Holm Darryl D, Ratnanather J Tilak, Trouvé Alain, Younes Laurent
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
Neuroimage. 2004;23 Suppl 1:S170-8. doi: 10.1016/j.neuroimage.2004.07.017.
Computational anatomy (CA) has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms. Among these geometric structures, landmarks and image outlines in CA are shown to be singular solutions of a partial differential equation that is called the geodesic EPDiff equation. A recently discovered momentum map for singular solutions of EPDiff yields their canonical Hamiltonian formulation, which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta. The momentum map provides an isomorphism between landmarks (and outlines) for images and singular soliton solutions of the EPDiff equation. This isomorphism suggests a new dynamical paradigm for CA, as well as new data representation.
计算解剖学（CA）引入了这样一种观点，即解剖结构可通过微分同胚群上的测地线变形进行变换。在这些几何结构中，CA中的地标点和图像轮廓被证明是一个偏微分方程的奇异解，该方程被称为测地线EPDiff方程。最近发现的EPDiff奇异解的动量映射给出了它们的规范哈密顿表述，这反过来又通过地标点的规范位置和动量对其进行了完整的参数化。动量映射在图像的地标点（和轮廓）与EPDiff方程的奇异孤子解之间提供了同构。这种同构为CA提出了一种新的动力学范式以及新的数据表示方式。
