In this paper, we introduce two novel techniques for digital color halftoning with green-noise--stochastic dither patterns generated by homogeneously distributing minority pixel clusters. The first technique employs error diffusion with output-dependent feedback where, unlike monochrome image halftoning, an interference term is added such that the overlapping of pixels of different colors can be regulated for increased color control. The second technique uses a green-noise mask, a dither array designed to create green-noise halftone patterns, which has been constructed to also regulate the overlapping of different colored pixels. As is the case with monochrome image halftoning, both techniques are tunable, allowing for large clusters in printers with high dot-gain characteristics, and small clusters in printers with low dot-gain characteristics.

We introduce the concept of green noise-the mid-frequency component of white noise-and its advantages over blue noise for digital halftoning. Unlike blue noise, which creates the illusion of continuous tone by spreading the minority pixels of a binary dither pattern as homogeneously as possible, green noise forms minority pixel clusters which are themselves distributed as homogeneously as possible. By clustering pixels, green noise patterns are less susceptible to image degradation from printer distortions such as dot-overlap (the overlapping of a printed dot with its nearest neighbors), and by adjusting the average number of pixels per cluster, green noise patterns are tunable to specific printer characteristics. Using both spectral and spatial statistics, are establish models for ideal green noise patterns.

In this paper, we introduce the concept of green noise-the multifrequency component of white noise-and its advantages over blue noise for digital halftoning. Unlike blue-noise dither patterns, which are composed exclusively of isolated pixels, green-noise dither patterns are composed of pixel-clusters making them less susceptible to image degradation from nonideal printing artifacts such as dot-gain. Although they are not the only techniques which generate clustered halftones, error-diffusion with output-dependent feedback and variations based on filter weight perturbation are shown to be good generators of green noise, thereby allowing for tunable coarseness. Using statistics developed for blue noise, we closely examine the spectral content of resulting dither patterns. We introduce two spatial-domain statistics for analyzing the spatial arrangement of pixels in aperiodic dither patterns, because green noise patterns may be anisotropic, and therefore spectral statistics based on radial averages may be inappropriate for the study of these patterns.

Donoho and Johnstone (1992) first introduced wavelet shrinkage as a denoising technique for signals embedded in Gaussian noise, but due to the linearity of wavelet decomposition, wavelet shrinkage is ineffective in non-Gaussian noise which exhibits outliers. We evaluate two schemes which have been developed to extend the denoising capabilities of wavelet shrinkage to signals corrupted by non-Gaussian noise. The first scheme introduced by Bruce et al. (see Proceedings SPIE Conference, 1994) smoother-cleaner wavelets integrates median filters into the wavelet decomposition. The second scheme, introduced by the authors, replaces the linear filters of wavelet decomposition with order statistic based Chameleon filters. We also show that a straight forward extension of these schemes to images does not offer the same effectiveness in denoising as they do with one dimensional signals.

It is noted that a necessary requirement of a strategic defense system is the detection of incoming nuclear warheads in an environment that may include nuclear detonations of undetected or missed target warheads. A computer model is described which simulates incoming warheads as distant endoatmospheric targets. A model of the expected electromagnetic noise present in the nuclear environment is developed; predicted atmospheric effects are included. Various morphological-based image-enhancement algorithms are examined with regard to their ability to suppress the noise and atmospheric effects of the nuclear environment. These algorithms are then tested, using the combined target and noise models, and evaluated in terms of noise removal and the ability to resolve closely spaced targets.>

Vector probability measure functions (density function) for recursively median filtered signals are found when the underlying input binary sequences are either independent identically distributed (i.i.d.) or Markov chains. The results are parametric in the window size of the filter and in the probability distribution of the input sequence. Using statistical threshold decomposition, the same results are found for discrete alphabet random sequences that are either i.i.d. or Markov chains. Some examples illustrating the efficacy of the recursive median filter relative to the nonrecursive implementation are presented. In particular, the breakdown probabilities are tabulated for both recursive and nonrecursive median filters.>

The properties of the median filter can be used to great advantage in the improvement of television images corrupted with "sparklie" noise. In this paper we describe the application of the median filter to a five second noisy movie sequence received from a satellite antenna with a weak signal. The sequence of two dimensional TV images is decomposed into a set of one dimensional signals. Each one dimensional signal consists of the succession of values a fixed picture element (pixel) takes as successive frames arrive for display. The median filter is applied separately to each of these one dimensional signals. It is shown that the portion of TV frames not containing noise almost always appears as a root signal to the median filter. Finally a real time implementation of the filtering scheme is proposed.

Median filters have been used with success in a number of image processing applications where they are often used to smooth noisy images without perturbing edge structures. In particular, square window median filters and separable median filters have been popular. This paper addresses the statistical properties of these operations. Specifically, the output distribution for the separable two dimensional median filter and the square window two dimensional median filter have been derived for several cases, including first order and multivariate output distributions with white noise, signals plus white noise, and general input images with multivariate distributions. These are then used to compute some specific statistics in several illustrative examples.

The distribution of the output of the one-dimensional median filter is derived for several cases including thekth-order output distribution with any input distribution. This is then used in several illustrative examples of median filtering a signal plus white noise.

