Vector median-type filters for color video signals are introduced. In the vector median approach the samples of the vector-valued input signal are processed as vectors as opposed to componentwise scalar processing. The VM-type filters utilize the correlation between different components in video signal. Different components in the color video signal correlate strongly. If there is a change in one component, a change is also likely to happen in the other components. This makes the VM-filters attractive in video signal processing. In this paper the performance of vector median filters for color video signals are investigated. Recursive vector median filters for cross luma and cross color cancellation are presented. The vector median operation is combined with linear filtering resulting in improved cross colour and cross luma attenuation and in very good noise attenuation.
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In this paper, we consider the extension of ranking a set of elements in R to ranking a set of vectors in a p'th dimensional space R^{p}. In the approach presented here vector ranking reduces to ordering vectors according to a sorted list of vector distances. A statistical analysis of this vector ranking is presented, and these vector ranking concepts are then used to develop ranked-order type estimators for multivariate image fields. We develop a class of vector filters which are efficient smoothers in additive noise and can be designed to have detail-preserving characteristics. A statistical analysis is developed for the class of filters and a number of simulations were performed in order to quantitatively evaluate their performance. These simulations involve the estimation of both stationary multivariate random signals and color images in additive noise.
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In this paper, based on the fact that the output of a weighted median (WM) filter is always one of the samples in the input window, rank and sample selection probabilities are defined. The former is the probability that a certain ranked sample will appear as the output and latter is the probability that the output equals one of the time-indexed samples. Using the rank selection probabilities, it is shown here that the output distribution of the WM filter of size N with independent identically distributed (i.i.d.) inputs is a weighted sum of the distributions of the i^{th}, i-1, 2, ... , N order statistics. The weights are given by the rank selection probabilities. The sample selection probabilities are the coefficients of the finite impulse response (FIR) filter whose output, of all linear filters, is closest to that of the WM filter. Several statistical properties of WM filters using selection probabilities are then derived. A method to compute the selection probabilities from the weights of the WM filter is also given.
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In this paper we discuss some hybrid filter structures which generalize linear filters and order statistic filters. The objective is to obtain multipurpose filters that can deal with noise and images of various type. Algorithms for computation of the coefficients are presented with simulations on two images.
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This paper discusses two examples of the design and use of order statistic filters in image compression. The first example is a pre-filter. The pre-filter is designed to gently smooth the image to promote better compression. However, this filter must not blur edges significantly. Various order statistic estimators, such as a simple median, are ideal at this. In the examples given, we are able to achieve between 10 and 20% fewer bits while actually improving image quality. The second example is as a postfilter. The post-filter, used at low bitrates when image degradation is present, is designed to "average out" contouring effects while not averaging across edges.
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The design of a linear filter is quite straight forward especially if the spectra of the required and unwanted signals are known and do not overlap with each other. Basically, a linear filter is implemented by tuning its passband to the required signal and adjusting its stopband to the unwanted signal. However, in case of two or higher dimensional image filtering, we normally only have knowledge about the shape of the unwanted feature and have little or no knowledge about its spectrum. For non-linear filters such as the 2-D standard median filters, their filtering characteristics can only be roughly controlled by the size and the shape of the filter window. The unwanted feature cannot be specified in details and hence some related but required features may also be removed. In this paper, a novel median based feature selective filtering technique is introduced. Unwanted features of any particular shape can be specified in details by a set of custom-tailored shells and all features, other than those specified features which are to be removed, can be preserved regardless of feature orientations. In particular, this technique can be used to remove very high density impulsive noise from corrupted images.
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Let X_{1}, X_{2}, ... be a stationary sequence of random variables with Pr(X_{t} less than or equal to x) = F(x), t = 1, 2, ... . Also let X_{i:n}^{(t)}, i = 1, ..., n, denote the i-th order statistic (OS) in the moving sample (X_{t-N}, ..., X_{t}, ..., X_{t+N}) of odd size n = 2N + 1. Then Y_{t} = (summation)a_{i}X_{i:n}^{(t)} with (summation)ai = 1 is an order statistics filter. In practice a_{i} greater than or equal to 0, i = 1, ..., n. For t > N, the sequence (Y_{t}) is also stationary. If X_{1}, X_{2}, ... are independent, the autocorrelation function ρ(r) = corr(Y_{t}, T_{t+r}) is zero for r > n-1 and for r less than or equal to n-l can be evaluated directly in terms of the means, variances, and covariances of the OS in random samples of size n + r from F(x). In special cases several authors have observed that (Y_{t}) produces a low-pass filter. It will be shown that this result holds generally under white noise. The effect of outliers (impulses) is also discussed.
