the input and 02 be the output when 12 is the input. Then the system
is linear when, if the input is al1+bI2 the output is a01+b02. This
property of linearity leads to a vast simplification in the
mathematical description of phenomena and represents the foundation of
a mathematical structure known as linear system theory. When the
system is linear, the input and output may be decomposed into a linear
combination of elementary components.
Another mathematical tool of great use is the Fourier transform.
The Fourier transform is defined by
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F(u,v) = fff(x,y) exp -j2rr(ux+vy) dx dy = F {f(x,y)}. (2.1)
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The transform is a complex valued function of u and v, the spatial
frequencies in the image plane. The Fourier transform provides the
continuous coefficients of each frequency component of the image. The
Fourier transform is a reversible process, and the inverse Fourier
transform is defined by
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f(x,y) = IfF(u,v) exp j27nux+vy) dx dy = F-1{F(u,v)}. (2.2)
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The transform and inverse transform are very similar, differing only
in the sign of the exponent appearing in the integrand. The magnitude
squared of the Fourier transform is called the power spectral density
Of = !F(u,v) 2 = F(u,v) F*(u,v). (2.3)
It is noteworthy that the phase information is lost from the Fourier
transform when the transform is squared and the image cannot, in
general, be reconstructed from the power spectral density. Several
useful properties of the Fourier transform are listed here.