Digital Darkfield Tableaus

Abstract

Like wavelet decompositions, power spectra "tiled with direct space maps" are versions of a lattice image intermediate between direct and reciprocal space [1]. They generally take about the same time to calculate as a power spectrum, and contain useful information of all sorts about "where periodicities lie". Questions of this sort will be of increasing importance e.g. in subångstrom resolution pictures. The information is useful in spite of the fact that members of the mathematical harmonic analysis community have long moved past the sharp edged Fourier apertures that electron optical darkfield requires [2]. By illustrating useful applications for sharp-edged tilings, we might thus inspire the math community to look at the much wider range of pattern recognition problems that today's high resolution electron microscopists face.The first tiling we discuss is the one most familiar to microscopists: Amplitude tableaus. It's familiar because the inverse transform (bandpass image) of an apertured Fourier transform (or diffraction pattern) has phase and amplitude components, but only the latter are detectable in optical darkfield images recorded on film. They are also optically tedious to set up. Digitally on the other hand, a 1024x1024 image yields a tableau of 32x32 complex bandpass images in only seconds on a modern computer, the amplitude version of which corresponds (in terms of bandpass) to 1023 darkfield and one brightfield image taken with the microscope. Because individual tiles with spatial correlations easily catch one's eye, one can quickly examine the tableau for regions of hidden periodicity, as well as for regions of characteristic shape (like the bowties and butterflies characteristic of icosahedral twins in the example of Fig. 1).The other tilings we mention here are Isotropic and Shear Strain tableaus. These exploit the idea that phase in the bandpass image measures displacement, so that phase gradient measures periodicity deviations from the bandpass reference [3]. This technique has been applied, one periodicity at a time, for example to measure interface and dislocation strain [4,5]. Fig. 2 illustrates how a tableau of isotropic (parallel to the reference g-vector) and shear (perpendicular to the g-vector) strain tiles in effect yield "diffraction contrast images" from a whole range of operating reflections. Here we use opposing hues red and cyan to represent relative compression and expansion, respectively, in the isotropic maps, and the orthogonal hues indigo and chartreuse to represent CCW and CW rotation, respectively, in the shear maps.

A modified cross-correlation algorithm for PIV image processing of particle-fluid two-phase flow

“…The short-time Fourier transform (STFT) [36] is more similar to the wavelet transform in that it is also time and frequency localized, but there are issues of the frequency/time resolution trade-off. Wavelets often give a better signal representation using multi-resolution analysis, with balanced resolution at any time and frequency [37]. Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, nonperiodic and/or non-stationary signals, making it very convenient for its application in numerous fields.…”

Digital Darkfield Analysis of Nanoparticle Defects

“…2 shows singleslice wedges for (200) and (220) reflections, in place of the "bowties" seen for icosahedral twins [7], and double-slice wedges for (111) reflections in place of icotwin "butterflies". Notice also the absence of a (111) discontinuity bisecting the double-slice wedges.…”

“…Now integrative views of lattice information in digital images yield new challenges for mathematical harmonic analysis [3]. Such challenges are illustrated by a series of notes [4][5][6][7] on the uses and shortcomings of "weakly convergent" Fourier transform window techniques. Here, we show how robust (if pedestrian) digital darkfield techniques help to characterize: (i) an Antimony-doped Tin Oxide nanoparticle screw dislocation, and (ii) twinning and strain in a gold decahedral twin.…”

Digital Darkfield Analysis of Lattice Fringe Images with ImageJ

“…This analysis involves a type of image decomposition which, like wavelets, is intermediate between direct and reciprocal (spatial-frequency) space [1]. Digital-darkfield analysis is of particular use on images that have well defined spatial-frequencies in them, like lattice-fringe images of atomic-planes encountered edge-on by high-energy electrons in crystalline materials [2][3][4][5][6][7][8][9][10].A set of java plugins is posted on the ImageJ wiki at http://imagejdocu.tudor.lu/, along with macros for putting them to use. The java plugins include routines for quantitatively converting complex-number arrays to and from RGB images with pixel-intensity proportional to coefficient log-amplitude and pixel-hue linked to coefficient phase.…”

“…This darkfield-tableau strategy has helped find patterns, like icosahedral twin"bow-ties" and "butterflies" in images of randomly-oriented nanoparticles [12][13]. …”