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The form of a remote sensing index is generally empirically defined, whether by choosing specific reflectance bands, equation forms or its coefficients. These spectral indices are used as preprocessing stage before object detection/classification. But no study seems to search for the best form through function approximation in order to optimize the classification and/or segmentation. The objective of this study is to develop a method to find the optimal index, using a statistical approach by gradient descent on different forms of generic equations. From six wavebands images, five equations have been tested, namely: linear, linear ratio, polynomial, universal function approximator and dense morphological. Few techniques in signal processing and image analysis are also deployed within a deep-learning framework. Performances of standard indices and DeepIndices were evaluated using two metrics, the dice (similar to f1-score) and the mean intersection over union (mIoU) scores. The study focuses on a specific multispectral camera used in near-field acquisition of soil and vegetation surfaces. These DeepIndices are built and compared to 89 common vegetation indices using the same vegetation dataset and metrics. As an illustration the most used index for vegetation, NDVI (Normalized Difference Vegetation Indices) offers a mIoU score of 63.98% whereas our best models gives an analytic solution to reconstruct an index with a mIoU of 82.19%. This difference is significant enough to improve the segmentation and robustness of the index from various external factors, as well as the shape of detected elements.
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remote sensing
Article
DeepIndices: Remote Sensing Indices Based on Approximation
of Functions through Deep-Learning, Application to
Uncalibrated Vegetation Images
Jehan-Antoine Vayssade * , Jean-Noël Paoli , Christelle Gée and Gawain Jones


Citation: Vayssade, J.-A.; Paoli, J.-N.;
Gée, C.; Jones, G. DeepIndices:
Remote Sensing Indices Based on
Approximation of Functions through
Deep-Learning, Application to
Uncalibrated Vegetation Images.
Remote Sens. 2021,13, 2261. https://
doi.org/10.3390/rs13122261
Academic Editors: Kuniaki Uto,
Nicola Falco and Mauro Dalla Mura
Received: 2 April 2021
Accepted: 16 May 2021
Published: 9 June 2021
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Agroécologie, AgroSup Dijon, INRA, University of Bourgogne-Franche-Comté, F-21000 Dijon, France;
jean-noel.paoli@agrosupdijon.fr (J.-N.P.); christelle.gee@agrosupdijon.fr (C.G.);
gawain.jones@agrosupdijon.fr (G.J.)
*Correspondence: jehan-antoine.vayssade@inra.fr
Abstract:
The form of a remote sensing index is generally empirically defined, whether by
choosing specific reflectance bands, equation forms or its coefficients. These spectral indices are
used as preprocessing stage before object detection/classification. But no study seems to search
for the best form through function approximation in order to optimize the classification and/or
segmentation. The objective of this study is to develop a method to find the optimal index, using a
statistical approach by gradient descent on different forms of generic equations. From six wavebands
images, five equations have been tested, namely: linear, linear ratio, polynomial, universal function
approximator and dense morphological. Few techniques in signal processing and image analysis are
also deployed within a deep-learning framework. Performances of standard indices and DeepIndices
were evaluated using two metrics, the dice (similar to f1-score) and the mean intersection over union
(mIoU) scores. The study focuses on a specific multispectral camera used in near-field acquisition of
soil and vegetation surfaces. These DeepIndices are built and compared to 89 common vegetation
indices using the same vegetation dataset and metrics. As an illustration the most used index for
vegetation, NDVI (Normalized Difference Vegetation Indices) offers a mIoU score of 63.98% whereas
our best models gives an analytic solution to reconstruct an index with a mIoU of 82.19%. This
difference is significant enough to improve the segmentation and robustness of the index from
various external factors, as well as the shape of detected elements.
Keywords:
image; precision agriculture; spectral indices; multi-spectral; deep-learning; vegetation
segmentation
1. Introduction
An important advance in the field of earth observation is the discovery of spectral
indices, they have proved their effectiveness in surface description. Several studies have
been conducted using remote sensing indices, often applied to a specific field of study like
evaluations of vegetation cover, vigor, or growth dynamics [
1
4
] for precision agriculture
using multi-spectral sensors. Some spectral indices have been developed using RGB or
HSV color space to detect vegetation from ground cameras [
5
7
]. Remote sensing indices
can also be used for other surfaces analysis like water, road, snow [
8
] cloud [
9
] or shadow [
10
].
There are two main problems with these indices. Firstly they are almost all empirically
defined, although the selection of wavelengths comes from observation, like NDVI for
vegetation indices. It is possible to obtain better spectral combinations or equations to
characterize a surface with specific acquisitions parameters. It is important to optimize
upstream the index, as the data transformation leads to a loss of essential information and
features for classification [
11
]. Most studies have tried to optimize some parameters of
existing indices. For example, an optimization of NDVI
(NIR Red)/(NIR +Red)
was
proposed by [
12
] under the name of WDRVI (Wide Dynamic Range Vegetation Index)
Remote Sens. 2021,13, 2261. https://doi.org/10.3390/rs13122261 https://www.mdpi.com/journal/remotesensing
Remote Sens. 2021,13, 2261 2 of 21
(αNIR Red)/(αNIR +Red)
. The author tested different values for
α
between 0 and 1.
The ROC curve was used to determine the best coefficient for a given ground truth. Another
optimized NDVI was designed and named EVI (Enhanced Vegetation Index). It takes
into account the blue band for atmospheric resistance by including various parameters
G(NIR Red)/(NIR +C1Red C2Blue +L)
, where
G
,
L
are respectively the gain factor
and the canopy background adjustment, in addition the coefficients
C1
,
C2
are used to
compensate for the influence of clouds and shadows. Many other indices can be found in
an online database of indices (IDB: www.indexdatabase.de accessed 10 August 2019) [
10
]
including the choice of wavelengths and coefficients depending on the selected sensors
or applications. But none of the presented indices are properly optimized. Thus, in the
standard approach, the best index is determined by testing all available indices against
the spectral bands of the selected sensor with a Pearson correlation between these indices
and a ground truth [
13
,
14
]. Furthermore, correlation is not the best estimator because it
neither considers the class ratio nor the shape of the obtained segmentation and may again
result in a non-optimal solution for a specific segmentation task. Finally, these indices are
generally not robust because they are still very sensitive to shadows [
11
]. For vegetation,
until recently, all of the referenced popular indices were man-made and used few spectral
bands (usually NIR, red, sometimes blue or RedEdge).
