Kwan F. Cheung

Imaging sampling below the Nyquist density without aliasing

For multidimensional band-limited functions, the Nyquist density is defined as that density corresponding to maximally packed spectral replications. Because of the shape of the support of the spectrum, however, sampling multidimensional functions at Nyquist densities can leave gaps among these replications. In this paper we show that, when such gaps exist, the image samples can periodically be deleted or decimated without information loss. The result is an overall lower sampling density. Recovery of the decimated samples by the remaining samples is a linear interpolation process. The interpolation kernels can generally be obtained in closed form. The interpolation noise level resulting from noisy data is related to the decimation geometry. The greater the clustering of the decimated samples, the higher the interpolation noise level is.

Synchronous vs asynchronous behavior of Hopfield's CAM neural net.

The performance of Hopfields neural net operating in synchronous and asynchronous modes is contrasted. Two interconnect matrices are considered: (1) the original Hopfield interconnect matrix; (2) the original Hopfield interconnect matrix with self-neural feedback. Specific attention is focused on techniques to maximize convergence rates and avoid steady-state oscillation. We identify two oscillation modes. Vertical oscillation occurs when the nets energy changes during each iteration. A neural net operated asynchronously cannot oscillate vertically. Synchronous operation, on the other hand, can change a nets energy either positively or negatively and vertical oscillation can occur. Horizontal oscillation occurs when the net alternates between two or more states of the same energy. Certain horizontal oscillations can be avoided by adopting appropriate thresholding rules. We demonstrate, for example, that when (1) the states of neurons with an input sum of zero are assigned the complement of their previous state, (2) the net is operated asynchronously, and (3) nonzero neural autoconnects are allowed, the net will not oscillate either vertically or horizontally.

Performance analysis of associative memories with nonlinearities in the correlation domain.

A matched filter-based architecture for associative memories (MFAMs) has been proposed by many researchers. The correlation from a leg of a matched filter bank, after being altered nonlinearly, weights its corresponding library vector. The weighted vectors are summed and clipped to give an estimate of the library vector closest to the input. We analyze the performance of such architectures for binary and/or bipolar inputs and libraries. Sufficient conditions are derived for the correlation nonlinearity so that the MFAM outputs the correct result. If, for example, N bipolar library vectors are stored, theicorrelation nonlinearity Z(x) = N(x/2) will always result in that library vector closest to the input in the Hamming sense.

Optical-processor architectures for alternating-projection neural networks

Optical-processor architectures for various forms of the alternating-projection neural network are considered. Required iteration is performed by passive optical feedback. No electronics or slow optics (e.g., phase conjugators) are used in the feedback path. The processor can be taught a new training vector by viewing it only once. If the desired outputs are trained to be either +/-1, then the network can be configured to converge in one iteration.

Conventional and composite matched filters with error correction: a comparison.

A common pattern recognition problem is finding a library object which most closely matches a received image. For additive white Gaussian input noise, optimal detection performance is obtained using a matched filter for each of the N possible library objects. The use of composite matched filters (CMFs) (also called synthetic discriminant functions or linear combination filters) is one technique of reducing the number of filters required for the recognition problem. For two-level composite matched filter outputs, the reduction is from N to Q = log(2) (N) filters. The CMFs performance, however, can be suboptimum. Using CMFs with bipolar (+1,-1) outputs, this paper examines the detection performance improvement obtained by using error correcting codes. Use of varying levels of error correction is shown to allow trade-off between detection probability and the number of bank filters. Also, we show that in the case of inexact processing, the CMF can perform better than the conventional matched filter.

Composite matched filter output partitioning

A common pattern recognition problem is finding a library element closest, in some sense, to a given reception. In many scenarios, optimal detection requires N matched filters for N library elements. Since N can often be quite large, there is a need for suboptimal techniques that base their decisions on a reduced number of filters. The use of composite matched filters (CMFs) (also called synthetic discriminant functions or linear combination filters) is one technique to achieve this reduction. For two level CMF outputs, the reduction is from N to log(2)N matched filters. Previously, the coefficients of the CMF output were restricted to positive values-often 0 and 1. We refer to such filters as binary CMFs. An alternative approach is to use -1 and +1 for filter coefficients. This alternative filter will be called a bipolar CMF. This paper demonstrates how the extension from a binary to a bipolar CMF greatly improves the detection performance while still maintaining the reduced computational requirements of the binary CMF. Furthermore, the bipolar CMF is invariant to scale: multiplying the input by a positive constant gives the same processor output. This desirable behavior does not exist for the binary CMF.

A Class of Continuous Level Neural Nets

A neural net capable of restoring continuous level library vectors from memory is considered. The vectors in the memory library are used to program the neural interconnects. Given a portion of one of the library vectors, the net extrapolates the remainder. Sufficient conditions for unique convergence are stated. An architecture for optical implementation of the network is proposed.

Convergence of Howard's minimum-negativity-constraint extrapolation algorithm

Howard’s minimum-negativity-constraint extrapolation algorithm is shown to be a special case of signal recovery by means of alternating convex set projections. Previously derived results in this richly developed field of analysis [ Appl. Opt.25, 1670 ( 1986); J. Opt. Soc. Am.71, 819 ( 1981)] are applied immediately to establish strong convergence for the extrapolation algorithm.

Architectures For A Continuous Level Neural Network Based On Alternating Orthogonal Projections

Optical processor architectures for various forms of the alternating projection neural network (APNN) are considered. Required iteration is performed by passive optical feedback using only free space and guided propagation. No electronics or slow optics (e.g. phase conjugators) are used. The processor can be taught a new training vector by viewing it only once.