Page 275 - Introduction to Information Optics

260 5. Transformation with Optics

3. The input is placed in a converging beam behind a converging lens of a
focal length / at any distance d < f from the focal plane of the lens, to
where the beam converges. The Fourier transform is obtained in the focal
plane of the lens [9] with the Fourier spectrum scaled as u — x'//J and
2
multiplied by the quadratic phase factor exp(mx' /M), where x' is the
coordinate in the Fourier plane.

The physics behind the optical Fourier transform is the Fraunhofer diffrac-
tion. The lens is used to remove the quadratic phase factor in the Fresnel
diffraction integral. However, some people used to call the lens as the Fourier
transform lens. Note that it is the Fraunhofer diffraction, not the lens, that
implements the optical Fourier transform.

5.4 WAVELET TRANSFORM

The wavelet transform has been introduced in the last fifteen years for
multiresolution and local signal analysis, and is widely used for nonstationary
signal processing, image compression, denoising, and processing [12, 13],

5.4.1. WAVELETS

The wavelet transform is an expansion of a signal into a basis function set
referred to as wavelets. The wavelets h Stt(t) are generated by dilations and
translations from a reference wavelet, also called the mother wavelet, h(t):

(5.8)

where s > 0 is dilation and r is the translation factors. The wavelet transform
2
of a signal f(f) is defined as the inner product in the Hilber space of L norm:
i r / \
=
W f(s, T) = </,„(*), /(0> = -7= U* ( - ~ I fit) dt, (5.9)

which can be considered a correlation between the signal and the dilated
wavelet, h(t/s). The normalization factor 1/^/s in the definition of the wavelet
in Eq. (5.8) is such that the energy of the wavelet, which is the integral of the
squared amplitude of the wavelet, does not change with the choice of the
dilation factor s.