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Jean-Luc Paul

A tool for monitoring agricultural policies : modelling farming systems in the Caribbean to build an agricultural supply model

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It is a great honour for me to speak on this occasion and I want to express my gratitude to the Abdus Salam International Centre for Theoretical Physics as well as the Selection Committee of the Dirac Medal. It is also a great pleasure to be here with Luciano Maiani and discuss the consequences of our common work with Sheldon Glashow on charmed particles [1]. I will argue in this talk that the same kind of reasoning, which led us to predict the opening of a new chapter in hadron physics, may shed some light on the existence of new physics at the as yet unexplored energy scales of LHC.

The argument is based on the observation that precision measurements at a given energy scale allow us to make predictions concerning the next energy scale. It is remarkable that the origin of this observation can be traced back to 1927, the two fundamental papers on the interaction of atoms with the electromagnetic field written by Dirac, which are among the cornerstones of quantum field theory. In the second of these papers [2] Dirac computes the scattering of light quanta by an atom γ(k 1 ) + A i → γ(k 2 ) + A f , where A i and A f are the initial and final atomic states, respectively. He obtains the perturbation theory result:

H f i = Σ j H 1 f j H 1 ji E i -E j (1)
where H are the amplitudes. For the significance of the rhs, Dirac notes: "...The scattered radiation thus appears as a result of the two processes i → j and j → f , one of which must be an absorption, the other an emission, in neither of which the total proper energy is even approximately conserved." This is the crux of the matter: In the calculation of a transition amplitude we find contributions from states whose energy may put them beyond our reach. The size of their contribution decreases with their energy, see (1), so, the highest the precision of our measurements, the further away we can see.

Let me illustrate the argument with two examples, one with a nonrenormalisable theory and one with a renormalisable one. A quantum field theory, whether renormalisable or not, should be viewed as an effective theory valid up to a given scale Λ. It makes no sense to assume a theory for all energies, because we know already that at very high energies entirely new physical phenomena appear (example: quantum gravity at the Planck scale). The first example is the Fermi four-fermion theory with a coupling constant G F ∼ 10 -5 GeV -2 . It is a non-renormalisable theory and, at the nth order of perturbation, the Λ dependence of a given quantity A is given by:

A (n) = C (n) 0 (G F Λ 2 ) n + C (n) 1 G F (G F Λ 2 ) n-1 + C (n) 2 G 2 F (G F Λ 2 ) n-2 + .... ( 2
)
where the C i 's are functions of the masses and external momenta, but their dependence on Λ is, at most, logarithmic. Perturbation theory breaks down obviously when A (n) ∼ A (n+1) and this happens when G F Λ 2 ∼ 1. This gives a scale of Λ ∼ 300GeV as an upper bound for the validity of the Fermi theory. Indeed, we know today that at 100GeV the W and Z bosons change the structure of the theory. But, in fact, we can do much better than that [3]. Weak interactions violate some of the conservation laws of strong interactions, such as parity and strangeness. The absence of such violations in precision measurements will tell us that G F Λ 2 ∼ ǫ with ǫ being the experimental precision. The resulting limit depends on the value of the C coefficient for the quantity under consideration. In this particular case it turned out that, under the assumption that the chiral symmetry of strong interactions is broken only by terms transforming like the quark mass terms, the coefficient C

(n) 0 for parity and/or strangeness violating amplitudes vanishes and no new limit is obtained [4]. However, the second order coefficient C

(n) 1 contributes to flavour changing neutral current transitions and the smallness of the K 1 -K 2 mass difference, or the K 0 L → µ + + µ -decay amplitude, give a limit of Λ ∼ 3GeV before new physics should appear. The new physics in this case turned out to be the charmed particles [1]. We see in this example that the scale Λ turned out to be rather low and this is due to the non-renormalisable nature of the effective theory which implies a power-law behaviour of the radiative corrections on Λ.

The second example in which new physics has been discovered through its effects in radiative corrections is the well-known "discovery" of the t quark at LEP, before its actual production at Fermilab. The effective theory is now the Standard Model, which is renormalisable. In this case the dependence of the radiative corrections on the scale Λ is, generically, logarithmic and the sensitivity of the low energy effective theory on the high scale is weak (there is an important exception to this rule for the Standard Model which we shall see presently). In spite of that, the discovery was made possible because of the special property of the Yukawa coupling constants in the Standard Model to be proportional to the fermion mass. Therefore, the effects of the top quark in the radiative corrections are quadratic in m t . The LEP precision measurements were able to extract a very accurate prediction for the top mass.

I claim that we are in a similar situation with the precision measurements of the Standard Model. Our confidence in this model is amply justified on the basis of its ability to accurately describe the bulk of our present day data and, especially, of its enormous success in predicting new phenomena. All these spectacular successes are in fact successes of renormalised perturbation theory. Indeed what we have learnt was how to apply the methods which had been proven so powerful in quantum electrodynamics, to other elementary particle interactions. The remarkable quality of modern High Energy Physics experiments, mostly at LEP, but also elsewhere, has provided us with a large amount of data of unprecedented accuracy. All can be fit using the Standard Model with the Higgs mass as the only free parameter. Let me show some examples: Figure 1 indicates the overall quality of such a fit. There are a couple of measurements which lay between 2 and 3 standard deviations away from the theoretical predictions, but it is too early to say whether this is accidental, a manifestation of new physics, or the result of incorrectly combining incompatible experiments.

Another impressive fit concerns the strong interaction effective coupling constant as a function of the momentum scale (Figure 2). This fit already shows the importance of taking into account the radiative corrections, since, in the tree approximation, α s is obviously a constant. Similarly, Figure 3 shows the importance of the weak radiative corrections in the framework of the Standard Model. Because of the special Yukawa couplings, the dependence of these corrections on the fermion masses is quadratic, while it is only logarithmic in the Higgs mass. The ǫ parameters are designed to disentangle the two. The ones we use in Figure 3 are defined by:

ǫ 1 = 3G F m 2 t 8 √ 2π 2 - 3G F m 2 W 4 √ 2π 2 tan 2 θ W ln m H m Z + ... (3)
ǫ 3 = G F m 2 W 12 √ 2π 2 ln m H m Z - G F m 2 W 6 √ 2π 2 ln m t m Z + ... (4)
where the dots stand for subleading corrections. As you can see, the ǫ s vanish in the absence of weak interaction radiative corrections, in other words, ǫ 1 = ǫ 3 = 0 are the values we get in the tree approximation of the Standard Model but after having included the purely QED and QCD radiative corrections. We see clearly in Figure 3 that this point is excluded by the data. The latest values for these parameters are ǫ 1 = 5.4 ± 1.0 and ǫ 3 = 5.34 ± 0.94 [START_REF] Altarelli | Private Communication[END_REF].

Using all combined data we can extract the predicted values for the Standard Model Higgs mass which are given in Figure 4. The data clearly σ The main conclusion I want to draw from this comparison can be stated as follows:

0123 ∆α had (m Z ) ∆α (5) 0.02761 ± 0.00036 0.02768 m Z [GeV] m Z [GeV] 91.1875 ± 0.0021 91.1873 Γ Z [GeV] Γ Z [GeV] 2.
Looking at all the data, from low energies to the Tevatron, we have learnt that perturbation theory is remarkably successful, outside the specific regions where strong interactions are important.

Let me explain this point better: At any given model with a coupling constant g we expect to have a weak coupling region g ≪ 1, in which weak coupling expansions, such as perturbation theory, are reliable, a strong coupling region with g ≫ 1, in which strong coupling expansions may be relevant, and a more or less large gray region g ∼ 1, in which no expansion is applicable. The remarkable conclusion is that this gray area appears to be extremely narrow. And this is achieved by an enlargement of the area in which weak coupling expansion applies. The perturbation expansion is reliable, not only for very small couplings, such as α em ∼ 1/137, but also for moderate QCD couplings α s ∼ 1/3, as shown in Figure 2. This is extremely important because without this property no calculation would have been possible. If we had to wait until α s drops to values as low as α em we could not use any available accelerator. Uncalculable QCD backgrounds would have washed out any signal. And this applies, not only to the Tevatron and LHC, but also to LEP. We can illustrate this observation using a qualitative argument first introduced by F. Dyson. He noted that in a field theory like QED, the contribution of the 2n-th order perturbation term to a physical amplitude A (2n) grows with n roughly as1

A (2n) ∼ α n (2n -1)!! ( 5
)
where α is (the square of) the coupling constant. Again, perturbation theory will break down when A (2n) ∼ A (2n+2) which gives

2n + 1 ∼ α -1 (6)
This leaves a comfortable margin for QED but leaves totally unexplained the successes of QCD at moderate energies.

