**3.1 Case study**

222 Remote Sensing of Biomass – Principles and Applications

where *g* is a (linear or not) observation operator and *vt* a Gaussian noise of varianceσ*<sup>v</sup>*

include:

where δ*xt*

*i*

sequential particle filtering.

samples *xt| t-1i*

result as in Eqn. (7):

1. Choose a set of *N* samples (or particles) *x1i*

=f(*xt-1|t-1i*

distribution *p*(*x*1) and compute *p*(*y1|x1*);

2. Prediction step: for all *t*∈[2,*T*] and the set of samples *x t-1| t-1 <sup>i</sup>*

)+ *nt i*

1:

is the dirac function in *xt*

of particles and their associated weights w(*xt| t-1i*

different particles. This is the so-called forward-backward smoother.

1: | 1

*px y px y*

(, ) (| ) ( )

*t t t t*

Eqn. (7) can be approximated from the set of particles using Eqn. (8):

Particle filtering, also known as Sequential Monte Carlo (Doucet et al., 2001), is an attractive way to recover the system state *xt* for all *t*∈[1,*T*]. It can be shown that a sequential estimation of *xt* can be obtained through the following system, starting from (a) an available sequential observation of sequence *y1:t = y1*,…, *yt* ; (b) an initial distribution of the system's state *p*(*x*1) ; (c) transition model *p*(*xt|xt-1*) and observation model *p*(*yt|xt*) respectively, related to the stochastic processes *f* and *g* presented above. See (Doucet et al., 2001) for details. Steps

for each particle and get Eqn. (6):

3. Correction step: from the distribution *p*(*xt| y1:t-1*) and the new observation *yt*, we have a

1:

1: 1

*py x px y wt px y p y x p x y dx*

( | )( | ) ( )( | ) ( | )( | )

1 (| ) ( ) *N*

1

Therefore, once the system at time *t*-1 has been obtained, the process consists in generating a prediction of all particles at time *t* thanks to the transition *p*(*xt|xt-1*) and the available observations *y1:t-1 = y1*,…, *yt-1*. These predictions are then corrected by taking into account the new observation *yt* in a second step. The final estimated distribution is obtained from the set

When the whole sequence is available, i.e. all observations *y1:T = y1*,…, *yT* are available for all time *t*, a smoothing version of the previous technique can be applied by taking into account future observations. From the first estimation issued from the previous process, the idea consists in performing a backward exploration in order to correct the weights of the

The idea consists in reweighting the particles recursively backward in time, starting from the end time *T* to the initial one. It can be shown that rewriting the distribution *p*(*xt|y1:T*) in

(| ) ( )

*t t t t <sup>N</sup>*

*py x w t*

 

*k*

*t t tt t i p x y wx x*

*tt t t tt t t*

1: 1

1:

*p y*

*i*


*tt t t t*

*t*

1: 1 1 1 1: 1 1 ( | ) ( | )( | ) *t t tt t t t <sup>p</sup> x y p x x p x y dx* (6)


(8)

(9)

). This is the main principle of the

*i i*

1

*k t t*

(| )

*py x*

*i*

1

and the weight function is as in Eqn. (9):

2.

, *i*=[1, …, *N*] randomly taken from the initial

, generate the predicted

(7)

We chose maize plant for a case study on application of the filtering method presented above. According to previous study on maize plant (Guo et al., 2006), we set model parameters in GreenScilab, an open source software for implementing GreenLab model. The LAI can is part of model output, as shown in Fig. 3 (a). By arranging the simulated 3D maize plant according to the given density, a virtual maize field can be simulated. Fig. 3 (b) shows such an image at a plant age. It is supposed that the aim is to recover this LAI sequence from noisy data of LAI obtained from remote sensing images at several different stages.

