Digital image warping

Convolution is a process which is difficult to visualize. Although a graphical con-

stmcfion is helpful in determining the output, it does not support the mathematical rigor

that is necessary to design and evaluate filter kernels. Moreover, .the convolution integral

is not a formulation that readily lends itself to analysis and efficient computation. These

problems are, in large part, attributed to the domain in which we are operating.

Thus far, our entire development has taken place in the spatial domain, where we

have represented signals as plots of amplitude versus spatial position. These signals can

just as well be represented in the frequency domain, where they are decomposed into a

sum of sinusoids of different frequencies, with each frequency having a particular ampli-

tude and phase shift. While this representation may seem alien for images, it is intuitive

for audio applications. Therefore, we shall first develop the rationale for the frequency

domain in terms of audio signals. Extensions to visual images will then follow naturally.