Сызықты емес сүзгілер: қағидалары мен қолданылуы

In signal processing, a nonlinear (or non linear) filter is a filter whose output is not a linear function of its input. That is, if the filter outputs signals R and S for two input signals r and s separately, but does not always output αR + βS when the input is a linear combination αr + βs. Both continuous domain and discrete domain filters may be nonlinear. A simple example of the former would be an electrical device whose output voltage R(t) at any moment is the square of the input voltage r(t); or which is the input clipped to a fixed range [a,b], namely R(t) = max(a, min(b, r(t))). An important example of the latter is the running median filter, such that every output sample Ri is the median of the last three input samples ri, ri−1, ri−2. Like linear filters, nonlinear filters may be shift invariant or not. Non linear filters have many applications, especially in the removal of certain types of noise that are not additive. For example, the median filter is widely used to remove spike noise — that affects only a small percentage of the samples, possibly by very large amounts. Indeed, all radio receivers use non linear filters to convert kilo to gigahertz signals to the audio frequency range; and all digital signal processing depends on non linear filters (analog to digital converters) to transform analog signals to binary numbers. However, nonlinear filters are considerably harder to use and design than linear ones, because the most powerful mathematical tools of signal analysis (such as the impulse response and the frequency response) cannot be used on them. Thus, for example, linear filters are often used to remove noise and distortion that was created by nonlinear processes, simply because the proper non linear filter would be too hard to design and construct. From the foregoing, we can know that the nonlinear filters have quite different behavior compared to linear filters. The most important characteristic is that, for nonlinear filters, the filter output or response of the filter does not obey the principles outlined earlier, particularly scaling and shift invariance. Furthermore, a nonlinear filter can produce results that vary in a non intuitive manner.

Several principles define a linear system. The basic definition of linearity is that the output must be a linear function of the inputs, that is

for any scalar values and This is a fundamental property of linear system design, and is known as superposition. So, a system is said to be nonlinear if this equation is not valid. That is to say, when the system is linear, the superposition principle can be applied. This important fact is the reason that the techniques of linear system analysis have been so well developed.

The context here is the formulation of the nonlinear filtering problem seen through the lens of the theory of stochastic processes. In this context, both the random signal and the noisy partial observations are described by continuous time stochastic processes. The unobserved random signal to be estimated is modeled through a non linear Ito stochastic differential equation and the observation function is a continuous time non linear transformation of the unobserved signal, an observation perturbed by continuous time observation noise. Given the nonlinear nature of the dynamics, familiar frequency domain concepts that can be applied to linear filters are not viable, and a theory based on the state space representation is formulated. The complete information on the nonlinear filter at a given time is the probability law of the unobserved signal at that time conditional on the history of observations up to that time. This law may have a density, and the infinite dimensional equation for the density of this law takes the form of a stochastic partial differential equation (SPDE). The problem of optimal nonlinear filtering in this context was solved in the late 1950s and early 1960s by Ruslan L. Stratonovich and Harold J. Kushner. The optimal filter SPDE is called Kushner Stratonovich equation. In 1969, Moshe Zakai introduced a simplified dynamics for the unnormalized conditional law of the filter known as Zakai equation. It has been proved by Mireille Chaleyat Maurel and Dominique Michel that the solution is infinite dimensional in general, and as such requires finite dimensional approximations. These may be heuristics based such as the extended Kalman filter or the assumed density filters described by Peter S. Maybeck or the projection filters introduced by Damiano Brigo, Bernard Hanzon and François Le Gland, some sub families of which are shown to coincide with the assumed density filters. Particle filters are another option, related to sequential Monte Carlo methods.

Energy transfer filters are a class of nonlinear dynamic filters that can be used to move energy in a designed manner. Energy can be moved to higher or lower frequency bands, spread over a designed range, or focused. Many energy transfer filter designs are possible, and these provide extra degrees of freedom in filter design that are just not possible using linear designs.

A min filter also known as erosion in morphological image processing, is a spatial domain filter used for image processing. It replaces each pixel in the image with the minimum value of its neighboring pixels. The size and shape of the neighborhood are defined by a structuring element, typically a square or circular mask. The transformation replaces the central pixel with the darkest one in the running window. For example, if you have text that is lightly printed, the minimum filter makes letters thicker.

A max filter, also known as dilation in morphological image processing, is another spatial domain filter used for image processing. It replaces each pixel in the image with the maximum value of its neighboring pixels, again defined by a structuring element. The maximum and minimum filters are shift invariant. Whereas the minimum filter replaces the central pixel with the darkest one in the running window, the maximum filter replaces it with the lightest one. For example, if you have a text string drawn with a thick pen, you can make the sign skinnier.