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The median filter has gained recognition as a filter that will edit impulses and retain edges. Order statistic (OS) filters such as the a-trimmed mean have also proven useful in providing smothers that also eliminate impulses. At first implementation was slow, but due to the introduction of the stack filter may now be implemented in real time. These filters have been extended to the generalized order statistic filters (GOS) which include the least median of squares (LMS) filter. The LMS filter averages the N+1 closest grouped (algebraic range) values in a window of size 2N+1 . We show the LMS eliminates impulses and preserves "perfect" edges exactly as the median. In addition the LMS is shown to enhance "nonperfect" edges.
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We derive the maximum likelihood (ML) estimators for estimating locally monotonic signals embedded in white additive noise, when the noise is assumed to have a density function that is a member of a family of generalized exponential densities with parameter p that includes the Laplacian (p = 1), Gaussian (p = 2) and, as a limiting case, the uniform (p = ∞) densities. The estimators are given by the so-called locally monotonic regression of the noisy signal, a tool of recent introduction in signal processing. The approach that is used in the paper results from a geometric interpretation of the likelihood function of the sample; it takes advantage of the fact that a term in the likelihood function is the p-distance between the vector formed by the data in the given signal (sample) and the vector formed by the elements in the desired signal (estimator). Isotonic regression is a technique used in statistical estimation theory when the data are assumed to obey certain order restrictions. Local monotonicity is a generalization of the concept of isotonicity which is useful for some problems in signal processing.
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In this paper we have proposed a new type of filter which has the most desirable properties of an image smoothing filter. These properties are (1) Robust smoothing efficiency. (2) Edge preservation. and (3) Thin-line detail preservation. The new filter is related to Hodges-Lehman D filter, which is the median of averages of symmetrically placed order statistics. Though it has robust smoothing efficiency, D filter cannot preserve edges or thin-line details. It is shown in this paper that by incorporating a subsampling scheme derived in this paper with the robust D filtering process, the edges as well as the thin-line details can be preserved. The new filter computes its output as the median of weighted averages, instead of plain averages of symmetrically placed order statistics. One particular weighting scheme is considered in details for experiments. The experimental and comparison results are included verifying the useful properties of the proposed filter. To carry out the comparison experiments some new measures for edge and detail preservation are also proposed in the paper.
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This paper considers an image filter to remove small features of low contrast based on a simple model of a quantum limited detector. That is, it removes image noise that can't be seen or that, in other circumstances, can't represent real information. The filtering scheme asks how large an area can be covered with one color without introducing visible departures from the original image. We use a quad tree structure to examine progressively larger image areas until we reach a point that setting the area to one color would obscure visible image features. We have applied these algorithms to a number of grey scale images, ranging from finely detailed images of high contrast to simple classroom video scenes without much fine detail. We have seen reductions by factors from four to twelve in the number of leaf nodes in the quad tree representation of the filtered images relative to the original images. We have also experimented with the filtering of difference images from the classroom video sequence which was made with a stationary camera and have seen substantial further reductions in quad tree complexity for the difference images by factors of two to four.
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A simple but efficient filtering method, suited to fit 3D ultrasonic images features, has been developed, using the order statistics filtering principle. The main idea is to use only a combination of 2D filters to process a 3D image, thus sparing a large amount of processing time. This method is possible because of some properties of the ultrasonic images. This paper presents results obtained using this method, and explains some techniques used to evaluate its efficiency and to analyze the effect of parameters variations.
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This paper reports on new results using three-dimensional binary ranking transforms. These transforms use the face-centered-cubic (FCC) tessellation which is the extension of the planar hexagonal transform in two dimensions. In this tessellation each voxel is surrounded by 12 equidistant neighbors. This leads to an interesting computational structure where any transform may be calculated by addressing a 8192-location lookup table. Thresholded medical images from CT or MR scanners have been processed to locate and measure volumes, both interior and exterior surface areas, surface convexity and concavity, tunnels, etc. Graylevel images may also be processed by column encoding where columns of voxels are erected at each x,y location whose height in the z-direction is proportional to the value of the graylevel image at that point. Using column-encoded graylevel images, three-dimensional mathem atical morphology transforms have been found which correspond to highpass, bandpass, and lowpass filters. These filters have the remarkable properties of sharp cutoffs (as steep as -60dB per octave) with no phase shifts. This paper presents several examples in military target detection, medical image analysis, and computer graphics.
