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

30 Chapter 2 Literature Review
In the applications that are interested in measuring the volume of a given anatom-
ical structure, the normalized volume difference
∆V = |DS| − |DR|
, (2.5)
|DR|
is also an important index assessing the accuracy of the segmentation.
Hausdorff distance measures the distance between the boundaries of two regions
dH = max
(
sup
y∈∂DS
inf d(x, y), sup
x∈∂DR
x∈∂DR
inf d(x, y)
y∈∂DS
)
(2.6)
where ∂DS and ∂DR are the boundaries of the DS and DR respectively, and d(·, ·)
is the Euclidean distance between the voxels.
2.2 Quantitative analysis of shapes
The quantitative study into the shape of biological objects may date back to
the publication of On Growth and Form (1917, 1942) by Sir D’Arcy Wentworth
Thompson, mathematician, zoologist and classicist, which laid the mathematical
ground for zoology and morphology, in a unified account of ‘the science of Form
and Number.’ Subjects such as the magnitude and surface/volume ratio, the forms
of cells, tissues and skeletons, shapes of horns and teeth, and their mechanical
aspects are investigated. One of the long withstanding contribution of the book
is the description of deformation and homologies among species using a Cartesian
grid or coordinates in the radial system, as well as the mathematical theory of
transformations, which inspired later researchers over the following decades.
Multivariate morphometrics (Reyment et al., 1984) is one traditional way in shape
analysis, which performs multivariate analysis on the measurements such as the
angles, distances between the landmarks, and the ratio between measurements.
The principal component analysis (PCA) is applied to measured covariables, with
the first principal component usually the size variation confounded with shape