A system with dimensional transducers typically consists of three parts: a dimensional transducer, a processor, and again a dimensional transducer. The first transducer may, for example, convert a two-dimensional (2-D) picture into a one-dimensional (1-D) temporal signal. That signal is processed by a digital computer. The computer output is again a 1-D temporal signal, which is converted back into a picture by the second dimensional transducer. The job of the first transducer is to adapt the format of the original signal to the capabilities of the processor. The second transducer converts the processor output into a format suitable for the user or receiver. Systems with dimensional transducers usually consist of more than one type of hardware: optics, TV electronics, digital electronics, and movie technology are all examples. We discuss the virtues of such systems and review briefly some historical examples that are not well known. Finally we present some new experiments with optical processors characterized by input and output dimensional transducers.

Abstract. We exploit the properties of the Dirac comb function to develop a pictorially appealing analysis of the relationship between the one-dimensional (1-D) spectrum of a signal waveform and the two-dimensional (2-D) spatial frequency spectrum of its raster record (generally referred to as the folded spectrum). Variations are considered, including the effects of recording with redundancy and of recording without appropriate "fall" of the raster lines. The use of the falling raster in time-integration folded spectrum analysis is analyzed, first from the stand-point of an important analogy with incoherent holography, then in terms of local oscillator arrays and moving comb functions. Finally, we consider the application of the falling raster to image processing.

We explore one-dimensional (1-D) to two-dimensional (2-D) transformations suitable for optical correlation of signals. An intuitive, geometrical development establishes and relates several 2-D formats including variations of the familiar falling raster format. Insight into the folded spectrum is gained by considering a matched spatial filter implementation of correlation between two signals in falling raster formats.

Abstract. This paper is concerned with the use of dimensionality changing transformations for the digital processing of signals that have been sampled on sampling lattices other than the familiar rectangular, or row-column one. After introducing the idea of nonrectangular sampling, the paper formally presents a particularly useful class of dimensionality-changing transformations and presents conditions under which they can be used for signal processing. It does this by means of a vector notation. The major result is that the use of such transformations with nonrectangularly sampled data is no less restrictive, no more difficult, nor substantially different than with rectangularly sampled data.

Abstract. We briefly review the dimensional mappings used in fast discrete Fourier transform and convolution algorithms and point out the equivalence between these discrete mappings and some of the mappings used in optical signal processing.

Abstract. Many signal processing architectures that exploit characteristics of current device technology can be devised by decomposing linear transform kernels and by employing chirp implementations of the Fourier transform. These methods allow complex algorithms to be implemented by devices with relatively fewer degrees of freedom. Dimensionality-changing transformations play an especially important role. Conditions for decomposition are described, and a variety of architectures, including those for the discrete Fourier transform, chirp-z transform, beam forming, and cross-ambiguity function calculation, are discussed.

Abstract. The description of a signal by means of a local frequency spec-trum resembles such things as the score in music, the phase space in mechanics, and the ray concept in geometrical optics. Two types of local frequency spectra are presented: the Wigner distribution function and the sliding-window spectrum, the latter having the form of a cross-ambiguity function. The Wigner distribution function in particular can provide a link between Fourier optics and geometrical optics; many properties of the Wigner distribution function, and the way in which it prop-agates through linear systems, can be interpreted in geometric-optical terms. The Wigner distribution function is linearly related to other signal representations like Woodward's ambiguity function, Rihaczek's complex energy density function, and Mark's physical spectrum. An advantage of the Wigner distribution function and its related signal representations is that they can be applied not only to deterministic signals but to stochastic signals as well, leading to such things as Walther's generalized radiance and Sudarshan's Wolf tensor. On the other hand, the sliding-window spectrum has the advantage that a sampling theorem can be formulated for it: the sliding-window spectrum is completely determined by its values at the points of a certain space-frequency lattice, which is exactly the lattice suggested by Gabor in 1946. The sliding-window spectrum thus leads naturally to Gabor's expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal, which is again another space-frequency signal representation, and which is related to the degrees of freedom of the signal.

