Classification of Deformable Smooth Shapes Through Geodesic Flows of Diffeomorphisms

Let \(\mathcal {D}\) $\mathcal{D}$ be a dataset of smooth 3D surfaces, partitioned into disjoint classes \( CL _j\) $C{L}_j$ , \(j= 1, \ldots , k\) $j=1,\dots ,k$ . We show how optimized diffeomorphic registration applied to large numbers of pairs \((S, S')\) $(S,S^{\prime })$ , \(S, S' \in \mathcal {D}\) $S,S^{\prime }\in \mathcal{D}$ can provide descriptive feature vectors to implement automatic classification on \(\mathcal {D}\) $\mathcal{D}$ and generate classifiers invariant by rigid motions in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ . To enhance the accuracy of shape classification, we enrich the smallest classes \( CL _j\) $C{L}_j$ by diffeomorphic interpolation of smooth surfaces between pairs \(S, S' \in CL _j\) $S,S^{\prime }\in C{L}_j$ . We also implement small random perturbations of surfaces \(S\in CL _j\) $S\in C{L}_j$ by random flows of smooth diffeomorphisms \(F_t:\mathbb {R}^3 \rightarrow \mathbb {R}^3\) $F_t:{\mathbb{R}}^3\to {\mathbb{R}}^3$ . Finally, we test our classification methods on a cardiology database of discretized mitral valve surfaces.