| A Fourier Tool
For Analysis of Coherent Scattering by Biological Nanostructures
In collaboration with Dr. Rodolfo Torres of
the University of Kansas Department of Mathematics, and with
programming by Dr. Scott Williamson (Cornell University),
Christopher Fallen, and Christopher Kovach (University of
Kansas), we have developed a new application of the discrete
Fourier analysis (DFT) to the investigation of coherent scattering
by biological nanostructures. Following originally on an electromagnetic
optical theory of corneal transparency by George Benedek of
MIT (Benedek, G.B. 1971, Applied Optics ), the Fourier Tool
uses the 2D Fast Fourier Transform (FFT) of transmission electron
micrographs of biological nanostructures to characterize the
spatial periodicity in variation in refractive index within
these structures. Using the 2D Fourier power spectrum (the
modulus of the coefficients of the Fourier component spatial
frequencies in all directions within the image), the tool
can be used to test whether spatial variation in refractive
index is random, as required for incoherent (Rayleigh or Tyndall)
scattering, or periodic as required for coherent scattering.
For periodic arrays, the 2D Fourier power spectrum can distinguish
between laminar, crystal-like, and quasi-ordered nanostructures.
Lastly, the 2D Fourier power spectra of color producing nanostructures
can be used to predict the reflectance spectrum due to coherent
scattering. The general justification and applications of
the Fourier Tool are reviewed in Prum and Torres (2003b).
Previous applications of the Fourier Tool are cited in the
references below
Description of the Fourier Tool
The Fourier Tool is implemented in MATLAB ,
a commonly available matrix algebra program (http://www.mathworks.com).
A current Beta version of the Fourier tool is available here
for free as a series of MATLAB commands. By unzipping the
archive of MATLAB m-files, putting in these files in the MATLAB
path, and typing the command "Image_gui" will bring
up the first GUI window (graphical user interface).
Matlab program files
The first input GUI, which allows users to select
an electron micrograph image for analysis, input image scale,
input refractive index values for two materials in the image,
select a square portion of the image for analysis, input other
data or comments, and save these data with the image. The
Input GUI also has some primitive and buggy image processing
capabilities, and a interesting thin-film simulation tool.
The second Fourier GUI produces the 2D Fourier
power spectrum of a selected portion of the image (or by default
the largest square portion). The four quadrants of the 2D
Fourier power spectrum are realigned so that the origin is
at the center. The power spectrum can be viewed in a color
or gray scale. A slide bar allows the user to adjust the power
values for the upper and lower limits of the color or gray
scale. Other buttons permit the user to zoom in on the power
spectrum, zoom in on a standard size section of the power
spectrum which covers spatial frequencies for the entire visible
spectrum (i.e. useful for all applications except transparency),
saving the results of the analysis, and saving a printable
output of power spectrum.
The third Spectrum Analysis GUI allows users
to produce radial averages of a power spectrum and predicted
reflectance spectra. The analyses are produced by averaging
the power values within a series of radial bins (or annuli)
of the power spectrum. The radial bins can be defined with
uniform wavelength or spatial frequency intervals. The number
of bins can also be defined, but the default for wavelength
analyses is 50, and for spectral averages is 100. Radial averages
of the power spectrum are useful for documenting the peak
spatial frequency of variation in refractive index, which
can be used to estimate the average distance between neighboring
objects. Predicted reflectance spectra are calculated from
a radial average by multiplying the inverse of each average
spatial frequency value by two and by the average refractive
index of the image.
Radial averages or predicted reflectance spectra
from multiple images can be combined into a composite average
to summarize the periodicity of multiple images of the same
structurally colored tissue.
(DOWN LOAD THE FOURIER TOOL HERE) coming soon
References
General description of the Fourier Tool:
Prum, R. O., and Torres, R. H. 2003. A Fourier
tool for the analysis of coherent light scattering by bio-optical
nanostructures. Integrative and Comparative Biology 43: 591-610.
Applications of the Fourier Tool:
Prum, R. O., Torres, R. H., Williamson, S.,
and Dyck, J. 1998. Coherent light scattering by blue bird
feather barbs. Nature 396: 28-29.
Prum, R. O., Torres, R. H., Williamson, S.,
and Dyck, J. 1999. Two-dimensional Fourier analysis of the
spongy medullary keratin of structurally coloured feather
barbs. Proceedings of the Royal Society, London: Biological
Sciences (B) 266: 13-22.
Prum, R. O., Torres, R. H., Kovach, C., Williamson,
S., and Goodman, S. M. 1999. Coherent Light Scattering by
Nanostructured Collagen Arrays in the Caruncles of the Malagasy
Asities (Eurylaimidae: Aves). Journal of Experimental Biology
202, 3507-3522.
Prum, R. O., Andersson, and S. F., Torres, R.
M. 2003. Coherent scattering of ultraviolet light by avian
feather barbs. Auk 120:163-170.
Prum, R. O., and Torres, R. H. 2003. Structural
colouration of avian skin: Convergent evolution of coherently
scattering dermal collagen arrays. Journal of Experimental
Biology. 206: 2409-2429.
Prum, R. O., and Torres, R. H. 2004. Structural
colouration of mammalian skin: Convergent evolution of coherently
scattering dermal collagen arrays. Journal of Experimental
Biology. 207: 2157-2172.
Prum, R. O., Cole, J. A., and Torres, R. H. 2004. Blue integumentary structural colours in dragonflies (Odonata) are not produced by incoherent Tyndall scattering. Journal of Experimental Biology 207:3999-4009.
Shawkey, M. D, , Saranathan, V., Pálsdóttir, H., Crum, J., Ellisman, M., Auer, M., Prum, R. O. 2009. Electron tomography, three-dimensional Fourier analysis and colour prediction of a three-dimensional amorphous biophotontic nanostructure. Journal of the Royal Society Interface doi:10.1098/rsif.2008.0374.focus
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