Digital image warping

(a) (b)

Figure 2.1: Image formats. (a) monochrome; (b) color.

operations which must smoothly blend images together [Porter 84]. In remote sensing,

many channels are used for multispectral image analysis in earth science applications

(e.g., the study of surface composition and structure, crop assessment, ocean monitoring,

and weather analysis). In all of these cases, it is important to note that the number of

variables used to index a signal is independent of the number of vector elements it yields.

That is, there is no relationship between the number of dimensions and channels. For

example, a two-dimensional function f (x,y) can yield a 3-tuple color vector, or a 4-tuple

(color, transparency) vector. Channels can even be used to encode spatially-varying sig-

nals that are not related to optical information. Typical examples include population and

elevation data.

Thus far, all of the examples referring to images have been two-dimensional. It is

possible to define higher-dimensional signals as well, although in these cases they are not

usually referred to as images. An animation, for instance, may be defined in terms of

function f (x,y,t) where (x,y) again refers to the spatial coordinate and t denotes time.

This produces a stack of 2-D images, whereby each slice in the stack is a snapshot of the

animation. Volumetric data, e.g., CAT scans, can be defined in a similar manner. These

are truly 3-D "images" that are denoted by f (x,y,z), where (x,y,z) are 3-D coordinates.

Animating volumetric data is possible by defining the 4-D function f (x,y,z,t) whereby

the spatial coordinates (x,y,z) are augmented by time t.

In the remainder of this book, we shall deal almost exclusively with 2-D color

images. It is important to remember that although warped output images may appear as

though they lie in 3-D space, they are in fact nothing more than 2-D functions. A direct

analogy can be made here to photographs, whereby 3-D world scenes are projected onto

flat images.

Our discussion thus far has focused on definitions related to images. We now turn

to a presentation of terminology for filters. This proves useful because digital image

warping is firmly grounded in digital filtering theory. Furthermore, the elements of an

image acquisition system are modeled as a cascade of filters. This review should help

put ou r discussion of image warping, including image acquisition, into more formal

terms.