# 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