Digital image warping

Images can be classified by whether or not they are defined over all points in the

spatial domain, and by whether their image values are represented with finite or infinite

precision. If we designate the labels "continuous" and "discrete" to classify the spatial

domain as well as the image values, then we can establish the following four image

categories: continuous-continuous, continuous-discrete, discrete-continuous, and

discrete-discrete. Note that the two halves of the labels refer to the spatial coordinates

and image values, respectively.

A continuous-continuous image is an infinite-precision image defined at a contin-

uum of positions in space. The literature sometimes refers to such images as analog

images, or simply continuous images. Images from this class may be represented with

finite-precision to yield continuous-discrete images. Such images result from discretiz-

ing a continuous-continuous image under a process known as quantization to map the

real image values onto a finite set (e.g., a range that can be accommodated by the numeri-

cal precision of the computer). Alternatively, images may continue to have their values

retained at infinite-precision, however these values may be defined at only a discrete set

of points. This form of spatial quantization is a manifestation of sampling, yielding

discrete-continuous images. Since digital computers operate exclusively on finite-

precision numbers, they deal with discrete-discrete images. In this manner, both the spa-

tial coordinates and the image values are quantized to the numerical precision of the

computer that will process them. This class is commonly known as digital images, or

simply discrete images, owing to the manner in which they are manipulated. Methods

for converting between analog and digital images will be described later.

We speak of monochrome images, or black-and-white images, when f is a single-

valued function representing shades of gray, or gray levels. Alternatively, we speak of

color images when f is a vector-valued function specifying multiple color components at

each spatial coordinate. Although various color spaces exist, color images are typically

defined in terms of three color components: red, green, and blue (RGB). That is, for

color images we have

f (x,y) fred(X,y), fgreen(X,Y), fOlue(X,Y) ) (2.1.1)

Such vector-valued functions can be readily interpreted as a stack of single-valued

images, called channels. Therefore, monochrome images have one channel while RGB

color images have three (see Fig. 2.1). Color images are instances of a general class

known as multispectral images. This refers to images of the same scene that are acquired

in different parts of the electromagnetic spectrum. In the case of color images, the scene

is passed through three spectral filters to separate the image into three RGB components.

Note tha* nothing requires image data to be acquired in spectral regions that fall in the

visible range. Many applications find uses for images in the ultraviolet, infrared,

microwave, and X-ray ranges. In all cases, though, each channel is devoted to a paxticu-

lar spectral band or, more generally, to an image attribute.

Depending on the application, any number of channels may be introduced to an

image. For instance, a fourth channel denoting opacity is useful for image compositing