Abstract. Because a Fourier transform relates time (t) and frequency (ν) of an optical signal, we expect and observe interrelationships between systems operating in the t domain and systems operating in the P domain. Indeed, it appears that each t system has a dual ν system that uses an essentially identical setup and measures the same underlying phenomenon. The same duality exists in the other direction as well (there is a t system dual to each ν system). This preliminary study of ν↔ interrelationships shows that many otherwise diverse optical information processors are readily related by the ν ↔ t duality concept.

Abstract. Intensity-to-spatial frequency transformations can be exploited for a variety of useful optical information processing operations. Resolution elements of an image are encoded with a grating structure whose spatial frequency and/or orientation is a function of the local image intensity. Assuming that certain sampling requirements are met, each intensity level is assigned to a different point in Fourier space. Various schemes of spatial filtering can then be used to alter the relative intensity levels. The procedure can be used for nonlinear analog point transformations and for numerical processing using binary or residue arithmetic. Theta modulation and frequency modulation are special cases of the technique. Off-line methods for implementing intensity-to-spatial frequency transformations include halftoning and grating techniques; real-time methods include new electro-optical systems such as the variable-grating-mode (VGM) liquid crystal devices. Details of these devices and experimental results are presented.

Three-dimensional (3-D) and four-dimensional (4-D) signals can be replaced by two-dimensional (2-D) sequences of sectional images if sampling in one or two dimensions is practicable. The convolution of such sequences results in a sequence that under certain conditions is the sampled version of the result of 3-D or 4-D convolution. Thus higher dimensional convolution can be performed in two dimensions, e.g., by coherent-optical filtering. Because of the usually large space-bandwidth products (SBP) of such sequences, the parallel computing ability of optical filtering makes it very feasible. A relation between input signal size, SBP of the filter, and parameters of the coherent-optical setup allows one to estimate limits in pixel numbers for the input signal size, e.g., for comparison with digital techniques. Based on linear systems theory, the necessary filter for sequence convolution can be calculated from a given higher dimensional point response or transfer function. Experimental results for 3-D bandpass filtering and a 4-D correlation are discussed.

We discuss two generalizations of the continuous Fourier transform performed by coherent optical systems. The first concerns the introduction of an appropriate exponential damping factor in the input plane, which leads to a processor that evaluates a two-dimensional slice through the four-dimensional complex Laplace transform domain. By performing Laplace filtering, rather than Fourier filtering, one can in principle trade off dynamic range in the filter plane for dynamic range in the input plane. Using a Laplace transform, it is also possible to find the complex roots of polynomials. The second generalization concerns modification of the continuous Fourier transform to behave as a discrete Fourier transform. With such a modification, it is in principle possible to find (in a single step, without iterations) the eigenvalues of any circulant matrix, or any circulant approximation to a Toeplitz matrix (including correlation matrices) using a coherent optical processor. Furthermore, if a light valve having a suitable nonlinear relation between amplitude transmittance and exposure is available, it is possible to obtain the inverse of any matrix in the class described above in a single pass through a coherent optical processor.

Abstract. This paper discusses the development of a noncoherent optical signal processing device, termed an electro-optical processor, and its application to performing discrete linear transformations. The device consists of a light-emitting diode (LED), a photographic matrix mask, and an area-array charge-coupled device (CCD). It is capable of performing matrix-vector products at high speed in a small, low-power, and potentially low-cost package. The operation of the processor and a sum-mary of methods for handling bipolar and complex arithmetic are briefly described. Then, consistent with the central theme of this book, em-phasis is placed upon the computation of linear transformations using this optical implementation. The transformations discussed are the cosine, Fourier, proportional-bandwidth spectral analysis, Karhunen-Loeve, sine, Mellin, Waish-Hadamard, Haar, and slant. Examples of the matrix masks for each of these transformations are shown along with a discussion of practical applications.

An iterative optical vector-matrix processor capable of solving systems of linear algebraic equations or vector-matrix equations is described. The basic system uses a vector-matrix multiplier, consisting of a linear light emitting diode (LED) input array and fiber optic interconnections to a matrix mask interfaced to a linear output photodetector ar-ray. When the photodetector's outputs are returned to the LED inputs via a microprocessor electronic feedback system, an iterative optical processor results. The operation, description, and fabrication of this system are described, and its use as a rather general purpose optical processor is emphasized.

