Image Processing Reference
In-Depth Information
4
Continuous Functions and Hilbert Spaces
We can reconstruct as well as decompose a discrete signal by means of a set of dis-
crete signals that we called the basis in the previous chapter. Just as a vector in E 3
can be expressed as a weighted sum of three orthogonal vectors, so can functions
defined on a continuous space be expressed as a weighted sum of orthogonal basis
functions with some coefficients. This will be instrumental when modeling nondis-
crete images. Examples of nondiscrete images are the images on photographic pa-
pers or films. Tools manipulating functions defined on a continium rather than on
a discrete set of points are necessary because without them, understanding and per-
forming many image processing operations would suffer, e.g., to enlarge or reduce
the size of a discrete image. In this chapter we deliberately chose to be generic about
functions as vector spaces. We will be more precise about the function families that
are of particular interest for image analysis in the subsequent chapters. First, we will
need to introduce functions, which are abstractions that define a rule, as points in
a vector space. Then, we define addition and scaling to establish that functions are
vector spaces. To make a Hilbert space of the function vector space we need a scalar
product, the generic form of which will be presented in the subsequent section. Fi-
nally, we present orthogonality and the angle concept for nondiscrete images.
4.1 Functions as a Vector Space
If one scrutinizes the theory of Hilbert space that we have studied so far, one con-
cludes that there are few difficult assumptions hindering its extension to cover quan-
tities other than arrays of discrete scalars. Continuous functions are quantities that we
will study in the framework of vector spaces. A major difference between the thus
far discussed arrays and the function spaces is that the latter are abstract quantites
that are continuous. A photograph is a continuous function defined on an ordinary
plane. In that sense it is called a continuous image . Its discrete version, a digital im-
age, is defined on a set of points on the same 2D plane. The continuous photograph
assumes many more values because it is defined even between the set of points where
the discrete image is undefined.
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