Geodesic Shooting for Computational Anatomy - PubMed (original) (raw)

Geodesic Shooting for Computational Anatomy

Michael I Miller et al. J Math Imaging Vis. 2006.

Abstract

Studying large deformations with a Riemannian approach has been an efficient point of view to generate metrics between deformable objects, and to provide accurate, non ambiguous and smooth matchings between images. In this paper, we study the geodesics of such large deformation diffeomorphisms, and more precisely, introduce a fundamental property that they satisfy, namely the conservation of momentum. This property allows us to generate and store complex deformations with the help of one initial "momentum" which serves as the initial state of a differential equation in the group of diffeomorphisms. Moreover, it is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint we introduce.

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Figures

Figure 1

Figure 1

Here is represented the deformation obtained by pulling back the Eulerian frame associated with ve and represents pictorially the adjoint action.

Figure 2

Figure 2

Three random deformations of an image.

Figure 3

Figure 3

Figure shows objects under translation (column 1), scale (column 2), and mitochondria 1 and 2.

Figure 4

Figure 4

Rows 1, 2: Results from translation experiment. Panels 1, 2, and 3 show _I_0, _I_1, _I_0 ◦ _φ_10. Panels 4, 5 and 6 show _α_0, _Lv_0 versus _α_0∇_I_0, and _V_0 and _L_−1_α_∇ _I_0.

Figure 5

Figure 5

Rows 1, 2: Results from scale experiment. Panels 1, 2, and 3 show _I_0, _I_1, _I_0 ◦; _φ_10. Panels 4, 5 and 6 show _α_0, _Lv_0 versus _α_0∇_I_0, and _V_0 and _L_−1_α_∇ _I_0.

Figure 6

Figure 6

Results from mitochondria 2. Panels 1, 2, and 3 show _I_1, _I_0, _I_1 ◦ _φ_10. Panels 4, 5 and 6 show _α_0, _LV_0 versus _α_0∇ _I_0, and _V_0.

Figure 7

Figure 7

Results from mitochondria 1. Panels 1, 2, and 3 show _I_0, _I_1, _I_0 ◦ _φ_10. Panels 4, 5 and 6 show _α_0, _LV_0 versus _α_0∇ _I_0, and _V_0.

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