A folded spectrum is a one-dimensional spectrum that has been reformatted into two dimensions by cutting it into segments of equal length and arranging the segments in sequential order into a two-dimensional array. Folded spectrum analysis in optical processing allows both spatial dimensions of the optical processor to be used effectively and therefore allows a greater number of spectrum elements to be displayed in parallel. The folded spectrum in optical processing is remarkably similar to the fast Fourier transform (FFT) algorithm in digital processing. The similarities unify many processing concepts and give physical and intuitive insights intc the FFT. More importantly, they show how folded spectrum analysis can be extended to more than two dimensions using either optical or other processing technologies. That result promises a rich variety of computing architectures for the immediate future.

Abstract. The bilinear transformation, which is a quadratic transformation with spread, characterizes the operations performed by many optical systems. This is due to the natural quadratic relation between the optical field and the optical intensity. The transmission of information through a bilinear system is examined. This includes frequency-domain analysis for space-invariant systems, and expansions in terms of orthogonal functions. Several methods of performing bilinear transformations using coherent or partially coherent processors are discussed. We demonstrate how dimensionality can be traded with order of nonlinearity. We also show how a bilinear system can be realized by use of a parallel bank of coherent spatial filters. The problem of inverting a bilinear transformation is addressed. This is the problem of image restoration when the distortion model is bilinear. Several methods of restoration are discussed.

Abstract. Several hardware methods for selecting the median value of a set of input signals are described. Two of the systems are based on the use of an analog multiplexer to select the channel containing the median value. The input levels are pairwise compared with comparators, and the comparator outputs provide logic levels to drive the address lines of a multiplexer. A third method involves the use of saturable amplifiers and a summing junction. An interesting feature of this system is that it can act as a median filter, a linear averaging filter, or a combination of the two. A circuit whose output automatically hunts around the median value is described next, and the final circuit relies on diode switching networks in the feedback loop of operational amplifiers to select the median input. Schematic circuits are provided.

Abstract. We discuss several interesting relationships between the tem-poral and the spatial dimensions which we encounter in image sequence processing: (i) equivalence of temporal filtering and spatial filtering; (ii) estimation of motion by relating time differences to spatial differences; and (iii) equivalence of time sequences and stereo (spatial) sequences.

Abstract. This paper reviews the Gerchberg-Saxton algorithm and variations thereof that have been used to solve a number of difficult reconstruction and synthesis problems in optics and related fields. It can be used on any problem in which only partial information (including both measurements and constraints) of the wavefront or signal is available in one domain and other partial information is available in another domain (usually the Fourier domain). The algorithm combines the information in both domains to arrive at the complete description of the wavefront or signal. Various applications are reviewed, including synthesis of Fourier transform pairs having desirable properties as well as reconstruction problems. Variations of the algorithm and the convergence properties of the algorithm are discussed.

Gerchberg's iterative extrapolation algorithm is generalized to two dimensions in two distinct ways. The first generalization is implemented on a coherent optical processor. Fundamental limitations are discussed. A second generalization is reformulated discretely and placed in closed form. A number of digital implementations are presented. A generalized methodology is then developed for a certain class of deconvolution problems. Gerchberg's algorithm and other deconvolution algorithms are shown to be special cases. Algorithm convergence and stability (posedness) are discussed and exemplified. Last, methods of incorporating further object information into the iterative algorithms are explored.

Abstract. The term planar projection refers to the integral of a three-dimensional (3-D) function over a plane, as opposed to the line integrals that form the basic data set of computed tomography. Planar projections arise naturally in nuclear imaging when a slit aperture is used, in imaging with nuclear spin resonance, and in time-domain scattering studies. If an appropriate set of translations and rotations of the plane of integration is carried out, a complete data set is generated and a reconstruction of the 3-D object is possible. Various reconstruction algorithms are discussed and compared to the more familiar case of computed tomography. Another potential application of planar projections is in general 3-D data processing, where it should be useful to preprocess the data by generating a set of planar integrals, even if the original data have nothing to do with projections. This has the effect of reducing 3-D operations such as convolutions and correlations to one-dimensional (1-D) operations, which are more readily performed. After a subsequent reconstruction from the filtered projections, the final 3-D result is equivalent to that which would have been obtained by 3-D operations. Several incoherent optical systems for performing the projection and reconstruction operations are described.

