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