A new adaptive algorithm is developed in this paper for determining optimal generalized stack (GS) filters under the mean absolute error criterion. This algorithm, based on the neural network representation of Boolean functions, is much more efficient than the traditional truth table based algorithms. This is because: (1) the number of variables to represent a GS filter is considerably reduced when a set of neurons is used to represent a GS filter, where the number of the variables is proportional to the filter window width, and (2) the procedure of enforcing the stacking constraints of GS filters is greatly simplified since a sufficient condition is derived under which the neurons satisfy the stacking property. Experimental results from image restoration are provided to demonstrate the performance of the new adaptive GS filters.

The general characterization of optimal morphological filters is based on the Matheron representation for morphological filters. As conceived in its most general form, optimal- morphological-filter design involves a search over potential bases of structuring elements that can be used to form the Matheron erosion expansion. The present paper provides expressions for the mean-absolute error of general binary morphological filters formed from erosion bases in terms of mean-absolute errors of single-erosion filters. The theorem has a recursive form and use of this form in the derivation of optimal-filter bases is demonstrated.

In this paper, statistical properties of standard and soft morphological filters are analyzed using stack filter representation. An asymptotically tight bounds are derived for the outputs of two-dimensional morphological filters. It is shown that soft morphological filters are less sensitive to noise than standard flat morphological filters. Simulation results illustrating this behavior are presented.

The median filter is a nonlinear filter that preserves edges and eliminates impulses. The initial papers on the median filter, by Tyan and Gallagher and Wise, concentrated on the development of their impulse rejection properties, the set of fixed points, and convergence. It was shown that the fixed points of the median filter are the class of LOMO (locally monotone) signals, that they converge within a finite number of iterations, and that they would reject burst of up to n aberrant values in each nonoverlapping segment of length 2n + 1. Their initial work has led to many research papers in robust signal processing in the presence of edges. The order-statistic (OS) filter is very similar to the FIR filter with the exception that it orders the values in each window before weighing them. The WMMRc filters weight the c ordered values in the window with minimum range. If more than one set of values in the window have the minimum range, the average of the possible outputs is taken. If c is not specified it is assumed to be N + 1 for a window of length 2N + 1. For OS and WMMR filters with convex (sum to one and are nonnegative) weights, fixed point results are derived similar to those of Gallagher and Wise for the median filter, i.e., the fixed points are completely classified under the assumption of a finite length signal with constant boundaries.

In this paper, an optimal multilevel stack filtering algorithm (the stack filters used at each level are designed to be optimal) is introduced. Optimal multilevel stack filter is capable of considering a larger window mask than an optimal stack filter. Multilevel unidirectional stack filters and multilevel bidirectional stack filters are studied in particular. Design of an optimal multilevel stack filter is similar to the design of an optimal stack filter where the compare-and- select algorithm can be used. However, the design procedure needs to be adapted to operation on more than one level. Several design examples are presented showing the good performance of the proposed method.

This paper considers the problem of prefiltering images to enhance edge detection. In particular, several order statistics based sharpeners are considered with a traditional linear technique, unsharp masking. The order statistics sharpeners include the LUM sharpener, the CS-filter, and the GOS filter.

This paper presents a case study of the design of a fully autonomous morphological detection algorithm. Grayscale input images contain objects to be detected among difficult clutter, replacement noise, and background tilt. The criteria for choosing algorithm structure is included, with associated grayscale and binary structuring elements based upon comparing the geometry of target and noise/clutter objects. Background cancellation is discussed, along with histogram-based techniques for final thresholding to binary detection images. Finally a performance characterization methodology for the detection algorithm is presented. In addition to conventional detection statistics, the authors consider the 'quality' of the hits and false alarms, vis-a-vis the feature set and classifier used in classification downstream of the detector in the overall system design.

The morphological pattern spectrum (or, granulometric size distribution) results from iteratively opening an image and at each step recording the area of the opened image. Owing to the manner in which the size distribution is normalized, it defines a probability distribution function, and possesses moments. If a binary image is considered as a random process, then the moments of the pattern spectrum are random variables, and it is these random moments that are employed as shape and texture signatures in image classification and segmentation. In the present paper, the moments of these moments are studied for a random Boolean model that has been employed in various applications. Both exact and asymptotic methods are developed to express the mean and variance of the pattern-spectrum mean and variance.

