Fig 1.
Sample image of S. albus (top half) with corresponding model output venation pattern (bottom half).
Fig 2.
(a) Section of input image. (b) Semantic image segmentation of the veins from background. (c) Binary segmentation and the network skeleton. (d) The final graph extracted, represented by the coordinates and incidences of all nodes and edges, as well as the lengths and widths of each edge.
Fig 3.
(a) Applying two different, but small perturbations, starting from a uniform edge width distribution, leads to very different topologies in the final minima. (b) The final solution can be biased by starting near the desired local minimum.
Fig 4.
(a) Loss values as a function of sink fluctuation amplitude for leaves from the three different species.
The average minimum loss values are indicated for each species. (b) Distributions of the optimal values of for each species. (c–e) Vein width distributions for specific leaf samples, one from each species. The distributions are plotted for image data and model output, using the species-specific optimal value of
, as well as zero sink fluctuation. Note that the histograms are shown with logarithmic axes. (Insets) The actual leaves analyzed. (f–h) Model output vein widths as a function of the corresponding image data vein widths.
Fig 5.
Murray’s law for loopy networks.
(a) Subsection of a leaf image, with model output edge widths (red lines) and fluxes (blue arrows) indicated. The arrow sizes indicate the relative average time the flow moves in the respective directions given by the arrows. (b) vs.
for all nodes in a sample of S. albus, for the optimal
and
. The points closely follow the line of equality. (Inset) The same data with both axes logarithmic. (c-d) Loss values for individual leaf samples (thin lines), with the mean value for each species (thick lines). The dashed vertical line is located at
. (c)
. The optimal value of
is predicted for all samples. (d) Best-fit values of
[Fig 4a–4b]. The optimal values have been shifted in the positive direction. (e–h): Residual plots with Gaussian weighted means and standard deviations (SD). (e–f)
. The residuals are (e) biased negatively, and (f) subsequently corrected by a term proportional to the associated sink area. (g–h)
. The residuals are (g) biased positively, have a larger SD compared to the case of no sink fluctuation, and are (h) corrected by a term proportional to the associated sink area.
Fig 6.
Running the model on a subsystem.
(a) Original network with the subnetwork indicated. (b) The subnetwork with redefined sources/sinks, the amplitudes of which are indicated by colors. (c) Output from model with sink fluctuation = 0. Note that most of the reticulate structure has disappeared.