Liu et al. Plant Methods (2017) 13:38
A method toestimate plant density
andplant spacing heterogeneity: application
Shouyang Liu1*, Fred Baret1, Denis Allard2, Xiuliang Jin1, Bruno Andrieu3, Philippe Burger4, Matthieu Hemmerlé5
and Alexis Comar5
Background: Plant density and its non-uniformity drive the competition among plants as well as with weeds. They
need thus to be estimated with small uncertainties accuracy. An optimal sampling method is proposed to estimate
the plant density in wheat crops from plant counting and reach a given precision.
Results: Three experiments were conducted in 2014 resulting in 14 plots across varied sowing density, cultivars and
environmental conditions. The coordinates of the plants along the row were measured over RGB high resolution
images taken from the ground level. Results show that the spacing between consecutive plants along the row direc-
tion are independent and follow a gamma distribution under the varied conditions experienced. A gamma count
model was then derived to deﬁne the optimal sample size required to estimate plant density for a given precision.
Results suggest that measuring the length of segments containing 90 plants will achieve a precision better than 10%,
independently from the plant density. This approach appears more eﬃcient than the usual method based on ﬁxed
length segments where the number of plants are counted: the optimal length for a given precision on the density
estimation will depend on the actual plant density. The gamma count model parameters may also be used to quan-
tify the heterogeneity of plant spacing along the row by exploiting the variability between replicated samples. Results
show that to achieve a 10% precision on the estimates of the 2 parameters of the gamma model, 200 elementary
samples corresponding to the spacing between 2 consecutive plants should be measured.
Conclusions: This method provides an optimal sampling strategy to estimate the plant density and quantify the
plant spacing heterogeneity along the row.
Keywords: Wheat, Gamma-count model, Density, RGB imagery, Sampling strategy, Plant spacing heterogeneity
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
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and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/
publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Plant density at emergence is governed by the sowing
density and the emergence rate. For a given plant den-
sity, the uniformity of plant distribution at emergence
may signiﬁcantly impact the competition among plants
as well as with weeds [1, 2]. Plant density and uniformity
is therefore a key factor explaining production, although
a number of species are able to compensate for low plant
densities by a comparatively signiﬁcant development
of individual plants during the growth cycle. For wheat
crops which are largely cultivated over the globe, tillering
is one of the main mechanisms used by the plant to adapt
its development to the available resources that are partly
controlled by the number of tillers per unit area. e till-
ering coeﬃcient therefore appears as an important trait
to be measured. It is usually computed as the ratio of the
number of tillers per unit area divided by the plant den-
sity . Plant density is therefore one of the ﬁrst variables
measured commonly in most agronomical trials.
Crops are generally sown in rows approximately evenly
spaced by seedling devices. Precision seedling systems
mostly used for crops with plants spaced on the row by
1 INRA, UMR-EMMAH, UMT-CAPTE, UAPV, 228 Route de l’aérodrome CS
40509, 84914 Avignon, France
Full list of author information is available at the end of the article
Page 2 of 11
Liu et al. Plant Methods (2017) 13:38
more than few centimeters (e.g. maize, sunﬂower or soy-
bean) distribute seeds relatively evenly along the row.
Conversely, for most crops with short distances among
plants on the row, e.g. wheat, barley or canola, seeds are
distributed non-evenly along the row. is can be attrib-
uted both to the mechanisms that free, at a variable fre-
quency, the seed from the seed tank, and the trajectory
of the seed that may also vary in the pipe that drives it
from the seed tank to the soil. Further, once reaching
the soil, the seed may also move with the soil displaced
by the sowing elements penetrating the soil surface.
Finally, some seeds may abort or some young plants
may die because of pests or too extreme local environ-
mental conditions (excess or deﬁcit of moisture, low
temperature etc.). e population density and its non-
uniformity are therefore recognized as key traits of inter-
ests to characterize the canopy at the emergence stage.
