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Liu et al. Plant Methods (2017) 13:38

DOI 10.1186/s13007-017-0187-1

METHODOLOGY

A method toestimate plant density

andplant spacing heterogeneity: application

towheat crops

Shouyang Liu1*, Fred Baret1, Denis Allard2, Xiuliang Jin1, Bruno Andrieu3, Philippe Burger4, Matthieu Hemmerlé5

and Alexis Comar5

Abstract

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,

provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license,

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.

Background

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 [3]. 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

Open Access

Plant Methods

*Correspondence: Shouyang.Liu@inra.fr

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

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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 [4]. Electromagnetic digitizers are very low

throughput and not well adapted to such ﬁeld measure-

ments [5]. 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 [11]. e elementary samples need to be

replicated in the plot to provide a more representative

value [12]. For wheat crops, [3] suggested that at least

a total of 3m of rows (0.5m segment length repeated 6

times) should be counted, while [13] proposed to sam-

ple a total of 6m (segments made of 2 consecutive rows

by one meter repeated 3 times in the plot). [14] 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.

Methods

Field experiment

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 [15], 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 [3]. 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.

Image processing

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

(http://matthewalunbrown.com/autostitch/autostitch.

html) [16]. 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.

e coordinates

xn

of plant n (noted

Plantn

) along the

row axis allow to compute the spacing

�xn=(xn−xn−1)

between

Plantn

and

Plantn−1.

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

model

Distribution ofplant spacing

e autocorrelation technique was used to explore the

spatial dependency of spacing between successive plants:

the linear correlation between

xn−m

and

xn

where m

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

2014

Fig. 2 Extraction of plants’ coordinates from the image

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Page 4 of 11

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) [17]. 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

[18], increasing the risk of overestimation for the larger

plant spacing. e gamma distribution was therefore

used in the following and writes [19]:

(1)

f

�x|a, b

=1

b

a

Γ(a)

�xa−1e−�x

b�x,a,b

∈

R

+

where a and b represent the shape and scale param-

eters respectively. e expectancy

E(�X)

and variance

Var(�X)

are simple expressions of the two parameters:

As a consequence, the coeﬃcient of variation

CV

(�X)

=√var

(�

X)

E(�X)

is a simple function of the shape

parameter:

Modeling the distribution ofthe number ofplants perrow

segment

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 [20] 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

(2)

E(�X)=a·b

(3)

Var(�X)=a·b2

(4)

CV(�X)=1/√a

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

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Liu et al. Plant Methods (2017) 13:38

representing the number of times an event occurs within

a given spatial or temporal domain [22]. e gamma-

count model suits well our problem with intervals inde-

pendently following a gamma distribution as in our case.

e probability,

P{Nl=n}

, 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

IG

a

·

n, l

b

is the incomplete gamma

function:

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:

(5)

P{N

l=

n}

=

1−IGa, l

bfor n=0

IG

a·n, l

b

−IG

a·n+a, l

b

for n=1, 2,

...

(6)

IG

a·n, l

b

=

1

Γ(a

·

n)

l/b

0

ta·n−1e−t

dt

(7)

E

(Nl)=

∞

n=1

IG

a·n, l

b

(8)

Var

(Nl)=

∞

n=1

(2n −1)IGa·n, l

b

−

∞

n=1

IG

a·n, l

b

2

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:

e expectation,

E(Dl)

, 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

(9)

E

(Dl)

=

E(N

l

)

l·r

(10)

Var

(Dl)

=

Var(N

l

)

(l·r)

2

(11)

CV

(Dl)

=

√

Var(Dl)

E(Dl)=

√

Var(Nl

)

E(Nl)

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

square test

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

sites.

Results

Optimal sample size toreach a givenprecision forplant

density estimation

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

measured

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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

onthe row

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

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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 [23]. is

property allows to compute the distribution of a segment

of length Ln corresponding to n plant spacing between

(n+ 1) consecutive plants with

Ln=n

i=1

xi,

as a

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

(12)

Ln∼Gamma(n·a,b)

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.

Discussion andconclusions

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]

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Page 9 of 11

Liu et al. Plant Methods (2017) 13:38

segments containing several consecutive plants [26]. 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 [25]. 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-

posed here.

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-

ment [30].

Optimal sampling requires a tradeoﬀ between mini-

mum sampling error obtained with maximum sampling

size and minimum cost obtained with minimum sam-

pling size [31]. 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

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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 [20]. 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.

Authors’ contributions

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

manuscript.

Author details

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.

Acknowledgements

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

Agriculture.

Competing interests

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.

Funding

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.

Publisher’s Note

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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