This paper outlines a methodology for characterizing finite subsets of (eta) d where (eta) is the set of integers and integer d >= 1. The schemes are based on the lattice-based mathematical morphology and on the notion of a pseudo-complement. Three different schemes for multidimensional objects are presented, two of which provide only pseudo- characterizations, while the third scheme provides a complete characterization of objects. The performance of the proposed pseudo-characterizations has been shown to be superior to the similar existing algorithms. The proposed methods lend themselves to efficient implementation on parallel machines.

Morphological operations applied to polygon representation of binary images are investigated, with special attention to the implementations of erosion and dilation. Binary images can be compressed by presenting only the outlines of the objects with 4- or 8-neighborhood chain code. The chain code representation of a binary image can be efficiently eroded (or dilated) with 8- or 4-neighborhood kernels, respectively. This method indirectly uses the following idea: applying morphological operations directly to the compressed (chain coded) images involves less data than applying them to the original binary image. The chain-coded binary image can be further compressed by identifying linear segments on the outline of the objects. If the polygon representation of an object requires less space than the chain code representation, then its morphological filtering should also be faster. The shape of the kernel is presented as a polygon. In morphological operations, the polygon node is either replaced by a new node defined by the intersection of the newly formed vertixes or replaced by some transformed nodes of the kernel. After erosion, the newly formed polygon may cross itself. These situations must be checked and the polygon must be broken accordingly to several smaller polygons. Several efficient clipping algorithms exist. However, clipping of polygons is commonly the most time-consuming part of the presented method.

In mathematical morphology, grey tone images are often considered as 3D Euclidean sets through their umbra or subgraph. This model allows one to extend measurements for sets to grey tone images. It has been shown that any valid measurement on Euclidean sets should satisfy some basic constraints such as invariance to displacements and to magnifications. However, when applied to subgraphs, these measurements may be meaningless as the image plane is not homogeneous with the grey tone axis. An additional constraint is introduced called dimensionality. This property holds for the inhomogeneity of image dimensions. A measurement on a grey tone image will be dimensional if the same measurement applied to this image after a magnification of its image plane and an affine transform of its grey tone axis can be related to the initial measure. The authors first recall valid measurements on sets and their properties. Then it is shown how to generalize to grey tone images and the dimensionality constraint is introduced. Set measurements are then reviewed to determine those satisfying the dimensionality criterion and consideration is given to the measure of the fractal dimension in the light of this new criterion. Eventually, dimensionality must also be considered when processing images. This is illustrated by a segmentation experiment.

Object boundaries are generally characterized by grey level intensity transitions. In order to detect these variations, gradient masks are widely used, and this paper surveys the morphological framework of gradient operators. Morphological gradients are based on the difference between extensive and anti-extensive transformations. For instance dilations and erosions with structuring elements containing their origin belong to this class of transformations. Generally, these gradients are used in segmentation applications with edge finders such as sequential searches, thresholdings or the watershed transformation. The robustness of this latter transformation allows more tolerances for the construction of a gradient operator. After a short introduction to gradients in digital images the gradients available in mathematical morphology are presented: Beucher, internal and external, thick, regularized, directional, and thinning/thickening gradients. Applicability and performance of each gradient are briefly evaluated, followed by a generalization of the morphological framework of gradient operators to other digital sources.

In this paper, a multichannel edge enhancing filter (MEEF) applied to color image processing is proposed to improve the degraded edges in color TV picture which are classified as the following cases: the blurred edges caused by unfocusing or fast shift of camera, the serrated edges caused by interlaced scanning, and the chroma-cross edges caused by channel dispersion Because of a high between-channel dependence in color image, individual edge enhancement in each channel is not a natural method without utilizing between-channel dependence. Thus, this technique is not recommended in color image processing. The proposed MEEF is successfully used in dealing with this sort of the degraded edges of multichannel image. The root signals and edge enhancement properties of MEEF are investigated in detail. The experimental results show that MEEF outperforms independent edge enhancing filter in improving the degraded edges caused by unfocusing, interlaced scanning and channel dispersion.