However, very little work documents the plant distribu-
tion pattern along the row, which is partly explained by
the lack of dedicated device for accurate plant position
measurement . Electromagnetic digitizers are very low
throughput and not well adapted to such ﬁeld measure-
ments . Alternatively, algorithms have been developed
to measure the inter-plant spacing along the row for
maize crops from top-view RGB (Red Green Blue) images
[6, 7]. Improvements were then proposed by using three
dimensional sensors [8–10]. However, these algorithms
were only validated on maize crops that show relatively
simple plant architecture with generally ﬁxed inter-plant
spacing along the row.
Manual ﬁeld counting in wheat crops is still exten-
sively employed as the reference method. Measurements
of plant population density should be completed when
the majority of plants have just emerged and before the
beginning of tillering when individual plants start to be
diﬃcult to be identiﬁed. Plants are counted over ele-
mentary samples corresponding either to a quadrat or
to a segment . e elementary samples need to be
replicated in the plot to provide a more representative
value . For wheat crops,  suggested that at least
a total of 3m of rows (0.5m segment length repeated 6
times) should be counted, while  proposed to sam-
ple a total of 6m (segments made of 2 consecutive rows
by one meter repeated 3 times in the plot).  pro-
posed to repeat at least 4 times the counting in 0.25m2
quadrats corresponding roughly to a total of 6.7m length
of rows (assuming the rows are spaced by 0.15 m). In
this case, quadrats may be considered as a set of con-
secutive row segments with the same length when the
quadrat is oriented parallel to the row direction or with
variable lengths when the quadrat is oriented diﬀerently.
Although these recommendations are simple and easy to
apply, they may not correspond to an optimal sampling
designed to target a given precision level. ey may either
provide low precision if under sampled or correspond to
a waste of human resources in the opposite case.
e sample size required to reach a given precision of
the plant density will depend on the population density
and the heterogeneity of plant positions along the row
that may be described by the distribution of the distances
between consecutive plants. is distribution is more
likely to be skewed, which could be described by an expo-
nential distribution or a more general one such as the
Weibull or the gamma distributions. Fitting such random
distribution functions provides not only access to the
plant density at the canopy level, but also to its local vari-
ation that may impact the development of neighboring
plants as discussed earlier.
e objective of this study is to propose an optimal
sampling method for plant density estimation and to
quantify the heterogeneity of plant spacing along the row.
For this purpose, a model is ﬁrst developed to describe
the distribution of the plants along the row. e model
is then calibrated over a number of ground experiments.
Further, the model is used to compare several plant
counting strategies and to evaluate the optimal sampling
size to reach a given precision. Finally, the model was also
exploited to design a method for quantifying the non-
uniformity of plant distribution.
ree sites in France were selected in 2014 (Table1): Avi-
gnon, Toulouse and Grignon. A mechanical seed drill
was used in the three sites, which represents the stand-
ard practice for wheat crops. In Grignon, ﬁve plots were
sampled, corresponding to diﬀerent cultivars with a sin-
gle sowing density. In Toulouse, ﬁve sowing densities were
sampled with the same “Apache” cultivar. In Avignon,
four sowing densities were sampled also with the same
Table 1 The experimental design in2014 overthe three sites
Sites Latitude Longitude Cultivar Density (seedsm−2)
Toulouse 43.5°N 1.5°E Apache 100, 200, 300, 400, 600
Grignon 48.8°N 1.9°E Premio; Attlass; Flamenko; Midas; Koréli 150
Avignon 43.9°N 4.8°E Apache 100, 200, 300, 400
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Liu et al. Plant Methods (2017) 13:38
“Apache” cultivar. All measurements were taken at around
1.5 Haun stage , when most plants already emerged
and were easy to identify visually. is stage is reached
approximately 10–14days after the germination for wheat
in France . A total of 14 plots are thus available over the
3 sites showing contrasted conditions in terms of soil, cli-
mate, cultivars, sowing density and sowing machine, with
however a ﬁxed row spacing of 17.5cm. All the plots were
at least 10m length and 2m width.
A Sigma SD14 RGB camera with a resolution of 4608
by 3072 pixels was installed on a light moving platform
(Fig.1). e camera was oriented at 45° inclination per-
pendicular to the row direction and was focused on the
central row from a distance of about 1.5m (Fig.1). e
50mm focal length allowed to sample about 0.9m of the
row with a resolution at the ground level close to 0.2mm.