The second problem with standard indices is that they work with reflectance-calibrated
data. Three calibration methods can be used in proximal-sensing. (i) The first method use
an image taken before acquisition containing a color patch as a reference [
15
,
16
], and is
used for correction. The problem with this approach is that if the image is partially shaded,
the calibration is only relevant on the non-shaded part. Moreover ideally the reference must
be updated to reduce the interference of weather change on the spectrum measurement,
which is not always possible since it’s a human task. (ii) An other method is the use of
an attached sunshine sensor [
17
], which also requires calibration but does not allow to
correct a partially shaded image. (iii) The last method is the use of a controlled lighting
environment [
18
,
19
], e.g., natural light is suppressed by a curtain and replaced by artificial
lighting. All of these approaches are sometimes difficult to implement for automatic,
outdoor use, and moreover in real time like detecting vegetation while a tractor is driving
through a crop field.
In recent years, machine learning algorithms have been increasingly used to improve
the definition of presented indices in the first main problem. Some studies favor the
use of multiple indices and advanced classification techniques (RandomForest, Boosting,
DecisionTree, etc.) [
4
,
20
24
]. Another study has proposed to optimize the weights in an
NDVI equation form based on a genetic algorithm [
25
] but does not optimize the equation
forms. An other approach has been proposed to automatically construct a vegetation
index using a genetic algorithm [
26
]. They optimize the equation forms by building a set
of arithmetic graphs with mutations, crossovers and replications to change the shape of
each equation during learning but it does not take into account the weights, since it’s use
calibrated data. Finally, with the emergence of deep learning, current studies try to adapt
popular CNN architectures (UNet, AlexNet, etc.) to earth observation applications: [
27
30
].
However there is no study that optimize both the equation forms and spectral bands
weights. The present study explicitly optimize both of them by looking for a form of
remote sensing indices by learning weights in functions approximators. These functions
approximators will then reconstruct any equation forms of the desired remote sensing index
for a given acquisition system. To solve the presented second problem, this study evaluates
the functions approximators on an uncalibrated dataset containing various acquisition
conditions. This is not a common approach but can be found in the literature [
31
,
32
].
This will lead to creating indices that do not require data calibration. The deep learning
framework has been used as a general regression toolkit. Thus, several CNN function
approximators architectures are proposed. DeepIndices is presented as a regression
problem, which is totally new, as is the use of signal and image processing.
Remote Sens. 2021,13, 2261 3 of 21
2. Material and Data
2.1. Instrument Details
The images were acquired with the Airphen (Hyphen, Avignon, France) six-band
multi-spectral camera (Figure 1). This is a multi-spectral scientific camera developed
by agronomists for agricultural applications. It can be integrated into different types of
platforms such as drones, phenotyping robots, etc.
Figure 1. AIRPHEN camera composed of 6 sensors
The camera has been configured using the 450/570/675/710/730/850 nm bands with
a 10 nm FWHM respectively denoted from
λ0
to
λ5
. These spectral bands have been
defined by a previous study [
33
] for crop/weed discrimination. The focal length of each
lens is 8 mm. The raw resolutions for each spectral band is 1280
×
960 px with 12 bit
precision. Finally, the camera is equipped with an internal GPS antenna.
2.2. Image Dataset
The dataset were acquired on the site of INRAe in Montoldre (Allier, France) within
the framework of the “RoSE challenge” founded by the French National Research Agency
(ANR) and in Dijon (Burgundy, France) within the site of AgroSup Dijon. Images of bean
and corn, containing various natural weeds (yarrows, amaranth, geranium, plantago,
etc) and sowed ones (mustards, goosefoots, mayweed and ryegrass) with very distinct
characteristics in terms of illumination (shadow, morning, evening, full sun, cloudy, rain,
etc) were acquired in top-down view at 1.8 m from the ground. The Table 1synthesis
the dataset.
Table 1. Acquisition sources and global illumination.
Source Year Corn Bean Illumination
Dijon 2019 - 9 full sun, evening
Montoldre 2019 20 22 shadow, sunny, cloudy
Montoldre 2020 18 22 morning, cloudy, rainy
total 38 53 =91
Manual annotation takes about 4 h per image to obtain the best quality of ground
truth, which is necessary for use in regression algorithms. Thus the ground truth size is
small and defined with very distinctive illumination condition. To simulate light variations
effect on the ground truth images a random brightness (20%) and a random saturation
(5%) are added to each spectral band during the training phase. As illustration the Figure 2
shows a false color reconstruction of corn crop in the field with various weeds and shadows
on the corners of the image (not vignetting).
Remote Sens. 2021,13, 2261 4 of 21
Figure 2. False color in the left and corresponding manual ground truth on the right.
2.3. Data Pre-Processing
2.3.1. Images Registration
Due to the nature of the camera (Figure 1), a spectral band registration is required
and performed with a registration method based on previous work [
34
] (with a sub-pixel
registration accuracy). The alignment is refined in two steps, with (i) a rough estimation
of the affine correction and (ii) a perspective correction for the refinement and accuracy
through the detection and matching of key points. The result shows that GoodFeatureToTrack
(GFTT) algorithm is the best key-point detector considering the
λ570
nm band as spectral
reference for the registration. After the registration, all spectral images are cropped to
1200
×
800 px and concatenated to channel-wise denoted
λ
where each dimension denoted
λdrefers to each of the six spectral bands.
2.3.2. Images Normalization
Spectral bands inherently have a high noise associated with the CCD sensor, which is
a potential problem during normalization [
35
]. To overcome this effect, 1% of the minimum
and maximum signal is suppressed by calculating the quantiles, the signal is clipped on the
given range and each band is rescaled in the interval
[
0, 1
]
using min-max normalization to
obtain ρd:
ρd=0λdmin(λd)
max(λd)min(λd)1 (1)
The method also reduces the lighting variation. According to [
36
] a little variation
is observed in the spectral correction factors between clear and cloudy days. Thus,
the correction has a limited impact on the scaling factor and should be managed by
this equation. However, the displacement factor could not be estimated, thus the output
images are not calibrated in reflectance.