A global view of the weak and strong coupling regions is given in Figure 5 which shows the R-ratio, i.e. the e + + e -total cross section to hadrons normalised to that of e + + e -→ µ + + µ -as a function of the centre-of-mass energy. The lowest order perturbation value for this ratio is a constant, equal to ΣQ 2 i , the sum of the squares of the quark charges accessible at this energy. We see clearly in this Figure the areas of applicability of perturbation theory: At very low energies, below 1 GeV, we are in the strong coupling regime characterised by resonance production. The strong interaction effective coupling constant becomes of order one (we can extrapolate from Figure 2), and perturbation breaks down. However, as soon as we go slightly above 1 GeV, R settles to a constant value and it remains such except for very narrow regions when new thresholds open. In these regions the cross section is again dominated by resonances and perturbation breaks down. But these areas are extremely well localised and threshold effects do not spread outside these small regions.

In this talk I want to exploit this observational fact and argue that the available precision tests of the Standard Model allow us to claim with confidence that new physics will be unravelled at the LHC, although we have no unique answer on the nature of this new physics. The argument assumes the validity of perturbation theory and it will fail if the latter fails. But, as we just saw, perturbation theory breaks down only when strong interactions become important. But new strong interactions do imply new physics.

The key is again the Higgs boson. As we explained above, the data favour a low mass Higgs. However, the opposite cannot be excluded, first ) which implies v ≈246 GeV. Therefore, any constraints will come from the allowed values of λ. A first set of such constraints is given by the classical requirement:

1 > λ > 0 ⇒ m H < 400 -500GeV (7)
The lower limit for λ comes from the classical stability of the theory. If λ is negative the Higgs potential is unbounded from below and there is no ground state. The upper limit comes from the requirement of keeping the theory in the weak coupling regime. If λ ≥ 1 the Higgs sector of the theory becomes strongly interacting and we expect to see plenty of resonances and bound states rather than a single elementary particle.

Going to higher orders is straightforward, using the renormalisation group equations. The running of the effective mass is determined by that of λ. Keeping only the dominant terms and assuming t = log(v 2 /µ 2 ) is small (µ ∼ v), we find

dλ dt = 3 4π 2 [λ 2 + 3λh 2 t -9h 4 t + ...] (8)
where h t is the coupling of the Higgs boson to the top quark. The dots stand for less important terms, such as the other Yukawa couplings to the fermions and the couplings with the gauge bosons. This equation is correct as long as all couplings remain smaller than one, so that perturbation theory is valid, and no new physics beyond the standard model becomes important. Now we can repeat the argument on the upper and lower bounds for λ but this time taking into account the full scale dependence λ(µ). We thus obtain for the Higgs mass an upper bound given by the requirement of weak coupling regime (λ(µ) < 1) all the way up to the scale µ, and a lower bound by the requirement of vacuum stability (λ(µ) > 0), again up to µ. Obviously, the bounds will be more stringent the larger the assumed value of µ. Figure 6 gives the allowed region for the Higgs mass as a function of the scale for scales up to the Planck mass. We see that for small µ ∼ 1TeV, the limits are, essentially, those of the tree approximation equation (7), while for µ ∼ M P we obtain only a narrow window of allowed masses 130GeV< m H <200GeV, remarkably similar to the experimental results. This analysis gives the first conclusion: If perturbation theory remains valid, in other words, if we have no new strong interactions, there exists at least one, relatively light, Higgs boson:

Conclusion 1:
The absence of a light Higgs boson implies New Physics.

Here "heavy Higgs" is not clearly distinguished from "no-Higgs", because a very heavy Higgs, above 1 TeV, is not expected to appear as an elementary particle. As we explained above, this will be accompanied by new strong interactions. A particular version of this possibility is the "Technicolor" model, which assumes the existence of a new type of fermions with strong interactions at the multi-hundred-GeV scale. The role of the Higgs is played by a fermion-antifermion bound state. "New Physics" is precisely the discovery of a completely new sector of elementary particles. Other strongly interacting models can and have been constructed. The general conclusion is that a heavy Higgs always implies new forces whose effects are expected to be visible at the LHC. 3The possibility which seems to be favoured by the data is the presence of a "light" Higgs particle. In this case new strong interactions are not needed and, therefore, we can assume that perturbation theory remains valid. But then we are faced with a new problem. The Standard Model is a renormalisable theory and the dependence on the high energy scale is expected to be only logarithmic. This is almost true, but with one notable exception: The radiative corrections to the Higgs mass are quadratic in whichever scale Λ we are using. The technical reason is that m H is the only parameter of the Standard Model which requires, by power counting, a quadratically divergent counterterm. The gauge bosons require no mass counterterm at all because they are protected by gauge invariance and the fermions need only a logarithmic one. The physical reason is that, if we put a fermion mass to zero we increase the symmetry of the model because now we can perform chiral transformations on this fermion field. Therefore the massless theory will require no counterterm, so the one needed for the massive theory will be proportional to the fermion mass and not the cut-off.

In contrast, putting m H = 0 does not increase the symmetry of the model. 4As a result the effective mass of the Higgs boson will be given by

(m 2 H ) ef f = m 2 H + Cα ef f Λ 2 (9)
where C is a calculable numerical coefficient of order one and α ef f some effective coupling constant. In practice it is dominated by the large coupling to the top quark. The moral of the story is that the Higgs particle cannot remain light unless there is a precise mechanism to cancel this quadratic dependence on the high scale. This is a particular aspect of a general problem called "scale hierarchy". The only known mechanism which reconciles a light Higgs and a high value of the scale Λ with the validity of perturbation theory is supersymmetry. In this case the Higgs mass is protected against the quadratic corrections of eq. ( 9) because it behaves like the mass of the companion fermion which, as we just said, receives only logarithmic corrections. It is closest in spirit to the charm mechanism, in the sense that a heavy effective cut-off is made compatible with the low energy data by the presence of new particles. The alternative is to have a low value of Λ, i.e. new physics, at a low scale. The models with large compact extra dimensions enter into this category. This brings us to our second conclusion: Both conclusions are good news for LHC. But the time for speculations is coming to an end. The LHC is coming. Never before a new experimental facility had such a rich discovery potential and never before was it loaded with so high expectations.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 1234 Figure 1: Various physical quantities measured and computed.

Figure 5 :

 5 Figure 5: The ratio R of e + + e -total cross section to hadrons normalised to that of e + + e -→ µ + + µ -as a function of the centre-of-mass energy. because it depends on the subset of the data one is looking at 2 , and, second, because the analysis is done taking the minimal Standard Model. Given this result, let us see what, if any, are the theoretical constraints. The Standard Model Higgs mass is given, at the classical level, by m 2 H = 2λv 2 , with v the vacuum expectation value of the Higgs field. v is fixed by the value of the Fermi coupling constant G F / √ 2 = 1/(2v 2 ) which implies v ≈246 GeV. Therefore, any constraints will come from the allowed values of λ. A first set of such constraints is given by the classical requirement:

Figure 6 :

 6 Figure 6: Bounds on the Higgs mass.

Conclusion 2 :

 2 A light Higgs boson is unstable without new physics.

The estimation is only heuristic. It is based on a rough counting of the number of diagrams and assumes that they all contribute equally and have the same sign, neither of which is exact. The estimation can be improved but the result remains the same.