Images were acquired along the row with at least 30%
overlap to allow stitching. A series of 20 pictures was col-
lected that correspond to three to ﬁve rows over about
5m length. e images were stitched using AutoStitch
html) . For each site, one picture was taken over a
reference chessboard put on the soil surface to calibrate
the image: the transformation matrix derived from the
chessboard image was applied to all the images acquired
within the same site. It enables to remove perspective
eﬀects and to scale the pixels projected on the soil sur-
face. e image correction and processing afterwards
was conducted using MATLAB R2016a (code available
on request). Coordinates of the plants correspond to the
intersection between the bottom of the plant and the soil
surface (Fig. 2). ey were interactively extracted from
the photos displayed on the screen. For each of the 14
plots, the coordinates of at least 150 successive plants
from the same row were measured along (X axis) and
across (Y axis) of the row. It took between 15 to 30min to
extract the plant coordinates, depending on the density.
e precision on the coordinates values along the row is
around 1.5mm as estimated by independent replicates
of the process over the same images. Some slightly larger
deviations are observed marginally in case of occlusions
by stones or straw in the ﬁeld.
of plant n (noted
) along the
row axis allow to compute the spacing
e actual plant density
expressed in plants per square meter horizontal ground
(plants m−2) was computed simply as the number of
plants counted on the segments, divided by the product
of the length of the segments and the row spacing.
Development andcalibration ofthe plant distribution
Distribution ofplant spacing
e autocorrelation technique was used to explore the
spatial dependency of spacing between successive plants:
the linear correlation between
is the lag is evaluated. Results illustrated in Fig.3 over the
Fig. 1 The moving platform used to take the images in the ﬁeld in
Fig. 2 Extraction of plants’ coordinates from the image
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Liu et al. Plant Methods (2017) 13:38
Toulouse site show that the autocorrelation coeﬃcient of
inter-plant distance is not signiﬁcant at 95% conﬁdence
interval. e same is observed over the other 13 plots
acquired. It is therefore concluded that the positions
among plants along the row direction are independent:
each observation ∆x could be considered as one inde-
pendent realization of the random variable ∆X.
e distribution of the plant spacing is positively
right-skewed (Fig. 4). A simple exponential distribu-
tion with only one scale parameter was ﬁrst tentatively
ﬁtted to the data using a maximum likelihood method.
However, the Chi square test at the 5% signiﬁcance level
showed that the majority of the 14 plots do not fol-
low this simple exponential distribution law. Weibull
and gamma distributions are both a generalization of
the exponential distribution requiring an extra shape
parameter. Results show that Weibull and gamma dis-
tributions describe well (Chi square test at the 5% sig-
niﬁcance level successful) the empirical distributions
over the 14 plots (Fig.4; Table2). However, the gamma
distribution will be preferred since it provides generally
higher p value of Chi square test (Table2) . Besides,
the tail of the Weibull distribution tends toward zero
less rapidly than that of the gamma distribution:
Weibull may show few samples with very large values
, increasing the risk of overestimation for the larger
plant spacing. e gamma distribution was therefore
used in the following and writes :
where a and b represent the shape and scale param-
eters respectively. e expectancy
are simple expressions of the two parameters:
As a consequence, the coeﬃcient of variation
is a simple function of the shape
Modeling the distribution ofthe number ofplants perrow
e plant density evaluated over row segments needs to
account for the uncertainties in row spacing. e vari-
ability of the row spacing is of the order of 10mm as
reported by  which corresponds to CV=6% using a
typical row spacing of 175mm. For the sake of simplic-
ity, the variability of row spacing will be neglected since
it is likely to be small. Further, it is relatively easy to get
precise row spacing measurements for each segment
and to actually account for the actual row spacing val-
ues. Considering a given row spacing, the plant density
depends only on the number of plants per unit linear row
length. Estimating the number of plants within a row seg-
ment is a count data problem analogous to the estimation
of the number of events during a speciﬁc time interval
[19, 21]. Counts are common random variables that are
assumed to be non-negative integer or continuous values
Fig. 3 The autocorrelation of the spacing among plants along the
row direction illustrated with sowing density of 300 seeds m−2
observed over the Toulouse experiment. The lag is expressed as the
number of plant spacing between 2 plants along the row direction
(X axis). Lags 1–20 are presented. The upper and lower horizontal line
represent the 95% conﬁdence interval around 0
Fig. 4 Empirical histogram of the spacing along the row (gray bars).
