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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
Chapter 2 Literature Review 31 . Pre-shape . Original . configuration . Centered configuration . Shape .remove translation .remove scale .remove rotation .remove rotation . Form, or, size-and-shape .remove scale Figure 2.4: Hierarchy of shape spaces. Adapted from Dryden and Mardia (1998). changes (Jolicoeur and Mosimann, 1960). Allometric study (Huxley, 1932; Mosi- mann, 1970) is another approach to shape analysis that assesses the shape varia- tions associated with the size and growth via the power-law relation. Recent advancement in mathematical shape theory arises from the seminal work by Kendall (1977) on the Brownian motion in the complex projective spaces, which is later developed in more details (Kendall, 1984, 1989), and the algorithmic develop- ment of Thompson’s grid method and the statistics of deformation by Bookstein (1978b,a, 1986). Up-to-date introduction on this subject may be found in the monographs by Bookstein (1991), Small (1996), Dryden and Mardia (1998), and Kendall et al. (1999). We follow the notions and definitions of shape spaces used by Dryden and Mardia (1998) and Kendall et al. (1999) in this thesis. A configuration X of k landmarks in the m-dimension Euclidean space can be represented by its k × m Cartesian coordinates. The configuration space comprises of all nondegenerate k-ads, which
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156 BIBLIOGRAPHY head in MR images
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BIBLIOGRAPHY 185 Woolrich, M. W., J
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