Introduction to Digital Image Processing

154 DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION

Because the expectation operator is linear,

. (6.5-1b)

The correlation function of the output image array is

(6.5-2a)

or in expanded form

. (6.5-2b)

After multiplication of the series, and performance of the expectation operation, one

obtains

(6.5-3)

where represents the correlation function of the input image array.

In a similar manner, the covariance function of the output image is found to be

. (6.5-4)

If the input and output image arrays are expressed in vector form, the formulation of

the moments of the transformed image becomes much more compact. The mean of

the output vector p is

(6.5-5)

and the correlation ...