Digital image warping

The response g (x) of a digital filter to an arbitrary input signal f (x) is expressed in

terms of the impulse response h (x) of the filter by means of the convolution integral

g(x) = f (x)* h(x) = I f O)h(x-))d? (2.1.11)

where * denotes the c0ff7oltion operation, h (x) is used as the convolution kernel, and 

is the dummy variable of_integration. The integration is always performed with respect

to a dummy variable (such as ) and x is a constant insofar as the integration is con-

cerned. Kernel h (x), also known as the filter kernel, is treated as a sliding window that is

shifted across the entire input signal. As it makes its way across f (x), a sum of the

pointwise products between the two functions is taken and assigned to output g (x). This

process, known as convolution, is of fundamental importance to linear filtering theory.

The convolution integral given in Eq. (2.1.11) is defined for continuous functions

f (x) and h (x). In our application, however, the input and convolution kernel are

discrete. This warrants a discrete convolution, defined as the following summation

g(x) = f (x)* h(x) =  f (?)h(x-?)d? (2.1.12)