# Riemannian Shape Analysis

Riemannian shape analysis studies the geometric properties of shapes by embedding them into Riemannian manifolds, where each shape corresponds to a point on a shape space manifold. A shape space manifold, $\mathcal{M}$, is equipped with a Riemannian metric $g$, which defines distances and enables geodesics—curves representing optimal shape transformations between shapes. Mathematically, given two shapes $S_1$ and $S_2$ on the manifold, their geodesic distance $d(S_1, S_2)$ reflects the minimal energy required for an optimal deformation to morph one shape into another:

$$d(S_1, S_2) = \inf_{\gamma \in \Gamma} \int_0^1 \sqrt{g(\dot{\gamma}(t), \dot{\gamma}(t))} , dt,$$

where $\Gamma$ is the set of all smooth paths connecting $S_1$ and $S_2$, and $\gamma(t)$ is a geodesic path parameterized by $t \in [0, 1]$.

Figure 1: Geodesic between source shape and target shape of left atrial appendages (LAAs).

In medical image analysis, Riemannian shape spaces are used to study anatomical variability. For example, clustering **left atrial appendages (LAAs)** based on their geometric properties can aid in stratifying patients by anatomical risk factors for stroke. By representing LAAs as points on a shape manifold, one can use geodesic distances to cluster similar shapes and detect outliers:

$${ S_1, S_2, \ldots, S_n } \mapsto \mathcal{C}_1, \mathcal{C}_2, \ldots, \mathcal{C}_k,$$

where each $\mathcal{C}_i$ is a cluster of shapes. This approach enables automated classification, patient-specific risk assessment, and treatment planning, such as determining the optimal occlusion device for stroke prevention.