Sparse projections onto the RKHS simplex

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Result on sequence of noisy observations

This kind of data represents a great challenge in remote sensing analysis since all observations are in general corrupted by noise. Many methods rely on signal processing filters (spatially or temporally) [Rabiner 1975][Burrus 1998] to re-move such noise. However this requires additional parameters and our method has the great advantage to get rid of this problem. In the inference process, the noises on the plant growth evolution as well as on observations ; are in-deed estimated along the procedure. This is to our opinion a great advantage compared to other denoising processes.
In order to make a quantitative evaluation, various levels of Gaussian zero-mean noise, with Signal-to-noise ratio(SNR) ranging from 2 to 10, were tested . The table 2.1 represents the root-mean-square error (RMSE) between re-constructed LAI (with particle filter and smoother and other reconstruction techniques introduced in section 2.4.1) and groundtruth data. It is interesting to observe in this table that the introduction of GreenLab plant growth model in the reconstruction process systematically reduces the error residual and out-performs all other methods. Particles smoother almost always gives a better performance than particle filter, which is quite rational as when estimating the LAI at any time point, particle filter only exploits information of up-to-date observations, while particle smoother would take into account all observations in time series. In order to have a visual inspection of the reconstructions, some illustrations are visible in Fig. 2.4 and confirm that our outputs are of reliable quality. Even if other curves are sometimes consistent with ground truth, par-ticle filter and/or smoother represent the best compromise since strong errors are likely to appear with other techniques (as the polynomial one very sensible to little variations). Let us now turn to sequences with sparse observations.

Sequence with sparse observations

As mentioned above, because of satellite orbit, weather condition, etc. it appears that observations are often very sparse in space or time. Therefore a robust remote sensing technique should be able to handle this common problem. In our dataset, we have tried to simulate this situation by removing a large number of data. In practice, we have randomly taken only six observations to reconstruct the whole time series. These observations are the blue dotted points in Fig. 2.5. As one can observe, they are more or less issued from a uniform sample along the time period. Quantitative results are depicted in table 2.2. and also prove, in this situation, the performance and the benefit of relying on GreenLab growth model compared with other techniques. Reconstructed curves are also visible in figure 2.5 and demonstrate the great ability of our technique to recover proper series. Let us now present the last experiment of this section: missing data.

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Sequence of missing observations

To model the fact that sometimes there are no observations during a long period of time (unavailability of the satellite due to technical problems), we have re-moved long period of consecutive data. Some illustrations are visible in Fig. 2.6 where one can observe that no observations during the growing period are avail-able, which is a very delicate problem. In spite of these difficulties, the proposed framework is able to provide very good reconstruction of LAI time series, as shown in Tab. 2.3 and in Fig. 2.6. Obviously here, only the dynamical model prior is able to estimate consistent values during the critical period. This is to our opinion a very interesting behavior of our estimators.

Table of contents :

1 Introduction générale 
1.1 Contexte scientifique : rôle des variables biophysiques
1.2 Reconstruction de variables biophysiques
1.3 Démélangeage de séries temporelles
1.4 Organisation du document
I Première partie : reconstruction de séries temporelles de LAI à partir d’images MODIS 
2 Biophysical variables reconstruction 
2.1 Introduction
2.2 Introduction to the Greenlab model
2.2.1 Biomass generation
2.2.2 Biomass repartition
2.2.3 Parameter Redundancy
2.2.3.1 Focus main organs
2.2.3.2 Plant structure simplification
2.2.3.3 Spoilage and organ aging
2.3 Data assimilation
2.3.1 State transition model
2.3.2 Observation model
2.3.3 Model inference and parameter estimation techniques
2.3.4 Application to the recovery of LAI using GreenLab
2.4 Experimental Results
2.4.1 Experimental settings
2.4.2 Data with ground truth
2.4.2.1 Result on sequence of noisy observations
2.4.2.2 Sequence with sparse observations
2.4.2.3 Sequence of missing observations
2.4.2.4 Overview of these experiments
2.4.3 Application on large scale study site
2.5 Conclusion
II Deuxième partie : démélangeage non-linéaire : aspects théoriques 
3 NMF, application to unmixing 
3.1 Introduction
3.1.1 Context
3.1.2 Notations
3.1.3 Related work
3.1.4 Proposal
3.1.5 Contributions
3.1.6 Outline
3.2 Geometry aware CSS procedure
3.2.1 CSS as a manifold sampling procedure
3.2.2 Original Simplex Volume Maximization
3.2.3 Exact simplex volume maximization
3.3 Sparse projections onto the RKHS simplex
3.3.1 Projection on the RKHS simplex
3.3.2 Convergence of the projector
3.4 Complete SAGA procedure
3.4.1 Algorithm
3.4.2 Computational complexity
3.5 Experiments and results
3.5.1 Experiments on toy datasets
3.5.2 Comparison on subsampling strategies
3.5.3 Computational complexity
3.5.4 Application to classification on real datasets
3.6 Conclusion
III Troisième partie : démélangeage de séries temporelles
4 Time series unmixing 
4.1 Introduction
4.2 Related work
4.2.1 Endmember extraction introduction
4.2.2 Abundance estimation
4.3 Methodology for time series unmixing
4.3.1 Kernel time series
4.3.2 Endmember extraction
4.3.3 Abundance estimation
4.3.4 Overall procedure
4.4 Experimental validation
4.4.1 Experimental settings
4.4.2 Dataset properties
4.4.3 Endmember extraction validation
4.4.4 Pixel unmixing result evaluation
4.5 Conclusion
5 Conclusion & Perspectives 
5.1 Résumé des contributions
5.2 Perspectives
IV Annexe 
6 GreenLab plant growth model 
7 Publications 
Bibliography 

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