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78 Chapter 4 Statistical shape model of Hippocampus . . . .1 .e4 .e3 .e1 .e2 Figure 4.2: Parameterization of quadrilateral. The square face (bold) on the physical surface (left) is mapped to the parameter space of unit sphere (right). The length of each edge ei in the quadrilateral equals to the corresponding center angle. Image adapted from Brechbühler et al. (1995). 4.1.2 Reparameterization by rotation Since the parameterization is performed on each surface Xi individually, the correspondence across the training set {Xi : i = 1, · · · , n} is not guaranteed. In the implementation by Styner et al. (2006), the first degree of SPHARM coeffi- cients for each shape is computed, and the first order ellipsoid is oriented to fit the shape surface. The parameterizations are rotated to coincide both poles of the parameterization to that of the first order ellipsoid. In our implementation, the parameterizations are rotated to minimize the shape difference between surfaces up to similarity transformations based on the parameterization correspondence. The L2 distance between parameterizations can be defined based on the standard Lebesgue measure on S 2 fi − fj 2 ∫ = S 2 fi(θ, φ) − fj(θ, φ)) 2 dΩ (4.16)
Chapter 4 Statistical shape model of Hippocampus 79 where dΩ = sin θdθdφ is the surface element on S 2 . The condition of isometry is satisfied for rotation Γ fi ◦ Γ − fj ◦ Γ 2 ∫ = ∫ = ∫ = S 2 S 2 S 2 fi(Γ(θ, φ)) − fj(Γ(θ, φ)) 2 dΩ fi(Γ(θ, φ)) − fj(Γ(θ, φ)) 2 J −1 (Γ(θ, φ))dΩ(Γ(θ, φ)) fi(Γ(θ, φ)) − fj(Γ(θ, φ)) 2 dΩ(Γ(θ, φ)) = fi − fj 2 (4.17) since the Jacobian J for rotation is 1 on S 2 . Assuming the shape surfaces {Xi} are aligned (e.g. by ICP algorithm) via similarity transforms to a common template, the rotation Γi can be found by minimizing the pairwise distance Γi = arg min fi ◦ Γ − f1 Γ∈SO(3) 2 , (4.18) where f1 is an arbitrarily selected parameterization as the template. Multiple runs with different choices of template may be performed in order to avoid bias towards the choice of template. In implementation, the L2 distance may be evaluated by sampling on S 2 fi − fj 2 4π k k∑ fi(θs, φs) − fj(θs, φs) s=1 2 , (4.19) where (θs, φs), s = 1, · · · k are uniformly sampled on S 2 by Platonic solids such as octahedron, dodecahedron or icosahedron. By parameterizing the rotation Γ using Rodrigues parameters or quaternions, the optimization problem in (4.18) thus becomes a least-square problem with closed form solution, which can be solved by Nelder-Mead simplex or other standard optimizers. A multi-resolution scheme can be easily implemented by subdividing the initial sampling (Figure 4.3). The multi-resolution algorithm for rotational reparameterization is listed in Algorithm 7.
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UNIVERSITE DE BOURGOGNE U.F.R. SCIE
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Résumé L’objectif de cette thè
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Abstract The aim of this thesis is
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TABLE DES MATIÈRES Acknowledgement
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TABLE DES MATIÈRES xi 4.1.4.3 Spec
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LISTE DES FIGURES 1.1 Cortical atro
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144 LIST OF PUBLICATIONS using dive
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146 BIBLIOGRAPHY Alzheimer, A. (191
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148 BIBLIOGRAPHY Baillard, C., Hell
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150 BIBLIOGRAPHY of post mortem mag
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152 BIBLIOGRAPHY B. (2008). Three-d
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154 BIBLIOGRAPHY Cootes, T. F., Tay
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156 BIBLIOGRAPHY head in MR images
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158 BIBLIOGRAPHY Evans, A. C., Coll
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BIBLIOGRAPHY 161 Ghanei, A., Soltan
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BIBLIOGRAPHY 165 R. C. (2003). MRI
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BIBLIOGRAPHY 167 Konrad, C., Ukas,
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BIBLIOGRAPHY 169 Mahony, R. and Man
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BIBLIOGRAPHY 171 disease, mild cogn
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BIBLIOGRAPHY 173 Petersen, R. C., S
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BIBLIOGRAPHY 175 Rorden, C. and Bre
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BIBLIOGRAPHY 177 Shen, K., Bourgeat
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BIBLIOGRAPHY 179 Barillot, C., Hayn
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BIBLIOGRAPHY 181 Tzourio-Mazoyer, N
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BIBLIOGRAPHY 183 Vos, F. M., de Bru
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BIBLIOGRAPHY 185 Woolrich, M. W., J
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