be resolved by a coherent system. Analysis of the power spectrum of the noisy spatial-intensity distribution in the image plane confirms that the bandwidth for coherent diffuse illumination corresponds to the incoherent transfer function of the system. Experimental verification is presented using a Ronchi ruling as a test object.

Previous studies have shown that the intensity of the correlation peaks in Vander Lugt optical correlators can change with the position of the object because of emulsion thickness effects. In this paper, it is shown that the intensity variations are caused by a combination of thickness effects and a position-dependent frequency cutoff due to space variance of the optical processor. Experiments showing the global effect, as well as experiments showing each effect acting alone, have been performed.

An optical processor has been used to filter aeromagnetic contour maps in order to obtain information on underground magnetic ore deposits. This was accomplished by directional filtering of the spatial Fourier transform of the contour map. The directional filtering yields maps of gradients having given directions. A digital analysis was also performed on the data so that the feasibility of the optical technique could be evaluated. The results obtained so far suggest that a systematic filtering of aeromagnetic maps can be carried out at low cost and that the filtered maps can yield useful information to the interpreter.

A point transformation is applied to images degraded by film-grain noise in order to transform the noise into approximately signal-independent additive Gaussian noise. The condition for validity of this technique is that the emulsion noise follow the square-root law predicted by the classical model of film-grain noise. We also assume that the scanning aperture is large compared with the grain size. Once the image has been transformed in this way, the appropriate least-squares filtering technique may be applied to the image. Experiments on two Kodak emulsions show that the technique works well over a range of diffuse densities up to densities of 2.

A new type of filter called the optimum circular symmetrical filter (OCF), described in a previous paper, is used for digital and optical pattern recognition. This rotation-invariant matched filter optimizes the signal-to-noise ratio. Optical experiments with letters of the alphabet gave results that agreed with the digital simulation.

In the paraxial approximation a symmetrical optical system may be represented by a 2 × 2 matrix. It has been the custom to describe each optical element by a transfer matrix representing propagation between the principal planes or through an interface for thin elements. If the focal-plane representation is used instead, any focusing element or combination of elements is represented by the same antidiagonal matrix whose nonzero elements are the focal lengths: The matrix represents propagation between the focal planes. For propagation between any two arbitrary planes, the system transfer matrix can be decomposed into the product of two upper triangular matrices and an antidiagonal matrix. This decomposition yields the above-mentioned focal-plane matrix, and the two upper triangular matrices represent propagation between the input and the output planes and the focal planes. Because the matrix decomposition directly yields the parameters of interest, the analysis and the synthesis of optical systems are simpler to carry out. Examples are given for lenses, diopters, mirrors, periodic sequences, resonators, lenslike media, and phase-conjugate mirror systems.

A new approach to the representation of nonsymmetrical optical systems by matrices is introduced. In the paraxial approximation each component of an optical system is represented by a 4 × 4 unitary matrix, and the product of those matrices yields the transfer matrix of the system. The transfer matrix that represents the propagation between two arbitrary planes through the system containing two independently rotated cylindrical lenses is decomposed into the product of three matrices. The eigenvalues of the submatrices in this factorized form determine the focal lengths of the equivalent system and the localization of the foci of the system with respect to these arbitrarily chosen planes.

When image plane speckle intensity integrated over a finite aperture is submitted to a logarithmic transformation, the noise becomes additive and signal independent. The first- and second-order moments of the probability distribution are derived. It is found that the logarithm of speckle noise approaches a normal distribution much faster than speckle intensity. The properties of speckle noise are different from those of film-grain noise; for example, neither Nutting’s law nor Selwyn’s law is satisfied by speckle.

An exact geometrical-optics theory of holography is worked out. A simple derivation of the holographic ray-tracing equations is given; it is used to determine the principal points of a hologram. The well-known paraxial conjugate equations of holography are shown to be exact relations, if the distance and the angles are measured in an appropriate manner. The intersection of all the principal rays determine the position of the aberration-free image.

A new type of filter called the optimum circular symmetrical filter is presented and its uses in optical pattern recognition are described. The filters can be realized by optical or computer-generated holography. They are space-invariant and rotation-invariant, and are optimized in the mean-square sense. This makes the recognition process fully parallel, without input scanning or coordinate transformation, no matter where and in what orientation the patterns to be recognized are located in the input plane. Multiple use of those filters can further improve system recognition ability. By using them jointly with other methods of linear combinations of matched filters, more advanced systems can be obtained.

An optodigital hybrid system using an incoherent circular correlator that has features suitable for space and industrial applications is introduced. The incoherent circular correlator uses a fiber-optics sampler and carries out the correlations in white light. The trade offs involved for efficient detection of objects are discussed. The system has been built, and experimental results on a real air photograph are presented.

A method of digital multiple correlation for rotation-invariant recognition has been developed. It is also a computer simulation of an incoherent optodigital hybrid system previously introduced. Speed and flexibility of the digital system are investigated.

Presented in this paper is an optical implementation of a shift- and rotation-invariant pattern recognition technique. A computer-generated hologram is designed to match with one of the circular harmonic components of the target. Experimental results with simple photographic objects show that targets with different locations and orientations can be simultaneously recognized by the optical system.

A new method has been developed for rotation-invariant pattern recognition. One component of the circular harmonic expansion of the target is used in the preparation of the reference. Correlations between the input and reference objects are accomplished by FFT and multiplication in the frequency domain. In an experience with targets from an image with 192 × 192 pixels, target orientations were detected with an accuracy of ~0.1°. This method is also suitable for optical implementation.