The two dimensional median filter has recently become popular as a technique to smooth images without rounding edges. The root structures (image segments invariant to filtering) of these nonlinear operations are important to the understanding of this filter's properties. The root structures of the two dimensional separable median filter have been developed and will be presented here. In addition, it has been proven that with some rare exceptions any two dimensional image will converge to such a root structure under repetitive passes of the separable median filter.

In this paper we source encode the truncated block used in block truncation coding. It is shown that the truncated block is well approximated by wide-sense Markoff statistics; a signal having these characteristics has a high probability of belonging to the root signal set of median filters. Because the root signal space is much smaller than the binary space, it takes fewer bits to specify the truncated block in the root signal space, obtaining in this manner rate compression. Using two-dimensional filtering we can reduce the standard BTC rate of 1.63 bits/pel to 1.31 bits/pel. Using one-dimensional filtering along with a trellis encoder, rates close to 1.1 bits/pel are obtained with this fixed-length coding method.

This paper considers the problems associated with computing optimal minimum mean-square error quantizers. Most computational methods in current use are iterative. These iterative schemes are extremely sensitive to initial conditions. Various methods of obtaining good initial conditions are presented and discussed.

Two results in minimum mean square error quantization theory are presented. The first section gives a simplified derivation of a well-known upper bound to the distortion introduced by ak-dimensional optimum quantizer. It is then shown that an optimum multidimensional quantizer preserves the mean vector of the input and that the mean square quantization error is given by the sum of the component variances of the input minus the sum of the variances of the output.

The problem of minimum mean-squared error prediction of a discrete-time random process using a nonlinear filter consisting of a zero-memory nonlinearity followed by a linear filter is studied. Classes of random processes for which the best predictor is realizable using a nonlinear filter of the above form are discussed. For those random processes for which the best predictor is not realizable using the above nonlinear filter, an iterative procedure is presented for finding a suboptimal nonlinear filter; special attention is directed to the case where the nonlinearity is a polynomial. Also, a noniterative approach based on nonlinear regression is presented.

The theoretical advantages of two-dimensional quantization over univariate quantization have been studied in the literature. However, in many cases there is no known implementation for the two-dimensional quantizer that can operate in real time. A new approach to the design of two-dimensional quantizers is presented. This technique, called prequantization, is used to design two-dimensional quantizers that operate in real time. The importance of prequantization is demonstrated by the design of the optimum uniform two-dimensional (hexagonal) quantizer. Additional examples are given to illustrate the flexibility of this design approach.

Two classes of random processes that exhibit one-step predictors with optimal nonlinear minimum mean-squared error (MMSE) are discussed, and conditions for membership to one of these classes are given. Examples of each class are presented, and the optimal one-step predictors are given.

Some properties of the optimal mean-square error uniform quantizer are treated. It is shown that the mean-square error (mse) is given by the input variance minus the output variance. Furthermore\lim_{N \rightarrow \infty}mse/(\Delta^{2}/12) \geq 1, whereNis the number of output levels and\Delta(a function ofM) is the step size of the uniform quantizer, with equality when the support of the random variable is contained in a finite interval. A class of probability densities is given for which the above limit is greater than one. It is shown that\lim_{N \rightarrow \infty}N^{2} \cdotmse=(b-a)^{2}/12, where(b-a)is the measure of the smallest interval that contains the support of the input random variable.

For a general class of optimal quantizers the variance of the output is less than that of the input. Also the mean value is preserved by the quantizing operation.

The problem of quantizing two-dimensional Gaussian random variables is considered. It is shown that, for all but a finite number of cases, a polar representation gives a smaller mean square quantization error than a Cartesian representation. Applications of the results to a transform coding scheme known as spectral phase coding are discussed.

The problem of quantizing a two-dimensional random variable whose bivariate density has circular symmetry is considered in detail. Two quantization methods are considered, leading to polar and rectangular representations. A simple necessary and sufficient condition is derived to determine which of these two quantization schemes is best. If polar quantization is deemed best, the question arises as to the ratio of the number of phase quantizer levels to that or magnitude quantizer levels when the product of these numbers is fixed. A simple expression is derived for this ratio that depends only upon the magnitude distribution. Several examples of common circularly symmetric bivariate densities are worked out in detail using these expressions.

The problem of quantizing a large-dynamic-range, possibly nonstationary signal after it has been transformed via the discrete Fourier transform (DFT) is investigated. It is demonstrated that, for purposes of d, the polar-form representation for these DFT coefficients is preferable to the Cartesian-form when fixed-information-rate quantization schemes are considered. A technique called spectral phase coding (SPC) is described for transforming the DFT coefficients into a bounded sequence\{\psi_{p}\}, where- \pi < \psi_{p} \leq \pi. In most cases, the terms\psi_{p}are uniformly distributed over this range. The results indicate that SPC is a robust suboptimum procedure for coding nonstationary or large-dynamic-range signals into digital form.

It is shown that a random process is spherically invariant if and only if it is equivalent to a zero-mean Gaussian process multiplied by an independent random variable. Several properties of spherically invariant random processes follow in a simple and direct fashion from this representation.

A novel approach to discrete signal encoding is presented. This approach is based on techniques employed in digital holography for the computation of phase-only holograms [1]. The decoded sequence exhibits a burst error immunity which is characteristic of holographic reproductions. Also, a nonrandom characterization of quantization error is presented that indicates a decoding procedure for reducing the quantization error in the decoded sequence.