A continuous time method is introduced to analyze the response of median, pseudomedian, average (mean), and midrange filters to certain periodic signals. The filter definitions are generalized to continuous time, and these definitions are applied to periodic signals such as triangle, square, and sinusoidal waves of varying frequencies. These operations yield 'amplitude response' measures which are analytic functions of the frequency of the input signal. In addition, a 'correlation' measure is defined to indicate the level of distortion introduced by each filter. Examples of this analysis for the median, pseudomedian, average, and midrange filters show similarities and differences among them. Although these theoretical measures do not perfectly demonstrate the performance of the discrete time filters, continuous time analysis does provide valuable insights into the filter behavior. The response of the continuous time median filter shows its susceptibility to high frequency periodic noise and proves, again, the existence of infinite-length bi-valued fast-fluctuating roots of this filter. The pseudomedian filter, in contrast, completely attenuates amplitude-symmetric periodic signals above a certain frequency, and has no infinite-length fast-fluctuating roots. Continuous time filter analogues are therefore an important theoretical tool for understanding the behavior of both linear and nonlinear filters.

One domain in which the ordering filters have not appeared is frequency analysis. Simultaneously one must note that the impulse rejection properties of the ordering filters could be very beneficial due to the lack of robustness of the DFT/FFT. Another problem with the DFT/FFT is the ambiguity of the estimate of frequency at a point (frequency localization). This paper introduces a transform (WMMR/MED/COUNT) that simultaneously solves both of the problems in some cases. The Gabor transform and various wavelet techniques have recently been reviewed as a substitute to FFT frequency analysis for spatial localization. While the Gabor transform optimally infers frequency content and spatial localization simultaneously, it suffers from the fact that it requires a full period within the window. This paper presents a transform based on the WMMR filters that will yield frequency analysis and spatial localization with a window width of 1/4 period or less. Experimentally, it has been shown that this technique can be used with impulsive noise of up to 40% and with random baseline shifts. The short-time Fourier, Gabor transform and the WMMR/MED/COUNT transforms (WMCT) are compared for their localization properties in noisy and noiseless situations.

A DPCM system employing a median predictor, which is called the predictive median-DPCM (PM-DPCM), is proposed. An interesting property that in PM-DPCM transmission noise is often isolated and not propagated over the reconstructed signals is observed and analyzed deterministically as well as statistically. In order to examine the performance characteristics of the PM-DPCM, it is applied to real image signals. The experimental results indicate that the PM-DPCM outperforms the standard DPCM when transmission errors occur, and the former performs like the latter under noise-free conditions.

This paper proposes a new nonlinear filtering structure based on a maximum a posteriori estimation criteria using a Markov random field model for the prior distribution. Estimates obtained with the proposed Markov random field model allow discontinuities in the signal to be accurately estimated while additive Gaussian noise is smoothed. A Markov random field based prior is chosen such that the filter has desirable analytical and computational properties. The estimate of the signal value is obtained at the unique minimum of the a posteriori log likelihood function. This function is convex so that the output of the filter can be easily computed using either digital or analog computational methods. Example outputs under various conditions are given.

Local histogram filtering utilizing feature selective templates consists of ordering the elements of the subimage histogram contained in the support of a nine element square template translating over the image, rather than ordering the subimage intensity values, as in standard order statistic filtering. Subsequently, the ordered local histogram is constructed from the nonzero subimage histogram elements arranged in descending order. If the subimage value contained in the center of the template support is equivalent to the most frequently occurring subimage intensity, corresponding to the 0th order statistic of the ordered local histogram, the central subimage value is preserved. Alternately, if the central subimage value differs from the most frequency subimage intensity, sequential morphological hit-miss transformations are performed on the subimage employing a subset of 36 feature selective templates specified according to the ordered local histogram element values. The feature selective template hit-miss transformations implement heuristic image smoothness criteria, determining if the central subimage pixel is a component of a valid intensity pattern that is preserved. An unsuccessful hit-miss transformation indicates an invalid intensity pattern, resulting in the modification of the central subimage value to the subimage value corresponding to the 0th order statistic of the ordered local histogram. Consequently, in simulated images consisting of linear, rectangular, circular, and spiral geometric patterns corrupted with uniformly distributed impulsive random noise, the resulting filtered images possess mean absolute errors between 2.0 and 3.9 times less than those of images employing median filtering.

