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     

Some representation properties of stack filters

Stack filters, a class of nonlinear filters which are based on positive Boolean functions as the window operators, are considered. The representations of these window operations are presented via the structures of on-set and off-set of the positive Boolean functions, which can be expressed as a Boolean expression containing no complements of the input variables. A fast algorithm for finding the representation of stack filters is designed. This algorithm can be easily extended to find the representation of morphological filters mentioned by P. Maragos (1989)     

A general weighted median filter structure admitting negativeweights

Weighted median smoothers, which were introduced by Edgemore in the context of least absolute regression over 100 years ago, have received considerable attention in signal processing during the past two decades. Although weighted median smoothers offer advantages over traditional linear finite impulse response (FIR) filters, it is shown in this paper that they lack the flexibility to adequately address a number of signal processing problems. In fact, weighted median smoothers are analogous to normalized FIR linear filters constrained to have only positive weights. It is also shown that much like the mean is generalized to the rich class of linear FIR filters, the median can be generalized to a richer class of filters admitting positive and negative weights. The generalization follows naturally and is surprisingly simple. In order to analyze and design this class of filters, a new threshold decomposition theory admitting real-valued input signals is developed. The new threshold decomposition framework is then used to develop fast adaptive algorithms to optimally design the real-valued filter coefficients. The new weighted median filter formulation leads to significantly more powerful estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior weighted median smoother structures     

基于形态学和OTSU算法的红外图像降噪及分割

为了在去除红外图像的脉冲噪声的同时,有效保持和恢复图像的边缘细节,提出了基于灰度特征和众数原则的迭代双边中值滤波方法。此方法根据脉冲噪声的灰度特征以及众数原则,将取最小和最大值、而在邻域的灰度分布上孤立的像素识别为噪声。根据基于空间距离和灰度相似的加权系数,对邻域中的无噪像素与已经去噪恢复的像素进行频次加权,用频次加权中值作为噪声像素的估计值。其中,以迭代遍历的方式执行去噪处理,充分利用前次遍历处理的结果,以去除高密度噪声。实验数据证明,此方法去噪所得的PSNR和EPI值以及视觉效果均优于现有方法,具有更好的去噪性能。     

Random-valued impulse noise removal using adaptive dual threshold median filter

Noise detection and its removal is very important in the image processing. Detection of noise is very crucial and significant in random valued impulse noise because it does not hamper the image pixels uniformly. This paper presents a novel and unique concept of adaptive dual threshold for the detection of random valued impulse noise along with simple median filter at noise removal stage. Simulation results shows that an efficient noise detection leads to a superior quality of de-noised image as compared to existing adaptive threshold based image de-noising techniques. Proposed threshold computation is based on averaging of pixel values of window which enhances the PSNR of our system as compared to existing median filter based image de-noising methods.     

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.     

On root structures and convergence properties of weighted median filters

A weighted median filter is a nonlinear digital filter consisting of a window of length 2N + 1 and a weight vector W=(W_? N,..., W₀,..., W_N). A root signal of a median type filter is a signal that is invariant to the filter. However, not all weighted median filters possess the convergence property. In this paper, we shall study the root structures and the convergence behavior of a subclass of weighted median filters, calledclass- 1 filters, which is symmetric in its weight vector. We shall introduce an important parameter, calledfeature value, and show that any one-dimensional unappended signal of lengthL will converge to a root signal in at most $$3\left\lceil {\frac{{L - 2}}{{2(2N + 2 - p)}}} \right\rceil$$ passes of aclass ?1 filter with window width 2N + 1 and thefeature value p.     