This prediction is, in fact, an average between a much lower value, around 50 GeV, given by the data from leptonic asymmetries, and a much higher one, of 400 GeV, obtained from the hadronic asymmetries. Although the difference sounds dramatic, the two are still mutually consistent at the level of 2-3 standard deviations.

We can build specific models in which the effects are well hidden and pushed above the LHC discovery potential, at least with the kind of accuracy one can hope to achieve in a hadron machine. In this case one would need very high precision measurements, probably with a multi-TeV e + -e -collider.

At the classical level, the Standard Model with a massless Higgs does acquire a new symmetry, namely scale invariance, but this symmetry is always broken for the quantum theory and offers no protection against the appearance of quadratic counterterms.


Sandrine Henry

Sandrine Pallaud

published or not. The documents may come


Mathieu Fauvel
email: mathieu.fauvel@gipsa-lab.inpg.fr

Student Member, IEEE Jón Atli

IEEE Benediktsson † Fellow

Jocelyn Chanussot
email: jocelyn.chanussot@gipsa-lab.inpg.fr

Senior Member, IEEE Johannes R Sveinsson
email: sveinsso@hi.is

Spectral and Spatial Classification of Hyperspectral Data Using SVMs and Morphological Profiles

Keywords: Data fusion, hyperspectral data, support vector machines, feature extraction, extended morphological profile, high spatial resolution

A method is proposed for the classification of urban hyperspectral data with high spatial resolution. The approach is an extension of previous approaches and uses both the spatial and spectral information for classification. One previous approach is based on using several principal components from the hyperspectral data and building several morphological profiles. These profiles can be used all together in one extended morphological profile. A shortcoming of that approach is that it was primarily designed for classification of urban structures and it does not fully utilize the spectral information in the data. Similarly, the commonly used pixel-wise classification of hyperspectral data is solely based on the spectral content and lacks information on the structure of the features in the image. The proposed method overcomes these problems and is based on the fusion of the morphological information and the original hyperspectral data, i.e., the two vectors of attributes are concatenated into one feature vector. After a reduction of the dimensionality the final classification is achieved using a Support Vector Machines classifier. The proposed approach is tested in experiments on ROSIS data from urban areas. Significant improvements are achieved in terms of accuracies when compared to results obtained for approaches based on the use of morphological profiles based on PCs only and conventional spectral classification. For instance, with one data set, the overall accuracy is increased from 79% to 83% without any feature reduction and to 87% with feature reduction. The proposed approach also shows excellent results with a limited training set.

I. INTRODUCTION

In classification of remote sensing data from urban areas, the identification of relatively small objects, e.g., houses and narrow streets is important. Therefore, high spatial resolution of the imagery is necessary for accurate classification. The most commonly available remote sensing data of high spatial resolution from urban areas are single-band panchromatic data. However, using only one high-resolution panchromatic data channel is usually not sufficient for accurate classification of structural information. To overcome that problem, Pesaresi and Benediktsson [START_REF] Pesaresi | A new approach for the morphological segmentation of high-resolution satellite imagery[END_REF] proposed the use of morphological transformations to build a Morphological Profile (MP). In [START_REF] Palmason | Classification of hyperspectral data from urban areas using morphological preprocessing and independent component analysis[END_REF] the method in [START_REF] Pesaresi | A new approach for the morphological segmentation of high-resolution satellite imagery[END_REF] was extended for hyperspectral data with high spatial resolution. The approach in [START_REF] Palmason | Classification of hyperspectral data from urban areas using morphological preprocessing and independent component analysis[END_REF] is based on using several Principal Components (PCs) from the hyperspectral data. From each of the PCs, a morphological profile is built. These profiles are used all together in one Extended Morphological Profile (EMP), which is then classified by a neural network. The method in [START_REF] Palmason | Classification of hyperspectral data from urban areas using morphological preprocessing and independent component analysis[END_REF] has been shown to perform well in terms of accuracies when compared to more conventional classification approaches. However, a shortcoming of the approach is that it is primarily designed for classification of urban structures and it does not fully utilize the spectral information in the multispectral or hyperspectral data.

However this type of data contains a lot of information about the spectral properties and the land cover of the data. A finer definition of the classes is possible and more classes can be considered. Based on the spectral signatures of the classes, many advanced pixel-based classifiers have been proposed including advanced statistical classifiers [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF] and distribution free approaches such as neural networks and support vector machines [START_REF] Vapnik | Statistical Learning Theory[END_REF]. The later one has shown remarkable abilities to deal with remote multispectral data, especially with hyperspectral data. However, if the spatial content of the image is not used the resulting thematic map sometimes looks noisy (salt and pepper classification noise). Approaches involving Markov Random Field (MRF) and Monte Carlo optimization have been proposed in [START_REF] Poggi | Supervised segmentation of remote sensing images based on a tree-structure MRF model[END_REF], [START_REF] Jackson | Adaptive bayesian contextual classification based on markov random field[END_REF]. These approaches use the contextual information. The main shortcoming of such algorithms is the computing time, which can be high even for small data sets. Regarding the high dimensionality of recently acquired data, both in the spectral and in the spatial domain, computationally light algorithm are of interest. In this sense, the MP has been proposed as an alternative way to use spatial information [START_REF] Pesaresi | A new approach for the morphological segmentation of high-resolution satellite imagery[END_REF], [START_REF] Benediktsson | Classification and feature extraction for remote sensing images from urban areas based on morphological transformations[END_REF]. Relatively to the MRF-based classifiers, the MP and its extension to a multiband image, the EMP, have the possibility to use geometrical contextual information (shape, size, etc) and perform well on many kinds of data (panchromatic, multispectral and hyperspectral data). However, as stated above, a shortcoming of this approach is it does not fully utilize the spectral information in the data, and consequently several approaches based on the MP/EMP have been proposed to fully exploit the spatial and the spectral information [START_REF] Dell'acqua | Exploiting spectral and spatial information in hyperspectral urban data with high resolution[END_REF]- [START_REF] Palmason | Fusion of morphological and spectral information for classification of hyperspectral urban remote sensing data[END_REF].

Each data set has its own properties, defining its ability to deal with different natures of classes. Table I sums up the properties of spectral and morphological/spatial data. The first main consideration is the complementary characteristics of the data. It has a consequence in the discrimination ability of such a feature, as will be seen in the experiments. The fusion of two types of information should clearly results in an increase of the classification in terms of global accuracy. The use of spectral information can be critical for classification of nonstructured information in urban areas, e.g., vegetation and soil classes while the use of spatial information can be useful for classification of structured objects, e.g., road and building classes.

The second consideration is the possible redundancy of each features set, see [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF] for the spectral features and [START_REF] Benediktsson | Classification of hyperspectral data from urban areas based on extended morphological profiles[END_REF] for the spatial features. Hence feature extraction (FE) algorithms could be of an interest.

To include both type of information an extension to the approach in [START_REF] Palmason | Classification of hyperspectral data from urban areas using morphological preprocessing and independent component analysis[END_REF] is proposed in this paper. The proposed method is based on the data fusion of the morphological information and the original data: First, an extended morphological profile is created based on the PCs from the hyperspectral data. Secondly, feature extraction is applied on the original hyperspectral data and the extended morphological profile. Finally, the extracted feature vectors from both the original data and the extended morphological profile are concatenated into one stacked vector and classified. The proposed approach is different from approaches in [START_REF] Camps-Valls | Composite kernels for hyperspectral image classification[END_REF]- [START_REF] Fauvel | A joint spatial and spectral SVM's classification of panchromatic images[END_REF], where the authors had extracted spatial information and used composite kernel to include both type of information. Here, feature extraction algorithms are used to select informative feature from the spectral and spatial domain.

For the multisource classification, Support Vector Machines (SVM) are used rather than a Neural Network, which was used in our previous experiment with MP/EMP. The superiority of SVM, implementing structural risk minimization, over the neural classifiers, implementing empirical risk minimization, has been discussed in [START_REF] Vapnik | Statistical Learning Theory[END_REF] (in Chapters 9.6 and 12) and in [START_REF] Melgani | Classification of hyperspectral remote sensing images with support vector machines[END_REF], [START_REF] Zheng | Performance analysis and comparison of neural networks and support vectors machines classifier[END_REF]. SVM aim to discriminate two classes by fitting an optimal separating hyperplane to the training data within a multidimensional feature space, by using only the closest training samples. Thus, the approach only considers samples close to the class boundary and works well with small training set, even when high dimensional data sets are classified. SVM have already been applied for multisource classification in [START_REF] Halldorsson | Support vector machines in multisource classification[END_REF] where several output coding methods were investigated.

