Van Cittert’s method

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. 

This is called point-simultaneous overrelaxation. If we set k = [s]nn, we have obtained the discrete formulation of Van Cittert s method. This connection between Van Cittert s method and the classic iterative methods of solving simultaneous equations was demonstrated in an earlier work (Jansson, 1968, 1970). 

The entire discussion of relaxation methods was conducted without examining Fourier space consequences. Van Cittert s method is easy to study this way and has been treated by Burger and Van Cittert (1933), Bracewell and Roberts (1954), Sakai (1962), and Frieden (1975). By applying the convolution theorem to Eq. (14), we may write... 

As with Van Cittert s method, we may taked(0) = i. We see that the successive estimates dik) cannot be negative, provided that s and i are everywhere positive. The reader should note that this assumption may be violated in a base-line region. Here the data i probably contain negative values arising from noise. A means of dealing with this problem is needed. 

We note that transposing s is equivalent to reversing the abscissa of s(x). The reader will be struck by the similarity of this treatment to the reblurring method of Kawata et al. (Chapter 3, Section IY.D.3). Their method is, in fact, an adaptation of the present use of sT to Van Cittert s method. 

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

It has been noted that deconvolution methods, most of which were linear, had a propensity to produce solutions that did not make good physical sense. Prominent examples were found when negative values were obtained for light intensity or particle flux. As noted previously, the need to eliminate these negative components was generally accepted. Accordingly, Gold (1964) developed a method of iteration similar to Van Cittert s but used multiplicative corrections instead of additive ones. 

The principle of the van Cittert method is simple (1) I = I0bS (2) In+ = 2In-In-kW (3) stop the iteration as soon as I0f,s / W. This convergence criterion guarantees that the correct solution is closely approximated. 

Constraining the solution to be positive can nevertheless provide the added benefits of reduced sensitivity to noise and improved resolution. Gold s iterative ratio method (Chapter 1, Section IV.A), for example, has been used successfully by a number of workers, including MacNeil and Dixon (1977) and Delwicke et al. (1980). MacNeil and Delwicke have compared it with the standard Van Cittert method, which is linear. 

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