An overview of median and stack filtering

Within the last two decades a small group of researchers has built a useful, nontrivial theory of nonlinear signal processing around the median-related filters known as rank-order filters, order-statistic filters, weighted median filters, and stack filters. This required significant effort to overcome the bias, both in education and research, toward linear theory, which has been dominant since the days of Fourier, Laplace, and Convolute.We trace the development of this theory of nonlinear filtering from its beginnings in the study of noise-removal properties and structural behavior of the median filter to the recently developed theory of optimal stack filtering.The theory of stack filtering provides a point of view which unifies many different filter classes, including morphological filters, so it is discussed in detail. Of particular importance is the way this theory has brought together, in a single analytical framework, both the estimation-based and the structural-based approaches to the design of these filters.Some recent applications of median and stack filters are provided to demonstrate the effectiveness of this approach to nonlinear filtering. They include: the design of an optimal stack filter for image restoration; the use of vector median filters to attenuate impulsive noise in color images and to eliminate cross luminance and cross color in TV images; and the use of median-based filters for image sequence coding, reconstruction, and scan rate conversion in normal TV and HDTV systems.     

Analysis of weighted median filters based on inequalities relating the weights

By considering all orderings of the input samples, which are discrete-time continuous-valued, it is shown here that a weighted median (WM) filter of spanN can be specified unambiguously by 2^N–1 consistent linear inequalities relating the weights. This specification is identical to that of a self-dual threshold function with the same weights. It is also shown that WM filters with symmetric weights can be specified by ternary threshold functions. Based on these inequalities, properties of WM filters which can be used to check equivalence of some WM filters are derived.This work was supported in part by the National Science Foundation under Grant DCI-8611859. A portion of this work was presented at the IEEE International Symposium on Circuits and Systems, Portland, Oregon, May 1989.     

实时中值滤波器的实现

针对二维中值滤波器的邻域数据处理的特点,提出了基于邻域图像帧存在体的并行处理方法,从而实现了实时的中值滤波器,论述了领域数据的形成及并行处理器结构,给出了３＊３实时的中值滤波器系统的逻辑框图。     

Analysis of the variance threshold of Kay's weighted linearpredictor frequency estimator

A theoretical approximation for the variance of Kay's weighted linear predictor frequency estimator is derived. From this expression, an inequality describing the variance threshold of the estimator is found. The window weights are then optimized to improve the variance. Numerical simulations demonstrate that the variance approximations are valid for medium to high signal-to-noise ratios or for large numbers of samples     

Accuracy of multichannel median filter

The question whether the output of the multichannel median filter should be one of the input vector samples is addressed. While the output is normally one of the inputs in the single-channel case, engineering it to be so in the multichannel case is found to lead to significant error in a large proportion of instances, a situation that should be avoided without good reason     

Local stability of the median LMS filter

Local stability properties of the median LMS adaptive filter are investigated by relating the behavior of the algorithm to the behavior of an associated ordinary differential equation. With independent inputs, the differential equation and the algorithm are shown to be locally stable. On the other hand, several classes of (periodic, nonindependent) inputs are described which cause the differential equation and the algorithm to be unstable about its equilibrium, even in the no disturbance case. This will help delineate those applications for which the median LMS is an appropriate adaptive algorithm     

Tuning the smoothness of the recursive median filter

The median filter is a special case of nonlinear filters used for smoothing signals. Since the output of the median filter is always one of the input samples, it is conceivable that certain signals could pass through the median filter unaltered. These signals define the signature of a filter and are referred to as root signals. Median filters are known to possess the convergence property, meaning that by repeating median filtering a root signal will be found, starting from any input signal. By associating the nonlinear operation of median filtering with a two terms cost function, an optimization process that minimizes that function is obtained. Cost functions of the same type are associated with different recursive median filtering schemes by replacing the actual values inside the filter's window with the original signal. The convergence behavior of these filters and their smoothness are studied. By changing the positions of the replacements during filtering, a tuning effect of the smoothness is obtained. Simulation results show that the proposed filtering schemes provide improved performance over the standard recursive median filter, succeeding in preserving. small details and fine textures     

