The large amounts of high-quality “multi-dimensional” data generated by modern microscopes open new avenues for quantitative nano-characterization. Quantitative analysis of spectra and images often involve fitting a model to experimental data and, indeed, the literature is rich in applications; examples include atom counting [1], time resolved microscopy [2], electron energy loss [3] and cathodoluminescence [4] spectroscopy. However, using conventional methods to fit large datasets is challenging and when applied to multi-dimensional models, they may become ill-suited. The dominant and most common problem is that conventional methods struggle with any non-linearity in the model and often require an estimate of the starting parameters that are close to the true values. Here we present a Smart Adaptive Multi-dimensional Fitting algorithm (SAMFire) designed to ease the task of fitting such data by automatically generating best estimates for the parameters as the fitting progresses. SAMFire can fit multi-dimensional spectra, images and data of higher dimensionality and will be available in open-source software package HyperSpy v1.0.0 [5].

   SAMFire enables quantitative analysis of large multi-dimensional datasets that would be very challenging—if not impossible—to analyse by other means. It provides multiple fitting strategies that consist of pixel selection and parameter estimation, each tailored to different data structures. Example pixel selection orders are shown in Figure 1(a) for conventional fitting algorithms and in Figure 1(b) for one of the SAMFire strategies. The “raster” order is only viable for unusually stable and constrained models. In contrast, SAMFire follows the “path of least resistance”, learned from already fitted parts of the data and hence is applicable to a much broader range of problems.

   As an example of a complex electron microscopy data analysis problem that can be easily addressed with SAMFire, Figure 2 shows a single spectrum and the result of EELS elemental and bonding quantification by curve fitting from a tilt-series of spectrum-images of a mixed phase nanoparticle. The model consists of eleven components to accurately describe the five elements and a background. Due to the complexity of the model, the geometry of the particle and the low signal-to-noise-ratio, the outcome of fitting individual pixels was highly dependent on the starting parameters, making the analysis very challenging using conventional fitting routines. In contrast, SAMFire was able to fit the whole tilt-series with minimal user input.

   Since SAMFire enables highly sophisticated models to be fitted to large multi-dimensional datasets significantly faster and more easily than previous algorithms, we anticipate it will become standard analysis practice, especially when quantitative analysis is required. Examples that we are currently considering include tracking motion in a time series and quantification of both light and trace elements in multiple-domain structures.

   We acknowledge the support received from the European Union Seventh Framework Program under Grant Agreement 312483 – ESTEEM2 (Integrated Infrastructure Initiative – I3) and under Grant Agreement 291522-3DIMAGE. We thank Raul Arenal and Rowan Leary for providing the raw data shown in Figure 2.

[1] Van Aert, S., et al. Nature 470.7334 (2011): 374-377.

[2] Yurtsever, A., van der Veen, R. M., & Zewail, A. H. (2012). Science, 335(6064), 59-64.

[3] Verbeeck, J., and Van Aert, S. Ultramicroscopy 101.2 (2004): 207-224.

[4] Zagonel, L. F., et al. Nano Letters 11.2 (2010): 568-573.

[5] www.hyperspy.org

Figures:

Figure 1. (a) Multi-dimensional data fitting pixel traversal example for a conventional algorithm. As the pixels are fitted in “raster” order, large jumps occur, and using the last solution as a new starting point may be invalid. An example of how SAMFire traverses such data is shown in (b). Pixels are fitted in the order “suggested” by the learned data structure, estimating starting parameters from neighbouring, already fitted, parts of the data (not shown). The starting pixel is darkened.

Figure 2. A raw EELS spectrum is plotted as black dots. The result of the fit is shown as the red line, with a power-law background as the black line, and individual element components highlighted in colours. Boron as an element, a nitride and an oxide are separated. In the inset a HAADF image, 90nm by 90nm in size, of the particle is shown.

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