Topological median filters

This paper describes the definition and testing of a new type of median filter for images. The topological median filter implements some existing ideas and some new ideas on fuzzy connectedness to improve, over a conventional median filter, the extraction of edges in noise. The concept of alpha-connectivity is defined and used to create an algorithm for computing the degree of connectedness of a pixel to all the other pixels in an arbitrary neighborhood. The resulting connectivity map of the neighborhood effectively disconnects peaks in the neighborhood that are separated from the center pixel by a valley in the brightness topology. The median of the connectivity map is an estimate of the median of the peak or plateau to which the center pixel belongs. Unlike the conventional median filter, the topological median is relatively unaffected by disconnected features in the neighborhood of the center pixel. Four topological median filters are defined. Qualitative and statistical analyses of the four filters are presented. It is demonstrated that edge detection can be more accurate on topologically median filtered images than on conventionally median filtered images.     

Vector median filters

Median filters (MF) are used both to filter ‘salt and pepper’ noise from signals and images and in other signal processing applications. In this paper, an extension of the MF, the vector median filter (VMF), is introduced. As opposed to the MF, the VMF outputs for each window location a number of data elements. By adjusting the VMF parameters, the MF is obtained as a VMF special case. Just like the MF, the VMF filters impulses while simultaneously preserving step changes in a signal. The VMF's principal advantage is that it reduces the total stored data signal computation time while it produces visual outputs comparable to that of an MF. Deterministic and statistical properties of the VMF are examined. Computer-generated experimental results are also presented.     

Vector median filters

Two nonlinear algorithms for processing vector-valued signals are introduced. The algorithms, called vector median operations, are derived from two multidimensional probability density functions using the maximum-likelihood-estimate approach. The underlying probability densities are exponential, and the resulting operations have properties very similar to those of the median filter. In the vector median approach, the samples of the vector-valued input signal are processed as vectors. The operation inherently utilizes the correlation between the signal components, giving the filters some desirable properties. General properties as well as the root signals of the vector median filters are studied. The vector median operation is combined with linear filtering, resulting in filters with improved noise attenuation and filters with very good edge response. An efficient algorithm for implementing long vector median filters is presented. The noise attenuation of the filters is discussed, and an application to velocity filtering is shown     

Convergence properties of median and weighted median filters

It has been shown that assuming the first and last value carry-on appending strategy, a finite number of passes of the same median filter to an arbitrary signal of finite length results in a root signal that will be invariant to additional filtering passes. This so-called convergence property is reproven using an extremely simple approach. In addition, the well-known idempotent property (i.e., where convergence is achieved with only one filtering pass) of a recursive median filter is reproven similarly, and the convergence behavior of weighted median filters is studied     

Wavelet transform based adaptive filters: analysis and new results

In this paper the wavelet transform is used in an adaptive filtering structure. The coefficients of the adaptive filter are updated by the help of the least mean square (LMS) algorithm. First, the wavelet transform based adaptive filter (WTAF) is described and it is analyzed for its Wiener optimal solution. Then the performance of the WTAF is studied by the help of learning curves for three different convergence factors: (1) constant convergence factor, (2) time-varying convergence factor, and (3) exponentially weighted convergence factor. The exponentially weighted convergence factor is proposed to introduce scale-based variation to the weight update equation. It is shown for two different sets of data that the rate of convergence increases significantly for all three WTAF structures as compared to that of time-domain LMS. The high convergence rates of the WTAF give us reason to expect that it will perform well in tracking rapid changes in a signal     

Sharpening vector median filters

A sharpening vector median (VM) filter for simultaneous denoising and enhancing vector-valued signals is introduced. This filter uses the trimmed aggregated distance minimization concept and robust vector order statistics to enhance edges and image details while retaining the noise removal characteristics of the standard VM operator. The procedure accommodates various design, implementation and application objectives by enhancing the vector-valued signals depending on the local image statistics and/or the user's needs. The filter properties discussed in this paper are proven and suggest that the proposed solution is a robust vector processing operator. The performance and efficiency of the filter are analyzed and commented upon. Examples from its application to color image filtering and virtual restoration of artworks are provided.     

Envelope-constrained filters: adaptive algorithms

We consider an envelope-constrained (EC) optimal filter design problem involving a quadratic cost function and a number of linear inequality constraints. Using the duality theory and the space transformation function, the optimal solution of the dual problem can be computed by finding the limiting point of an ordinary differential equation given in terms of the gradient flow. An iterative algorithm is developed via discretizing the differential equation. From the primal-dual relationship, the corresponding sequence of approximate solutions in the original EC filtering problem is obtained. Based on these results, an adaptive algorithm is constructed for solving the stochastic EC filtering problem in which the input signal is corrupted by an additive random noise. For illustration,a practical example is solved for both noise-free and noisy cases     

Weighted median filters: An analog implementation

The feasibility of implementing analog CMOS VLSI weighted median filters for image and signal processing is discussed. The proposed weighted median filter uses a transconductance comparator as a basic cell, where the output saturation current is used as the weight parameter in the median filter. Experimental results of the proposed analog weighted median filter for an ON Semiconductor 0.5 μm technology through MOSIS fabricated prototype are shown.     