In this paper, the proposed approach has been compared to statistical classification methods and SVM classification. Experiments were conducted on two different high resolution remote sensing data sets from urban areas. The effectiveness of the proposed methodology with a limited training set has been also assessed.

The paper is organized as follows. Section II reviews the use of morphological transformations for processing of hyperspectral imagery in urban areas. In Section III, the considered supervised feature extraction approaches are introduced. Support Vectors Machines (SVM) are discussed in Section IV. The applied data fusion schemes are discussed in Section V. Experimental results obtained on two ROSIS data sets from urban areas are presented in Section VI. Finally, conclusions are drawn in Section VII.

II. EXTENDED MORPHOLOGICAL PROFILE

Mathematical Morphology is a theory aiming to analyze spatial relationship between pixels. For a remote sensing application, several morphological operators are available for extracting geometrical information. An overview of operators can be found in [START_REF] Soille | Advances in mathematical morphology applied to geoscience and remote sensing[END_REF]. In the following sub-section, some basic notions of mathematical morphology are reviewed. Then, the concept of the Morphological Profile and its extension to multivalued data are detailed.

A. Mathematical Morphology

The two fundamental operators in mathematical morphology are erosion and dilation [START_REF] Soille | Morphological Image Analysis, Principles and Applications-2[END_REF]. These operators are applied to an image with a set of known shape, called a structuring element (SE). To erode an image consists of finding where the SE fits the objects in the image. The dilation, which is dual to the erosion, shows where the SE hits the objects.

Opening and closing are combinations of erosion and dilation. These operators remove from an original image structures of size less than the SE. But they also modify structures which are still present in the image after the opening/closing. Thus, they can introduce fake objects in the image. To avoid this problem, geodesic morphology and reconstruction should be used [START_REF] Soille | Morphological Image Analysis, Principles and Applications-2[END_REF]. Opening and closing by reconstructions are connected operators that satisfy the following assertion: If the structure of the image cannot contain the SE, then it is totally removed, else it is totally preserved. For a given SE, geodesic opening or geodesic closing allows to know the size or shape of some objects present in the image: The objects that are smaller than the SE are deleted while the other (that are bigger than the SE) are preserved. To determine the shape or size of all elements present in an image, it is necessary to use a range of different SE sizes. This concept is called Granulometry.

Granulometries are typically used for the analysis of the size distribution of the structures in the images. Classical granulometry by opening is built by successive opening operations with an SE of an increasing size. By doing so, the image is progressively simplified. Using connected operators, like opening by reconstruction, no shape noise is introduced.

Morphological Profiles (MPs) are defined using the granulometry. An MP is composed of the opening profile (OP) and the closing profile (CP). The OP at the pixel x of the image I is defined as an n-dimensional vector:

OP i (x) = γ (i) R (x), ∀i ∈ [0, n] (1)
where γ

(i)
R is the opening by reconstruction with an SE of a size i and n is the total number of openings. Also, the CP at the pixel x of image I is defined as an n-dimensional vector:

CP i (x) = φ (i) R (x), ∀i ∈ [0, n] (2)
where φ

(i)
R is the closing by reconstruction with an SE of a size i. Clearly we have CP 0 (x) = OP 0 (x) = I(x). By collating the OP and the CP , the MP of image I is defined as 2n + 1dimensional vector:

M P (x) = {CP n (x), . . . , I(x), . . . , OP n (x)} (3)
Example of MP is shown in Fig. 1. Thus, from a single image results a multiband image, whose dimension corresponds to the number of transformations and spatial information is now contained in the MP for each pixel. However, an MP is built with only one band. Therefore, the spectral information is lost. One approach to deal with this problem is to extract several images that contain some parts of the spectral information and then build the MP on each of the individual images. This approach, namely the Extended Morphological Profile (EMP), is discussed in the following.

B. Extended Morphological Profile

In order to apply this approach to hyperspectral data, characteristic images need to be extracted. In [START_REF] Benediktsson | Classification of hyperspectral data from urban areas based on extended morphological profiles[END_REF], it was suggested to use several principal components of the hyperspectral data for such a purpose. Hence, the MP is applied on the first PCs, corresponding to a certain amount of the cumulative variance and a stacked vector is built with the MP on each PC. This yields to the extended morphological profile (EMP). Following the previous notation, the EMP is an m(2n + 1)-dimensional vector:

M P ext (x) = {M P P C 1 (x), . . . , M P P C m } ( 4
)
where m is the number of retaining PCs. Example of EMP is shown in Fig. 2.

As with multispectral data, the MP/EMP may include some redundancy. Classical feature reduction algorithm can be applied, as detailed in the following section.

III. SUPERVISED FEATURE EXTRACTION

Feature extraction can be viewed as finding a set of vectors that represents an observation while reducing the dimensionality. In pattern recognition, it is desirable to extract features that are focused on discriminating between classes of interest. Although a reduction in dimensionality is desirable, the error increment due to the reduction in dimension has to be without sacrificing the discriminative power of classifiers. In linear feature extraction, the number of input dimensions corresponds to the number of selected eigenvectors [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF]. The transformed data are determined by

x = Φ T x ( 5
)
where Φ is the transformation matrix composed of the eigenvectors of the feature matrix, x is the data in the input space and x is the transformed data in the feature space. We have in general dim(x) ≥ dim(x). Several statistical extraction approaches have been proposed for remote sensing data [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF], including Decision Boundary Feature Extraction (DBFE) and Nonparametric Weighted Feature Extraction (NWFE).

A. Decision Boundary Feature Extraction

It was shown in [START_REF] Lee | Feature extraction based on decision boundaries[END_REF], that both discriminantly informative features and redundant features can be extracted from the decision boundary between two classes. The features are extracted from the decision boundary feature matrix (DBFM). The eigenvectors of the DBFM corresponding to non-zero eigenvalues are the necessary feature vectors to achieve the same classification accuracy as in the original space. The efficiency of the DBFE is related to the training set and can be computationally intensive.

B. Nonparametric Weighted Feature Extraction

To overcome the limitations of the DBFE, Kuo and Landgrebe [START_REF] Kuo | A robust classification procedure based on mixture classifiers and nonparametric weighted feature extraction[END_REF] proposed the nonparametric weighted feature extraction. NWFE is based on the Discriminant Analysis Feature Extraction by focusing on samples near the eventual decision boundary. The main ideas of the NWFE are 1) putting different weights on every sample to compute the local means and 2) defining nonparametric between-class and within-class scatter matrices [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF].

Many experiments have shown the effectiveness of these approaches for the classification of hyperspectral data [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF]. They are usually applied on the spectral data, but it was successfully applied to the EMP [START_REF] Benediktsson | Classification of hyperspectral data from urban areas based on extended morphological profiles[END_REF].

IV. CLASSIFICATION BY THE SUPPORT VECTOR MACHINE

So far, in our previous approach [START_REF] Palmason | Classification of hyperspectral data from urban areas using morphological preprocessing and independent component analysis[END_REF], [START_REF] Benediktsson | Classification and feature extraction for remote sensing images from urban areas based on morphological transformations[END_REF], [START_REF] Benediktsson | Classification of hyperspectral data from urban areas based on extended morphological profiles[END_REF], [START_REF] Chanussot | Classification of remote sensing images from urban areas using a fuzzy possibilistic model[END_REF] the classification was done with either a statistical classifier (Gaussian Maximum Likelihood), a neural network or a fuzzy classifier.

