Resolution with constrained optimization

Figure 10. Improved but still unrealistic resolution with constrained optimization. Corrected equatorial trace of a polypropylene fiber specimen cold drawn X-5.5. Total theoretical trace still comprised of unrealistic profiles. Compare with Figure 9.

For measurement adjustment, a constrained optimization problem with model equations as constraints is resolved at a fixed interval. In this context, variable classification is applied to reduce the set of constraints, by eliminating the unmeasured variables and the nonredundant measurements. The dimensional reduction of the set of constraints allows an easier and quicker mathematical resolution of the problem. 

Peak resolution is usually easier if well chosen background parameters are input and if constrained optimization methods are utilised. Misleading results can be obtained if the constraints are too limited and tests with unconstrained optimization are desirable if at all possible. In particular, the possible presence of paracrystalline or intermediate phase peaks must be tested with extreme care in order to avoid ambiguity. It is not sufficient to have a good mathematical resolution alone, all peaks must be significant in crystallographic or structural terms. The incidental measurement of peak-area crystallinity is considered to be of secondary importance to the resolution of overlapping peaks. 

Exploration of a data set before resolution is a golden rule fully applicable to image analysis. In this context, there are two important domains of information in the data set the spectral domain and the spatial domain. Using a method for the selection of pure variables like SIMPLISMA [53], we can select the pixels with the most dissimilar spectra. As in the resolution of other types of data sets, these spectra are good initial estimates to start the constrained optimization of matrices C and ST. The spatial dimension of an image is what makes these types of measurement different from other chemical data sets, since it provides local information about the sample through pixel-to-pixel spectral variations. This local character can be exploited with chemometric tools based on local-rank analysis, like FSMW-EFA [30, 31], explained in Section 11.3. 

Previous lower-frequency electron spin echo envelope modulation (ESEEM) studies showed a histidine nitrogen interaction with the Mn cluster in the S2 state, but the amplitude and resolution of the spectra were relatively poor at these low frequencies. With the intermediate frequency instruments we are much closer to the exact cancellation limit, which optimizes ESEEM spectra for hyperfine-coupled nuclei such as 14N and 15N. We will report the results on 14N and 15N labeled PSII at these two frequencies, along with simulations constrained by both isotope datasets at both frequencies, with a focus on high-resolution spectral determination of the histidine ligation to the cluster in the S2 state. 

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