Optical flow on evolving sphere-like surfaces

Figure 1.  Frames 140 (left) and 141 (right) of the volumetric zebrafish microscopy images recorded during early embryogenesis. The sequence contains a total number of 151 frames recorded at time intervals of $120 \, \mathrm{s}$. Fluorescence response is indicated by blue colour and is proportional to the observed intensity. The embryonic axis of the animal forms around the clearly visible dent. All dimensions are in micrometer ($\mu$m).

Figure 2.  Depicted are frames no. 140 (left) and 141 (right) of the processed zebrafish microscopy sequence. Top and bottom row differ by a rotation of 180 degrees around the $x^{3}$-axis. All dimensions are in micrometer ($\mu$m).

Figure 11.  From left to right, top to bottom: a) colour-coded surface velocity $\mathbf{\hat{V}}$, b) signed norm $\text{sign}(\partial_{t} \tilde{\rho}) \left\| {{\bf{\hat V}}} \right\|$, c) optical flow $\mathbf{\hat{v}}$ for $\alpha = 1$, and d) total motion $\mathbf{\hat{M}} = \mathbf{\hat{V}} + \mathbf{\hat{v}}$. Values are computed for the interval between frames 140 and 141. All surfaces are depicted in a top view.

Figure 12.  Top view of the optical flow field computed on a static spherical geometry. The same values of α as in Fig. 10 were used.

Figure 3.  Schematic illustration of a cut through the surfaces $\mathcal{S}^2$ and $\mathcal{M}_{t}$ intersecting the origin. In addition, we show a radial line along which the extension $\bar{f}(t, \cdot)$ is constant. The surface normals are shown in grey.

Figure 5.  Commutative diagram relating spaces $\Omega$, $\mathcal{S}^{2}$, and $\mathcal{M}_{t}$, and tangent vector fields. We highlight that $\mathbf{y}_{p}$ is the coordinate representation, see Sec. 2.1, of a particular tangential vector spherical harmonic $\mathbf{\tilde{y}}_{p}$ and $\mathbf{\hat{y}}_{p}$ is its uniquely identified tangent vector field on $\mathcal{M}_{t}$.

Figure 4.  Illustration of trajectories through the evolving surface. Their corresponding velocities are shown in grey.

Figure 6.  Illustration of a triangular face (filled gray) intersecting the sphere $\mathcal{S}^{2}$ at the vertices (hollow circles). The six nodal points consist of the vertices of the triangle together the edge midpoints (filled black dots). The approximated sphere-like surface is shown by the hatched gray area. A radial line passing through the vertex $v_{i}$ is shown. The hollow circle indicates the intersection with $\mathcal{S}^{2}$ at which $\bar{f}(v_{i})$ in (49) is taken. $\bar{f}$ itself, as described in Sec. 5.2, is assigned by taking the maximum image intensity along the drawn radial line between the two cross marks.

Figure 7.  Frames no. 140 (left) and 141 (right) of the processed image sequence in a top view. The embryo's body axis is oriented from bottom left to top right.

Figure 9.  Tangent vector field minimising $\mathcal{E}_{\alpha}$. Depicted is the colour-coded optical flow field computed between frames 140 and 141 for different values of $\alpha$. The bottom row differs from the top view by a rotation of 180 degrees around the $x^{3}$-axis. From left to right: a) $\alpha = 10^{-2}$, b) $\alpha = 10^{-1}$, c) $\alpha = 1$, and d) $\alpha = 10$.

Figure 8.  Function $\tilde{\rho}_{h}$ obtained by minimising $\mathcal{F}_{\beta}$ for frames 140 (left column) and 141 (right column). Colour corresponds to the radius ($\mu$m) of the fitted surface. The top row depicts $\mathcal{S}_{h}^{2}$ in a top view.

Figure 10.  Top view of the optical flow field computed for different values of $\alpha$. From left to right, top to bottom: a) $\alpha = 10^{-2}$, b) $\alpha = 10^{-1}$, c) $\alpha = 1$, and d) $\alpha = 10$.