Digital image warping

A .filter is any system that processes an input signal f (x) to produce an output sig-

nal, or a response, g (x). We shall deootc this as

f (x) -- g(x) (2.1.2)

Although we arc ultimately interested in 2-D signals (e.g., images), we use 1-D signals

here for notational convenience. Extensions to additional dimensions will be handled by

considering each dimension independently.

Filters are classified by the nature of their responses. Two important criteria used to

distinguish filters re linearity and spatiabinvariance. A filter is said to be linear if it

satisfies the following two conditions:

q.f (x) -- xg(x) (2.1.3)

fl(x) + f2(x) --> gl(x)+ g2(x)

for all values of {t and all inputs ft (x) and f2(x). The first condition implies that the out-

put response of a linear filter is proportional to the input. The second condition states

that a linear filter responds to additional input independently of other signals present.

These conditions can be expressed more compactly as

{tf t(x) + o;2f 2(x) ' {tlgt(x) + o;292(x) (2.1.4)

which restates the following two linear properties: scaling and superposition at the input

produces equivalent scaling and superposition at the output.

A filter is said to be space-invariant, or shift-invariant, if a spatial shift in the input

causes an identical shift in the output:

f (x-a) --> g(x-a) (2.1.5)

In terms of 2-D images, this means that the filter behaves the same way across the entire

image, i.e., with no spatial dependencies. Similar consU'alnts can be imposed on a filter

in the temporal domain to qualify it as time-variant or time-invariant. In the remainder

of this discussion, we shall avoid mention of the temporal domain although the same

statements regarding th s..tial domain apply there as well.

In practice, most physically realizable filters (e.g., lenses) are not entirely linear or

space-invariant. For instance, most optical systems are limited in their maximum

response and thus cannot be strictly linear. Furthermore, brightness, which is power per

unit area, cannot be negative, thereby limiting the system's minimum response. This pre-

cludes an arbitrary range of values for the input and output images. Most optical imaging

systems are prevented from being snfctly space-invariant by finite image area and lens

aberrations.

Despite these deviations, we often choose to approximate such systems as linear and

space-invariant. As a byproduct of these modeling assumptions, we can adopt a rich set

of analytical tools from linear filtering theory. This leads to useful algorithms for pro-

cessing images. In contrast, nonlinear and space-variant filtering is not well-understood