Here it is proposed to use the Support Vector Machines (SVM). Early work in classification of remotely sensed images by SVM showed excellent results [START_REF] Halldorsson | Support vector machines in multisource classification[END_REF], [START_REF] Gualtieri | Support vector machines for hyperspectral remote sensing classification[END_REF], [START_REF] Gualtieri | Support vector machines for classification of hyperspectral data[END_REF]. In [START_REF] Melgani | Classification of hyperspectral remote sensing images with support vector machines[END_REF], several SVMbased classifiers were compared to other classical classifiers such as a K-nearest neighbors classifier and a neural network classifier and the SVM using the kernel method outperformed the other classifiers in terms of accuracy. Multiclass SVM performances were also positively compared with a discriminant analysis classifier, a decision tree classifier and a feedforward neural network classifier with a limited training set [START_REF] Foody | A relative evaluation of multiclass image classification by support vector machines[END_REF]. SVM show good results in the situation of limited training set in [START_REF] Fauvel | Evaluation of kernels for multiclass classification of hyperspectral remote sensing data[END_REF]. Semisupervised SVM were also investigated for multi-spectral data classification [START_REF] Bruzzone | A novel transductive SVM for semisupervised classification of remote-sensing images[END_REF], [START_REF] Chi | Semisupervised classification of hyperspectral images by SVMs optimized in the primal[END_REF].

SVM are surely among the most used kernel learning algorithm. It performs robust non-linear classification of samples using the kernel trick. The idea is to find a separating hyperplane in some feature space induced by the kernel function while all the computations are done in the orignal space [START_REF] Vapnik | Statistical Learning Theory[END_REF]. A good introduction to SVM for pattern recognition can be found in [START_REF] Burges | A tutorial on support vector machines for pattern recognition[END_REF]. Given a training set S = {(x 1 , y 1 ), . . . , (x ℓ , y ℓ )} ∈ R n × {-1; 1}, the decision function is found by solving the convex optimization problem:

max α g(α) = ℓ i=1 α i - 1 2 ℓ i,j=1 α i α j y i y j k(x i , x j ) subject to 0 ≤ α i ≤ C and ℓ i=1 α i y i = 0 (6)
where α are the Lagrange coefficients, C a constant that is used to penalize the training errors, and k the kernel function. To be an acceptable kernel, k should be a positive semi-definite function [START_REF] Aronszajn | Theory of reproducing kernel[END_REF]. One classical effective kernel is the Gaussian kernel:

k σ (x i , x j ) = exp - x i -x j 2 2σ 2 (7)
where the norm is the Euclidean-norm and σ ∈ R + tunes the flexibility of the kernel. A short comparison of kernels for remotely sensed image classification can be found in [START_REF] Fauvel | Evaluation of kernels for multiclass classification of hyperspectral remote sensing data[END_REF].

When the optimal solution of ( 6) is found, i.e., the α i , the classification of a sample x is achieved by looking to which side of the hyperplane it belongs:

y = sgn ℓ i=1 α i y i k(x i , x) + b . (8)
To deal with multiclass classification problem, the pairwise approach was used in our experiments [START_REF] Hsu | A comparison of methods for multiclass support vector machines[END_REF]. More advanced multiclass approaches applied to remote sensing data can be found in [START_REF] Melgani | Classification of hyperspectral remote sensing images with support vector machines[END_REF]. For the particular case of one-class-classification, a dedicated methodology is proposed in [START_REF] Sanchez-Hernandez | One-class classification for mapping a specific land-cover class: SVDD classification of fenland[END_REF]. The SVM are mainly a non-parametric method, yet some parameters need to be tuned before the optimization. In the Gaussian kernel case, there are two parameters: C, the penalty term, and σ, the with of the exponential. It is usually done by a cross-validation step, where several values are tested. In our experiments, C was fixed to 200 and σ 2 ∈ {0.5, 1, 2, 4} was selected using a 5-fold cross validation. The SVM optimization problem was solved using the LIBSVM [START_REF] Chang | LIBSVM: a library for support vector machines[END_REF].

V. DATA FUSION

The proposed method is based on the data fusion of the morphological information and the original data. In a previous work [START_REF] Fauvel | Decision fusion for the classification of urban remote sensing images[END_REF], it was proposed to fuse the classification results of two SVM classifiers, each one working with either the spectral or the EMP data. It consisted in an appropriate adaptive fusion scheme based on the output's characteristics of the SVM. The results in terms of accuracy were increased but it needed two training of SVM, that could be time consuming.

Here it is proposed to use a multisource strategy to fuse spectral and spatial information. First, an EMP is created based on applying the PCA on the hyperspectral data. Secondly, feature extraction is applied on both the EMP and the original hyperspectral data. Finally, the extracted feature vectors are concatenated into one stacked vector and classified by the SVM.

In the morphological processing we usually retain PCs corresponding to 99% of the cumulative variance. This is done in order to reduce the redundancy in the data but keep most of the variation. The EMP is built using the m PCs that correspond to the 99% variance. Each MP is composed of n geodesic openings, n geodesic closing and the corresponding PC. The SE is a disk with initial radius of r pixels. The size increment is s. Hence, each MP has 2n + 1 features and the EMP has m(2n + 1) features. Noting x ϕ the features associated to the spectral bands and x ω the features associated to the EMP, the corresponding extracted features from the FE algorithm are:

x ϕ = Φ T ϕ x ϕ (9)
and

x ω = Φ T φ x ω . ( 10
)
The stack vector is finally x = [x ϕ , x ω ] T . Fig. 3 presents the data fusion scheme. Note that in this work, only morphological information is extracted, but it is possible to extract other types of spatial information with other processing and include them in the stacked vector.

VI. EXPERIMENTS

A. Data set

Airborne data from the ROSIS-03 (Reflective Optics System Imaging Spectrometer) optical sensor are used for the experiments. The flight over the city of Pavia, Italy, was operated by the Deutschen Zentrum fur Luft-und Raumfahrt (DLR, the German Aerospace Agency) in the framework of the HySens project, managed and sponsored by the European Union. According to specifications, the number of bands of the ROSIS-03 sensor is 115 with a spectral coverage ranging from 0.43 to age of agreement corrected by the amount of agreement that could be expected due to chance alone [START_REF] Richards | Remote Sensing Digital Image Analysis: An Introduction[END_REF].

These criteria were used to compare classification results and were computed using the confusion matrix. Furthermore, the statistical significance of differences was computed using Mc-Nemar's test, which is based upon the standardized normal test statistic [START_REF] Foody | Thematic map comparison: Evaluating the statistical significance of differences in classification accuracy[END_REF]:

Z = f 12 -f 21 √ f 12 + f 21 (11)
where f 12 indicates the number of samples classified correctly by classifier 1 and incorrectly by classifier 2. The difference in accuracy between classifiers 1 and 2 is said to be statistically significant if |Z| > 1.96. The sign of Z indicates whether 1 is more accurate than classifier 2 (Z 0) or viceversa (Z < 0). This test assumes related testing samples and thus is adapted to our situation since the training and testing set were the same for each experiment.

The feature extractions were done with MultiSpec c [3] while the morphological operations were done with the Image Processing Toolbox of Matlab c . The SVM classification was done using the LIBSVM through its Matlab c interface [START_REF] Chang | LIBSVM: a library for support vector machines[END_REF]. From previous experiments on the same data set, the Gaussian kernel provides the best results and was used for the experiments [START_REF] Fauvel | Evaluation of kernels for multiclass classification of hyperspectral remote sensing data[END_REF]. The range of each feature, be it spectral or morphological, was stretched between 0 and 1.

To obtain a baseline result for comparison, the classification was also done using the Gaussian Maximum Likelihood (ML) classifier on the hyperspectral data using MultiSpec c 1 . Feature extraction was done using the two FE algorithms (DBFE and NWFE) but only the best results have been reported for both data sets. The results were compared to those obtained by the proposed approach.

B. University Area data set

PCs were computed from the hyperspectral data. The results for the eigenvalues are shown in Table IV. The left column gives the component number, the center column the eigenvalues in percentage of the total amount of variance and the right column the cumulative amount of variance. From the table, three PCs were necessary to retain 99% of the variance criterion. EMPs were built according to the scheme presented in the Section V: A circular SE with a step size increment of 2 was used. Four openings and closings were computed for each PC, resulting in an EMP of dimension 27.

