User Manual - pancroma

62. FFT Image Analysis_______________________________________________________________IntroductionFourier Analysis has many applications to the field of satellite image processing.The idea behind Fourier analysis is as follows: a variable quantity, say a functionof time, f(t), can be approximated by a linear combination of harmonic functions(sines and cosines) of increasing frequencies, specifically, positive and negativemultiples of the fundamental harmonic. If you know the right contributions of allfrequencies (Fourier coefficients), that knowledge is equivalent to knowing thefunction. Put another way, a time dependent function f(t) can be transformed andwritten equivalently as a frequency-dependent function F(ω). The FourierTransform is a mathematical equation that computes the coefficients that,multiplied by cosine and sine functions expresses f(t) in terms of F(ω).The Fourier transformation has an inverse, which is very similar mathematically,except that it transforms F(ω) back into f(t). Both transformations are linear, sothat, if a and b are any constants, the transformations take a f ( t)+ b g(t)intoa F( ω)+ b G(ω)and back again.Intuitively, it is best to think of the Fourier transformation as approximating afunction by sines and cosines. However, the actual formulas become simpler ifthe transformation is re-cast as an approximation by a sum of complexexponentials according to Euler's equation:exp( ix ) = cos( x)+ isin(x)This notation is also essential for formulating the Fourier matrices used forcomputing the Fast Fourier Transform (FFT). In this new expression, which iscalled the complex Fourier transformation, the transformed function "f" can becomplex, and its transformation "F" is generally a complex function as well. Allthe basic properties (the inverse, linearity, convolution etc.) remain the same, asexpressing the Fourier transformation in complex notation is mathematicallyequivalent to the more cumbersome sine and cosine version.IMPORTANT NOTE: In image processing, the image is considered to only havea real component, i.e. the imaginary component is always zero. However, theFourier transform of an image resides in the complex plane and has both types ofcomponents.One property of the Fourier transformation that is very useful in graphics andimaging is known as the convolution theorem. Technically, a convolution of twofunctions is an integral of their product, where one function is displaced relative283