Bayesian estimation of templates in computational anatomy: Difference between revisions

Revision as of 19:40, 14 January 2017

Revision as of 19:41, 14 January 2017

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Flows were first introduced<ref>GE Christensen,RD Rabbitt,MI Miller, Deformable templates using large deformation kinematics, IEEE Trans Image Process. 1996;5(10):1435-47.</ref><ref>GE Christensen, SC Joshi, MI Miller, Volumetric transformation of brain anatomy

IEEE Transactions on Medical Imaging,1997.</ref> for large deformations in image matching; <math>\dot \phi_t(x)</math> is the instantaneous velocity of particle <math>x</math> at time <math>t</math>. with the vector fields termed the Eulerian velocity of the particles at position of the flow. The modelling approach used in CA enforces a continuous differentiability condition on the vector fields by modelling the space of vector fields <math>(V, \| \cdot \|_V )</math> as a [[reproducing kernel Hilbert space]] (RKHS), with the norm defined by a 1-1, differential operator<math> A: V \rightarrow V^* </math>, Green's inverse <math>K = A^{-1}</math>. The norm according to <math> \| v\|_V^2 \doteq (\int_X Av \midcdot v) dx , v \in V,

</math> where for <math> \sigma(v) \doteq Av \in V^*

</math> a generalized function or distribution, then <math> (\sigma\mid w)\doteq \int_{{\mathbb R}^3} \sum_i w_i(x) \sigma_i(dx)

</math>. Since <math> A </math> is a differential operator, finiteness of the norm-square <math> \int_0^1 \int_X Av \cdot v dx < \infty

</math> includes derivatives from the differential operator implying smoothness of the vector fields.