Propagation of priors for more accurate and efficient spectroscopic functional fits and their application to ferroelectric hysteresis

Multi-dimensional spectral-imaging is a mainstay of the scanning probe and electron microscopies, micro-Raman, and various forms of chemical imaging. In many cases, individual spectra can be fit to a specific functional form, with the model parameter maps, providing direct insight into material properties. Since spectra are often acquired across a spatial grid of points, spatially adjacent spectra are likely to be similar to one another; yet, this fact is almost never used when considering parameter estimation for functional fits. On datasets tried here, we show that by utilizing proximal information, whether it be in the spatial or spectral domains, it is possible to improve the reliability and increase the speed of such functional fits by ∼2–3×, as compared to random priors. We explore and compare three distinct new methods: (a) spatially averaging neighborhood spectra, and propagating priors based on functional fits to the averaged case, (b) hierarchical clustering-based methods where spectra are grouped hierarchically based on response, with the priors propagated progressively down the hierarchy, and (c) regular clustering without hierarchical methods with priors propagated from fits to cluster means. Our results highlight that utilizing spatial and spectral neighborhood information is often critical for accurate parameter estimation in noisy environments, which we show for ferroelectric hysteresis loops acquired on a prototypical PbTiO3 thin film with piezoresponse spectroscopy. This method is general and applicable to any spatially measured spectra where functional forms are available. Examples include exploring the superconducting gap with tunneling spectroscopy, using the Dynes formula, or current–voltage curve fits in conductive atomic force microscopy mapping. Here we explore the problem for ferroelectric hysteresis, which, given its large parameter space, constitutes a more difficult task than, for example, fitting current–voltage curves with a Schottky emission formula (Chiu 2014 Adv. Mater. Sci. Eng. 2014 578168).