Image Processing Reference
In-Depth Information
Principal components analysis is an established mathematical tool unfortunately beyond
the scope of this text, but help is available in
Numerical Recipes
(Press, 1992). Essentially,
it rotates a co-ordinate system so as to achieve maximal discriminatory capability: we
might not be able to see something if we view it from two distinct points, but if we view
it from some point in between then it is quite clear. That is what is done here: the co-
ordinate system is rotated so as to work out the most significant variations in the morass
of data. Given a set of
N
training examples where each example is a set of
n
points, for the
i
th training example
x
i
we have
x
i
= (
x
1
i
,
x
2
i
, . . .
x
ni
)
i
∈ 1,
N
(6.53)
where
x
ki
is the
k
th variable in the
i
th training example. When this is applied to shapes,
each element is the two co-ordinates of each point. The average is then computed over the
whole set of training examples as
N
=
1
=1
x
x
(6.54)
i
N
i
The deviation of each example from the mean δ
x
i
is then
xxx
=
-
(6.55)
i
i
This difference reflects how far each example is from the mean at a point. The 2
n
2
n
covariance matrix
S
shows how far all the differences are from the mean as
N
=
1
=1
T
S
x
x
(6.56)
i
N
i
i
Principal components analysis of this covariance matrix shows how much these examples,
and hence a shape, can change. In fact, any of the exemplars of the shape can be approximated
as
i
= +
(6.57)
where
P
= (
p
1
,
p
2
, . . .
p
t
) is a matrix of the first
t
eigenvectors, and
w
= (
w
1
,
w
2
, . . .
w
t
)
T
is a corresponding vector of weights where each weight value controls the contribution of
a particular eigenvector. Different values in
w
give different occurrences of the model, or
shape. Given that these changes are within specified limits, then the new model or shape
will be similar to the basic (mean) shape. This is because the modes of variation are
described by the (unit) eigenvectors of
S
, as
Sp
k
=
xxPw
λ
k
p
k
(6.58)
where
λ
k
denotes the eigenvalues and the eigenvectors obey orthogonality such that
pp
T
= 1
(6.59)
k
k
and where the eigenvalues are rank ordered such that
λ
k
+1
. Here, the largest eigenvalues
correspond to the most significant modes of variation in the data. The proportion of the
variance in the training data, corresponding to each eigenvector, is proportional to the
corresponding eigenvalue. As such, a limited number of eigenvalues (and eigenvectors) can
be used to encompass the majority of the data. The remaining eigenvalues (and eigenvectors)
correspond to modes of variation that are hardly present in the data (like the proportion of
very high frequency contribution of an image; we can reconstruct an image mainly from
λ
k
≥
