Digital image warping

by many engineers and scientists, although it is currently the subject of much active

research [Marvasti 87]. We will revisit this topic later when we discuss nonlinear image

warping.

In the continuous domain, we define

a (x) = (2.1.6)

0, x0

can be used to sample a continuous function f (x) as follows

f (xo) = i f (?)5(x-?)d? (2.1.7)

If we are operating in the discrete (integer) domain, then the Kronecker delta function is

used:

1, x=0

for integer values ofx. The two-dimensional versions of the Dirac and Kinnecker delta

functions are obtained in a separable fashion by taking the product of their 1-D coonter-

parts:

Kronecker: 8(m,n) = 8(m)(n)

When an impulse is applied to a filter, an altered impulse, referred to as the impulse

response, is generated at the output. The first direct outcome of linearity and spatial-

invariance is that the filter can be uniquely characterized by its impulse response. The

significance of the impulse and impulse response function becomes apparent when we

realize that any input signal can be represented in the limit by an infinite sum of shifted

and scaled impulses. This is an outcome of the sifting integral

f(x) = i f(?)(x-?)d? (2.1.10)

which uses the actual signal f (x) to scale the collection of impulses. Accordingly, the

output of a linear and space-invariant filter will be a superposition of shifted and scaled

impulse responses.