Wiener filtering approach

In this work, an inverse filtering technique based on Wiener s optimal theory (1-3) is presented. This approach is valid for time-varying systems, and is solved in the time domain in mtrix form. Also, it is in many respects equivalent to the numerically "effl- lent" Kalman filtering approach described in ( ). For this reason, a... 

Adopting Eu=ql and Ey=0, then Equation l6 reduces to Equation 5 With Eu=ql and Ey=rl, Equation l6 has a format which is identical to the solution derived in (2T) through a deterministic minimum least squares approach for time-invariant systems. This is to be expected, because the Wiener filtering technique may be in fact Included as part of the general theory of least squares. 

How can this approach be adapted to deconvolution The problem is similar, but now we ask that y(x) also incorporate the inverse of s(x). Both Bracewell (1958) and Helstrom (1967) have derived this variant of the Wiener filter. Accordingly, we may minimize... 

L. M. Gugliotta, D. Alba, and G. R. Meira, Correction for instrumental broadening in SEC through a stochastic matrix approach based on Wiener filtering theory, ACSSymp. Ser. 352 287 (1987). 

A more general process known as least-squares filtering or Wiener filtering can be used when noise is present, provided the statistical properties of the noise are known. In this approach, g is deblurred by convolving it with a filter m, chosen to minimize the expected squared difference between / and m g. It can be shown that the Fourier transform M of m is of the form (1///)[1/(1 - - j], where S is related to the spectral density of the noise note that in the absence of noise this reduces to the inverse filter M = /H. A. number of other restoration criteria lead to similar filter designs. 

In addition to recursive filters, other model-based estimation-theoretic approaches have been developed. For example, in the Wiener filter described above, one can use random field models (see Section in) to estimate the power spectra needed. Alternatively, one can use MRF models to characterize the degraded images and develop deterministic or stochastic estimation techniques that maximize the posterior probability density function. 

In this section we present the principles of the signal subspace approach and its relations to Wiener filtering and spectral subtraction. Our presentation follows [7] and [11]. This approach assumes that the signal and noise are noncorrelated, and that their second-order statistics are available. It makes no assumptions about the distributions of the two processes. 

There is a major flaw with the inverse filter which renders it useless when B(u, v) falls to near zero, the correction becomes large, and any noise present is substantially amplified. Even computer rounding error can be substantial. An alternative approach, which avoids this problem, is based on the approach of Wiener. This approach models the image and noise as stochastic processes, and asks the question What re-weighting in the Fourier domain will produce the minimum mean squared error between the tme image and our estimate of it The Wiener solution has the form... 

Here the spectral density functions of the true image and noise are R/(u, v) and R/ /(u, v), respectively. Note that at spatial frequencies where the signal-to-noise is very high, the ratio Rn(u, v)/Ri(u, v) approaches zero, and the Wiener filter reduces to the inverse filter. However, when the signal-to-noise ratio is very poor (i.e., v)/... 

P - pixei this solution is identical to the one given by Wiener inverse-filter in Eq. (11). This shows that Wiener approach is a particular case in MAP framework. 

There are several ways of detecting peaks in such noisy signals. The Wiener-Hopf filter minimizes the expectation value of the noise power spectrum and may be used to optimally smooth the original noisy profile [19]. An alternative approach described by Hindeleh and Johnson employs knowledge of the peak shape. It synthesizes a simulated diffraction profile from peaks of known width and shape, for all possible peak amplitudes and positions, and selects that combination of peaks that minimizes the mean square error between the synthesized and measured profiles [20], This procedure is illustrated... 

An important hybrid approach has also developed in recent years that makes use of block-structured or modular models. These models are composed of parametric and/or nonparametric components properly connected to represent reliably the input-output relation. The model specification task for this class of models is more demanding and may utilize previous parametric and/or nonparametric modeling results. A promising variant of this approach, which derives from the general Volterra-Wiener formulation, employs principal dynamic modes as a canonical set of filters to represent a broad class of nonlinear dynamic systems. Another variant of the modular approach that has recently acquired considerable popularity but will not be covered in this review is the use of artificial neural networks to represent input-output nonlinear mappings in the form of connectionist models. These connectionist models are often fully parametrized, making this approach affine to parametric modeling, as well. 

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