The solid (respectively dashed) line represents the ﬁtted gamma (resp.
Weibull) distribution. Case of the sowing density 300 seeds m−2
observed over the Toulouse experiment. a and b represent the shape
and scale parameters respectively
Page 5 of 11
Liu et al. Plant Methods (2017) 13:38
representing the number of times an event occurs within
a given spatial or temporal domain . e gamma-
count model suits well our problem with intervals inde-
pendently following a gamma distribution as in our case.
, to get n plants over a segment
of length l, writes (Eqs.5–8 were cited from [19, 21]):
where N1 is the number of plants over the segment
of length l, and
is the incomplete gamma
where Γ is the gamma Euler function. e expectation
and variance of the number of plants over a segment of
length l is given by:
for n=1, 2,
(2n −1)IGa·n, l
Finally, the expectation and variance of the plant den-
sity, D1, estimated over a segment of length l can be
expressed by introducing the row spacing distance, r,
assumed to be known:
, converges toward the actual
density of the population when 1→∞.
e transformed gamma-count model allows evaluating
the uncertainty of plant density estimation as a function
of the sampling size. e uncertainty can be characterized
by the coeﬃcient of variation (CV) as follows:
Several combinations of values of a and b may lead to
the same plant density, but with variations in their dis-
tribution along the row (Fig.5). e ﬁtting of parameters
a and b over the 14 plots using the transformed gamma-
count model (Eq.9) shows that the shape parameter, a,
varies from 0.96 to 1.39 and is quite stable. Conversely,
the scale parameter, b, appears to vary widely from 0.96
to 6.38, mainly controlling the plant density (Fig. 5).
Since the CV depends only on the shape parameter a
(Eq.4), it should not vary much across the 14 plots con-
sidered. is was conﬁrmed by applying a one-way analy-
sis of variance on the CV values of the 14 plots available
(F = 1.09, P = 0.3685): no signiﬁcant diﬀerences are
Table 2 Parameters ofthe tted distributions
Sites Sowing density
(seedsm−2)Cultivar Gamma Weibull
a b p value ofChi
square test a b p value ofChi
Avignon 100 Apache 1.14 6.38 0.27 7.44 1.07 0.29
200 Apache 1.25 4.04 0.62 5.29 1.13 0.05
300 Apache 0.99 2.53 0.38 2.51 1.00 0.56
400 Apache 0.96 1.50 0.22 1.39 0.94 0.57
Toulouse 100 Apache 1.07 5.01 0.12 5.32 0.99 0.10
200 Apache 1.39 1.95 0.17 2.86 1.15 0.12
300 Apache 1.21 2.28 0.94 2.89 1.12 0.94
400 Apache 1.24 1.37 0.51 1.76 1.10 0.40
600 Apache 1.16 0.96 0.37 1.14 1.09 0.21
Grignon 150 Premio 1.12 3.37 0.70 3.85 1.06 0.68
150 Attlass 1.13 2.48 0.69 2.87 1.05 0.67
150 Flamenko 1.11 3.3 0.92 3.75 1.05 0.92
150 Midas 1.24 3.03 0.21 3.92 1.12 0.24
150 Koréli 1.15 2.89 0.24 3.48 1.15 0.18
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Liu et al. Plant Methods (2017) 13:38
observed. is result may be partly explained by the fact
that the same type of seed drill was used for all the three
Optimal sample size toreach a givenprecision forplant
e transformed gamma-count model provides a con-
venient way to investigate the eﬀect of the sampling size
on the precision of the density estimates. e precision
will be quantiﬁed here using the coeﬃcient of varia-
tion (CV). e sample size can be expressed either as a
given length of the segments where the (variable) num-
ber of plants should be counted, or as a (variable) length
of the segment to be measured corresponding to a given
number of consecutive plants. e two alternative sam-
pling approaches will be termed FLS (Fixed Length of
Segments) for the ﬁrst one, and FNP (Fixed Number of
Plants) for the second one.
