Greater Realization of Constraint Benefits

Just as others who have used linear methods, this author was disappointed to note the appearance of spurious nonphysical components when he applied the linear relaxation methods (Chapter 3) to the deconvolution of infrared spectra. Infrared absorption spectra, and other types of spectra as well, must lie in the transmittance range of zero to one. Spurious peaks appeared to nucleate on specific noise fluctuations in the data and grow with successive iterations, even though the mean-square error 

the author considered forcing the solution to lie within the physical bounds. Besides eliminating the objectionable nonphysical result, this approach, it was hoped, would also improve the accuracy of the solution within the physical bounds. Because of the limited performance evidenced in previous literature available on deconvolution, the author was unprepared for the magnitude of the improvement that resulted. 

The nonlinear iterative methods described here are based on the linear relaxation methods developed in Sections III. C.l and III.C.2 of Chapter 3. Initially, the correction term was set equal to zero in regions where o(k) was nonphysical. To illustrate this, we may rewrite the point-simultaneous equation [Chapter 3, Eq. (23)] with a relaxation parameter that depends on the estimate d(k)  

For clipping or simple truncation of the nonphysical part, we define the relaxation function 

With the present definition of r, however, an overcorrection that would normally disappear gradually through ensuing iterations results in a value of d(k)(x) that vanishes for all subsequent iterations. This behavior occurs because further corrections to that value are prohibited. To use the method, the investigator is compelled to take small values for r0. Even in this case, erroneously nonphysical values of o(k) that have been forced to zero are never allowed to return to the finite range that might better represent the true spectrum o(x). This form of the method therefore demands excessive computation and yields a solution that, although physically realizable, is not the best achievable estimate. 

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V. Greater Realization of Constraint Benefits A Jansson s Method... 

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