Stack Filters Research Articles

Fuzzy logic and mathematical morphology: implementation by stack filters

Mathematical morphology has proven invaluable for signal/image processing. Fuzzy theory is enjoying great success in control and signal processing. We show that stack filters, which are already related to morphology, require slight modification to implement basic fuzzy operators. The article therefore provides a fast errorless method for implementation of basic fuzzy operations.

Genetic algorithm implementation of stack filter design for image restoration

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.

PIXE ANALYSIS VS. GRAVIMETRY TO MONITOR THE MASS CONCENTRATION OF ATMOSPHERIC AEROSOLS IN THE RESPIRABLE FRACTION

It is demonstrated in this work, that PIXE is a good mass indicator for suspended particles in urban environments. Results are presented for several samplings performed in urban and other environments in Chile and other countries. Aerosols were collected on filters operated by a Stacked Filter Unit (SFU) in the range of particles < 15 µ m (PM15) and < 10 µ m (PM10), mainly the respirable fraction. For samplings performed in downtown Santiago, the sum of the concentrations of the elements detected by PIXE correlated well with the indices of respirable suspended particles (RSP) measured by gravimetry. There was no high correlation for other samplings, near a crushing plant and near mineral storage. On the average, the sum of the PIXE results for Santiago’s aerosols represents about the 15% of the total mass collected in the respirable fraction. A measurement of collection efficiency for the 0.4 µm Nuclepore filter was performed by means of PIXE and gravimetry. These results were η PIXE =0.98 and η balance =0.96, for the PIXE and the gravimetric method.

Formation of chloroaromatics at a metal reclamation plant and efficiency of stack filter in their removal from emission

Sampling and analyses of chloroaromatic compounds emitted from the metal reclamation plant of Kuusakoski Ltd in Heinola, South Finland, were continued in order to find out formation pathways and to follow the impact of process improvements done in 1991–1992. Especially, process gas cleaning by baghouse filter at aluminum smelter plant (ALS) and by scrubber at car schredding (CSH) were successful in emission control. At ALS, reductions of polychlorinated dioxins, furans, planar PCB's and dibenzothiophenes were better that 90, naphthalenes 78, other PCB's 73, chlorobenzenes 51 and chlorophenols 43 per cent. CSH scrubber gave similar results.

Computational Gray-scale Mathematical Morphology on Lattices (A Comparator-based Image Algebra) Part II: Image Operators

In this second part of a two-part study, the lattice-based computational representational structure is applied to image operators. The computational representations are first applied to some common non-linear window (vector) operators used in image processing — for instance, flat erosion, flat dilation, fuzzy erosion, and flat filters, in general. For these, application of the representations is direct. Representations are then developed for gray-to-binary and gray-to-gray image operators. In all cases, images are assumed to be lattice-valued. It is shown that under appropriate circumstances the representations can be viewed as structural generalizations of classical representations. Flat (stack) filters are treated in their own fight (as operators on lattice-valued images) and it is seen that for these the lattice representations can be interpreted in terms of threshold decomposition.

Soft morphological filtering

Stack filters are widely used nonlinear filters based on threshold decomposition and positive Boolean functions. They have shown to form a very large class of filters which includes rank-order operations as well as standard morphological operations. The stack filter representation of an order statistic filter provides an efficient tool for the theoretical analysis of the filter. Soft morphological filters form a large subclass of stack filters. They were introduced to improve the behavior of standard morphological filters in noisy conditions. In this paper, different properties of soft morphological filters are analysed and illustrated. Their connection to stack filters is established, and that connection is used in the statistical analysis of soft morphological filters. Soft morphological filters are less sensitive to additive noise than standard morphological filters. The deterministic properties of soft morphological filters are also analysed and it is shown that soft morphological filters form a class of filters with many desirable properties. For example, they preserve well details of images.

Biogenic sulfur emissions and aerosols over the tropical South Atlantic: 3. Atmospheric dimethylsulfide, aerosols and cloud condensation nuclei