When considering the FLS approach, the sample size is
deﬁned by the length of segment, L, where plants need to
be counted. e optimal L value for a given target preci-
sion quantiﬁed by the CV will mainly depend on the cur-
rent density as demonstrated in Fig.6a: longer segments
are required for the low densities. Conversely, shorter
segments are needed for high values of the plant density
to reach the same precision. e scale parameter, b, that
controls the plant density drives therefore the optimal
segment length L (Fig.6a). Counting plants over L=5m
(500cm) provides a precision better than 10% for den-
sities larger than 150 plants·m−2 for the most common
conditions characterized by a shape coeﬃcient a >0.9.
ese ﬁgures agree well with the usual practice for plant
counting as reviewed in the introduction [3, 13, 14].
Increasing the precision quantiﬁed by the CV will require
longer segments L to be sampled (Fig.7a).
When considering the FNP approach, the sample
size is driven by the number, N, of consecutive plants
that deﬁnes to a row segment whose length need to be
Fig. 5 Relationship between parameters a and b of the gamma-
count model for a range of plant density (from Eqs. 6, 7, 9). The lines
correspond to, 100, 150, 200, 300, 400 and 600 plants m−2. The dots’
color corresponds to the experimental sites
Fig. 6 a The optimal sampling size length (the horizontal solid lines, the length being indicated in cm) used in the FLS approach as a function of
parameters a and b to get CV = 10% for the density estimation. b Idem as on the left but the sample is deﬁned by the number of plants to be
counted (the vertical solid lines with number of plants indicated) for the FNP approach. The gray dashed lines correspond to the actual plant density
depending also on parameters a and b. The row spacing is assumed perfectly known and equal to 17.5 cm. The gray points represent the 14 plots
Page 7 of 11
Liu et al. Plant Methods (2017) 13:38
measured. e simulations of the model (Fig.6b) show
that N is mainly independent from the plant density.
For the 14 plots considered in this study, segments with
70<N<110 plants should be measured to reach a pre-
cision of CV =10%. e shape parameter a inﬂuences
dominantly the sample size: more heterogeneous dis-
tribution of plants characterized by small values of the
shape parameter will require more plants to be counted
(Fig. 6b). To increase the precision (lower CV), more
plants will also need to be counted (Fig.7b).
e sampling approach FLS (Fixed Length of segments)
is extensively used to estimate the plant density. e
600cm segment length recommended by [13, 14] agrees
well with our results (Figs.6a, 7a) demonstrating that a
precision better than 10% is ensured over large range of
densities and non-uniformities. e optimal sampling
length (FSL) and optimal number of plants sampled
(FNP) was computed for other precision levels for a range
of plant densities (Table3). Results show that the FNP
method provides very stable values of the sampling size:
it is easy to propose an optimal number of consecutive
plants to count to reach a given precision. Conversely, the
optimal length of the segment used in the FSL approach
varies strongly with the plant density (Table3): the FLS
approach when applied with a segment length chosen a
priori without knowing the plant density will result in a
variable precision level.