Adaptive reduced-rank LCMV beamforming algorithms based on joint iterative optimization of filters: Design and analysis

This paper presents reduced-rank linearly constrained minimum variance (LCMV) beamforming algorithms based on joint iterative optimization of filters. The proposed reduced-rank scheme is based on a constrained joint iterative optimization of filters according to the minimum variance criterion. The proposed optimization procedure adjusts the parameters of a projection matrix and an adaptive reduced-rank filter that operates at the output of the bank of filters. We describe LCMV expressions for the design of the projection matrix and the reduced-rank filter. We then describe stochastic gradient and develop recursive least-squares adaptive algorithms for their efficient implementation along with automatic rank selection techniques. An analysis of the stability and the convergence properties of the proposed algorithms is presented and semi-analytical expressions are derived for predicting their mean squared error (MSE) performance. Simulations for a beamforming application show that the proposed scheme and algorithms outperform in convergence and tracking the existing full-rank and reduced-rank algorithms while requiring comparable complexity.     

Adaptive reduced-rank LCMV beamforming algorithms based on joint iterative optimisation of filters

A novel approach to linearly constrained minimum variance (LCMV) beamforming based on reduced-rank processing is proposed. The method is based on a constrained joint iterative optimisation of an adaptive projection matrix and a reduced-rank filter according to the minimum variance criterion. We derive LCMV expressions for the design of the projection matrix and the reduced-rank filter and present low-complexity adaptive algorithms for their efficient implementation. Simulations show that the proposed scheme outperforms the full-rank and existing reduced-rank methods with low complexity.     

Generalized multistage median filters.

As an extension of the multistage median filters, a class of r -filters is presented. Some properties of these filters are discussed. It is shown that the filters can smooth noise and preserve details of images efficiently.     

Deterministic properties of analog median filters

Two key deterministic properties of analog median filters are established. The first is that root signals are characterized by a special class of locally monotone functions, and the second is that the repeated application of a median filter produces a root signal which retains significant features in the original data. Both results are extensions of well known facts for discrete median filters. In addition, it is shown that these properties do not extend to a natural multidimensional version of the median filter     

二值FIR中值混合滤波

本文对传统的FIR中值混合(FMH)滤波,给出了一种二值FMH滤波的概念,并在此基础上,首次建立了VLSI实现的实时信号处理电路结构。     

Adaptive mixed-rank filters

Proposes an adaptive median-type filtering scheme which is both computationally efficient and effective in suppressing impulsive noise from images without loss of image detail. A fast implementation is briefly discussed and comparisons with other filters are presented.<>     

Adaptive polynomial filters

Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described     

Adaptive Laguerre-lattice filters

Adaptive Laguerre-based filters provide an attractive alternative to adaptive FIR filters in the sense that they require fewer parameters to model a linear time-invariant system with a long impulse response. We present an adaptive Laguerre-lattice structure that combines the desirable features of the Laguerre structure (i.e., guaranteed stability, unique global minimum, and small number of parameters M for a prescribed level of modeling error) with the numerical robustness and low computational complexity of adaptive FIR lattice structures. The proposed configuration is based on an extension to the IIR case of the FIR lattice filter; it is a cascade of identical sections but with a single-pole all-pass filter replacing the delay element used in the conventional (FIR) lattice filter. We utilize this structure to obtain computationally efficient adaptive algorithms (O(M) computations per time instant). Our adaptive Laguerre-lattice filter is an extension of the gradient adaptive lattice (GAL) technique, and it demonstrates the same desirable properties, namely, (1) excellent steady-state behavior, (2) relatively fast initial convergence (comparable with that of an RLS algorithm for Laguerre structure), and good numerical stability. Simulation results indicate that for systems with poles close to the unit circle, where an (adaptive) FIR model of very high order would be required to meet a prescribed modeling error, an adaptive Laguerre-lattice model of relatively low order achieves the prescribed bound after just a few updates of the recursions in the adaptive algorithm     

A new family of concurrent algorithms for adaptive Volterra andlinear filters

A novel idea for introducing concurrency in least squares (LS) adaptive algorithms by sacrificing optimality has been proposed. The resultant class of algorithms provides schemes to fill the wide gap in the convergence rates of LS and stochastic gradient (SG) algorithms. It will be particularly useful in the real time implementations of large-order linear and Volterra filters for which both the LS and SG algorithms are unsuited     

Weighted median filters with sigma-delta modulation encoding

Digital decimation filters play a fundamental role in oversampled sigma-delta A/D decoders. In this paper, we first show that weighted median (WM) filtering of a demodulated sequence (at the Nyquist rate) can be implemented concurrently in the A/D decoder. Through a simple modification of the binary time-series outputted by the A/D modulator, the sequence obtained after the sigma-delta modulation (SDM) decoder is shown to be equivalent to WM filtering the multilevel sequence at the Nyquist rate. Second, we show that WM filters can be used for SDM decimation filters and that these filters are readily implemented in the SDM binary domain. A very promising characteristic of SDM converters equipped with WM decimating filters is that sharp discontinuities (edges) can be preserved and acquired. Thus, the bandlimited constraint imposed on the input signals can be relaxed making SDM more attractive to A/D conversion of signals containing sharp transitions. The proposed signal processing algorithms, in essence, combine A/D sigma-delta converters and WM filters into a single programmable system     

Analysis of two-dimensional center weighted median filters

Center weighted median (CWM) filters, which have been recognized as detail preserving filters, are an important and the simplest subclass of weighted median (WM) filters. In this paper, we analyze the root signals of two-dimensional (2-D) CWM filters. In particular, we derive the required form for a signal to be a root of a 2-D CWM filter. The required form of signals to be roots is then used to evaluate the detail preserving properties of 2-D CWM filters. As examples, the detail preserving properties of some 2-D CWM filters are compared with other detail preserving filters, i.e. multilevel median filters. The generation of binary root signals of some 2-D CWM filters is treated in the term of the smallest surviving object (SSO). It is illustrated by some examples that CWM filters with different orientation of windows can be useful in image segmentation.