First, the classification with SVM was done using the spectral information and the extended morphological profile. The best ML accuracy was obtained using 8 features extracted with the NWFE, following Landgrebe's recommendations in [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF]. The results are reported in Table V. Regarding the global accuracies, both SVM approaches perform equally well, for instance the difference between the classification using the spectral information and the EMP is not statistically significant, see Table VI. Note that it is consistent with the characteristics of the scene: The University Area is a mix between man-made structures and natural materials. Therefore, the morphological information is not as useful as it could be in a very dense urban area. When a careful analysis is done on the class-specific accuracies, we can see from Table V that each approach performed well for complementary classes, e.g. the spectral approach performed better for classes 3, 6, 9 while the EMP approach performed better for classes 1, 2, 7, 8. After the data fusion we have to look at these classes and see if the best information was used, i.e., if the classification accuracy increased for these classes.

The experiment was then performed with the concatenated vector. The vector was made of the 103 spectral bands and the 27 features of the EMP. The vector was directly used as an input to the SVM. The classification results are reported in Table V. As can be seen from the table, the global accuracies increased. The κ value in percentage is 79.13% against 74.47% for the spectral approach and 73.25% for the EMP and the differences are statistically significant (see Table VI). Regarding the classspecific accuracies, the results in terms of accuracies have increased for classes 1, 7, 8 when compared to both individual approaches. In fact all the classes are more accurately classified than the worst respective cases for the individual approaches.

In the last experiment, feature reduction was applied on the morphological data and the original data before the concatenation. Then the stacked vector was classified by the SVM. Considering the class-specific accuracies, the DBFE approach improved the classification for class 2 while class 3 was less accurately classified than with the concatenated full hyperspectral data and EMP. However, the DBFE outperformed the individual classifications of the spatial or spectral information. On this data set, the classification of the DBFE feature extracted data gave the best classification results. Similar comments can be made for the accuracies obtained with classifications of the NWFE. Still, the number of features needed to achieve the same accuracy is significantly lower for the NWFE approach than for the DBFE. Since the SVM is linearly related to the dimensionality of the data, lower dimensional data reduced the training time and increased the speed of the classification.

To assess this increase, comparison of the processing time (training and classification process) for the different approaches was made. Table VIII the results, which are clearly different according to the features used. The training time could depend on several factors:

1) The dimension of the data;

2) The size of the training set;

3) The number of parameters for the kernel.

For our given problem, items 2) and 3) are the same. Reducing the size of the data is beneficial for the processing time, since data with lower dimensionality (EMP and NWFE) have the shortest processing time. For the best case (NWFE), the gain is about 73%.

For the classification processing time, two factors have an influence: The dimension of the data and the number of support vectors (non-zero α i in ( 8)). Thus, approaches with low dimensionality and few support vectors perform the classification task of the whole image faster (EMP and NWFE). Nevertheless, the classification processing is really fast by comparison to the training time, in all the cases.

Classification maps for the different approaches are shown in Fig. 5.

C. Pavia Center data set

For the second test, the scene is a very dense urban area in the center of the city of Pavia. Because of that, morphological information should be useful for the discrimination. PCs were computed from the hyperspectral data. The results for the eigenvalues are shown in Table IX. From the Table, three PCs were necessary to retain 99% of the variance criterion. The EMP was built according to the scheme presented in the section V: A circular SE with a step size increment of 2 was used. Four openings and closings were computed for each PC, resulting in an EMP of dimension 27. SVM classification was applied to the original hyperspectral data and the EMP. The best ML accuracy was obtained using 29 features extracted with the DBFE. The results are reported in Table X. From the Table, it can be seen that SVM classifier achieved excellent global accuracies. In these experiments, the morphological approach performs better than the spectral based approach. Table XI shows the statistical significance of differences between the classification accuracies for the different approaches. This is consistent with the characteristics of the picture: it is a very dense urban area and morphological processing provides discriminative information. In terms of accuracies, the main improvement in the classification is achieved for class 4. The other classes are classified equally accurately. The data fusion should thus improve the classification of class 4 while preserving very good results for the others classes.

Next, the experiment was performed using the concatened vector. The vector was made of the 102 spectral bands and the 27 features of the EMP. This vector was used as an input for the SVM without any additional processing. The classification results are reported in Table X. The differences of classification accuracies between the EMP and the concatened vector are not statistically significant, since the McNemar's test is almost equal to zero, see Table XI. Thus, both EMP and concatened vector perform equally well.

As in the previous experiment, feature reduction was applied both on the morphological data and on the original data before the concatenation. Then, the stacked vector was classified by the SVM. Table XII summarizes the test accuracies for several values for the variance criterion for the DBFE and NWFE. The best results are obtained with 99% variance criterion for both DBFE and NWFE. Using 99% of the variance with the DBFE, the hyperspectral data is reduced to 51 features and the EMP is reduced to 15 features. With the NWFE and 99% of the variance criterion, 44 features were extracted from the hyperspectral data and 20 from the EMP. The results are given in Table X.

For this experiment, the DBFE does not help for the classification since the Z test is not significant. On the other hand, similar classification accuracy is reached with far less features, nearly half the size of the previous feature set, thus decreasing the total training and classification time. The NWFE leads to a significant increase of the classification accuracies, |Z|= 7.75 by comparison to the best results obtained with the concatenation vector, which is contrary to the previous experiment. Classification maps for the different approaches are shown in Fig. 6. Visually, the thematic map produced with the classification of the NWFE features seems less noisy than the one obtained with the classification of the DBFE features. This is especially true in the top-left corner which correspond to a Classification maps for the different approaches are shown in Fig. 6.

D. Small training set experiment: University Area

To assess the effectiveness of the proposed methodology for a limited training set, we have randomly extracted a few training samples from the training set. For this experiment, we used 20 samples for each class, which represents less than 5% of the original training set. We have used the same EMP but had some problems with the DBFE, the covariance matrix was noninvertible (the NWFE does not suffer from this problem). In order to overcome this shortcoming and to apply the DBFE anyway, we use the leave on out covariance (LOOC) to estimate the covariance matrix and perform a statistical enhancement with unlabeled samples, both algorithms were implemented in the MultiSpec c [START_REF] Hoffbeck | Covariance matrix estimation and classification with limited data[END_REF], [START_REF] Shahshahani | The effect of unlabeled samples in reducing the small sample size problem and mitigating the Hughes phenomenon[END_REF]. We have repeated the training samples selection and the classification process five times, and the mean classification results are reported in the paper. As with the previous experiments, we perform the classification using the spectral or the morphological feature with SVM. The ML produced very poor results, simply close to random classification and hence not reported. The global accuracies are reported in Table XIV. Statistical significance of differences is reported in Table XV.

First of all, the test results are lower than those in Table V and VII, due to the limited training set. For instance, with the concatened feature vector, the overall accuracy and the κ are respectively, 83.53% and 79.13% for the original training set while using a limited training set, the overall accuracy and the κ are respectively, 75.35% and 68.66%. Nevertheless, with a very small training set, the results are still good.

For the feature extraction, NWFE with 99% of the cumulative variance provides the best results: The obtained overall accuracy is 85.42% and the κ is 80.87%, which is closed to the best results obtained with the full training set (OA=97.87% and κ = 84.40%, see Table V). The |Z| between the best results with limited training and the best results with full training set is equal to 13.65.

Furthermore, the accuracies are better than those obtained with the full training set with the spectral or morphological information alone. It is also important to note that NWFE outperforms better DBFE without any statistical enhancement.

Considering the processing time, with only 20 samples for each class, the training as well as the classification of the entire data set are done in 1 or 2 seconds.

VII. CONCLUSION

Classification of hyperspectral data with a fine spatial resolution has been investigated. The contribution of this work is a methodology to include both spatial and spectral information in the classification process by a data fusion scheme. Experimental results on two ROSIS data sets showed excellent accuracies and improvements compared to those obtained with pixel-based classifiers and the EMP-based classifier.

