Multilevel stack filtering for image processing applications

Optimal multistage (MS) stack filtering under the mean absolute error (MAE) criterion is discussed, with image restoration in mind. Using the recently developed techniques, the optimal MS stack filters are designed efficiently. Compared to single-stage (SS) stack filtering, an MS filter provides an additional design flexibility by choosing the window masks at different stages to provide a good compromise between the noise removal and the image-detail preservation. To achieve a comparable performance, MS filters can select considerably smaller window masks. The relationship between the filtering complexity/memory requirement and the window size suggests that the MS stack filtering is much faster and requires much less memory than SS stack filtering. Results of image recovery experiments are reported. >

A training framework is developed in this paper to design optimal nonlinear filters for various signal and image processing tasks. The targeted families of nonlinear filters are the Boolean filters and stack filters. The main merit of this framework at the implementation level is perhaps the absence of constraining models, making it nearly universal in terms of application areas. We develop fast procedures to design optimal or close to optimal filters, based on some representative training set. Furthermore, the training framework shows explicitly the essential part of the initial specification and how it affects the resulting optimal solution. Symmetry constraints are imposed on the data and, consequently, on the resulting optimal solutions for improved performance and ease of implementation. The case study is dedicated to natural images. The properties of optimal Boolean and stack filters, when the desired signal in the training set is the image of a natural scene, are analyzed. Specifically, the effect of changing the desired signal (using various natural images) and the characteristics of the noise (the probability distribution function, the mean, and the variance) is analyzed. Elaborate experimental conditions were selected to investigate the robustness of the optimal solutions using a sensitivity measure computed on data sets. A remarkably low sensitivity and, consequently, a good generalization power of Boolean and stack filters are revealed. Boolean-based filters are thus shown to be not only suitable for image restoration but also robust, making it possible to build libraries of "optimal" filters, which are suitable for a set of applications.

A deterministic algorithm is proposed to design the optimal stack filter. The proposed algorithm can generate the optimal stack filter in one second for a window size of 9 and it can still generate the optimal stack filter for a window size of 21 although it takes about 4 hours. Experimental results reveal the feasibility and efficiency of the proposed algorithm in finding the optimal stack filter.

Two interpolative BTC (block truncation coding) image coding schemes with optimal stack filters (in the minimum MAE sense) as the interpolator are suggested. Compared to the standard BTC, the new schemes yield >25% and >37.5% reductions in bit rate, respectively, with little performance degradation. Compared to a previous interpolative BTC coding method, the quality of coded images has been greatly improved with the new schemes (at the same or even a lower rate).

Stack filters are a class of nonlinear filters for removing noise that is uncorrelated with signal. Their design is formulated as a high nonlinear optimization problem. An improved immune memory clonal selection algorithm (IMCSA) is suitable for designing stack filters. But as many algorithms for designing stack filters have to face that evaluation of each candidate solution is still takes the most computation complexity. By representing the positive Boolean functions (PBF) in objective function, a hybrid computation approach (software and hardware) for the calculation tasks has been proposed. Compared with available methods, the proposed method can improve the efficiencies of design stack filters and the design time is significantly reduced. The procedure of design and implementation for stack filters can be reached at the same time by the method

The stack filter is the largest class of nonlinear digital filters, including weighted median filters and morphology filters. The design of the stack filter is reduced to the problem of determination of the Boolean function defining its filter and is divided into design methods of estimation type using a training signal, and design methods maximizing the noise elimination capability under the constraint of preserving a specific configuration. With a view to expanding the class of stack filters, the authors have proposed the fuzzy stack filter, a stack filter defined by the fuzzy Boolean function. Also, for the fuzzy weighted median filter, as the most fundamental fuzzy stack filter, a design method based on the estimation standpoint is provided. In this paper, a new fuzzy stack filter preserving a specific structure is explained. The noise elimination characteristics of the fuzzy stack filter preserving a specific structure and the preservation characteristics of the specific structure under the effect of the noise are clarified. Further, a procedure is presented for deriving a filter suitable for restoration of the degraded images in a wide range without departing from the standpoint of a design method preserving the specific structure (for which empirical information on the signal to be processed cannot be obtained). The effectiveness of the filter obtained by this procedure is presented. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 83(2): 11–21, 2000

Nonlinear signal processing elements are increasingly needed in current signal processing systems. Stack filters form a large class of nonlinear filters, which have found a range of applications, giving excellent results in the area of noise filtering. Naturally, the development of fast procedures for the optimization of stack filters is one of the major aims in the research in the field. The objective of the optimization method presented in this paper is to find the stack filter producing optimal noise attenuation and satisfying given constraints. The constraints may limit the search into a set of stack filters with some common statistical description or they may describe certain structures which must be preserved or deleted.

An analytic solution for real optimal filters is known, and the special case of optimal binary phase-only filters can be solved by a fast binning algorithm but no analytic solution is known. We establish a geometric solution for the design of optimal binary amplitude filters (OBAF's) and optimal binary phase-only filters (OBPOF's) for any object. The optimal filter is found in terms of maximizing the field strength at the origin in the correlation plane. We found that it is possible to construct a unique convex polygon by using an ordered set of phasors from the filter object's Fourier transform. This process leads eventually to an exact solution for the filter-design problem. We show that the maximum distance across the polygon divides the phasors into two groups: For the OBAF, it determines the group that is passed or blocked; for the OBPOF, it determines which group is passed with a zero or a pi phase shift. The shape of the convex polygon gives qualitative information on the criticalness and the tightness needed in the design process. It provides good insight into the binning-process algorithm and permits us to bound the error in the binning process. Design examples through computer simulation and applications in fingerprint identification are presented.