Fig. 3. Synthetic result from GreenLab model. (a) simulated LAI dynamics of virtual maize from GreenLab model; (b) top view of a virtual maize field;

Reconstructing LAI Series by Filtering Technique and a Dynamic Plant Model 225

Fig. 5. Recovering LAI series from strongly noisy data, with a synthetic LAI sequence (blue

From Fig. 5, one can immediately observe that during the period corresponding to the main evolution of LAI where no data are acquired, a simple interpolation technique without any prior model would result in incorrect values and dramatically underestimate the LAI. At the opposite, using the forward-backward smoother, the recovered series of Fig. 5 (green line) is very closed to the original one, despite this difficult testing situation. The RMS error is slightly higher than the previous situation (when observations are less noisy) but is still very competitive. These results demonstrate the great benefit of recovering the data under the

RMSE Original data RMSE Reconstructed data

0.09341 for 256 recovered time steps

0.10055 for 256 recovered time steps

line), a strongly noisy version (black points), and a recovered time series under the

for 25 observed values

0.9951 for 18 observed values

These experiments on noisy and strongly noisy synthetic data demonstrate the possibility of the framework presented in this chapter (schematized in Fig. 1) to recover time consistent series of LAI with fine time resolution from a small set of noisy image observations. The

Table 1. Root Mean Square Errors for the two noisy sequences tested

practical issue with real images is under development.

GreenLab model (green line)

constraint of a dynamic model.

Strongly noisy series

(Fig. 5)

Noisy series (Fig. 4) 0.6145

#### **3.2 Recovering consistent series of LAI**

We tested our smoothing approach on the sequence of synthetic values of LAI generated from the GreenLab model, see Fig. 4 (blue line). From this sequence, we have randomly extracted some points on which we have added noise (black points). We assume that these points correspond to measurements obtained on the corresponding field from remote sensing images. They represent an incomplete time series of noisy values of LAI. From these inconsistent series, we have generated a smooth version from the forward-backward smoother presented in the previous section, using GreenLab as a dynamic model. This is depicted in Fig. 4 (green line). From this synthetic example, it is obvious to observe that the new series is consistent with the expected ground truth. This is confirmed when observing the quantitative values of the Root Mean Square Error between the ground truth, the noisy and reconstructed data shown in Table 1.

In order to highlight the benefit of the use of a dynamical model, we have blurred in a stronger way the series. We have assumed that during a long time period in which the variation of LAI is maximal, no observations are available (due, for instance, to the maintenance of the sensor, a too large cloud covering during the winter, etc.). In addition, to take into account the errors related to the acquisition process itself, we have also blurred the remaining data. This results in a strongly noisy sequence of LAI, as shown with the black points in Fig. 5.

Fig. 4. Recovering LAI series from noisy data, with a synthetic LAI sequence (blue line), a noisy version (black points), and a recovered time series under the GreenLab model (green line)

We tested our smoothing approach on the sequence of synthetic values of LAI generated from the GreenLab model, see Fig. 4 (blue line). From this sequence, we have randomly extracted some points on which we have added noise (black points). We assume that these points correspond to measurements obtained on the corresponding field from remote sensing images. They represent an incomplete time series of noisy values of LAI. From these inconsistent series, we have generated a smooth version from the forward-backward smoother presented in the previous section, using GreenLab as a dynamic model. This is depicted in Fig. 4 (green line). From this synthetic example, it is obvious to observe that the new series is consistent with the expected ground truth. This is confirmed when observing the quantitative values of the Root Mean Square Error between the ground truth, the noisy

In order to highlight the benefit of the use of a dynamical model, we have blurred in a stronger way the series. We have assumed that during a long time period in which the variation of LAI is maximal, no observations are available (due, for instance, to the maintenance of the sensor, a too large cloud covering during the winter, etc.). In addition, to take into account the errors related to the acquisition process itself, we have also blurred the remaining data. This results in a strongly noisy sequence of LAI, as shown with the black

Fig. 4. Recovering LAI series from noisy data, with a synthetic LAI sequence (blue line), a noisy version (black points), and a recovered time series under the GreenLab model (green

**3.2 Recovering consistent series of LAI** 

and reconstructed data shown in Table 1.

points in Fig. 5.

line)

Fig. 5. Recovering LAI series from strongly noisy data, with a synthetic LAI sequence (blue line), a strongly noisy version (black points), and a recovered time series under the GreenLab model (green line)

From Fig. 5, one can immediately observe that during the period corresponding to the main evolution of LAI where no data are acquired, a simple interpolation technique without any prior model would result in incorrect values and dramatically underestimate the LAI. At the opposite, using the forward-backward smoother, the recovered series of Fig. 5 (green line) is very closed to the original one, despite this difficult testing situation. The RMS error is slightly higher than the previous situation (when observations are less noisy) but is still very competitive. These results demonstrate the great benefit of recovering the data under the constraint of a dynamic model.


