Docteur de l'université Automatic Segmentation and Shape Analysis ...

32 Chapter 2 Literature Review
is identified with R k×m \ {0}. Centering the configuration at its gravity centre
filters out the translation, such that the centered landmarks in XC do not depend
on the choice of the origin for the coordinate systems. The centered pre-shape ZC
is defined by normalizing XC of its centroid size S(XC). The pre-shape space S k m is
equivalent to the unit sphere S (k−1)m−1 in the (k − 1)m-space. The size-and-shape
[X]S, also known as the form, is the orbit of the centered landmarks under the
rotation group SO(m)
[X]S = {Γ(XC) : Γ ∈ SO(m)}, (2.7)
with the size-and-shape space SΣ k m defined as the quotient space R (k−1)m /SO(m).
The shape [X] is defined as the remaining geometrical information when both scale
S(X) and the rotation are filtered
[X] = {Γ(ZC) : Γ ∈ SO(m)}, (2.8)
and the shape space Σ k m ≡ S (k−1)m−1 /SO(m) is the orbit of configurations under
the action of Euclidean similarity transformations. The hierarchy of different shape
spaces are shown in Figure 2.4. Due to the scope of this thesis and the application
in biomedical imaging, the dimension m will be limited to 3 here.
2.2.1 Representation of shapes
In biomedical imaging applications, statistical shape analysis is usually carried
out on the surface and on landmarks extracted from volume data. Marching
cubes algorithm is most commonly used to convert the segmented image volume
into surface mesh (Lorensen and Cline, 1987; for a survey of the methods, see
Newman and Yi, 2006). For a set X of k landmarks in the 3D Euclidean space, it
can be represented by a concatenation of the coordinates of its k landmarks as a
3k-vector
X = (
x 1 , y 1 , z 1 , x 2 , y 2 , z 2 , · · · , x k , y k , z k) T
∈ R 3k . (2.9)