Accurate extraction of reciprocal space information from transmission electron microscopy images. Edward Rosten and Susan Cox LA-UR

Transcription

1 Accurate extraction of reciprocal space information from transmission electron microscopy images Edward Rosten and Susan Cox LA-UR

2 Diffraction patterns Diffraction pattern is the Power Spectral Density of the crystal structure.

3 Diffraction patterns TEM, X-ray diffraction,... Manganites, Uranium, Nicklates, Cuprates,...

4 Diffraction patterns

5 Diffraction patterns

6 Diffraction patterns a* c*

7 Diffraction patterns a* c* Parent lattice reflection

8 Diffraction patterns a* c* Parent lattice reflection

9 Diffraction patterns a* c* Parent lattice reflection

10 Diffraction patterns a* c* Superlattice reflections Parent lattice reflection

11 Diffraction patterns a* c* Superlattice reflections Parent lattice reflection

12 Diffraction patterns a* c* Superlattice reflections Parent lattice reflection

13 TEM imaging Electron gun Condenser lens Objective lens Sample Projective lens Viewer Magnifier Flourescent screen Camera Image can be recorded on: CCD Linear response Not very robust Short exposure times only Film Nonlinear response Response varies Very robust Long exposure times possible Images must be acquired on film.

14 Film response correction Noise is constant over the image Shot noise is small Noise amplitide is small η df de f(e) = film intensity at electron intensity e df de = film sensitivity η = noise intensity

15 Computing noise intensity 1. Smooth image Horizontal smoothing by fitting local polynomial Polynomial prevents spreading of spots 2. Smoothed image contains true pixel values 3. Bin pixels according to true intensity 4. η i = σ i 25 Noise standard deviation Pixel value

16 Film response Electron intensity (a.u.) r = , n = 3r r=5, n=15 Range of r, n Pixel value

17 Film response Electron intensity (a.u.) r = , n = 3r r=5, n=15 Range of r, n Pixel value

18 Film response Electron intensity (a.u.) r = , n = 3r r=5, n=15 Range of r, n Pixel value Electron intensity (arbitraty units) Uncorrected film Corrected film CCD Position along scanline

19 Parent lattice localisation Mean shift: Compute centroid of pixels in window Move window centre to computed centroid

20 Parent lattice localisation Mean shift: Removing background by filtering improves accuracy

21 Parent lattice localisation Two points along a define grid Grid is inaccurate away from points

22 Parent lattice localisation Mean shift optimizes all points

23 Parent lattice refinement Model of imaging system: 0 sx sy s 1 C A = 0 h 1, h 1,2n+1 h 2, h 2,2n h 3,2n 1 h 3,2n C A X n Y n. X Y 1 C A 1 (X, Y ) = pixel coordinate (x, y) = coordinate on grid Find h 1,1,..., h 3,2n using reweighted least squares n = 2 is sufficient

24 Parent lattice refinement

25 Intensity is un-normalized probability distribution Spots are roughly Gaussian Background is roughly constant Model as three component mixture model Fit mixture model with EM

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66 Robust mean computation

67 Robust mean computation Use EM to compute the mean

68 Evaluation: Uncorrelated noise 0.45 Computed q/a* Mean Robust mean (uniform) Robust mean (histogram) Ground truth Asymptotic noise Noise intensity / spot intensity PDF of measured positions with very high noise Probability density Computed q/a*

69 Evaluation: Uncorrelated noise 0.45 Computed q/a* Mean Robust mean (uniform) Robust mean (histogram) Ground truth Asymptotic noise Accuracy of measurement (3σ) Noise intensity / spot intensity Noise intensity / spot intensity Uniform Histogram

70 Evaluation: Correlated noise 0.45 Computed q/a* Mean Robust mean (uniform) Robust mean (histogram) Ground truth Ramp intensity (R) / spot intensity

71 Results on La 0.5 Ca 0.5 MnO q/a* σ = 0.02 pixels

72 Any Questions?