Fast implementation of generalised median filter

The vector median filter (VMF) is a widely used means of eliminating impulse noise in colour images, and has the advantage of largely eliminating colour bleeding. The generalised median filter (GMF) has the capability for eliminating even more noise than the VMF, while not being targeted quite so specifically at colour bleeding. Ways of implementing the GMF are demonstrated so that it runs sufficiently rapidly to make it a useful alternative to the VMF, thereby opening the way for much wider use of this highly effective type of filter     

Performance characteristics of the median LMS adaptive filter

The median least-mean-square (MLMS) adaptive filter alleviates the problem of degradation of performance when inputs are corrupted by impulsive noise by protecting the filter coefficients from the impact of the impulses. MLMS is obtained from the least mean square (LMS) by applying a median operation to the raw gradient estimates of the mean-squared-error performance surface. The algorithm is analyzed for the class of independent and identically distributed inputs, establishing exponential convergence. The rate of convergence is shown to depend on order statistics of the input but shows little dependence on characteristics of the impulsive interference. Analysis of the steady-state performance indicates a significantly improved performance for MLMS compared to LMS. Analytic predictions for both convergence and steady-state behavior are supported by simulations     

Feasibility of substrate fed threshold logic

Proposes a modified integrated injection logic (I/SUP 2/L) structure called substrate fed threshold logic (SFTL) for the implementation of multivalued logic. In this structure, threshold current is supplied from the p/SUP +/ substrate to the top p region through the windows in the n/SUP +/ buried layer, and the magnitude of the threshold current is determined by the number of injection windows. The feasibility of the structure is discussed theoretically and experimentally. The results show that four-valued threshold logic is possible with the proposed structure.     

On the LP which finds a MMAE stack filter

Two methods are proposed to modify the linear program (LP) developed by E.J. Coyle and J.-H. Lin (1988) to find a stack filter which minimizes the mean absolute error (MAE). In the first approach, the number of constraints is substantially reduced at the expense of requiring a zero-one LP to solve for an optimal filter. This scheme reduces the number of constraints from O(n2ⁿ) to O(28ⁿ), which is exactly the cardinality of the set of possible binary vectors which can appear in the window of the filter. In the second approach, the LP is transformed into a max-flow problem. This guarantees that the problem can be solved in time which is a polynomial function of the number of variables in the LP, as opposed to the worst-case exponential time that may occur with the simplex method. It also allows the many fast algorithms for the max-flow problem to be used to find an optimal stack filter. Recursive algorithms for construction of the window width n constraint matrix for both the original LP and the max-flow modification are also provided     

Analysis of the properties of three-terminal transferred-electron logic devices

A two-dimensional analysis of Schottky-barrier gate GaAs logic devices utilizing the transferred-electron effect is described. The analysis is used to study the basic properties of two devices with anode to cathode spacing of 13 µm and 33 µm. The reduction in current drop due to the presence of the gate is discussed. The switching properties of the 13-µm device are studied for operation with either anode or cathode resistance. A gate delay of 30 ps and a total gate power of 180 mW is estimated.     

Fast adaptive optimization of weighted vector median filters

Weighted vector median (WVM) filters are effective tools for multichannel signal processing. To obtain the desired filtering behavior and characteristic, the WVM filter weights must be determined in an appropriate manner. In this paper, we first analyze previously defined approaches for WVM filter optimization and show their drawbacks related to derivative computation and vector direction information utilization. Based on this analysis, we propose two fast adaptive algorithms for WVM filter design. Proposed Algorithm I computes locally optimal weight changes at each iteration and updates the filter weights accordingly. This algorithm does not involve derivative computation, thus eliminating the instability caused by derivative approximations utilized in previous approaches. Proposed Algorithm II extends the results from established marginal weighted median optimization methods to the vector case by error metric generalization. Both algorithms can be applied to WVM filters using the L/sub p/ norm, while Algorithm I can operate on more general distance metrics. The presented simulation results show that both algorithms are effective, fast, and stable; they perform well under a wide range of circumstances.