Table 1. Root Mean Square Errors for the two noisy sequences tested

These experiments on noisy and strongly noisy synthetic data demonstrate the possibility of the framework presented in this chapter (schematized in Fig. 1) to recover time consistent series of LAI with fine time resolution from a small set of noisy image observations. The practical issue with real images is under development.

Reconstructing LAI Series by Filtering Technique and a Dynamic Plant Model 227

structure and optical properties of individual organs, it provides a possibility of validating the reconstructed canopy dynamics by comparing the virtual canopy with the obtained high resolution source images. The development of remote sensing technique and advance in

Baret F., Hagolle O., Geiger B., Bicheron P., Miras B., Huc M., Berthelot B., Nino F., Weiss

Congalton R. G., A review of assessing the accuracy of classifications of remotely sensed

Dong Q.X., Louarn G., Wang Y.M., Barczi J.F., de Reffye P. 2008. Does the Structure-

Doucet A., Freitas N., and Gordon N. Sequential Monte Carlo Methods in Practice.

Feng L., Mailhol J. C., Rey H., Griffon S., Auclair D. and de Reffye P. Combining a process

Models (FSPM 10), Sept. 12-17, 2010. University of California, Davis, USA. Guo Y., Ma Y.T., Zhan Z.G., Li B.G., Dingkuhn M., Luquet D., de Reffye P. 2006. Parameter

Jacquemoud, S. and Baret F., 1990. PROSPECT: A model of leaf optical properties spectra.

Kang M.Z., Heuvelink E., and de Reffye P. 2006. Building virtual chrysanthemum based on sink-source relationships: Preliminary results. Acta Horticulturae 718: 129–136. Kang M.Z., Evers J.-B., Vos J., de Reffye P. 2008. The derivation of sink functions of wheat organs using the GreenLab model. Annals of Botany 101: 1099-1108. Kang M.Z., Yang L.L., Zhang B.G., de Reffye P. 2011. Correlation between dynamic tomato

Kitagawa G. Monte carlo filter and smoother for non-gaussian nonlinear state space models.Journal of Computational and Graphical Statistics, 5(1):1–25, 1996 Lecerf R., Hubert-Moy L., Baret F., Abdel-Latif B., Corpetti T., Nicolas H.. Estimating

Lefebvre A., Corpetti T., Hubert-Moy L. Segmentation of very high spatial resolution

Ma Y, Wen M, Guo Y, Li B, Cournède P-H, de Reffye P. 2008. Parameter optimization and

IGARSS '08, Volume 2, Pages 954-957, Boston, USA, July 2008.

Lions J.-L. 1971. Optimal control of systems governed by PDEs. Springer-Verlag.

different population densities. Annals of Botany 101: 1185–1194.

spacing? A case study on tomato. Annals of Botany 101: 1195-1206.

M., Samain O., Roujean J.L., and Leroy M.. "LAI, FAPAR, and FCover CYCLOPES global products derived from Vegetation. Part 1: principles of the algorithm",

Function Model GREENLAB deal with crop phenotypic plasticity induced by plant

based model with a functional structural plant model for production partitioning and visualization. 6th International workshop on Functioanl-Structural Plant

optimization and field validation of the Functional-Structural Model GREENLAB

fruit set and source sink ratio: a common relationship for different plant densities

biophysical variables at 250m with reconstructed EOS/MODIS time series to monitor fragment landscapes. In IEEE Int. Geoscience and Remote Sensing Symp,

panchromatic images based on wavelets and evidence theory. In SPIE International Conference on Image and Signal Processing for Remote Sensing, 78300E Proc. SPIE

field validation of the functional structural model GREENLAB for maize at

plant modelling are increasing the interdisciplinary research of these two areas.

data, Remote Sensing of Environment, vol. 37, p. 35-46, 1991.

Remote Sensing of Environment, 110:305-316, 2007.

SpringerVerlag, first edition, 2001.

for maize. Annals of Botany 97: 217-230.

Remote Sensing of Environment, 34: 75-91.

and seasons? Annals of Botany 107: 805-815.

7830 (ed.), Toulouse, France, September 2010

**5. References** 
