Wiener filter inverse

The Wiener filter therefore avoids noise amplification and provides the best solution according to some quality criterion. We will see that these features are common to all other methods which correctly solve the deconvolution inverse problem. The result of applying Wiener inverse-filter to the simulated image is shown in Fig. 3b. 

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... 

For a basic deconvolution problem involving band-limited data, the trial solution d(0) may be the inverse- or Wiener-filtered estimate y(x) (x) i(x). Application of a typical constraint may involve chopping off the nonphysical parts. Transforming then reveals frequency components beyond the cutoff, which are retained. The new values within the bandpass are discarded and replaced by the previously obtained filtered estimate. The resulting function, comprising the filtered estimate and the new superresolving frequencies, is then inverse transformed, and so forth. 

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. 

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)/... 

Figure 2b. Profiles of the modulation transfer function (MTF), its inverse and Wiener inverse-filter.

Figure 3b. Image obtained by using Wiener inverse-filter.

The Wiener inverse-filter is derived from the following two criteria ... 

To summarize, Wiener inverse-filter is the linear filter which insures that the result is as close as possible, on average and in the least squares sense, to the true object brightness distribution. 

Figure 2b and Eq. (10) show that the Wiener inverse-filter is close to the direct inverse-hlter for frequencies of high signal-to-noise ratio (SNR), but is strongly attenuated where the SNR is poor ... 

Wiener inverse-filter however yields, possibly, unphysical solution with negative values and ripples around sharp features (e.g. bright stars) as can be seen in Fig. 3b. Another drawback of Wiener inverse-filter is that spectral densities of noise and signal are usually unknown and must be guessed from the data. For instance, for white noise and assuming that the spectral density of object brightness distribution follows a simple parametric law, e.g. a power law, then ... 

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. 

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... 

Focusing our attention once again on Fourier space, recall that the Wiener inverse filter yw(co) is obtained by finding the function 7w(co) that minimizes the mean-square error... 

Schell (1965) recognized that the major deficiency of the Wiener inverse filter is the nonphysical nature of the partially negative solutions that it is prone to generate. He sought to extrapolate the band-limited transform O(co) by seeking a nonnegative physical solution 6 + (x) through minimization of... 

We are permitted to specify the integrals for positive co only, because of the even property of the integrand. This simplication, in turn, stems from the real nature of all the x-space components of the integrand. Minimizing expression (9) is equivalent to asking that the physical solution conform to the Wiener inverse-filter estimate in the sense of minimum mean-square error after suitable weighting of the positive solution to ensure best conformance at frequencies of greatest certainty. 

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... 

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