The use of feature extraction was motivated by the fact that the full stacked vector contains a lot of redundancy, because there is a redundancy in the hyperspectral data [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF] as well as in the EMP [START_REF] Benediktsson | Classification of hyperspectral data from urban areas based on extended morphological profiles[END_REF], which was confirmed by the experiments. On the other hand, SVM are known to be robust to dimensionality. Therefore, the use of feature reduction for SVM could be disputable. However, in the experiments lower dimensional data decreased the processing time, which can be crucial for some applications, and more important it has been showed that SVM can suffer from the dimensionality when many features are irrelevant [START_REF] Chapelle | Support vector machines: Induction principle, adaptive tuning and prior knowledge[END_REF]. By construction, the stacked vector may contain many copies of the same information and a feature extraction step may finally be needed to ensure correct classification on every data set, which is confirmed by the experiments (Usefulness of features reduction for the classification of remote sensing data with SVM was also assessed in [START_REF] Bazi | Toward an optimal SVM classification system for hyperspectral remote sensing images[END_REF]).

It is clear that feature extraction helps for the classification of hyperspectral data but it is not clear which one of the feature extraction methods should be used for the fusion of morphological and spectral features. From a theoretical point of view, the NWFE was derived because of some intrinsic problems with the DBFE [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF], i.e., "DBFE can involve lengthly calculations and more significantly it does not perform as well for small numbers of training samples". Hence, the NWFE might be more preferable, especially when a small training set is available. The experiments performed with a limited training set confirmed that.

In conclusion, the proposed fusion method succeed in taking advantage of the spatial and the spectral information simultaneously. It outperformed previous results [START_REF] Palmason | Classification of hyperspectral data from urban areas using morphological preprocessing and independent component analysis[END_REF], [START_REF] Palmason | Fusion of morphological and spectral information for classification of hyperspectral urban remote sensing data[END_REF]. Our current research is oriented to the definition of additional spatial features, such as textural characteristics [START_REF] Epifanio | Morphological texture features for unsupervised and supervised segmentations of natural landscapes[END_REF], to be include in the feature vectors.

Fig. 1 .

 1 Fig. 1. Simple morphological profile with two opening and two closings. In the shown profile, circular structuring elements are used with radius increment 4 (r = 4, 8 pixels). The processed image is a part of Fig. 4.(a).

Fig. 2 .

 2 Fig. 2. Extended morphological profile of two images. Each of the original profile has two openings and two closings. Circular structuring element with radius increment 4 was used (r = 4, 8). The processed image is a part of Fig. 4.(a).

Fig. 3 .

 3 Fig. 3. Proposed data fusion scheme.

Fig. 4 Fig. 4 .

 44 Fig. 4. ROSIS data: (a) University Area, (b) Pavia Center. Three-channel color composite of the areas used for the classification. Data characteristics are detailed in Sub-Sections VI-B and VI-C.

Fig. 5 .

 5 Fig. 5. University Area: Classification map obtained with SVM from: (a) the original hyperspectral data, (b) the EMP, (c) 37 DBFE features and (d) 13 NWFE features. Classification accuracies are reported in the TableV.

Fig. 6 .

 6 Fig. 6. Pavia Center: Classification map obtained with SVM from: (a) the original hyperspectral data, (b) the EMP, (c) 66 DBFE features and (d) 64 NWFE features. Classification accuracies are reported in the TableX.

TABLE I SPECTRAL

 I AND SPATIAL DATA PROPERTIES. 'ր' INDICATES A GOOD PROPERTY, '∼' INDICATES THAT THE PROPERTY MIGHT BE HARMFUL AND 'ց' INDICATES A CRITICAL PROPERTY.

		Spectral features		Morphological features
	ր Fine physical description	ր Geometrical information
	ր Directly accessible	∼	Needs to be extracted
	∼	Redundancy	∼	Redundancy
	ց No spatial information	ց Reduced spectral information

TABLE II INFORMATION

 II CLASSES AND TRAINING-TEST SAMPLES FOR THE UNIVERSITY AREA DATA SET.

		Class	Samples
	No	Name	Train	Test
	1	Asphalt	548	6641
	2	Meadows	540	18649
	3	Gravel	392	2099
	4	Trees	524	3064
	5	Metal Sheets	265	1345
	6	Soil	532	5029
	7	Bitumen	375	1330
	8	Bricks	514	3682
	9	Shadows	231	947
		Total	3921	42776

TABLE III INFORMATION

 III CLASSES AND TRAINING-TEST SAMPLES FOR THE PAVIA CENTER DATA SET. The first test set is around the Engineering School at the University of Pavia. It is 610 by 340 pixels. Some channels (twelve) have been removed due to noise. The remaining 103 spectral dimensions are processed. Nine classes of interest are considered, i.e., trees, asphalt, bitumen, gravel, metal sheets, shadow, bricks, meadows and soil. 2) Pavia Center: The second test set is the center of Pavia. The Pavia center image was originally 1096 by 1096 pixels. A 381 pixel wide black stripe in the left part of image was removed, resulting in a "two parts" image. This "two parts" image is 1096 by 715 pixels. Some channels (thirteen) have been removed due to noise. The remaining 102 spectral dimensions are processed. Nine classes of interest are considered, i.e., water, trees, meadows, bricks, soil, asphalt, bitumen, tiles and shadows. Available training and testing set for each data set are given in Table II and III and

	Class	Samples
	No	Name	Train	Test
	1	Water	824	65971
	2	Trees	820	7598
	3	Meadows	824	3090
	4	Bricks	808	2685
	5	Soil	820	6584
	6	Asphalt	816	9248
	7	Bitumen	808	7287
	8	Tiles	1260	42826
	9	Shadows	476	2863
	Total	7456	148152
	0.86µm. The data have been atmospherically corrected but not
	geometrically corrected. The spatial resolution is 1.3m per pixel.
	Two data sets were used in the experiment:
	1) University Area:

  Table VII summarizes the test accuracies for several values of the variance criterion for the DBFE and NWFE. Best results were obtained with 95% and 80% variance criterion for the

TABLE VII UNIVERSITY

 VII AREA. GLOBAL ACCURACIES IN PERCENTAGE WITH DIFFERENT FEATURE EXTRACTION METHODS. THE NUMBERS OF FEATURES FROM THE SPECTRAL DATA AND THE MORPHOLOGICAL DATA, RESPECTIVELY, ARE GIVEN IN BRACKETS. Using 95% of the variance criterion with DBFE, the hyperspectral data were reduced to 27 features and the EMP to 10 features. With NWFE and 80%, 7 features were extracted from the hyperspectral data and 6 from the EMP. Again, as can be seen in Table VI, differences between the classification accuracies are statistically significant.

	Feature extraction	Cum. Var.	Features	OA	AA	κ
		99	60 (45,15)	84.77	89.98	80.43
		95	27 (27,10)	87.97	88.94	84.40
	DBFE	90	28 (20,8)	86.49	88.94	82.50
		80	19 (14,5)	82.95	87.51	77.27
		70	14 (10,4)	76.93	84.63	71.20
		99	62 (42,20)	84.15	88.89	79.61
		95	28 (16,12)	82.90	87.25	77.98
	NWFE	90	18 (10,8)	82.64	86.77	77.65
		80	13 (7,6)	87.59	88.93	83.89
		70	10 (5,5)	79.57	87.39	74.49
	DBFE and NWFE, respectively.

TABLE V

 V UNIVERSITY AREA. SUMMARY OF THE GLOBAL AND THE CLASS-SPECIFIC TEST ACCURACIES IN PERCENTAGE FOR THE CLASSIFICATION. THE NUMBERS OF FEATURES FROM THE SPECTRAL DATA AND THE MORPHOLOGICAL DATA, RESPECTIVELY, ARE GIVEN IN BRACKETS.

