Appendix S1. Light-field camera: from grey-scale depth images to metric distance
information.
In a conventional camera, the main lens focuses directly onto the image sensor,
with each pixel capturing the intensity of incoming light, resulting in a 2D image. In
a light-field single-lens 3D camera, information about the position, the direction and
the intensity of incident light rays is captured by an additional array of microlenses
with various focal lengths located in front of the image sensor.
The functional principle of the light-field camera and its optical path are shown in
Figure 1b. The thin lens equation,
1
1
𝐿
𝐿
1
(S1.1) 𝑓 = 𝑎 + 𝐵 ,
𝐿
describes how light coming from an object at distance 𝑎𝐿 from the main lens will
form a sharp main image at distance 𝐵𝐿 behind the main lens, with the focal length
𝑓𝐿 .
The microlens array (MLA) located in front of the image plane contains
approximately 50,000 microlenses. Each microlens images a subsection of the full
image. The number of microlenses imaging an object depends on the distance,
whereby closer objects are imaged in more microlenses. This redundancy enables
the extraction of local distance information. Since this redundancy depends on the
distance of the imaged object, the virtual depth 𝑣 is introduced:
(S1.2) 𝑣 =
𝑎𝐿
𝐵
,
1
where 𝑎 is the distance between the MLA and the main image and 𝐵 is the
distance between the MLA and the image plane. The range of distance
measurements is further extended by using microlenses of three different focal
lengths (Figure 1c). Similar to the focus image, the virtual depth is stored as a 16
bit image, and the grey values of each pixel depend on its virtual depth:
(S1.3) 𝑧𝑔𝑟𝑒𝑦 = (1 − 𝑣 −1 )(216 − 1).
In the next step, this virtual depth is linked to the actual camera-object distance.
The total focus distance, 𝑇𝐿 , corresponding to the farthest camera-object distance,
is given by
(S1.4) 𝑇𝐿 = 𝑎𝐿 + 𝐵𝐿 ,
where 𝐵𝐿 is the distance between the plane of the main lens and the total covering
plane (TCP), which is the closest plane where each point on the plane is projected
by at least one microlens. From Eqs. (S1.1) and (S1.4) we obtain
(S1.5) 𝐵𝐿 =
𝑇𝐿
2
𝑓
(1 − √1 − 4 𝑇𝐿).
𝐿
Therefore, the constant distance 𝐵𝐿 = 226.31 mm between the main lens and the
TCP can be calculated for given parameters of our set-up. These were determined
in a calibration and validation step with 𝑇𝐿 = 809 mm, 𝑓𝐿 =163 mm, and 𝐵 = 2.75
mm. However, other parameters, such as 𝑎𝐿 and 𝑏𝐿 , depend on the distance of the
object to the camera. The distance between the main lens and the main image 𝑏𝐿
can be calculated via
2
(S1.6) 𝑏𝐿 = 𝐵𝐿 + 𝑎 − 2𝐵.
As the ratio of 𝑎 and 𝐵 is provided via the virtual depth 𝑣 (Eq. (S1.2)), we can
calculate 𝑏𝐿 for every pixel:
(S1.7) 𝑏𝐿 = 𝐵𝐿 + (𝑣 − 2)𝐵.
Thus, the distance 𝑎𝐿 from the object to the lens (metric distance) can be
calculated by Eq. (S1.1) and ranges from 474.46 mm (closest structures with high
virtual depth of 10) to 582.69 mm (farthest structures with low virtual depth of 2).
Given the metric distance 𝑎𝐿 for each pixel, we calculate their magnification via
𝑏
(S1.8) 𝑀 = 𝑎𝐿,
𝐿
which ranges from 0.39 to 0.52, indicating that the photographed plant structures in
our set-up have a projected size at the sensor/image plane of 39-52% of the
original. The sensor of the camera has dimensions of 36 x 24 mm and 3288 x 2192
pixels resulting in an effective sensor pixel size 𝑝 = 11 µm. Dividing the pixel size
by the magnification yields the metric size of the pixel (edge length) which ranges
from 21 µm for farthest, to 28 µm for closest structures. Therefore, the resolution of
the camera is distance-dependent and ranges from 35 - 47 px/mm.
Combining equations (S1.1), (S1.3), (S1.5) and (S1.7) yields the metric distance
(𝑧𝑚𝑒𝑡𝑟𝑖𝑐 ) and the metric size (𝑧𝑝𝑖𝑥𝑒𝑙𝑠𝑖𝑧𝑒 ) of the pixels depending on its depth grey
value (𝑧𝑔𝑟𝑒𝑦 ):
3
𝑇
𝑓
𝑓𝐿 −( 𝐿 (1−√1−4 𝐿 )+(
2
1
𝑧𝑔𝑟𝑒𝑦 −2)𝐵)
1− 16
2 −1
𝑇𝐿
(S1.9) 𝑧𝑚𝑒𝑡𝑟𝑖𝑐 (𝑓𝐿 , 𝑇𝐿 , 𝐵, 𝑧𝑔𝑟𝑒𝑦 ) = 𝑎𝐿 =
𝑇
𝑓
𝑓𝐿 ( 𝐿 (1−√1−4 𝐿 )+(
2
𝑝
(S1.10) 𝑧𝑝𝑖𝑥𝑒𝑙𝑠𝑖𝑧𝑒 (𝑓𝐿 , 𝑇𝐿 , 𝐵, 𝑧𝑔𝑟𝑒𝑦 ) = 𝑀 =
𝑇𝐿
1
𝑧𝑔𝑟𝑒𝑦 −2)𝐵)
1− 16
2 −1
11µ𝑚∗𝑎𝐿
𝑇𝐿
𝑓
1
(1−√1−4 𝐿 )+( 𝑧𝑔𝑟𝑒𝑦 −2)𝐵
2
𝑇𝐿
1− 16
2 −1
The parameters 𝑓𝐿 , 𝑇𝐿 and 𝐵 are obtained in a calibration step as described in the
Results.
4