Darkfield microscopy, and sinc-let decompositions in nature

This is working draft of a short note* on one particular ``space and frequency specific'' decomposition found in nature.  Comments, corrections, and contributions both on this communication, and on ways to optimize the performance of these and related tools for future application, are invited. /pf

Abstract

Introduction

The word wavelet is less familiar in microscopy than in other areas of applied math.  However, microscopists have long been versed at interpreting images which trade spatial resolution for information on transverse momentum (e.g.on illumination direction, for example).  As a result, remarkably and perhaps without knowing it microscopists (particularly those involved in electron microscopy of structures whose periodicity is large compared to electron wavelength) have been developing skills connected to the use of wavelets since well before the coining of the word in the 1980's [Grossman and Morlet].  This is of course because microscopists do wavelet analysis with help from optical (rather than digital) computation, wherein their analysis hides behind the more familiar appellation: darkfield imaging.  Here we discuss a few ways microscopists, and other periodicity analysts as well, might enjoy bringing wavelets to bear on recorded datasets with help from their computers, in ways inspired by practices long established on live images with help from time-honored optical darkfield tools.

Darkfield Images and Wavelets

The elementary wavelet transform in optics and signal processing can be viewed as a windowed Fourier transform, in which the window and exponential kernel of the original Fourier transform together combine to become the new kernel of the wavelet transform [cf. Boone, 1998].  In this context, elementary wavelet transforms in 1D might be written as

where is the input function, c is a shift parameter, and a is a scale parameter.  If the kernel function is then written as the product of a window function w[x], and a modulation function , then the Fourier transform of the wavelet becomes simply

where S[u] and W[u] are the Fourier transforms of s[x] and w[x], respectively.

In the specific case when , then the Fourier transform of the window function is simply W[u] = Rect[u] If[ |u| < 1/2, 1, 0].  The resulting wavelet window has a width in frequency space of 1/a, and an offset from the origin of n/a.  The resulting wavelet , for integer values of n, can thus be used to "sample" all of frequency space with a set of non-overlapping and sharp-edged "tiles".  In the brightfield case when , these regions include the DC peak in the Fourier transform. Otherwise, offers what microscopists would call a darkfield image.  This particular example lends itself to rapid calculation of a series of fast Fourier darkfield decompositions, to which we return below, that bridges the gap in a discrete fashion between direct and reciprocal versions of a given dataset.

In two dimensions, the window function becomes w[x,y] and the modulation function becomes , so that kernel .  Then the wavelet is written as

where s[x,y] is the input image, {c,d} are shift parameters, and {a,b} are scale parameters.  It's Fourier Transform is

In the special case mentioned above, the 2D window function w[x,y] becomes the inverse Fourier transform of W[u,v] = Rect[u]*Rect[v].  Hence the window becomes a rectangular tile in reciprocal space, of width 1/a in the x-direction, and 1/b in the y-direction.  However, almost any function for w[x,y] will serve for darkfield imaging, provided W[u,v] is zero whenever , or , so that the DC peak is excluded from all except the image.  If a tiling is also to cover reciprocal space completely, e.g. for heirarchical applications as in the case above, then a conservation of energy constraint will also need to apply.

Non-tiled Circular, Gaussian, and Bayesian Background-Subtracted Darkfields

One feature of the wavelets defined above, that may at first glance be unfamiliar to microscopists, is that they in general are complex.  Fourier filtering of initially real images normally preserves the conjugate symmetry in reciprocal space, by selecting regions (e.g. to window) which are symmetric about the DC peak.  The darkfield images described here do not.  As a result, however, they reclaim something that is normally lost in Fourier filtering of images: The continuous (fringe-free) illumination of scattering centers (e.g. of crystals in the field diffracting electrons into the selected Fourier aperture).  This is because asymmetric tile darkfield imaging, of the sort described here, relegates periodicity fringes in the images largely into phase, rather than amplitude, variations in the image.  By simply examining amplitude (or amplitude squared) across the images, these variations disappear just as they do in the microscope when image intensities (and not phases) are recorded in the imaging process.  These advantages of asymmetric darkfield are illustrated in the Fig. 1.

Fig. 1:  Amplitude images of a two dimensional spatially-delimited periodic array (e.g. a small crystal in projection).  Inset a) is the starting brightfield dataset; b) is an image formed from a conjugate symmetric pair of diffracted beams and the DC peak; c) is the same as inset b without the DC peak -- note the doubled periodicity because of the doubled spacing between interfering tiles; d) is the amplitude of a complex darkfield formed using a rather small single tile; e) is the same as inset d using a larger tile in frequency space; f) is the same as inset d using an even larger tile in frequency space -- note the uncertainty principle in action in this last series of 3.

As long as tiling is not a requirement of the analysis process, for example when one simply wants to see where in the image the source of a peak in the power spectrum is located, any imaginable window function (not including the DC peak) may be used to form a darkfield image.  Given the growing ability of microscopes to provide large digital images containing abundant crystal lattice fringe information, one can thus use these wavelets in the computer to do darkfield imaging just as in a transmission electron microscope, with the added advantage that one can be much more creative, and less limited by the physical challenges of aperture design, than is possible in the microscope.  

For example, microscope apertures are usually circular (at least by design)...

In addition to use of different individual "apertures" of the form H[u,v], this strategy also facilitates a number of "decoherence" strategies, some of which (like hollow cone darkfield) have been implemented optically, and others which have not.  This is done by forming separate darkfield images in more than one chosen (typically local) range of frequencies, and then either combining or cross-correlating the intensities (square amplitudes) found therein.  Thus annular darkfield images might be formed by adding intensities in patches forming one or more annular rings in the power spectrum, thus targeting regions responsible for a given "powder diffraction" signature.  Alternatively, one might cross-correlate intensities found in patches associated with a specific pair of reflections, thus searching for cross-fringes from a particular single crystal lattice orientation in a large image.

The Fast Fourier Darkfield (Sinc Wavelet) Series

Sinc wavelet (square-aperture darkfield) direct <=> reciprocal glide (.5MB).  Hexagonal, and symmetric Penrose, tiling may be other natural options in this regard.  A template for extension to non-orthogonal 2-fold symmetries, and logarithmic frequency maps, is provided as well.

Some Practical Applications

Recent application to a project with Jay Switzer at UM-Rolla on a paper for Science about electrochemical epitaxy on integrated circuit silicon.

Challenges and Conclusions

References

Bradley G. Boone, Signal Processing Using Optics (Oxford University Press, 1998)

Context

* This material contains information on work in progress at the UM-StL Center for Molecular Electronics' Scanned Tip and Electron Image Lab.  Collaborators include faculty, staff and students in Departments of Physics/Astronomy, and Mathematics, at UM-StL.  For legal purposes, you should consider this material Copyright 2000, by P.Fraundorf.  Until primary elements of this work have been submitted for publication, at least to a preprint archive, we also request that you consult with us in sharing the information on, or about, this page with others.