2.3.3. Enriching Information
In order to enrich the pool of information, some spectral band transformations are
added, which allow to take into account spatial gradients and spectral mixing [
6
] in the
image. The choice is oriented towards seven important information in different respects:
The standard deviation between spectral band, noted
ρstd
can help to detect the
spectral mixture. For example between two different surface like ground and leave which
have opposite spectral radiance the spectral mixing make a pixel with linear combination,
thus the standard deviation tend to zero [
33
]. Three Gaussian derivatives on different
orientations are computed
Gxx
,
Gxy
and
Gyy
over the standard deviation
ρstd
which give
an important spatial information about the gradients breaks corresponding to the outer
limits of surfaces. These Gaussian derivatives are computed with a fixed
Sigma =
1.
The Laplacian computed over the standard deviation
ρstd
, the minimum and maximum
eigenvalues of the Hessian matrix (obtained by Gaussian derivation
Gxx
,
Gxy
and
Gyy
),
also called ridge are included. These transformations sould improve the detection of fine
elements [37] such as monocotyledons for vegetation images.
Remote Sens. 2021,13, 2261 5 of 21
All these transformations are concatenated to the channel-wise normalized spectral
band input and build the final input image. In total seven transformations are added to the
six spectral images for a final image of 13 channels, which will probably help the convergence.
2.4. Training and Validation Datasets
The input dataset is composed by spectral images
I
of size 1200
×
800
×
13 (or 6
if the “Enriching information” part is disabled) and a manual ground truth
p
of size
1200
×
800
×
1 where
p∈ {
0, 1
}
. The desired output
ˆ
p
is a probability vegetation map of
size 1200
×
800
×
1 where
ˆ
p[
0, 1
]
. This input dataset is randomly split into two sub-sets
respectively training (80%) and validation (20%). All random seed is fixed at the start-up to
keep the same training/validation dataset across all trained models which help to compare
them. Keeping the same random seed also results in the same starting point between
different new runs, making results reproducible on the same hardware.
3. Methodology
3.1. Existing Spectral Indices
From the indices database, 89 vegetation indices have been identified (Table 2) as
compatible with the wavelengths used in this study (as near as possible), they will be
tested and compared to the designed DeepIndices. Five forms of simple equations have
been extracted from this database (a wide variety of indices are derived from these forms,
generally a combination of 2 or 3 bands):
band reflectance =ρi(2)
two bands difference =ρiρj(3)
two bands ratio =ρi÷ρj(4)
normalized difference two bands = (ρiρj)÷(ρi+ρj)(5)
normalized difference three bands = (2ρiρjρk)÷(2ρi+ρj+ρk)(6)
By analyzing these five equations we can synthesize them into two generic equations
(Linear combination and Linear ratio) which take into account all spectral bands. Three
other models can generalize any function: the polynomial fitting, the continuous function
approximations by Taylor development, and the piecewise continuous function
approximations trough morphological operators. These forms are interesting to optimize
because they can approximate any function. This optimization will lead to automatically
defining new indices (DeepIndices). The following subsections present these different models.
3.2. Deepindices: Baseline Models
3.2.1. Linear Combination
To synthesize Equations (2) and
(3)
, a simple linear equation such as
y=N
d=0αdρd
can be defined. This equation can be generalized to the 2D domain using a 2D convolution
allowing consider the neighboring pixels. For a pixel at the position
[i
,
j]
the convolution is
defined by:
y[i,j] =
D
d=0
N
h=0
N
w=0
ρd[iN/2 +h,jN/2 +w]H[h,w,d](7)
where
H
defines neighborhood weights (corresponding to
αi
).
D
is the number of dimensions
(6 spectral bands + 5 transformations) and
N
is the kernel size. The linear combination
is given by
N=
1,
D=
12. The kernel weights are initialized by a truncated normal
distribution centered on zero [
38
], weights are updated during the training of the CNN
trough back-propagation and unnecessary bands should be set to zero. The interesting
part is that increasing the kernel size
N
allows to take into account the neighborhood of
a pixel and should estimate more accurately the spectral mixing [
33
]. Figure 3shows the
corresponding network.
Remote Sens. 2021,13, 2261 6 of 21
Figure 3. Linear combination model.
3.2.2. Linear Ratio
To generalize Equations (4)–(6), a simple model based on the division of two linear
combination is set. In the same way, this form is generalizable to the 2D domain and then
corresponds to two 2D convolutions, one for the numerator, the other for the denominator.
When the denominator is zero, the result is set to zero as well, to leverage the “not a
number” output. The Figure 4shows the corresponding network.
Figure 4. Linear ratio model.
3.2.3. Polynomial
According to the Stone-Weierstrass theorem any continuous function defined on a
segment can be uniformly approximated by a polynomial function. Thus all forms of
color indices can be approximated by a polynomial
y=N
d=0αdρdδd
of degree
N
. Setting
the degree is a difficult task which may imply under-fitting or over-fitting. In addition
un-stability can be caused by near-zero
δd
. But since the segment is restricted to the domain
[
0, 1
]
the Bernstein polynomials are a common demonstration and the equation can be
wrote as a weighted sum of Bernstein basis polynomials
BN,i= (
1
ρ)iρNi
which are
more stable during the training. Moreover Bernstein Neural Network can solve partially
differentiable equations [
39
]. For implementation reasons, two different layers are defined
in the network (visible in the Figure 5). One for the Bernstein expansion limited to
B11,11
which takes the input image and produces different Bernstein basis polynomial, then each
Bernstein basis is concatenated to the channel-wise and the linear combination is defined
by a 2D convolution.
Figure 5. Polynomial model with Bernstein expansions between B4,1 and B4,4.
3.2.4. Universal Function Approximation
The Gaussian color space model proposed by [
40
] shows that the spatio-spectral
energy distribution of the incident light
E
is the weighted integration of the spectrum
ρd
denoted
E(ρd)
. Where
E
can be described as a Taylor series and the energy function is
convolved by different derivatives of a Gaussian kernel or structured receptive fields [
41
].
