Page 271 - Introduction to Information Optics

256 5. Transformation with Optics
Fourier transform, which describe optical propagation and diffraction in
free-space optical systems, we introduce three transforms: the physical wavelet
transform, the Wigner distribution function, and the fractional Fourier trans-
form. These recently developed transforms also describe optical propagation
and diffraction but from different perspectives. The Wigner distribution func-
tion was discussed in Sec. 1.5.5 as a signal analysis tool. In Sec. 5.6 we show
the Wigner distribution function with a geometrical interpretation; this leads
to Wigner distribution optics in Sec. 5.8. Also, we discuss the Hankel transform
in terms of the Fourier transform in the polar coordinate system.
There are a large number of optical processors and systems developed for
implementation of a variety of transforms for optical signal processing and
pattern recognition. These optical transforms are discussed in different chapters
of the book, such as the Mellin transform in Sec. 2.6.1, the circular harmonic
transform in Sec. 2.6.2, the homomorphic transform in Sec. 2.6.3 and many
other optical processing algorithms and neural networks implemented with
optical correlators and other optical processors.
In this chapter we include the radon transform, which is widely used in
medical image processing; and the geometric transform, which has a variety of
applications. The Hough transform is a type of geometric transform, also
discussed.
The collection of optical transforms in this chapter is far from complete.
Readers interested in transformation optics can find many books [8, 9] and
technical journals on the subject.

5.1. HUYGENS-FRESNEL DIFFRACTION

As early as 1678, Huygens suggested for interpreting optical diffraction that
each element on a wavefront could be the center of a secondary disturbance,
which gives rise to a spherical wavelet [10], and that the wave front at any
later time is the envelope of all such wavelets. Later, in 1818, Fresnel extended
Huygens's hypothesis by suggesting that the wavelets can interfere with one
another, resulting in the Huygens- Fresnel principle, which is formulated as

E(r) = - dsEM cos(n, (r _
.MJi '

where JE(r) is the complex amplitude of the optical field in the three-dimen-
sional (3D) space, r is the position vector in the 3D space, and r' is the position
vector at the aperture E, where the integral is taken over, k = 2n/A with the
wavelength A, and cos(w, (r — r')) is a directional factor, which is the cosine of