The mathematical structure of binary nonlinear filtering is expressed in the context of binary cellular logic and the relevance of existing image algebras is discussed. Operator properties such as antiextensively and idempotence are examined from a discrete logical perspective, as are the classical Matheron representations. The simplicity of the operational properties is exposed by such an approach, as is the use of commonplace logic design methods for the composition and decomposition of nonlinear filters, in particular, binary morphological filters.

A novel method for shape representation based on the multiresolution representation and morphological residues of the binary image is presented in this paper. A shape description method must have three properties: translation, rotation, and size invariance. The binary image which contains the object to be described is represented by multiresolution pyramid. A set of structuring elements is selected and used to compute morphological residues. The structuring elements are rotated versions of the one initial structuring element. This enables us to achieve rotational invariance of the shape description method. The morphological residue is the difference between the original image area and the eroded image area. It is a basic descriptor used in this method. The morphological residues for each of the structuring elements and each of the multiresolution pyramid levels are computed. The obtained set of morphological residues (numbers) is then sorted by order. The sort will enable comparison of rotated versions of objects. The residues for each pyramid level are normalized to enable more accurate comparison (distance measure) between object descriptors. Real vectors obtained in such a way are used as the shape descriptors. Experiments have been performed to investigate noise robustness of the proposed method, and have shown that method is not sensitive to noise. A set of twenty test images has been created and corrupted by noise. The Euclidean distance between the original and the corrupted images has been computed, and has shown that the proposed method has good shape matching properties.

This paper introduces a two-stage adaptive filter for digital image enhancement. This filter has good edge preservation as well as optical enhancement characteristics. The first stage of the filter uses a set of three-layer perceptron neural networks to predict the location of edges. As a result, a piecewise linear model for the image is obtained. With that knowledge, the second stage uses a neural network similar to Hopfield-Tank network to optimally smooth the image to fit the model.

In some industrial optical character recognition applications, the background of the image surrounding the characters is very confusing and contains clutter that often overlays and connects the characters. Characters are found amidst the clutter by applying a number of morphological structuring elements to the image. Each structuring element is responsible for locating a specific class of characters where all characters in that class are similar to each other. The set of all structuring elements efficiently covers the entire character set. To further reduce noise, other checks such as colinearity and equidistance of the characters in the string are applied. This paper describes an automatic training method for defining efficient classes of structuring elements.

This paper introduces a neural network implementation of gray scale operators. In this structure, synaptic weights are represented by a gray scale structuring element and trained by a learning algorithm based on an optimal criterion called the overall equality index. The proposed algorithm leads to a computationally simple implementation, with numerical examples to illustrate its performance.

There is much work concerning morphological image processing, both binary and gray scale. Almost all implementations to date are performed electronically on standard computers, specialized processors, or specialized hardware. Prior work has described implementation of binary morphology on an optical processor, as well as indicating the relative merits of using an optical system. However, the restriction to binary morphology on an optical system has required that gray scale problems be reduced to binary morphology solutions using judiciously chosen binarization thresholds. This paper describes how gray scale morphology can be implemented on an optical correlator system using a threshold decomposition algorithm. A series of thresholded binary correlations are formed optically and summed on a CCD detector array or spatial light modulator, to produce the output morphologically processed gray scale image. The speed this optical system is much faster than 30 gray scale images per second. The details of the architecture used to implement threshold decomposition on an optical system is described, and issues relating to the implementation of binary morphology on an optical system are discussed. The threshold decomposition algorithm is discussed with attention to ways to reduce the number of intermediate processing steps required.