Remote Sens. 2021,13, 2261 7 of 21
This important point shows that Taylor expansions can decompose any function
f(x)
,
especially for color decomposition and remapping, into :
f(x) = f(0) + f0(x)x+1
2! f00 (x)x2+1
3! f000 (x)x3+o(x3)(8)
Here, the signature of the incident energies distribution of a remote sensing index
associated to a surface can be reconstructed. An approach to learn this form of development
is proposed by [
42
] which is commonly called DenseNet and then corresponds to the sum
of the concatenation of the signal and these spatio-spectral derivatives
x[x,f1(x),f2(x,f1(x)), . . .](9)
Various convolutions allow to learn receptive fields and derivatives in spectral domain
when the kernel size
k
is 1, and in spatio-spectral domain when
k
is higher. Batch-Normalization
are used to reduces the covariate shift across convolution output by re-scaling it and speed up
the convergence. Finally the Sigmoid activation function is used and defined by
Sigmoid(x) = 1
1+ex(10)
Sigmoid function allows to learn more complex structures and non-linearity of the
reconstructed function. The number of derivative and receptive field are configurable with
two parameters. The
depth
which corresponds to the number of layers in the network.
And the
width
which refers to the number of outputs for each convolution. By default,
the
depth
is fixed to 3 and the
width
is fixed to 5. The Figure 6shows the corresponding
universal function approximator network.
Figure 6. Universal function approximation model (depth = 3, width = 5).
3.2.5. Dense Morphological Function Approximation
As for the Taylor series, an approximation of any piecewise continuous function can
be established by morphological operators such as dilatation and erosion [
43
], respectively
denoted
ρs
and
ρs
where
s
are the corresponding erosion or dilatation coefficients.
Several erosion and dilation are defined for each spectral band
i
, then the expanded layer
is defined as the channel concatenation of
z+
i
and in the same way for the erosion layer via
z
i. Both are defined by
z+
i=ρsi=max
k(ρksk,i, 0)(11)
z
i=ρsi=max
k(sk,iρk, 0)(12)
To obtain the output
I=N
i=0z+
iw+
i+N
i=0z
iw
i
of which the
w+
i
and the
w
i
are the
linear combination coefficients obtained by a 2D convolution. We chose to set the number
of dilation and erosion neurons at 6. The Figure 7shows the corresponding network.
Remote Sens. 2021,13, 2261 8 of 21
Figure 7. Dense-morphological model.
3.3. Enhancing Baseline Models
3.3.1. Input Band Filter (IBF)
To remove parts of the signal that may be dispensable, the addition of a low-pass,
high-pass and band-pass filter upstream of the network are studied. A good example
is provided by vegetation indices, only the high values in the green and near infra-red,
and the low values in the red and blue characterize the vegetation.
This is the principle of the NDVI index. Due to the internal structure, the leaves
reflect a lot of light in the near infrared, which is in sharp contrast to most non-vegetable
surfaces. When the plant is dehydrated or stressed, the spongy layer collapse and the
leaves reflect less light in the near-infrared, reaching red values in the visible range [
44
].
Thus, the mathematical combination of these two signals can help to differentiate plants
from non-plant objects and healthy plants from diseased plants. However, this index is
then less interesting when detecting only vegetation and is strongly influenced by shade or heat.
We will therefore add a filter in the previous equations to remove undesirable spectral
energies of each
ρd
by using two thresholds a and b, which will also be learned. If it turns
out that the whole signal is interesting, these two parameters will not change and their
values will be a=0 and b=1. To apply the low-pass filter the equation
z=max(ρa
, 0
)÷
(
1
a)
is used and thus allows to suppress low values. For the high-pass filter the equation
w=max(bρ
, 0
)÷b
is applied to suppress high values. The band-pass filter it’s the
product of low and high-pass filters
y=zx
. The output layer is the concatenation in
the channel-wise of the input images, the low-pass, the high-pass and the band-pass filter
which produce 4
×
13
=
52 channels. Finally to reduce the output data for the rest of the
network, a bottleneck is inserted using a convolution layer, and generate a new image with
6 channels. This image is used by the rest of the network defined previously in Section 3.2.
The Figure 8shows the corresponding module inserted upstream of the network.
Figure 8. Input Band Filter inserted at the beginning of the model.
3.3.2. Spatial Pyramid Refinement Block (SPRB)
To take into account different scales in the image, the addition of a “Spatial Pyramid
Refinement Block” at the downstream part of the network is studied. [
45
] showed that
fusing the low to high-level features improved the segmentation task. It consists in the sum
of different 2D convolutions whose core sizes have been set to 3, 5, 7 and 9. The results of
all convolutions are concatenated and the final image output is given by a 2D convolution.
The Figure 9shows the corresponding module inserted downstream of the network.
Remote Sens. 2021,13, 2261 9 of 21
Figure 9. Spatial refinement block inserted at the end of a model.
3.4. Last Activation Function
To obtain an index and facilitate convergence, we will only be interested in the values
between 0 and 1 at the output of the last layer with the help of an activation function of
type clipped ReLU defined by
ClippedReLU(x) =
1 if x>1
xif 0 <x<1
0 if x<0
(13)
where x is a pixel of the output image. Each negative or null pixel will then be the unwanted
class, greater or equal to 1 will be the searched class. The indecision border is the values
between 0 and 1 which will be optimized. And then correspond to the probability that
the pixel is the searched surface
P(Y=
1
)
or not
P(Y=
0
)
. This is valid for the output
prediction denoted ˆ
p[0, 1]and the ground truth denoted p∈ {0, 1}.
3.5. Loss Function
A wide variety of loss functions have been developed during the emergence of
deep-learning (MSE, MAE, Hinge, Tversky, etc). A cross-entropy loss function is usually
used when optimizing binary classification [
46
]. This loss function is not optimized for
the shape. Recently, with deep neural network and for semantic segmentation [
47
] has
proposed a solution to optimize an approximation of the mean intersection over union
(mIoU) and defined by
mIoU_Loss =1pˆ
p
p+ˆ
ppˆ
p(14)
The performance of this loss function seems more efficient than previous methods [
48
50
].
We will then use it as a loss function.