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This paper presents a dual architecture for the high-speed realization of basic morphological operations. Since morphological filtering can be described as a combination of erosion and dilation, two basic building blocks are required for the realization of any morphological filter. Architectures for the two basic units, namely the erosion unit and the dilation unit, are proposed and studied in terms of cycle time, hardware complexity, and cost. These basic units are similar in structure to the systolic array architecture used in the implementation of linear digital filters. Correspondingly, the proposed units are highly modular and are suitable for efficient VLSI implementation. These basic units allow the processing of either binary or gray-scale images. They are particularly suitable for applications in robotics, where speed, size and cost are of critical importance.
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The present paper introduces a digital gray-scale morphological filtering technique that is based on postfiltering a given filter so the invariant class of the new filter is larger than the invariant class of the original filter. More specifically, the invariant class of the new filter contains the original invariant class together with all signals whose variation is below a chosen threshold. The postfiltering includes a single erosion and a single dilation by a one-parameter structuring element, the choice of parameter determining the resulting invariant class. While the methodology is quite general, the two applications considered pertain to moving averages with nonnegative weights and moving means, both of which are morphological filters. Both suppress noise in a signal and each possesses specific advantages and disadvantages. In brief, means tend to give better noise suppression while blurring edges, whereas medians preserve edges, while at the same time flattening small background variation in the underlying signal. The new one parameter family of filters derived from a moving average, called pseudoconvolutions, preserve steps beneath a predetermined threshold. The filters derived from medians, called pseudomedians, preserve uncorrupted low background variation. Because they preserve small variation while at the same time behaving like their mother filters, both filters are especially effective when the noise occurs in bursts, rather than uniformly across the signal.
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Mathematical Morphology is a new branch of mathematics powerful enough to study some vision problems like multiscale filtering. Due to the fact morphological openings smooth the signal while preserving the edges, and using the three Matheron's axioms, an important result is obtained: morphological openings do not introduce additional zero-crossing as one moves to a coarser scales. With these results a multiscale filtering scheme is developed. The choice of the structuring element is constrained to the sub-space of convex, compact and homothetic ones. In this paper we will report a procedure for choosing the structuring element based on the pre-filtering effects of morphological openings and the subsequent detection of edges.
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An adaptive filtering algorithm is developed for the class of stack filters, which is a class of nonlinear filters obeying a weak superposition property. The adaptation algorithm can be interpreted as a learning algorithm for a group of decision-making units, the decisions of which are subject to a set of constraints called the stacking constraints. Under a rather weak statistical assumption on the training inputs, the decision strategy adopted by the group, which evolves according to the proposed learning algorithm, can be shown to converge asymptotically to an optimal strategy in the sense that it corresponds to an optimal stack filter under the mean absolute error criterion. This adaptive algorithm requires only increment, decrement and comparison operations and only local interconnections between the learning units. Implementation of the algorithm in hardware is therefore very feasible.
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We extend the current theory of stack filters by determining fixed points for the infmite length signals using positive Boolean functions. Fixed point analysis is basically determining root structure. We describe the fixed point structure for three and four variable positive Boolean functions and describe how one may generate larger positive Boolean functions and simultaneously generate their fixed point structure. We give two explicit formulations for the stack filter that correspond to any positive Boolean function. We then discuss explicit algorithms for generalizing positive Boolean functions that will "stack" on each other. We define a generalized stack filter as the filter that has varying positive Boolean function at various levels. We provide sufficient conditions so that positive Boolean functions in a generalized stack filter can be varied at levels and still uphold threshold decomposition and stacking properties. We also describe a rule which must be followed in the stacking of rank order filters if the stacking property is to be satisfied. Design of stack filters is also presented.
