Abstract. An experimental study of the possibility of spatial frequencies
spectra multiplication for the black and white images of flat fractal objects by
using their preliminary transformation into rarefied ones is carried out. The
multiplication algorithm, based on well-known method of
sampling with a comb function, is illustrated for L-system
based geometrical fractal with definite production rules and with seed set
(axiom) in a form of unit black square. Realisation of used procedure within
the frames of chosen fixed production rules bases on the substitution of
initial axiom by unit white square centred with black
square with side equals 1/q, where q is any positive number greater than one. For any integer
positive odd number multiplication factor of spectra equals q2.
A first step for the experimental
determination of spatial frequencies spectra of objects by observation of the
diffraction pattern forming in the Fraunhofer zone is the creation of
black-and-white raster pictures of the selected fractals on the computer using
specially designed software. Then the resulting
pictures were printed onto a transparent film with imagesetter having
resolution of 1333 points / cm and the spot size about of 7.5 µm.
Forming after the passage of the
light beam through the transparent film with fractal picture diffraction
patterns were compared with the fast Fourier transform data of digitized images,
obtained by a uniform grid function approximation with
sufficiently large (up to 4096x4096) number of nodes in order to ensure
adequate approximation of the smallest image details.
It was shown that a good accordance the
experimental results with digital calculations data is achieved for
p>2, where parameter
p equals to the ratio of the smallest element of
the fractal image to the approximating grid function period.
Key words:
approximation, 2D Fourier transform, diffraction pattern, spectra
multiplication, spatial frequencies, sparse fractal, grid function, L-system
fractal.
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