[Here, we have rewritten the elementary image (A1.2) by substituting ui by l/(IAx) and x by iAx, etc.) To understand equation (A1.4), consider one of its elementary images, indexed (i = 1, m = 0): as i increases from 1 to I, the argument of the sine function goes from 0 to 2n exactly once: this elementary image is a horizontally running sine wave that has the largest wavelength (or smallest spatial frequency) fitting the image frame in this direction.
The 2D sine transform is a 2D scheme, indexed (i, m), that gives the amplitude aim and the phase shifts 'im of all elementary sine waves with spatial frequencies (ui, vm), as defined in equation (A1.3), that are required to represent the image.
The Fourier representation generally used differs from the definition (A1.4) in that it makes use of ''circular'' complex exponential waves:
2n ii mk
2n ii mk
(Note that "i" is the symbol for the imaginary unit, to be distinguished from the index i.) The reason is that this representation is more general and leads to a more tractable mathematical formulation. Hence, the Fourier representation of an image is
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