model, a spatial carrier frequency on the Fourier transform of the
image must be induced. The image is encoded as f(mAx,nAy) with a SBP
of M x N where M and N are the number of points in the image in each
direction. If the Fast Fourier Transform (FFT) is applied to the
image, a digital representation of the Fourier transform of the image
is obtained. This transformed image F(mAu,nAv) contains the same
number of points as the image and obviously the same SBP. If the
image contained M points along the x direction, the highest spatial
frequency possible in this image would be M/2 cycles/frame. This
situation would exist when the pixels alternated between 0 and 1 at
every pixel. That is, the image consisted of {0,1,0,1, ...}. The
maximum frequency in the transform is M/2 cycles/frame in both the
positive and negative directions. The FFT algorithm provides the real
and imaginary weights of each frequency component ranging from -M/2+1
cycles/frame to +M/2 cycles/frame in one cycle/frame steps. This
provides M points in the u direction. The same is true for N points
in the v direction. Thus, the first point in the FFT matrix is
(-M/2+1,-N/2+1), the D.C. term is in the M/2 column and N/2 row, and
the last term in the FFT matrix is (M/2,N/2).
It is useful to point out that the FFT describes the frequency
components of the image f(x,y). The FFT pattern also contains
structure which can also be represented by a Fourier series. That is,
the FFT pattern or image has specific frequency components. Because
the image and the FFT are Fourier transform pairs, the image describes
the frequencies in the FFT pattern. For example, a spike in the image
implies the FFT will be sinusoidal. A spike in the FFT implies the
image is sinusoidal. The existence of a non-zero value on the outer