We measured dimethylsulfide in air (DMSa) and the number concentration, size distribution, and chemical composition of atmospheric aerosols, including the concentration of cloud condensation nuclei (CCN), during February‐March 1991 over the tropical South Atlantic along 19°S (F/S Meteor, cruise 15/3). Aerosol number/size distributions were determined with a laser‐optical particle counter, condensation nuclei (CN) concentrations with a TSI 3020, and cloud condensation nuclei (CCN) with a Hudson‐type supersaturation chamber. Aerosol samples were collected on two‐stage stacked filters and analyzed by ion chromatography for soluble ion concentrations. Black carbon in aerosols was measured by visible light absorption and used to identify and eliminate periods with anthropogenic pollution from the data set. Meteorological analysis shows that most of the air masses sampled had spent extended periods over remote marine areas in the tropical and subtropical region. DMSa was closely correlated with the sea‐to‐ air DMS flux calculated from DMS concentrations in seawater and meteorological data. Sea salt made the largest contribution to aerosol mass and volume but provided only a small fraction of the aerosol number concentration. The submicron aerosol had a mean composition close to ammonium bisulfate, with the addition of some methanesulfonate. Aerosol (CN and CCN) number and non‐sea‐salt sulfate concentrations were significantly correlated with DMS concentration and flux. This suggests that DMS oxidation followed by aerosol nucleation and growth in the marine boundary layer is an important, if not dominating, source of CN and possibly CCN. The degree of correlation between DMS and particle concentrations in the marine boundary layer may be strongly influenced by the different time scales of the processes regulating these concentrations. Our results provide strong support for several aspects of the CLAW hypothesis, which proposes the existence of a feedback loop linking DMS emission from marine plankton to sulfate aerosol and global climate.

Processors for generalized stack filters

New processor structures for generalized stack filters are proposed. They can be implemented using different numbers of Boolean function units. The class of pipeline-parallel structures for generalized stack filters is simple and modular in structure, and suitable for VLSI implementation. Coder and decoder networks are developed for the mutual transform of binary-weighted and unary-weighted codes as the thresholding and addition units of the processors. The area time complexity A/spl middot/T of the proposed filters is O(k/sup 3/) due to the feedback, instead of O(2/sup k/) for the threshold decomposition structure, where k is the number of bits for sample representation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stack filter design: a structural approach

A new approach is developed for finding the optimal stack filter that minimizes noise subject to constraints on its structural behavior. Based on the output moments of stack filters, it is proven that the optimal stack filter is a combination of the median filter, which has the same window width as the stack filter, and a set of maximum and minimum filters, which are attributed to the structural constraints. Design examples for 1-D signal processing and image processing are provided. >

Optimal stack filters under rank selection and structural constraints

A new expression for the moments about the origin of the output of stack filtered data is derived in this paper. This expression is based on the A and M vectors containing the well-known coefficients A i of stack filters and numbers M( Φ, γ, N, i) defined in this paper. The noise attenuation capability of any stack filter can now be calculated using the A and M vector parameters in the new expression. The connection between the coefficients A i and so called rank selection probabilities r i is reviewed and new constraints, called rank selection constraints, for stack filters are defined. The major contribution of the paper is the development of an extension of the optimality theory for stack filters presented by Yang et al. and Yin. This theory is based on the expression for the moments about the origin of the output, and combines the noise attenuation, rank selection constraints, and structural constraints on the filter's behaviour. For self-dual stack filters it is proved that the optimal stack filter which achieves the best noise attenuation subject to rank selection and structural constraints can usually be obtained in closed form. An algorithm for finding this form is given and several design examples in which this algorithm is used are presented in this paper.

Decompositional methods for stack filtering using Fibonacci p-codes

Stack filters form a wide class of nonlinear filters which has received a great deal of attention during recent years. In this paper new decompositional methods based on Fibonacci p-codes for computing the output for different stack filters are presented. The computational complexities of these methods are also studied and numerical examples illustrating the benefits of using different values of p for different situations.

Parallel algorithms and VLSI architectures for stack filtering using Fibonacci p-codes

A parallel decompositional algorithm and VLSI architecture is proposed for computation of the output of a stack filter over a single window of input samples using Fibonacci p-codes. For a subclass of positive Boolean functions, a more efficient parallel algorithm and VLSI architecture for running stack filtering is also presented. The area-time complexities of the proposed designs are estimated. >

Permutation weighted order statistic filter lattices

We introduce and analyze a new class of nonlinear filters called permutation weighted order statistic (PWOS) filters. These filters extend the concept of weighted order statistic (WOS) filters, in which filter weights associated with the input samples are used to replicate the corresponding samples, and an order statistic is chosen as the filter output. PWOS filters replicate each input sample according to weights determined by the temporal-order and rank-order of samples within a window. Hence, PWOS filters are in essence time-varying WOS filters. By varying the amount of temporal-rank order information used in selecting the output for a given observation window size, we obtain a wide range of filters that are shown to comprise a complete lattice structure. At the simplest level in the lattice, PWOS filters reduce to the well-known WOS filter, but for higher levels in the lattice, the obtained selection filters can model complex nonlinear systems and signal distortions. It is shown that PWOS filters are realizable by a N! piecewise linear threshold logic gate where the coefficients within each partition can be easily optimized using stack filter theory. Simulations are included to show the advantages of PWOS filters for the processing of image and video signals.

