Resolution enhancement for infrared spectroscopy data

The analysis and utilization of spectroscopic data represent an emergent technology of increas- ing importance from both practical and theoretical perspectives. The reasons for this include: (i) It is now relatively inexpensive to rapidly collect, for the material being examined, a highly accurate measurement on a (very) fine wavelength grid of its spectroscopic response to a given electromag- netic stimulus. (ii) Because of the availability of such data, sophisticated algorithms can be applied to recover informa- tion of relevance to the application context within which the spectroscopic data have been recorded. They include: (a) Derivative Spectroscopy (Anderssen and Hegland (2010); Wiley et al. (2009)). The classical numerical analysis mantra about the need to explicitly invoke regularization methodology to stabilize the application of finite difference differentiators related to the fact that the available observational data was sparse and noisy. The sparseness of the data precluded the use of finite difference differentiators with large footprints. For accurate data on a fine grid, the stablization can be invoked explicitly by choosing finite difference differentiators with large footprints. As explained in Anderssen and Hegland (1999), and earlier in Anderssen and de Hoog (1984), large footprint finite difference differentiators implicitly perform averaging with the size of the footprint taking on the role of the regularization parameter. (b) Resolution Enhancement (Hegland and Anderssen (2005)). In the analysis of measured spectra, the goal is the identification of the positions and heights of the spectral lines which correspond to different molecular aspect of the material being studied. In some situations, as in mass spectroscopy, the associated technology can yield highly accurate positions and heights for the spectral lines. In others, such as near infrared (NIR) and mid infrared (mid-IR) spectroscopy, the technology only recovers a smooth approximation of the lines. When the lines are closely spaced and broadened, overlapping betweem them occurs. The recovery of the positions and heights of the lines then becomes a challenging problem especially when explicit models for the measured shape of the peaks approximating the lines are unknown. Numerous methods have been proposed for resolving the fine scale structure in the latter situ- ations. They include a variety of non-linear least squares methods. Their disadvantage is that good starting solutions for the subsequent numerical iterations are required. The alternative approach of resolution enhancement aims to undo the broadening occuring as a result of the measurement process in order to yield an informative reconstruction of the actual positions and heights of the lines. Here, we investigate the recovery of molecular information from mid-IR data using the peak sharpening (narrowing) methodology of Hegland and Anderssen (2005), and compare it with the traditional resolu- tion enhancement techniques of derivative spectroscopy Anderssen and Hegland (2010).