U. Glaeser

The first hypothesis (the less plausible of the two) can be easily modeled with temporal linear filters. The
second, more interesting behavior can be modeled with a simple comparator network.
28.4 Image Sequence Representation
What Does “Representation” Mean?
The term “representation” may require some explanation. Perhaps the best way to do so is to consider
some examples of familiar representations. For simplicity, 2-D examples will be used. Extension to 3-D
is relatively straightforward.
The Pixel Representation
The pixel representation is so common and intuitive that it is usually considered to be “the image.” More
precisely, however, it is a linear sum of weighted impulses:
© 2002 by CRC Press LLC
(28.25)
where u(m, n) is the image, u(m′, n′) are the coefficients of the representation (numerically equal to the
pixel values in this case), and the δ(m − m′, n − n′) play the role of basis functions.
The DFT
The next most familiar representations (at least to engineers) is the DFT, in which the image is expressed
in terms of complex exponentials:
where 0 ≤ m, n ≤ N − 1 and
N−1 N−1
∑ ∑
u( m, n)
= u( m′, n′ )δ( m– m′, n– n′ )
u( m, n)
m′=0 n′=0
N−1 N−1
1
= ------ v( h, k)WN
∑∑
N 2
h=0 k=0
W N
e j2π/N –
(28.26)
(28.27)
In this case v(h, k) are the coefficients of the representation and the 2-D complex exponentials
are the basis functions.
The choice of one representation over the other (pixel vs. Fourier) for a given application depends on
the image characteristics that are of most interest. The pixel representation makes the spatial organization
of intensities in the image explicit. Because this is the basis of the visual stimulus, it seems more “natural.”
The Fourier representation makes the composition of the image in terms of complex exponentials
(“frequency components”) explicit. The two representations emphasize their respective characteristics
(spatial vs. frequency) to the exclusion of all others. If a mixture of characteristics is desired, different
representations must be used.
Spatial/Spatial-Frequency Representations
=
– hm −kn
WN
hm
WN – –
kn
WN
A natural mixture is to combine frequency analysis with spatial location. An example of a 1-D representation
of this type (a time/frequency representation) is a musical score. The need to know not only what the
frequency content of a signal is, but where in the signal the frequency components exist is common to many
signal, image, and image sequence processing tasks [7]. A variety of approaches [8,9] can be used to develop
a representation to facilitate these tasks. The most intuitive approach is the finite-support Fourier
transform.