A new efficient implementation of the oddly stacked Princen-Bradley filter bank

An improved implementation of the oddly stacked Princen-Bradley filter bank is proposed. The new implementation is based on the same approach as suggested by Duhamel et al. [1991], where the complex fast Fourier transform algorithm of length equal to half of the number of subbands is used in order to obtain a significant reduction of number of arithmetic operations. In the implementation, the analysis filter bank is similar to Duhamel's, with some minor differences. However, the synthesis filter bank is quite different, having easy and elegant merging of output rotations and windows and reduced arithmetic complexity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Elemental concentrations in atmospheric particulate matter sampled on the North Sea and the English Channel

Air sampling on a series of 10 cruises, spanning the whole area of the North Sea, yielded detailed spatial distributions of atmospheric concentrations of Al, Si, S, Cl, P, K, Ca, Ti, V, Cr, Mn, Fe, Ni, Cu, Zn and Pb determined by EDXRF. A strong influence of the air mass history on the heavy metal concentrations was demonstrated for the whole sampling period of 5 years. Factor analysis performed on all samples collected with a stack filter unit resulted in the identification of three factors for the coarse particle fraction (sulphate particles with trace metals, sea salt particles and soil dust or metallurgic particles containing Fe) and four factors for the fine particle fraction (sea salt, sulphate with Pb and Zn), trace metal particles with Cu, Ni, Zn and fly-ash particles. The same statistics performed on all samples collected above the Southern Bight of the North Sea yielded three factors, namely: sea salt particles, particles enriched in Ni and V, originating from natural oil combustion and particles containing a variety of elements such as S, K, Ca, Fe, Pb, Cu and Zn. Compared with relevant measurements of trace elements in this area, a relatively good agreement can be found.

Formula for the joint distribution of stack filters

A formula for the joint distribution function of two stack filters with common arguments is derived. Using this formula, the joint distribution function of any two samples in the output of a stack filter can be obtained. Here, it is presumed that the input signal is modelled as a sequence of independent random variables. This also allows analytic determination of the autocorrelation function of the output of an arbitrary stack filter.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optical threshold-decomposed Boolean-based gray-scale filter

Because of the advances made in optical technology, an extension can be made from the well-known stack filter to a new kind of gray-scale filter by using arbitrary Boolean functions instead of positive Boolean functions in the second step of a three-step process. Applications of these threshold-decomposed Boolean-based gray-scale filters in image processing include image smoothing and feature detection. A general optical implementation approach of nonlinear threshold-decomposed Boolean-based gray-scale filters is also described that utilizes area coding, gray scale correlation and lookup table thresholding techniques. Some experimental results are demonstrated.

Stack filter lattices

Lattice theory, rank selection probabilities (RSPs), and sample selection probabilities (SSPs) are used to develop methods for comparing and ordering stack filters. New recursive algorithms for calculating RSPs and SSPs are developed. The structure of these algorithms assists in the derivation of several interesting properties of RSPs and SSPs. The identification of some important sublattices of the stack filter lattice leads to an improved characterization of the impulsive noise suppression capabilities of stack filters.

Input compression and efficient VLSI architectures for rank order and stack filters

Rank-order-based filters include rank-order filters, stack filters, and weighted order statistic filters. The output of a rank-order-based filter is always one of the sample points in its input window; which one is chosen depends only upon the ranks and positions of the samples within the window. This paper introduces new architectures for rank-order-based filters. They all achieve fast, efficient operation by exploiting an algorithm called input compression. Under this algorithm, the sample points in the input window are first mapped to their relative ranks — the sample points in a window of size N + 1 would thus be mapped to the integers 0− N. The rank-order-based filter to be implemented is then applied directly to this compressed input, and the rank chosen is then mapped back to the sample of that rank in the original data to obtain the final output. This approach has been used to implement rank-order filters, in which case the same rank is always chosen from the compressed data. In this paper, which rank is chosen also depends on the positions of the ranks in the compressed data. Implementations employing input compression have several advantages. They are computationally efficient like running order sorters, yet can be pipelined to a fine degree like sorting networks. In stack filter implementations, the threshold decomposition circuitry can be eliminated when input compression is combined with unary encoding of the ranks. Weighted order statistic filter implementations based on input compression can support programmable, noninteger weights.

Stack filter phase lock loops

The basic design of a general purpose digital phase lock loop is improved upon by incorporating a stack filter into the feedback path of the loop. The addition of a stack filter provides enhanced performance in the presence of impulsive noise while providing the same steady-state performance of a conventional digital phase lock loop. Both first- and second-order loops benefit from the addition of stack filters. The particular case of using a median filter is examined in detail. In addition, a linear regression model is used in conjuction with the absolute error criterion to provide robust estimates of phase and frequency. The resulting loop structure incorporates median and weighted median filters in its feedback path.