3.6. Performance Evaluation
Commonly, accuracy and Pearson correlation are used to quantify the performance of
remote sensing indices [
13
,
14
]. However this type of metrics does not take into account
either the class ratio nor the shape of the segmentation. Correlation is also highly sensitive
to non-linear relationship, noise, subgroups and outliers [
51
,
52
] making incorrect evaluation.
According to [
53
,
54
], the dice score and the mean intersection over union (mIoU) are more
adapted to evaluate the segmentation mask. Defined by:
Dice =2pˆ
p
p+ˆ
p(15)
mIoU =pˆ
p
p+ˆ
ppˆ
p(16)
Remote Sens. 2021,13, 2261 10 of 21
We will then used these two metrics for the performance evaluation. Prior to quantization,
a threshold of 0.5 is applied to the output of the network to transform the probability into a
segmentation mask. When
ˆ
p
is lower than 0.5, it is considered as the background, otherwise it
is considered as the object mask we are looking for. Other metrics are not considered because
they are not always appropriate in case of segmentation or use in unbalanced data.
3.7. Comparison with Standard Indices
In order to make a fair comparison it is necessary to optimize each standard index.
A minimal neural network is used to learn a linear regression. The network is thus
composed of the spectral index, followed by a normalization
x= (xmin)/(min
max)
, then a 2D convolution with a kernel size of
k=
1 is used for the linear regression.
To perform the classification in the same way as our method, a ClippedReLU activation
function is used. This tiny network is presented in the next Figure 10. Obviously the same
metrics and loss function are used.
Figure 10. Optimized model for standard indices
3.8. Training Setup
The training is done through Keras module within Tensorflow 2.2.0 framework. All
computation is done on an NVidia GTX 1080 which have 8111MiB of memory, this limits
the number of simultaneous layers on the memory and so the size of the model. Each
model is compiled with Adam optimizer. This optimization algorithm is primarily used
with lookahead mechanism proposed by [
55
]. It iteratively updates two sets of weights:
the search directions for the
fast weights
are chosen by inner optimizer, while the
slow
weights
are updated every
k
steps based on the direction of the
fast weights
and the
two sets of weights are synchronized. This method improves the learning stability and
lowers the covariance of its inner optimizer. The initial learning rate is fixed to 2
3
. Batch
size is fixed to 1 due to memory limitation. And the learning rate is decreased using
ReduceLROnPlateau with
f actor =
0.2,
patience =
5,
min_lr =
2
e6
. The training is done
through 300 iterations. Finally an EarlyStopping callback is used to stop the training when
there is no improvement in the training loss after 50 consecutive epochs.
4. Results and Discussion
4.1. Fixed Models
All standard vegetation models have been optimized using the same training and
validation datasets. Each of them has been optimized using a min-max normalization
followed by a single 1
×
1 2D convolution layer and a last clipped ReLU activation function
is used like the generic models implemented. The top nine standard indices are presented
in Table 2. Their respective equations are available in Table A1 in Appendix A.
Table 2. Synthesized standard indices performances: the nine best models are presented.
Standard Index Used ρmIoU Dice
Modified Triangular Vegetation Index 1 3 73.71 83.23
Modified Chlorophyll Absorption In Reflectance
Index 1 3 73.68 83.22
Enhanced Vegetation Index 2 2 67.94 79.20
Soil Adjusted Vegetation Index 2 67.28 78.65
Soil And Atmospherically Resistant VI 3 2 65.86 77.61
Enhanced Vegetation Index 3 2 65.05 77.07
Global Environment Monitoring Index 2 65.04 77.01
Adjusted Transformed Soil Adjusted VI 3 64.96 77.00
NDVI 2 63.98 75.97
Remote Sens. 2021,13, 2261 11 of 21
It is interesting to note that most of them are very similar to NDVI indices in their
form. This shows that according to all previous studies, these forms based on a ratio of
linear combination are the most stable against light variation. For example the following
NDVI based indices are tested and show very different performances, highlighting the
importance of weight optimization:
NDVI = (ρ5ρ2)÷(ρ5+ρ2)(17)
Enhanced Vegetation Index =2.5 (ρ5ρ2)÷(ρ5+6ρ27.5 ρ0+1)(18)
Enhanced Vegetation Index 2 =2.4 (ρ5ρ2)÷(ρ5+ρ2+1)(19)
Enhanced Vegetation Index 3 =2.5 (ρ5ρ2)÷(ρ5+2.4 ρ2+1)(20)
Soil Adjusted Vegetation Index = (ρ5ρ2)÷(ρ5+ρ2+1)2 (21)
Soil And Atmospherically Resistant VI 3 =1.5 (ρ5ρ2)÷(ρ5+ρ2+0.5)(22)
The Modified Triangular Vegetation Index 1 is given by
vi =
1.2
(
1.2
(ρ5ρ1)
2.5
(ρ2ρ1))
which shows that a simple linear combination can be as much efficient as
NDVI like indices by taking one additional spectral band (
ρ2=green
) and more adapted
coefficients. However, the other 80 spectral indices do not seem to be stable against of light
variation and saturation. It is thus not relevant to present them.
4.2. Deepindices
Finally, each baseline model such as linear,linear ratio,polynomial,universal function
approximation and dense morphological function approximation are evaluated with 4 different
modalities of each kernel size
N=
1,
N=
3,
N=
5 and
N=
7. In addition input band
filter (ibf) and spatial pyramid refinement block (sprb) are put respectively at the upstream and
downstream of the network. Figure 11 shows that network synthesis. To deal with lighting
variation and saturation a BatchNormalization is put in the upstream of the network in all
cases. The ibf and sprb modules are optional and can be disabled.
Figure 11. Network synthesis with ibf,evalated index equation, and sprb.
When the input band filter (ibf) is enabled, the incoming tensor size of 1200
×
800
×
13
is transformed to a tensor of size 1200
×
800
×
6 and passed to the generic equation. When
it is not, the generic equations get the raw input tensor of size 1200
×
800
×
13. In all cases
the baseline model output a tensor of shape 1200
×
800
×
1. The spatial pyramid refinement
block transforms the output tensor of the baseline model to a new tensor of the same size.
All models are evaluated with two metrics, respectively the dice and mIoU score.
For each kernel size, the results are presented in Tables 36. All models are also evaluated
with and without ibf and sprb for each kernel size.
