# 1. ImagesΒΆ

An image is a mathematical representation of a physical observation as
a function over a spatial domain. At every point on the retina (the
photosensitive surface in the eye or camera) the electromagnetic
energy is measured. For a computer scientist the physical laws that
govern image formation and image observation are just that: *laws*. We
can’t change them; we have to live with them. Needless to say that
this does *not* mean that we need not familiarize ourselves with these
physical laws. It is senseless to develop programs that are not in
accordance with physical reality. Especially in the field of machine
vision the physical (and psycho-physical) description of the sensors
is the starting point in utilizing these sensor signals in information
systems. Therefore we would like to treat an image as defined over a
*continuous* spatial domain, not as a mere collection of pixel values.
The fact that we need to discretize an image by sampling should be
considered a practical inconvenience that should be banned from our
thinking once we have dealt with the matter in the very
beginning. Only on the lowest level in an image information system
should we be concerned with the fact that the individual pixels make
up the discrete representation of the image. Most of our image
processing work and image analysis work should regard images as
mappings over a continuous domain and treat them accordingly. For a
computer scientist this might seem a rather awkward and
counterintuitive starting point. He or she is probably familiar with
images being large arrays of say \(800\times 600\) pixels, each
representing a color specified as an RGB triplet.

Fig. 1.11 above illustrates the difference between what
the human visual brain perceives as an image (the photograph), its
mathematical model (the image function visualized as the luminance
surface) and its discrete representation (the array of numbers). In
this chapter we will develop the mathematical tools to think in terms
of the mathematical model (the image *function*) when only the image
*samples* (the array of numbers) are given.

In this chapter we look at:

- Image Formation: The optical formation of images is briefly discussed.
- Image Definition: Images as mappings from a continuous domain (most often 2D space) to the set of real numbers (most often the luminance measurements).
- Image Discretization: A pixel is
*not*a little square such that all pixels in an image ‘tile’ the image domain, a pixel is the outcome of the measurement of electromagnetic wave energy in a small (but not infinitessimally small) spot on the retina (either in the human eye or in a camera). A discrete image is a sampled and quantized version of the spatial distribution of the electromagnetic energy. - Image Interpolation: whereas sampling takes us from a function defined on a continuous domain to a set of samples, interpolation does the opposite. Given a set of samples (like a discrete image) it generates a function defined on the continuous domain. It is seldomly true that sampling and interpolation are truly inverse operators.
- Image Representation: a digital image can be represented as an array of all image samples. This section hopefully sheds some light on the confusion between the different choices to be made for coordinate axes and indices.