Abstract. The general effects of imperfections in measured data on image reconstruction are considered. The particular problem of phase recovery is discussed in some detail, both direct and iterative restoration procedures being examined. In several important applications of image reconstruction, measured data is sampled at a grid of points distinct from the grid most convenient for the reconstruction procedure. The difficulties likely to be encountered when carrying out the necessary interpolations are itemized and ways of ameliorating them are outlined. It is shown that the phenomenon of speckle has a bearing on both the phase problem and the interpolation difficulty, thereby forming a conceptual bridge between various computational niceties of phase recovery and image reconstruction.

Abstract. The formation of speckle images and the more practically important aspects of their statistics are reviewed. The ways in which fixed aberrations can be absorbed within randomly fluctuating ones are detailed. The processing methods of Labeyrie and Knox-Thompson are briefly described. The shift-and-add principle and its extensions are treated in some detail, and the results of optical laboratory simulations are presented. Experimental precautions found to be particularly important are emphasized.

The ambiguity of the complex logarithm, whereby its imaginary part is only specified modulo 2r, poses a problem that often arises in diverse contexts. This article reports on the inadequacy of phase unwrapping in two separate contexts. One example occurs with image restoration that is intended to counteract the deleterious effects of at-mospheric turbulence. By tracking the positions (phases) of all the interference fringes that compose the image, the mean positions might be estimated. When only a few photons are available however, the resulting uncertainties sometimes lead to 27r slips that drastically upset the mean positions. A second example occurs in rotational synthesis, where the only available information lies on a single circle in the 2-D space-frequency complex domain. If it were possible to reliably unwrap the imaginary part of the logarithms around that circle, we could easily find the brightest point of the picture domain. The procedure attempts to trace a path as the source traverses interference fringes, without ever accidently slipping a fringe. In both examples, subsidiary contributions occasionally conspire to give the 27 slips. Smooth continuity is not quite sufficient as a basis for phase unwrapping. The problem is especially important because the character of images is more dependent on the phases of their Fourier components than on the amplitudes. Perhaps it will become possible to invoke other criteria than continuity for more reliable phase unwrapping.

Abstract. Image construction by Fourier inversion may be hindered by various deficiencies in the data. Examples of defects include data presented on a nonrectangular array of points and missing data values. Among special cases that are commonly encountered are measurements made in polar coordinates and the existence of sectors where data may be entirely missing. An exact method of interpolation analogous to sincfunction interpolation in the rectangular situation is described. However, inverting such data involves not merely a problem of mathematical interpolation, but also account must be taken of the character of the object, the kind of information desired, and the penalties to be associated with errors, considerations that go beyond what is involved under conditions where the one-to-one correspondence between function and transform may be invoked and the Fourier inversion formula is applicable.

Abstract. The median window operation is being increasingly used to process images. Although the deterministic properties of the median are fairly well known, its statistical properties are not. Consider a median window of width N scanning a noisy background image with white power spectrum. We present here the probability law for the median outputs, its mean, variance, and signal-to-noise ratio, and the probability that two successive median outputs are equal. Specialization is made to speckle imagery. Key results are as follows: the probability law is of a Bernoulli multinomial form; the mean is asymptotic with N to the average background times In 2, and hence is about 30% less than the background value; the variance is asymptotic with N to a 1/N dependence; signal-to-noise ratio is asymptotic with N to Ni171 In 2. Finally, the probability that two successive median outputs are equal is 2^-1(N^-1)/N, or slightly less than 0.5 for N ⪆ 7. This is independent of the type of image data at hand, i.e., whether speckle, Poisson, or normal, provided that it has a white power spectrum.