Sampling strategy toquantify plant spacing variability
e previous sections demonstrated that the scale and
shape parameters could be estimated from the observed
distribution of the plant spacing. However, the measure-
ment of individual plant spacing is tedious and prone to
errors as outlined earlier. e estimation of these param-
eters from the variability observed between small row
segments containing a ﬁxed number of plants will there-
fore be investigated here. is FNP approach is preferred
Fig. 7 The optimal sampling length for the FLS approach (a) and the number of plants for the FNP approach (b). The dominant parameter is used
(the scale parameter for FLS and the shape coeﬃcient for FNP). The precision is evaluated with the CV = 5, 10, 15 and 20%
Table 3 Optimal sampling size forFSL and FNP overdierent densities (100, 150, 200, 300, 400 and600 seeds m−2)
andprecisions (5, 10 and15%)
This was calculated using the average values of the parameters a and b of the gamma distribution derived for each density over the 14 plots available
Sowing density (seedsm−2) Parameters CV=5% CV=10% CV=15%
a b FSL (cm) FNP (Nb. Plt) FSL (cm) FNP (Nb. Plt) FSL (cm) FNP (Nb. Plt)
100 1.11 5.70 2478 363 620 90 250 39
150 1.15 3.01 1406 348 351 88 130 37
200 1.32 3.00 1398 308 350 78 130 33
300 1.10 2.41 1162 363 291 90 110 39
400 1.10 1.44 774 363 194 90 60 39
600 1.16 0.96 584 348 146 85 50 37
Page 8 of 11
Liu et al. Plant Methods (2017) 13:38
here to the FSL one because there will be no additional
uncertainties introduced by the position of the ﬁrst and
last plants of the segment with the corresponding start
and end of the segment. ese uncertainties may be sig-
niﬁcant in case of small segments in the FLS approach.
e probability distribution of a gamma distribution
can be expressed as the sum of an arbitrary number of
independent individual gamma distributions . is
property allows to compute the distribution of a segment
of length Ln corresponding to n plant spacing between
(n+ 1) consecutive plants with
gamma distribution with n·a as shape parameter and
the same scale parameter b as the one describing the dis-
tribution of ∆X.
e parameters a and b will therefore be estimated by
adjusting the gamma model described in Eq. 12 for the
given value of n+1 consecutive plants.
e eﬀect of the sampling size on the precision of a
and b parameters estimation was further investigated. A
numerical experiment based on a Monte-Carlo approach
was conducted considering a standard case correspond-
ing to the average of the 14 plots sampled in 2014 with
a=1.10 and b=2.27. e sampling size is deﬁned by
the number of consecutive plants for the FNP approach
considered here and by the number of replicates. For
each sampling size 300 samples were generated by ran-
domly drawing in the gamma distribution (Eq.12) and
parameters a and b were estimated. e standard devi-
ation between the 300 estimates of a and b parameters
was ﬁnally used to compute the corresponding CV. is
process was applied to a number of replicates varying
between 20 to 300 by steps of 10 and a number of plants
per segment varying between 2 (i.e. spacing between two
consecutive plants) to 250 within 12 steps. is allows
describing the variation of the coeﬃcient of estimated
values of parameters a and b as a function of the number
of replicates and the number of plants (Fig.8).
Results show that the sensitivity of the CV of estimates
of parameters a and b are very similar (Fig.8). e sensi-
tivity of parameters a and b is dominated by the number
of replicates: very little variation of CV is observed when
the number of plants per segment varies (Fig.8). Param-
eters a and b require about 200 replicates independently
from the number of plants per segment. It seems there-
fore more interesting to make very small segments to
decrease the total number of plants to count.
Additional investigations not shown here for the sake
of brevity, conﬁrmed the independency of the number
of replicates to the number of plants per segment when
parameters a and b are varying. Further, the number of
replicates need to be increased as expected when the
shape parameter a decreases (i.e. when the plant spacing
is more variable) to keep the same precision on estimates
of a and b parameters.
A method was proposed to estimate plant density and
sowing pattern from high resolution RGB images taken
from the ground. e method appears to be much more
comfortable as compared with the standard outdoor
methods based on plant counting in the ﬁeld. Images
should ideally be taken around Haun stage 1.5 for wheat
crops when most plants have already emerged and tiller-
ing has not yet started. Great attention should be paid to
the geometric correction in order to get accurate ortho-
images where distances can be measured accurately. e
processing of images here was automatic except the last
step corresponding to the interactive visual extraction
of the plants’ coordinates in the image. However, recent
work [24, 25] suggests that it will be possible to automa-
tize this last step to get a fully high-throughput method.
e method proposed is based on the modeling of
the plant distribution along the row. It was ﬁrst dem-
onstrated that the plant spacing between consecutive
plants are independent which corresponds to a very
useful simplifying assumption. e distribution of plant
spacing was then proved to follow a gamma distribution.