Table 3. Scores of DeepIndices with/without ibf and sprb for a kernel size of 1.
mIoU dice
Model Baseline ibf sprb ibf + sprb Baseline ibf sprb ibf + sprb
linear 78.58 79.63 78.88 78.12 87.56 88.34 87.57 86.93
linear-ratio 79.01 78.86 77.73 79.67 87.85 87.87 86.55 88.28
polynomial 70.08 80.03 74.47 79.32 80.53 88.61 84.07 88.03
universal-function 78.39 76.59 79.04 80.15 87.27 85.36 87.63 88.53
dense-morphological 76.15 78.86 75.96 80.00 85.26 87.80 85.15 88.54
diff to baseline 2.35 0.78 3.01 1.90 0.50 2.37
Best models, higher than 80% of mIoU are highlighted in bold and the last row of tables corresponds to the difference to the baseline model (without
ibf
and sprb).
Remote Sens. 2021,13, 2261 12 of 21
Table 4. Scores of DeepIndices with/without ibf and sprb for a kernel size of 3.
mIoU dice
Model Baseline ibf sprb ibf + sprb Baseline ibf sprb ibf + sprb
linear 78.89 78.21 78.53 79.76 87.66 87.16 87.35 88.36
linear-ratio 76.63 78.21 74.90 78.17 85.49 87.37 83.89 86.92
polynomial 72.83 79.31 73.20 79.13 83.06 88.13 82.78 87.82
universal-function 76.67 79.63 77.81 81.08 85.57 88.28 86.67 89.22
dense-morphological 76.54 79.39 75.65 80.29 85.43 88.17 84.40 88.66
diff to baseline 2.64 0.29 3.37 2.38 0.42 2.75
Best models, higher than 80% of mIoU are highlighted in bold and the last row of tables corresponds to the difference to the baseline model (without
ibf
and sprb).
Table 5. Scores of DeepIndices with/without ibf and sprb for a kernel size of 5
mIoU dice
Model Baseline ibf sprb ibf + sprb Baseline ibf sprb ibf + sprb
linear 77.80 78.83 78.92 79.92 86.91 87.67 87.61 88.24
linear-ratio 75.72 77.94 77.36 80.08 84.87 87.26 86.33 88.43
polynomial 73.11 79.92 73.69 80.67 83.29 88.58 83.31 88.83
universal-function 77.60 80.63 80.31 80.63 86.38 89.02 88.53 88.71
dense-morphological 74.89 79.74 76.04 81.92 83.84 88.42 85.09 89.80
diff to baseline 3.59 1.44 4.82 3.13 1.12 3.74
Best models, higher than 80% of mIoU are highlighted in bold and the last row of tables corresponds to the difference to the baseline model (without
ibf
and sprb).
Table 6. Scores of DeepIndices with/without ibf and sprb for a kernel size of 7.
mIoU dice
Model Baseline ibf sprb ibf + sprb Baseline ibf sprb ibf + sprb
linear 79.08 80.29 79.25 81.49 87.75 88.57 87.80 89.42
linear-ratio 78.43 80.58 77.85 81.35 87.04 88.78 86.68 89.45
polynomial 72.49 80.79 74.14 81.21 82.92 88.99 83.77 89.27
universal-function 78.49 80.20 80.21 80.36 87.38 88.72 88.35 88.70
dense-morphological 75.70 80.35 76.34 82.19 84.48 88.70 85.61 89.94
diff to baseline 3.60 0.72 4.48 2.84 0.53 3.44
Best models, higher than 80% of mIoU are highlighted in bold and the last row of tables corresponds to the difference to the baseline model (without
ibf
and sprb).
For all baseline models, the results (in term of mIoU) show that increasing the kernel
size also increases performances. The gain performance between best models in kernel size
1 and 7 are approximately 2% and then correspond to the influence of spectral mixing. So
searching for spectral mixing 3 pixels farther (kernel size 7) still increases performance. It
could also be possible that function approximation allows to spatially reconstruct some
missing information.
For all kernel sizes, the
ibf
module enhance the mIoU score up to 3.6%. So the
ibf
greatly prune the unneeded part of the input signal which increases the separability and
the performances of all models. The
sprb
module allows to smooth the output by taking
into account neighborhood indices, but their performance are not always better or generally
negligible when it is used alone with the baseline model.
The baseline
polynomial
model is probably over-fitted, because it’s hard to find the
good polynomial order. But enabling the
ibf
fixes this issue. However further study should
be done to setup the order of Bernstein expansion.
The
dense morphological
with a kernel size of 5 and 7 using both
ibf
and
sprb
modules is the best model in term of dice (
90%) and mIoU score (
82%). Followed
by
universal function approximator
with a kernel size of 1 or 3 with both
ibf
and
sprb
Remote Sens. 2021,13, 2261 13 of 21
modules (dice up to 89% and mIoU up to 81%). Further studies on the width of the universal
function approximator could probably increase performance. According to [
43
] it seems
normal that the potential of
dense morphological
is higher although the hyper-parameters
optimization of universal function approximator could increase their performance.
4.3. Initial Image Processing
To show the importance of the initial image processing, each model has been trained
without the various input transformations, such as
ρstd
, Gxx, Gxy, Gyy filters, Laplacian
filter, minimum and maximum Eigen values. Table 7shows the score of DeepIndices
considering only kernel size of 1 in different model.
Table 7. Scores of DeepIndices in different modalities for a kernel size of 1 without initial image processing
mIoU dice
Model Baseline ibf sprb ibf + sprb Baseline ibf sprb ibf + sprb
linear 72.34 74.29 72.94 76.97 83.15 84.66 83.03 86.50
linear-ratio 73.72 70.51 73.30 71.55 84.10 82.36 83.19 81.57
polynomial 74.33 74.14 77.88 76.42 85.07 84.49 87.19 85.94
universal-function 74.24 74.42 75.46 76.25 84.36 84.49 85.16 85.86
dense-morphological 72.04 73.72 71.03 74.69 82.27 84.00 81.33 84.72
diff to baseline 0.08 0.79 1.84 0.21 0.19 1.13
Best models, higher than 80% of mIoU are highlighted in bold and the last row of tables corresponds to the difference to the baseline model (without
ibf
and sprb).