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Stack filters are generalizations of median filters; a stack filter is a composition of local minimum and maximum operators. A major question in the study of these filters is whether a particular stack filter will make any signal of finite extent converge to an invariant signal--i.e. whether it will "filter out all the noise." Here, we consider the class of stack filters that reduce to symmetric threshold functions for binary inputs. We show that, if we surround an n-dimensional signal with an arbitrary fixed boundary, then any n-dimensional stack filter from the class will make the signal converge to an invariant signal, or a cycle of period 2, in a finite number of iterations. If we make the stack filter recursive, it will always filter the signal to an invariant signal, no matter how the filter moves over the signal. Our results follow from similar theorems on the convergence of neural networks. Many known useful filters are governed by our results. They include all 1-dimensional ranked-order filters with symmetric windows, and all 2-dimensional ranked-order filters with windows that are invariant under a 180° rotation.
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Two techniques for image restoration are compared in this paper. One is a technique based on the theory of optimal adaptive stack filtering; the other is a recently developed vector detection approach to image restoration. The primary difference between these two techniques is that the optimal detection technique exploits multilevel a priori information, while the stack filter uses only single level information. Both approaches have very similar design constraints: (a) both rely on the existence of a training sequence for the image source in order to obtain optimal processing; (b) the underlying random fields need not be stationary and a direct computation of the statistics of the desired images is not required. Adaptive stack filters do, however, require a training set of the noise while the optimal detection approach only needs a multivariate parametric representation. The image restoration performance of these two methods is compared in a signal dependent noise environment characterizing imaging systems with speckle, film-grain, and Poisson shot noise. Comparisons are made using the Mean Absolute Error measure as well as a subjective measure.
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Several wraparound image processing environments are introduced. Operations in these environments are given along with equational identities. Mappings between wraparound environments such as triple convolution and Volterra convolution are also presented in pointwise and parallel algorithms.
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This paper presents the implementation of MSHELL, a new interactive image processing language which facilitates the development of linear and non-linear image processing algorithms. MSHELL permits professionals to concentrate on algorithm development rather than on programming specifics. Using an off-the-shelf array processor and a frame grabber board, a PC-AT implementation of MSHELL which provides floating-point real-time performance is demonstrated.
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The application of an adaptive plan to the design of a class of nonlinear digital image processing operators known as stack filters is presented in this paper. The adaptive plan is based on the mechanics found in genetics and natural selection. Such learning mechanisms have become known as genetic algorithms. A stack filter is characterized by the coefficients of its underlying positive Boolean function. This set of coefficients constitute a binary string, referred to as a chromosome in a genetic algorithm, that represents that particular filter configuration. A fitness value for each chromosome is computed based on the performance of the associated filter in specific tasks such as noise suppression. A population of chromosomes is maintained by the genetic algorithm, and new generations are formed by selecting mating pairs based on their fitness values. Genetic operators such as crossover or mutation are applied to the mating pairs to form offsprings. By exchanging some substrings of the two parent-chromosomes, the crossover operator can bring different blocks of genes that result in good performance together into one chromosome that yields the best performance. Empirical results show that this method is capable of configuring stack filters that are effective in impulsive noise suppression.
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We describe the design of an image-recognition system and its performance on multi-sensor imagery. The system satisfies a list of natural requirements, which includes locality of inferences (for efficient VLSI implementation), incorporation of prior knowledge, multi-level hierarchies, and iterative improvement. Two of the most important new features are: a uniform parallel architecture for low-, mid- and high- level vision; and achievement of recognition through short-, as opposed to its long-time behavior, of a dynamical system. Robustness depends on collective effects rather than high precision of the processing elements. The resulting network displays a balance of high speed and small size. We also indicate how this architecture is related to the Dempster-Shafer calculus for combining evidence from multiple sources, and present novel methods of learning in such networks, including one that addresses the integration of model-based and data-driven approaches.
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A holographic implementation for neural networks is proposed and demonstrated as an alternative to the optical matrix-vector multiplier architecture. In comparison, the holographic architecture makes more efficient use of the system space-bandwidth product for certain types of neural networks. The principal network component is a thermoplastic hologram, used to provide both interconnection weights and beam redirection. Given the updatable nature of this type of hologram, adaptivity or network learning is possible in the optical system. Two networks with fixed weights are experimentally implemented and verified, and for one of these examples we demonstrate the advantage of the holographic implementation with respect to the matrix-vector processor.
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