The principles of detection theory [1] have been used to explore the limits with which two atom types of only slightly different atomic number can be distinguished from STEM images [2]. If one has to choose between two hypotheses corresponding to the presence of atom type 1 or 2, there is a probability of assigning the wrong hypothesis. Using repetitive image simulations with induced Poisson noise this probability of error, P e , can be computed and minimised as a function of the experimental design. Here the inner detector radius of an annular detector is considered as a tunable microscope parameter. The Kullback-Leibler divergence is an alternative performance measure to this probability of error which does not require repetitive simulations. The results of both measures will be compared in the case where one has to choose between the presence of an Al or Ti atom. Atomic number estimation from STEM images: what are the limits? J. Gonnissen 1 , A.J. den Dekker 2 , A. De Backer 1 , J. Sijbers 3 and S. Van Aert 1 1 Electron Microscopy for Materials Science (EMAT), University of Antwerp, Antwerp, Belgium 2 Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands 3 Vision Lab, University of Antwerp, Wilrijk, Belgium References 1. S.M. Kay. “Fundamentals of Statistical Signal Processing. Volume II Detection Theory”, (Prentice-Hall, New Jersey) (2009). 2. A.J. den Dekker, J. Gonnissen, A. De Backer, J. Sijbers and S. Van Aert, Ultramicroscopy, http://dx.doi.org/10.1016/j.ultramic.2013.05.017 (2013). 3. S. Kullback and R.A. Leibler, Annals of Mathematical Statistics 22 (1951), p. 79-86. 4. S. Kullback. “Information theory and statistics”, John Wiley and Sons (1959). 5. J. Gonnissen kindly acknowledges EMS for receiving a scholarship for EMC2013 and financial support from the Research Foundation Flanders (FWO, Belgium) through project fundings (G.0393.11, G.0064.10 and G.0374.13). P  = Pr   ଴ , ଵ ݐݑ + Pr   ଵ , ଴ ݐݑ • Kullback-Leibler (KL) divergence The KL divergence quantifies the difference between two probability distributions [3,4]. The KL divergence from   భ =p  ሺ;  ଵ ሻ to   బ =p  ሺ;  ଴ ሻ is defined as: For a binary hypothesis problem, where each hypothesis corresponds to the assumption of a specific Z value, the probability of error to assign an incorrect hypothesis is computed using repetitive image simulations. For the experimental case where the question was to decide between the presence of a Al or Ti atom, this error has been minimised as a function of the inner detector radius of an annular detector using image simulations. Furthermore, the KL divergence has been proposed as an alternative measure for experiment design which does not require the use of repetitive simulations. A maximum in the KL divergence as a function of the experimental settings corresponds to a minimum in P e . From the optimisation of the experiment design, it turns out that for the considered experimental case, the optimal inner detector radius lies in the LAADF regime, corresponding to a trade-off between signal-to-noise ratio and contrast. Detection Theory Introduction Conclusion • Probability of error When identifying the atomic number Z from STEM images, the probability of choosing the wrong hypothesis is defined as: Simulation study Julie.Gonnissen@ua.ac.be Figure 1. Histograms of the log likelihood ratio ln ሺሻ for three different inner detector radii of 0.7Å -1 , 1.1Å -1 and 2.5Å -1 from left to right. Figure 2. P e (blue) and sum of KL divergences (red) as a function of the inner detector radius. =P  ଴  ଵ P  ଵ +P  ଵ  ଴ Pሺ ଴ ሻ   ሺ; భ ሻ   ሺ; బ ሻ > ሺ బ ሻ ሺ భ ሻ = ln ሺሻ = ln   ሺ; భ ሻ   ሺ; బ ሻ >0 Equal prior probabilities ሺ ଴ ሻ = ሺ ଵ ሻ = ½  ሺ  భ ,  బ ሻ≡ ॱ   భ [ln   ሺ; భ ሻ   ሺ; బ ሻ ] ≡ ॱ   భ [ln ሺሻ] KL-divergence KL-divergence KL-divergence Hypotheses: H 0 : Z=Z 0 =13 (Al) and H 1 : Z=Z 1 =22 (Ti) The sum of KL divergences corresponds to the difference of the expected or mean log-likelihood ratio under H 1 and the corresponding value when assuming H 0 to be true (see Figure 1). Perform repetitive simulations under both hypotheses Image simulations of an Al and Ti atom with Poisson noise, for an inner detector radius of 2.5 Å -1 , a probe forming aperture angle of 1.1 Å -1 , a dose of 100 incident electrons/pixel and acceleration voltage of 300 kV are shown for two different noise realisations: Aluminium Titanium Which hypothesis to choose? Decide H 1 if : with   ሺ;   ሻ the joint probability function evaluated at the simulated observations. Histograms of the log likelihood ratio ln ሺሻ are obtained:    భ ,  బ +   బ ,  భ =ॱ   భ [ln ሺሻ]- ॱ   బ [ln ሺሻ] The sum of KL divergences and the probability of error P e are calculated as a function of the inner radius of an annular detector: 1 2 3 choose H 0 choose H 1 choose H 0 choose H 1 choose H 0 choose H 1