Although the Weibull distribution showed similar good
performance, it was not selected because of the com-
paratively heavier tails of the distribution that may cre-
ate artefacts. Further the Weibull model does not allow
to simply derive the distribution law of the length of
Fig. 8 Contour plot of the CV associated to the estimates of param-
eters a (solid line) and b (dashed line) as a function of the number of
replicates of individual samples made of n plants (the y axis). The solid
(respectively dashed) isolines correspond to the CV of parameter a
(respectively parameter b). These simulations were conducted with [a,
b] = [1.10, 2.27]
Page 9 of 11
Liu et al. Plant Methods (2017) 13:38
segments containing several consecutive plants . e
gamma model needs a scale parameter that drives mostly
the intensity of the process, i.e. the plant density, and a
shape parameter that governs the heterogeneity of plant
spacing. is model was transformed into a count data
model to investigate the optimal sampling required to get
an estimate of plant density for a given precision level.
e adjustment of the gamma-count model on the
measured plant spacing using a maximum likelihood
method provides an estimate of the plant density (Eq.9).
e comparison to the actual plant density (Fig.9) sim-
ply computed as the number of plants per segment
divided by the area of the segments (segment length by
row spacing), shows a good agreement, with RMSE≈50
plantsm−2 over the 14 plots available. e model per-
forms better for the low density with a RMSE of 21
plantsm−2 for density lower than 400 plantsm−2. ese
discrepancies may be mainly explained by the accuracy
in the measurement of the position of individual plants
(around 1–2 mm). Uncertainties on individual plant
spacing will be high in relative values as compared to
that associated with the measurement of the length of the
segment used in the simple method to get the ‘reference’
plant density. Hence it is obviously even more diﬃcult to
get a good accuracy in plant spacing measurements for
high density, i.e. with a small distance among plants. In
addition, small deviations from the gamma-count model
are still possible, although the previous results were
showing very good performance.
e model proposed here concerns mainly relatively
nominal sowing, i.e. when the sowing was successful
on average on the row segments considered: portions
of rows with no plants due to sowing problems or local
damaging conditions (pests, temperature and moisture).
e sowing was considered as nominal on most of the
plots investigated in this study, with no obvious ‘acci-
dents’. However, it is possible to automatically identify
from the images the unusual row segments with missing
plants or excessive concentration of plants . Rather
than describing blindly the bulk plant density, it would
be then preferred to get a nested sampling strategy: the
unusual segments could be mapped extensively, and the
plant density of nominal and unusual segments could
be described separately using the optimal sampling pro-
is study investigated the sampling strategy to esti-
mate the plant density with emphasis on the variability of
plant spacing along the row, corresponding to the sam-
pling error. However additional sources of error should
be accounted for including measurement biases, uncer-
tainties in row spacing or non-randomness in the sample
selection [27– 29]. Unlike sampling error, it could not be
minimized by increasing sampling size. e non-sam-
pling error may be reduced by combining a random sam-
pling selection procedure with a measurement method
ensuring high accuracy including accounting for the
actual values of the row spacing measured over each seg-
Optimal sampling requires a tradeoﬀ between mini-
mum sampling error obtained with maximum sampling
size and minimum cost obtained with minimum sam-
pling size . e optimal sampling strategy should
ﬁrst be designed according to the precision targeted
here quantiﬁed by the coeﬃcient of variation (CV) char-
acterizing the relative variability of the estimated plant
density between several replicates of the sampling pro-
cedure. e term ‘optimal’ should therefore be under-
stood as the minimum sampling eﬀort to be spent to
achieve the targeted precision. Two approaches were
proposed: the ﬁrst one considers a ﬁxed segment length
(FSL) over which the plants have to be counted; the sec-
ond one considers a ﬁxed number of successive plants
(FNP) deﬁning a row segment, the length of which needs
to be measured. e ﬁrst method (FLS) is the one gener-
ally applied within most ﬁeld experiments. However, we
demonstrated that it is generally sub-optimal: since the
segment length required to achieve a given CV depends
mainly on the actual plant density: the sampling will
be either too large for the targeted precision, or con-
versely too small, leading to possible degradation of the
precision of plant density estimates. Nevertheless, for
the plant density (>100 plantsm−2) and shape param-
eter (a>0.9) usually experienced, a segment length of
6m will ensure a precision better than 10%. e second
approach (FNP) appears generally more optimal: it aims
at measuring the length of the segment corresponding to
a number of consecutive plants that will depend mainly
on the targeted precision. Results demonstrate that in
Fig. 9 Comparison between the actual density and that estimated
from the gamma-count model
Page 10 of 11
Liu et al. Plant Methods (2017) 13:38
our conditions, the density should be evaluated over seg-
ments containing 90 plants to achieve a 10% precision.