A rigorous proof of quantum parallelism cannot be based on the Heisenberg inequality because the standard deviation of self-adjoint operators in complex Hilbert space is insensitive to fine structures of the collective stationary interference distribution generated by a Mach-Zehnder interferometer from a coherent primary beam. Actually Niels Bohr's indeterminacy principle of spatio-temporal quantum electrodynamics cannot be based on any of the known uncertainty principles. It is shown how the holographic transform allows to circumvent the difficulties with the standard deviation by using a group theoretical implementation of the canonical commutation relations of quantum electrodynamics. The geometric quantization approach combined with the emitter-absorber transaction model of quantum dynamics on the whole real line R allows one to describe by a Liouville density the flow and counter-flow of single optical photons in split fan-in/fan-out coherent photonic channels. It makes the heuristic arguments concerning quantum parallelism rigorous by considering the collective stationary interference distributions of coherent wavepacket densities as symplectic spinors over the linear symplectic manifold modelled on the hologram plane. Consequently the symplectic spinorial organizational form governs photonic holograms. It implies the existence of single-photon holograms and includes the standard uncertainty inequality as a special case.

Acoustic signal tracking is formulated as a simulated annealing problem in this paper. Detection of the track is based on a global optimization of a cost function which relates to the signal to noise ratio. Track can move only 0, -1, or +1 position from time line to time line. The constraint is built into the next state generation of the algorithm. Emphasis is on the small scale localization of track with known starting position. The procedure can detect 1 8db monotone data which is superior to any conventional techniques. Dependency of equilibrium iterations with respect to S/N was revealed in the experimental results. This procedure is also tested with inaccurate starting position in the image which yielded acceptable performance.

The neural network model based on the theory proposed by Wi1son—owan has been simulated using digitized real images. The Wilson—Cowan net can operate in different modes depending on the parameter selection, and it is shown to store images in reduced form and to recognize edges of object. Examples how the net process the input images are shown. Due to large number of neurons in this model, the preferable technique to simulate it should be parallel processing one. Optics serve highly parallelism and we propose a basic hybrid—optical processing unit for the Wilson—Cowan net.

(Xi) -filters were introduced in the mid-1980s by the author and have been used in many applications, including industrial inspection, remote sensing, and medicine. This paper reviews some new developments in each of these areas. The (Xi) -filter is based on high- speed LUT (lookup table) operations wherein small-kernel, nonlinear convolutions are performed on three-dimensional data. In order to execute 3D transforms by LUT, it is necessary to utilize as compact a kernel as possible. This has led to the FCC (face-centered- cubic) tessellation where the kernel comprises only 13 binary data elements or voxels. (This is in contradistinction to the Cartesian tessellation where the kernel comprises 27 voxels.) Since each LUT contains at each of 8192 locations the transformed value of the voxel, the program word that defines a single (Xi) -filter transform is 8192 bits in length. There are, therefore, an essentially infinite number of program words and therefore an infinite number of transforms. In order to delimit the number of transforms to a set that are both useful and manageable, program works limited to the various ranking filters have been generated. In a kernel of 13 elements, there are, of course, only 13 ranks. By iterating (Xi) -filters based on these ranking transforms many interesting operations are possible as illustrated in this paper.

This paper explores the use of morphological operators for feature extraction in range images and curvature maps of the human face. Two general procedures are described. The first is the identification of connected part boundaries for convex structures, which is used to extract the nose outline and the eye socket outlines of the face. The part boundaries are defined locally based on minima of minimum principal curvature on the surface. The locus of these points suggests boundary lines which surround most convex regions on the surface. However, most of these boundaries are not completely connected. To remedy this problem, a general two-step connection procedure is developed: the partial boundaries are first dilated in such a way that the gaps between them are filled. Second, the resulting dilated outlines are skeletonized with the constraint that the pixels belonging to the original boundary parts cannot be removed. A marker which identifies the convex region being described is then used to select the region enclosed by the new connected outline. Examples are given of this procedure in the extraction of the nose boundary and eye socket boundary. The second general procedure discussed is the identification of connected ridge lines, which is demonstrated in the extraction of the browline and the chin/jaw line. Ridge lines are defined as local maxima of maximum curvature in the direction of maximum curvature. The same skeleton-based procedure as above is first used to connect the ridge lines. Skeletonization is then used again to reduce these lines to simply connected ones. The last step primarily consists in extracting the longest path within the obtained components: this is achieved by using the propagation function to find the extremities of these paths and then connecting them within the components by means of geodesic distance functions. The entire process provides a robust and accurate extraction of brow and chin/jaw lines.