The results shows that none of optimized models outperforms the previous performance
with the initial image processing (best mIoU at 80.15%). The maximum benefit is approximately
6% for mIoU score depending on the model and module, especially when using combination
of
ibf
,
sprb
and small kernel size. Meaning that signal processing is much more important
than spectral mixing and texture.
4.4. Discussion
Further improvements can be set on hyper-parameters of the previously defined
equations, such as the degree of the polynomial (set to 11), the CNN depth and width for
Taylor series (set to 3) and the number of operations in morphological network (set to 10).
In particular the learning of 2D convolution kernel of Taylor series may be replaced by a
structured receptive field [
41
]. In addition it would be interesting to transpose our study
with new data for other surfaces such as shadows, waters, clouds or snows.
The training dataset is randomly split with a fixed seed, which is used for every
learned models. As previously noted, this is important to ensure reproducible results but
could also favor specific models. Further work to evaluate the impact of varying training
datasets could be conducted.
4.4.1. Model Convergence
Another way to estimate the robustness of a model against its initialization is to
compare the model’s convergence speed. Models with faster convergence should be less
sensitive to the training dataset. As an example, the convergence speed of few different
models is shown in Figure 12. The baseline model convergence is the same, as well as
sprb
module. However the speed of convergence also increases with the size of the kernel
but does not alter subsequent observations. For greater readability only models with
ibf
are presented.
Remote Sens. 2021,13, 2261 14 of 21
Figure 12. first 80 epochs of loss of generic models with ibf in kernel size of 1.
An important difference in the speed of convergence between models is observed. An
analysis of this figure allows the aggregation of model types and speed:
Slow converging models: polynomials models converge slowly as well as the majority
of linear or linear-ratio models.
Fast converging models: universal-functions and dense-morphological are the fastest
to converge (less than 30 iterations)
A subset of slow and fast converging models could be evaluated in term of sensitivity
against initialization. It shows that the dense morphological followed by universal function
approximator convergence faster than the other. Regardless of the used module nor kernel size.
4.4.2. Limits of Deepindices
Shadows can be a relatively hard problem to solve in image processing, the proposed
models are able to correctly separate vegetation from soil even with shadowy images,
as shown in Figure 13. In addition, the Figure A1 in the Appendix Ashows the impacts of
various acquisition factors, such as shadow, noise, specular or thin vegetation features.
Figure 13. Correct vegetation/soil discrimination despite shadows.
Some problem occurs when there are abrupt transitions between shadowed and light
areas of an image as shown in Figure 14.
Remote Sens. 2021,13, 2261 15 of 21
Figure 14. Vegetation/soil discrimination issue with abrupt transition between shadow and light.
It appears that the discrimination error appears where the shadow is cast by a solid
object, resulting in edge diffraction that creates small fringes on the soil and vegetation.
A lack of such images in the training dataset could explain the model failure. Data
augmentation could be used to obtain a training model containing such images, from cloud
shadows to solid objects shadows. Further work is needed to estimate the benefit of such a
data augmentation on the developed models.
The smallest parts of the vegetation (less than 1 pixel, such as small monocotyledon
leaves or plant stems) cannot be detected because of a strong spectral mixture. This
limitation is due to the acquisition conditions (optics, CCD resolution and elevation) and
should be considered as is. As vegetation with a width over 1 pixel is correctly segmented
by our approach, the acquisition parameters should be chosen so that the smallest parts of
vegetation that are required by an application are larger than 1 pixel in the resulting image.
A few spots of specular light can also be observed on images, particularly on leaves.
These spots are often unclassified (or classified as soil). This modifies the shape of the
leaves by creating holes inside them. This problem can be seen on Figure 15. Leaves with
holes are visible on the left and the middle of the top bean row. It would be interesting to
train the network to detect and assign them to a dedicated class.
Figure 15. Vegetation/soil discrimination issue caused by specular lights on leaves.
Next the location of the detected spots could be studied to re-assign them to two
classes: specular-soil and specular-vegetation. To perform this step, a semantic segmentation
could be set up to identify the surrounding objects of the holes specifically. It would be
based on the UNet model, which performs a multi-scale approach by calculating, treating
and re-convolving images of lower resolutions.
More generally, the quality of the segmentation between soil and vegetation strongly
influences the discrimination between crop and weed, which remains a major application
following this segmentation task. Three categories of troubles have been identified: the
plants size, the ambient light variations (shades, specular light spots), and the morphological
complexity of the studied objects.
Remote Sens. 2021,13, 2261 16 of 21
The size of the plants mainly impacts their visibility on the acquired images. It is not
obviously related to the ability of the algorithm to classify them. However, it leads to the
absence of essential elements such as monocotyledon weeds at an early vegetation stage.
A solution is proposed by setting the acquisition conditions to let the smallest vegetation
part be over 1 pixel.
Conversely, the variations of ambient light should be treated by the classification
algorithm. As previously mentioned, shadow management needs an improvement of the
learning base, and specular light spots could be treated by a multi-scale approach. Their
influence on the discrimination step should be major. Indeed, they influence the shape of
the objects classified as plants, which is a useful criterion to discriminate crops from weeds.
The morphological complexity of the plants can be illustrated by the presence of stems.
In our case, bean stems are similar to weed leaves. This problem should be treated by the
discrimination step. The creation of a stem class (in addition to the weed and crop classes)
will be studied in particular.
5. Conclusions
In this work, different standard vegetation indices have been evaluated as well as
different methods to estimate new DeepIndices through different types of equations that
can reconstruct the others. Among the 89 standard vegetation indices tested, the MTVI
(Modified Triangular Vegetation Index 1) gives the best vegetation segmentation. Standard
indices remain sub-optimal even if they are downstream optimized with a linear regression
because they are usually used on calibrated reflectance data. The results allow us to
conclude that any simple linear combination is just more efficient (
+
4.87% mIoU) than
any standard indices by taking into account all spectral bands and few transformations.
The results also suggest that un-calibrated data can be used in proximal sensing applications
for both standard indices and DeepIndices with good performances.
We therefore agree that it is important to optimize both the arithmetic structure of
the equation and the coefficients of the spectral bands, that is why our automatically
generated indices are much more accurate. The best model is much more efficient by
+
8.48% compared to the best standard indices and by
+
18.21% compared to NDVI. Also the
two modules
ibf
,
sprb
and the initial image transformation show a significant improvement.
