Discrete Images and Hilbert Spaces - Vision with Direction - page 33

3

Discrete Images and Hilbert Spaces

A Hilbert space is a mathematical space that is equipped with a scalar product. A
Hilbert space is also referred to as an inner product space . The elements of Hilbert
spaces are typically sequences of scalars or functions, also called vectors . Provided
that certain rules are respected, matrices, which are studied in linear algebra, can be
viewed as vectors, and their space as a vector space. A scalar product for matrices,
which can easily represent discrete images, can also be defined. Most important,
however, is that the space of all images can be viewed as a Hilbert space. By using the
scalar product the angles between vector pairs of the Hilbert space can be determined.
This allows us to quantify how similar two images are. A Hilbert space contains
numerous orthonormal bases. Once a basis is given, every element of the Hilbert
space can be written uniquely as a sum of multiples of the basis elements. In this
way, it is feasible to represent an image as a sum of the basis vectors weighted with
appropriate coefficients.

3.1 Vector Spaces

In linear algebra courses one studies vectors that represent points in spaces. A vector
is often visualized as an arrow from the center of the coordinate system, the origin ,
to the point. The point itself can be seen as equivalent to the vector, or vice versa.
Although we will work with other vector spaces mostly in this topic, we start with
a vector space with which we have the most familiarity, the environment in which
we move about, i.e., ordinary space. Without the time aspect, it is referred to as
a3D Euclidean space, or simply as E 3 . The points in this space are numerically
represented as an ordered set of real numbers x (1) , x (2) , x (3) standing one on the
top of the other:

⎛ ⎞

x (1)
x (2)
x (3)

⎝ ⎠

x = (3.1)

The elements, x ( i ) are to be interpreted relative to a set of abstract quantities, e 1 , e 2 ,
and e 3 , called the basis , such that they are coordinates, i.e., x ( i ) act as coefficients