Riemannian methods for optimization in a shape space of triangular meshes

Figures & data

Table 1. Definitions of some frequently used notations.

Notation	Definition
$C$	Set of all pairs of neighbouring points on a triangulated mesh
$M$	Triangulated mesh
$N$	Number of nodes of a triangulated mesh
$n_p$	Local normal vector to a mesh at the point $p$
$\overrightarrow{\overrightarrow{n}}$	Concatenation of the normal vectors to a triangulated mesh
$\mathcal{N}(p)$	Points of a mesh neighbouring to the point $p$ including $p$
$P$	Set of all points of a triangulated mesh
$\mathcal{S}$	Shape space of triangulated meshes in ${\mathbb{R}}^3$ with fixed connectivity
$\mathcal{T}(p)$	Triangles of a mesh adjacent to the point $p$

Figure 1. A free geodetic propagation with different Riemannian metrics. From top to bottom and left to right: Initial shape, Euclidiean, $H^0$-, $H^1$-, $H^2$-, $H^3$-, $H^5$- and $H^8$-metric.

Figure 2. A shape together with the descent direction for different Riemannian metrics. From top to bottom and left to right: Euclidiean, $H^0$-, $H^1$- and $H^2$-metric.

Figure 3. The synthetic shape, its shading image, the initial shape and its reconstruction using the SSD-method.

Figure 4. Reconstruction of the synthetic shape with the GSD-method using different Riemannian metrics. From left to right: Euclidean, $H^0$- and $H^2$-metric.

Figure 5. Reconstruction of the synthetic shape with the GNCG-method using different Riemannian metrics. From left to right: Euclidean, $H^0$- and $H^2$-metric.

Figure 6. Reconstruction of the synthetic shape using the GSD-method, the Euclidean metric and an oblique light source $l/\Vert l\Vert$. From left to right: $l=(0.1,0,1)$, $l=(0,0.1,1)$.

Figure 7. The bottom of the ceramic cup, its shading image, the initial shape and its reconstruction using the SSD-method.

Figure 8. Reconstruction of the bottom of the ceramic cup with the GSD-method using different Riemannian metrics. From left to right: Euclidean, $H^0$- and $H^2$-metric.

Figure 9. Reconstruction of the bottom of the ceramic cup with the GNCG-method using different Riemannian metrics. From left to right: Euclidean, $H^0$- and $H^2$-metric.

Figure 10. The shading image of the face, the initial shape and its reconstruction using the SSD-method.

Figure 11. Reconstruction of the face with the GSD-method using different Riemannian metrics. From left to right: Euclidean, $H^0$- and $H^2$-metric.

Figure 12. Reconstruction of the face with the GNCG-method using different Riemannian metrics. From left to right: Euclidean, $H^0$- and $H^2$-metric.

Figure 13. Convergence of several optimization techniques in different situations. Top left: Synthetic shape (coarse-grid optimization). Top right: Synthetic shape (fine-grid optimization). Bottom left: Ceramic cup. Bottom right: Face. The legend applies to all four plots.