The developed DeepIndices allow to take into account the lighting variation within the
equation. It makes possible to abstract from a difficult problem which is the data calibration.
Thus, partially shaded images are correctly evaluated, which is not possible with standard
indices since they use sprectum measurement that change with shades. However, it
would be interesting to evaluate the performance of standard indices and DeepIndices on
calibrated reflectance data.
These results suggest that deep learning algorithms are a useful tool to discover
the spectral band combinations that identify the vegetation in multi spectral camera.
Another conclusion from this research is about the genericity of the methodology developed.
This study presents a first experiment employed in field images with the objective of
finding deep vegetation indices and demonstrates their effectiveness compared to standard
vegetation indices. This paper ’s contribution improves the classical methods of vegetation
index development and allows the generation of more precise indices (i.e., DeepIndices).
The same kind of conclusion may arise from this methodology applied on remote sensing
indices to discriminates other surfaces (roads, water, snow, shadows, etc).
Remote Sens. 2021,13, 2261 17 of 21
Author Contributions:
Conceptualization, J.-A.V.; data curation, J.-A.V.; formal analysis, J.-A.V.;
funding acquisition, G.J. and J.-N.P.; investigation, J.-A.V.; methodology, J.-A.V.; project administration,
Paoli.J-N. and G.J.; resources, Paoli.J-N. and G.J.; software, J.-A.V.; supervision, J.-N.P. and G.J.;
validation, J.-N.P.; C.G. and G.J.; visualization, J.-A.V.; writing—original draft preparation, J.-A.V.;
writing—review and editing, J.-A.V.; J.-N.P.; C.G. and G.J. All authors have read and agreed to the
published version of the manuscript.
Funding:
This project is funded by ANR Challenge RoSE and the Horizon 2020 project IWMPRAISE.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable: this studies did not involve humans or animals.
Data Availability Statement:
Data in this study is publicly available at https://data.inrae.fr/
dataset.xhtml?persistentId=doi:10.15454/DSQC8N, using Creative Common CC0 1.0 Public Domain
Dedication licence
Acknowledgments:
We would like to thank Masson Jean-Benoit for the realization of the metal
gantry which allowed us to position the camera at different heights, it was used in particular for the
calibration of the camera and the band registration. We also thank Djemai Mehdi for the spelling
correction of the English. And we thank Aubry Clément and Cozic Thibault of the company SITIA
for their help in interfacing the camera with the used robot “Trecktor”.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or
in the decision to publish the results.
Appendix A
Table A1. Top optimized fixed vegetation model equations. b=ρ0,g=ρ1,r=ρ2,e=ρ3,u=ρ4,n=ρ5.
Model Equation
Modified Triangular Vegetation Index 1 1.2 (1.2 (ng)2.5 (rg))
Modified Chlorophyll Absorption In Reflectance Index 1 1.2 (2.5 (nr)1.3 (ng))
Enhanced Vegetation Index 2 2.4 (nr)/(n+r+1)
Soil Adjusted Vegetation Index 2.0 (nr)/(n+r+1.0)
Soil And Atmospherically Resistant VI 3 1.5 (nr)/(n+r+0.5)
Enhanced Vegetation Index 3 2.5 (nr)/(n+2.4 r+1)
Global Environment Monitoring Index 2(n2r2)+1.5n+0.5r
n+r+0.5 (1n/4)r0.125
1+r
Adjusted Transformed Soil Adjusted VI anar0.03
an+ra0.03+0.08(1+a2)a=1.22
NDVI (nr)/(n+r)
Remote Sens. 2021,13, 2261 18 of 21
Noise Shadow Thin Specular
RGB
Ground Truth
dense 7 ibf+sprb
linear 1 baseline
MTVI1
NDVI
Figure A1.
Visual comparison between some relevant models. NDVI (63.98 mIoU), MTVI1 (73.71 mIoU), linear 1 baseline
(78.58 mIoU), dense 7 ibf-sprb (82.19 mIoU). Blue indicates sure soil, red indicates sure vegetation, and the other colors
indicate uncertainty.
Remote Sens. 2021,13, 2261 19 of 21
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... Second, a new loss function is also introduced to limit under and over-segmentation. Third output is refined using a specific vegetation index based on previous work (Vayssade et al., 2021) and a watershed algorithm. This method is applied to dense leaf segmentation, on images containing mixes of plant species and acquired in natural light. ...
... Thus, the architecture (Fig. 5) is composed of three upstream modules (IIT, IBF, UFA) that improves the input data and eliminates unnecessary information. This step composed of 3 upstream modules, was proposed in a previous work to construct a vegetation index (Vayssade et al. (2021)). It is used to identify relevant spectral features on the input data to exploit the inter-channel relationships. ...
... The UFA is based on Taylor expansion theorem, an approach to learn this form of development in deep-learning is called DenseNet and then corresponds to the sum of the concatenation of the signal with these spatiospectral derivatives. This was successfully used for vegetation segmentation Vayssade et al. (2021). Three parameters, such as the depth (number of convolutions), the width (number of filters denoted W) and k (kernel size) configure the network and were empirically fixed to depth = 3, width = 16 and k = 1. ...
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... The next step of this research would be to assess the portable mid-IR and vis-NIR soil spectra obtained under field conditions and further eliminate the environmental and systematic factors [44,63] affecting both soil spectra in a similar way to [46,47,72]. In addition, a variety of calibration models (ranging from simple to complex), e.g., partial least squares regression (PLSR) [1,73,74], principal component regression (PCR) [7,75], artificial neural networks (ANN) [76], convolution neural networks (CNN) [77][78][79], support vector machines (SVM) [80], and deep learning [81][82][83][84], to name a few, could be studied. ...
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... This approach is appropriate when analyzing other soil properties and PSS systems geared to investigate the spatial heterogeneity of specific soil attributes. However, the goal and the purpose of this paper was to use the simplest method that focused on the reproducibility of soil reflectance spectral elements and to assess direct relationships with soil attributes rather than model development [38][39][40][41]68]. The latter is the subject of follow-up work. ...
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