Biomedical Engineering Reference
In-Depth Information
provided by the Hessian matrix indicates local curvature and bending [17, 21].
We argue that all this information can be relevant in characterizing a region.
5.2.2
Non-parametric Tissue Probability Estimation
Pattern recognition techniques can be used to obtain non-parametric PDF esti-
mators. In this framework, the PDF is computed using classifiers. The estima-
tion is usually done using a sparse set of neighboring samples. For example,
the Parzen density estimation method computes the probability for a point
x
to
belong to a region by computing the number of samples belonging to that region
in a fixed neighborhood of radius
k
. The k-Nearest Neighbor (kNN) estimation
technique can be interpreted as a Parzen approach with a neighborhood size
adjusted automatically depending on the location of the point. It is reasonable
to assume that points with similar local image structure belong to the same tis-
sue class. For this reason, we use kNN rule for the generation of the PDF in our
algorithm.
The nearest neighbor principle is one of the best studied techniques for
pattern classification [22]. A classifier based on this principle uses a training set
of vectors as a collection of labelled cases. For a given pattern, it searches for
the nearest vector in this learning set according to a given metric. The traditional
classification consists in assigning to the pattern the label of the most voted class.
Combinations of number of votes (kNN rule) and distances can be also used for
classification. A PDF estimation can be derived from these voting systems.
5.2.2.1
Training Set Construction
We use the kNN rule to estimate the probability density function for each tissue
class as follows. For the construction of the training set, some representative
CTA images are selected from the whole data base. Then,
N
points are manually
picked from these images, and labelled with one of these three tissue classes:
vessel, background or bone. A label and a feature vector is associated with each
point in the training set. The feature vector is derived from the local differential
structure of the image at a small scale
σ
(Eq. (5.1)). For a point
x
in the training
set, we associate the feature vector
f
(
x
)
=
(
I
σ
,
|∇
I
σ
|
,λ
1
σ
,λ
2
σ
,λ
3
σ
)
(5.2)
