Filter inverse

Because the pseudo-inverse filter is chosen from the class of additive filters, the regularization can be done without taking into account the noise, (n). At the end of this procedure the noise is transformed to the output of the pseudo-inverse filter (long dashed lines on Fig. 1). The regularization criteria F(a,a) has to fulfill the next conditions (i) leading to an additive filter algorithm, (ii) having the asymptotic property a, —> a, for K,M... 

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 ... 

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. 

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. 

Fig. 40.32. Deconvolution (result in solid line) of a Gaussian peak (dashed line) for peak broadening ((M i/,)prf/(H vi)G = 1). (a) Without noise, (b) With coloured noise (A((0,1%), Tx = 1.5) inverse filter in combination with a low-pass filter, (c) With coloured noise (A (0,1 %), Ta = 1.5) inverse filter without low-pass filter.

With respect to the spreading calibration, several methods have been suggested e.g. (6-1 ) Numerous techniques have been proposed for solving the inverse filtering problem represented by Equation 1, with different degrees of success e.g. (it,15-19) Only references (M, (l8) and (I9) make no assumptions on the shape of g(t,x). 

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... 

Finally, it should be noted that apart from its use in chromatographic data treatment, inverse filtering techniques such as that described in this work have also potential applications in other areas of polymerization engineering, (see for example (30) and (31)). 

B(co) factor describing modification of inverse-filter trans-... 

We have presented two deconvolution methods from an intuitive point of view. The approach that suits the reader s intuition best depends, of course, on the reader s background. For those versed in linear algebra, methods that stem from a basic matrix formulation of the problem may lend particular insight. In this section we demonstrate a matrix approach that can be related to Van Cittert s method. In Section IV.D, both approaches will be shown to be equivalent to Fourier inverse filtering. Similar connections can be made for all linear methods, and many limitations of a given linear method are common to all. 

We defer additional analysis of the relaxation method until we have properly introduced the concept of Fourier inverse filtering. 

This filter is not an inverse filter of the type that we seek, being intended only for noise reduction. It does not undo any spreading introduced by s(x). It is, however, an optimum filter in the sense that no better linear filter can be found for noise reduction alone, provided that we are restricted to the knowledge that the noise is additive and Gaussian distributed. 

For oo, (co) approaches unity, provided that l — t(co) < 1. In this case, 0 k) is just the object estimated by inverse filtering. For k finite, the inverse-filter estimate is modified by a factor that suppresses frequencies for which t (co) is small. The larger k is, the less is this suppression. For typical transfer functions t(co) that suppress high frequencies, the factor B(co) controls the high-frequency content of o k In the spectrum domain, it is also possible to derive simple expressions for filters y(x) that are fully equivalent to an arbitrary number of relaxation iterations. Blass and Halsey (1981) have done so, but the highly useful nonlinear modifications of these methods cannot be incorporated. 

Let us not forget, at this point, that well-designed bandpass filtering can only prevent the appearance of solution frequencies totally rejected by z(eo). Noise in the frequency range that we wish to restore cannot be thus rejected, however. In the limit of simple inverse filtering we find that... 

We have noted the noise-sensitivity problem of the simple inverse filter and introduced modifications to alleviate these difficulties. Modifications yielded different functional forms for y(co). The convenient single-step property of the basic method was nevertheless retained. This property contrasts with the need for possibly arbitrary stopping criteria when we use iterative methods, which are computationally more expensive. The iterative methods do, however, allow the user to control the signal-to-noise versus resolution tradeoff by stopping the process when the growth of spurious... 

The composite filter 7(g)) may either be the true inverse filter, truncated for oo large if necessary, or any of the variations described in Section IV. In their original work, Rendina and Larson chose 7(g)) = (co)/t(co), where //(co) is a Gaussian line-broadening function that limits the ultimate resolution obtainable but yields a manageable 7(g)). For their studies Rendina and Larson used Ns = 4. 

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. 

The solutions illustrated in Schell s original publication were indeed entirely positive and showed some resolution improvement over the inverse-filter estimates. The improvement in these examples was not, however, as great as we have come to expect from the best of the newer methods and may not in fact demonstrate the method s real potential. The method does bring with it in a very explicit way, however, the idea that the Fourier spectrum may be extended, on the basis of a knowledge of positivity. Previous studies had focused on the finite extent constraint to achieve this objective. 

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