e sampling size will always be close to optimal as com-
pared to the ﬁrst approach where optimality requires the
knowledge of the plant density that is to be estimated.
Further, the FNP approach is probably more easy to
implement with higher reliability: as a matter of facts,
measuring the length of a segment deﬁned by plants at
its two extremities is easier than counting the number of
plants in a ﬁxed length segment, where the extremities
could be in the vicinity of a plant and its inclusion or not
in the counting could be prone to interpretation biases
by the operator. e total number of plants required
in a segment could be split into subsamples containing
smaller number of plants that will be replicated to get
the total number of plants targeted. is will improve
the spatial representativeness. Overall, the method pro-
posed meets the requirements deﬁned by [32, 33] for
the next genearation of phenotyping tools: increase the
accuracy, the precision and the throughput while reduc-
ing the labor and budgetary costs.
e gamma-count model proved to be well suited to
describe the plant spacing distribution along the row
over our contrasted experimental situations. It can thus
be used to describe the heterogeneity of plant spacing
as suggested by . is may be applied for detailed
canopy architecture studies or to quantify the impact of
the sowing pattern heterogeneity on inter-plant com-
petition [1, 2]. e heterogeneity of plant spacing may
be described by the scale and shape parameters of the
gamma model. Quantiﬁcation of the heterogeneity of
plant spacing requires repeated measurements over seg-
ments deﬁned by a ﬁxed number of plants. Our results
clearly show that the precision on estimates of the
gamma count parameters depends only marginally on
the number of plants in each segment. Conversely, it
depends mainly on the number of segments (replicates)
to be measured. For the standard conditions experienced
in this study, the optimal sampling strategy to get a CV
lower than 10% on the two parameters of the gamma dis-
tribution would be to repeat 200 times the measurement
of plant spacing between 2 consecutive plants.
SL and FB designed the experiment and XJ, BA, PB, MH and AC contributed
to the ﬁeld measurement in diﬀerent experimental sites. DA signiﬁcantly
contributed to the method development. The manuscript was written by
SL and signiﬁcantly improved by FB. All authors read and approved the ﬁnal
1 INRA, UMR-EMMAH, UMT-CAPTE, UAPV, 228 Route de l’aérodrome CS 40509,
84914 Avignon, France. 2 UMR BioSP, INRA, UAPV, 84914 Avignon, France.
3 UMR ECOSYS, INRA, AgroParisTech, Université Paris-Saclay, 78850 Thiver-
val-Grignon, France. 4 UMR AGIR, INRA, INPT, 31326 Toulouse, France. 5 Hi-Phen,
84914 Avignon, France.
We thank the people from Grignon, Toulouse and Avignon who participated
to the experiments. Great thanks to Paul Bataillon and Jean-Michel Berceron
from UE802 Toulouse INRA for their help in ﬁeld experiment. The work
was completed within the UMT-CAPTE funded by the French Ministry of
The authors declare that they have no competing interests.
Availability of data and materials
All data analyzed during this study are presented in this published article.
This study was supported by “Programme d’investissement d’Avenir” PHE-
NOME (ANR-11-INBS-012) and Breedwheat (ANR-10-BTR-03) with participation
of France Agrimer and “Fonds de Soutien à l’Obtention Végétale”. The grant of
the principal author was funded by the Chinese Scholarship Council.
Springer Nature remains neutral with regard to jurisdictional claims in pub-
lished maps and institutional aﬃliations.
Received: 25 September 2016 Accepted: 2 May 2017
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