An efficient method is proposed for the design of finite impulse response (FIR) filter with arbitrary pass band edge, stop band edge frequencies and transition width. The proposed FIR band stop filter is designed using craziness based particle swarm optimization (CRPSO) approach. Given the filter specifications to be realized, the CRPSO algorithm generates a set of optimal filter coefficients and tries to meet the ideal frequency response characteristics. In this paper, for the given problem, the realizations of the optimal FIR band pass filters of different orders have been performed. The simulation results have been compared with those obtained by the well accepted evolutionary algorithms, such as Parks and McClellan algorithm (PMA), genetic algorithm (GA) and classical particle swarm optimization (PSO). Several numerical design examples justify that the proposed optimal filter design approach using CRPSO outperforms PMA and PSO, not only in the accuracy of the designed filter but also in the convergence speed and solution quality.

In this paper multistage weighted order statistic (MWOS) filters are introduced. With the aid of threshold decomposition, it is shown that any MWOS filter in real domain corresponds to a multistage threshold logic gate, or a multilayer perceptron in binary domain, which can be viewed as another representation of a stack filter. The MWOS filter requires much fewer parameters to represent a stack filter than the truth table of the positive Boolean function. An adaptive filtering algorithm, named as constrained backpropagation (CBP) algorithm, is developed for finding the optimal MWOS filters under the mean absolute error (MAE) criterion. The CBP algorithm is the same as the backpropagation algorithm used in multilayer perceptrons except the positivity of the parameters of MWOS filters, equal to the stacking property of stack filters, is imposed. Simulation results on image restoration are provided to compare the performance of the adaptive MWOS filters and the adaptive stack filters.

Selection probabilities, viz. rank and sample selection probabilities, quantify the impulse suppression, robustness, lowpass filtering ability, and detail preserving ability of stack filters. Closed-form expressions exist for computing selection probabilities from positive Boolean function describing the stack filter. We first present an analysis framework that interprets a Boolean vector to denote membership in a partition set. Using this framework a new algorithm to compute selection probabilities from positive Boolean functions is derived. Next a method to synthesize a stack filter from specified selection probabilities is presented. An underdetermined system of linear Diophantine equations is derived in the unknown elements of the C-matrix and the desired selection probabilities. This system of equations is solved to obtain the candidate solutions for the C-matrix. For each candidate C-matrix, an algorithm based on partition sets is used to derive one or more positive Boolean functions of the stack filter or show that none exists.

A recently developed approach to synthesizing optimal stack filters under the mean absolute error (MAE) criterion (see B. Zong et al., 1991) is extended to FIR (finite impulse response) stack hybrid (FSH) filters, where FIR filters are assumed to be the average filter. This approach is demonstrated through synthesizing a group of optimal FSH filters in the minimum MAE sense for the standard image 'Bridge', where the task is to restore the image from impulsive noise. Practical testing results show that optimal FSH filters can do much better than the corresponding FIR median hybrid filters. >

Filters are important building blocks in signal processing circuits. Some researcher had successfully utilized genetic algorithm (GA) in design of filters. Nevertheless, filters obtained by GA are always complex and require lengthy computations. Ant colony optimization (ACO) is a novel searching technique used in optimization problems. In this paper, an ACO approach for optimization of analog filters is presented. In a design example, the order of a lowpass filter and the parameters of its components have been optimized in a discrete search space. AC analysis of the optimized filter has been conducted, and the results have been compared with a filter obtained by GA. The results show that filters obtained by ACO have simpler structures and better performance.

A new fuzzy filter, called fuzzy stack filter (FSF), is proposed to extend the filtering capability of conventional stack filter (SF), which is based on the positive Boolean function (PBF) as its window operator. We fuzzify the onset and off-set of a given PBF to obtain two types of fuzzy PBFs. Then, we adopt the architecture of threshold decomposition to develop this new fuzzy filter with a fuzzy PBF as its window operator. Each fuzzy PBF is associated with a set of control parameters. Therefore, the original PBF can be estimated from above and below by two fuzzy PBFs with appropriate control parameters. Furthermore, we can apply the fuzzy modifiers to modify the fuzzy PBFs such that the PBFs can be completely estimated by the fuzzy PBFs. Hence, the stack filter is a special case of fuzzy stack filter. Since some control parameters are added in this new filter, the neural learning algorithms can be easily developed under the flexibility of the given control parameters. We first propose the fuzzy (m,n) rank-order filter to test our proposed learning algorithm. In this simple learning algorithm, we can remove the noise-corrupted images very well in contrast to the filtering behavior of rank-order filters. We believe that the results presented will lead to more fruitful research on more advanced and powerful learning algorithms dedicated to the appropriate applications.

Stack filters are a class of nonlinear spatial operators used for noise suppression. Their design is formulated as an optimisation problem and genetic algorithms (GAs) are used to perform the configuration. Applying the mean absolute error (MAE) as the basis of an objective function, the stack filter is used to restore magnetic resonance (MR) images corrupted with uncorrelated additive noise from 10% and 50%. The filter is trained on corresponding patches of the original and noisy image and then applied to the whole image. The outcomes are compared with the median filter and return a smaller MAE for all noise levels. The dependency of MAE on training window size and GA early termination is examined, showing that a reduction of 75% in computational complexity can be achieved by a 10% relaxation in MAE. The design is then extended from 9-point to 13-point filters and by training on Poisson noise, the filter is applied to nuclear medicine bone scans where no absolute truth exists. Surface topology, image profiles and the measurement of relative contrast show its value in reducing noise whilst preserving contrast. Because of its computational complexity the process has been implemented as a distributed GA using the parallel virtual machine (PVM) software.