		ML (NWFE)	Spectral	EMP	Spec. EMP	DBFE 95%	NWFE 80%
	Features	8	103	27	130	37 (27,10)	13 (7,6)
	OA	80.10	79.48	79.14	83.53	87.97	87.59
	AA	87.00	88.14	84.30	89.39	89.43	88.93
	κ	-	74.47	73.25	79.13	84.40	83.89
	Class 1	76.00	84.36	94.50	95.33	90.92	86.80
	Class 2	73.90	66.20	72.82	73.46	85.91	86.95
	Class 3	70.80	71.99	53.22	65.89	57.88	63.26
	Class 4	96.70	98.01	98.89	99.18	99.22	98.53
	Class 5	99.90	99.48	99.55	99.48	99.48	99.88
	Class 6	87.60	93.12	58.11	84.15	85.32	82.62
	Class 7	92.00	91.20	96.09	97.22	95.19	96.61
	Class 8	87.30	92.26	95.27	96.12	95.84	95.38
	Class 9	99.10	96.62	91.24	93.66	95.14	90.60

TABLE VI UNIVERSITY

 VI AREA. STATISTICAL SIGNIFICANCE OF DIFFERENCES IN CLASSIFICATION ACCURACIES.

		EMP/Spectral	Spectral/Spec. EMP	Spectral/DBFE 95%	Spectral/NWFE 80%	Spec. EMP/DBFE 95%	Spec. EMP/NWFE 80%	DBFE 95%/NWFE 80%
	Z	-1.36	-19.55	-37.98	-36.61	-25.19	-21.79	2.32

TABLE VIII UNIVERSITY AREA .

 VIIIAREA PROCESSING TIME IN SECONDS AS FUNCTION OF DIMENSIONALITY AND NUMBER OF SUPPORT VECTORS.

		Spectral	EMP	Spec. EMP	DBFE 95%	NWFE 80%
	Dimension	103	27	130	37	13
	Training	3074	850	3257	1184	859
	Number of SVs	1085	406	529	727	572
	Classification	76	9	41	18	9

TABLE XII PAVIA

 XII CENTER. GLOBAL ACCURACIES IN PERCENTAGE WITH DIFFERENT FEATURE EXTRACTION METHODS. THE NUMBERS OF FEATURES FROM THE SPECTRAL DATA AND THE MORPHOLOGICAL DATA, RESPECTIVELY, ARE GIVEN IN BRACKETS.

	Feature extraction	Cum. Var.	Features	OA	AA	κ
		99	66 (51,15)	98.65	97.30	98.10
		95	41 (31,10)	98.37	96.86	97.70
	DBFE	90	31 (23,8)	98.08	96.71	97.29
		80	22 (17,5)	98.53	97.22	97.42
		70	17 (13,4)	98.53	97.31	97.42
		99	66 (44,20)	98.87	97.95	98.41
		95	31 (19,12)	98.58	96.66	97.99
	NWFE	90	21 (12,9)	98.41	97.28	97.71
		80	14 (8,6)	98.24	96.63	97.52
		70	10 (6,4)	98.39	96.39	97.73

  Regarding the computing time, the results for the training and the classification are reported in TableXIII. As expected, using feature extraction methods reduces the processing time for both the training and the classification.

			TABLE XIV
	UNIVERSITY AREA. SUMMARY OF THE GLOBAL TEST ACCURACIES IN
	PERCENTAGE FOR SVM CLASSIFICATION USING A LIMITED TRAINING
	SET.THE NUMBERS IN BRACKETS ARE THE NUMBERS OF FEATURES FROM
	THE SPECTRAL DATA AND FROM THE MORPHOLOGICAL DATA,
			RESPECTIVELY.
		Cum. Var.	Features	OA	AA	κ
	Spectral	-	103	71.25	75.79	63.70
	EMP	-	27	75.79	80.72	69.00
	Spec. EMP	-	130	75.25	81.23	68.66
		99	66 (44,12)	77.33	83.77	71.31
		95	32 (25,7)	81.34	82.84	75.65
	DBFE	90	23 (18,5)	74.55	79.15	67.45
		80	16 (12,4)	72.28	80.35	65.25
		70	11 (9 ,2)	61.31	72.04	52.26
		99	53 (35,18)	85.42	87.48	80.87
		95	25 (14,11)	81.04	85.42	75.84
	NWFE	90	17 (10,7)	79.05	83.72	73.27
		80	13 (7,6)	84.77	85.65	80.09
		70	9 (5,4)	83.10	84.35	77.84
	very dense urban area.

TABLE X

 X PAVIA CENTER. SUMMARY OF THE GLOBAL AND THE CLASS-SPECIFIC TEST ACCURACIES IN PERCENTAGE FOR SVM CLASSIFICATION. THE NUMBERS OF FEATURES FROM THE SPECTRAL DATA AND THE MORPHOLOGICAL DATA, RESPECTIVELY, ARE GIVEN IN BRACKETS.

		ML (DBFE)	Spectral	EMP	Spec. EMP	DBFE 99%	NWFE 99%
	Features	29	102	27	129	66 (51,15)	64 (44,20)
	OA	94.50	97.67	98.69	99.69	98.65	98.87
	AA	94.00	95.60	97.69	98.07	97.30	97.95
	κ	-	96.71	98.15	98.15	98.10	98.41
	Class 1	91.50	98.35	99.08	98.66	99.17	99.21
	Class 2	92.00	91.23	91.62	93.52	90.00	92.49
	Class 3	97.70	96.76	96.18	95.95	96.54	96.76
	Class 4	86.90	88.45	98.40	98.77	98.92	99.55
	Class 5	95.60	96.97	98.81	99.42	99.27	99.74
	Class 6	94.40	96.32	97.98	98.36	98.45	98.70
	Class 7	96.40	96.01	97.89	98.22	97.91	98.41
	Class 8	99.30	99.40	99.74	99.79	99.81	99.72
	Class 9	92.30	99.93	99.44	99.93	98.60	96.93

TABLE XI PAVIA CENTER .

 XICENTER STATISTICAL SIGNIFICANCE OF DIFFERENCES IN CLASSIFICATION ACCURACIES.

		EMP/Spectral	EMP/Spec. EMP	EMP/DBFE 99%	EMP/NWFE 99%	Spec. EMP/DBFE 95%	Spec. EMP/NWFE 80%	DBFE 95%/NWFE 80%
	Z	27.84	-0.06	1.44	-8.14	1.42	-7.75	-9.44

TABLE XIII PAVIA CENTER .

 XIIICENTER PROCESSING TIME IN SECONDS AS FUNCTION OF DIMENSIONALITY AND NUMBER OF SUPPORT VECTORS.

		Spectral	EMP	Spec. EMP	DBFE 95%	NWFE 80%
	Dimension	102	27	129	66	66
	Training (s)	5178	1569	5909	3520	3962
	Number of SVs	691	265	401	558	408
	Classification (s)	143	20	105	73	47

TABLE XV UNIVERSITY

 XV AREA. STATISTICAL SIGNIFICANCE OF DIFFERENCES IN CLASSIFICATION ACCURACY FOR A LIMITED TRAINING SET.

		EMP/Spectral	EMP/Spec. EMP	EMP/DBFE 95%	EMP/NWFE 99%	NWFE 99%/DBFE 95%
	Z	17.5	2.67	-25.4	-51.14	21.32

For the ML, the kappa coefficient was not accessible.

ACKNOWLEDGMENT

The authors appreciate the contributions of Mr. Jon Aevar Palmason to this research. Furthermore, the authors would like to thank the IAPR -TC7 for providing the data and Prof. Paolo Gamba and Prof. Fabio Dell'Acqua of the University of Pavia, Italy, for providing reference data. The authors also thank the reviewers for their many helpful comments. This research was supported in part by the Research Fund of the University of Iceland and the Jules Verne Program of the French and Icelandic governments (PAI EGIDE).


Mathieu Fauvel

Jocelyn Chanussot

Jon Atli Benediktsson

ADAPTIVE PIXEL NEIGHBORHOOD DEFINITION FOR THE CLASSIFICATION OF HYPERSPECTRAL IMAGES WITH SUPPORT VECTOR MACHINES AND COMPOSITE KERNEL

Keywords: Support vectors machines, spatial information, hyperspectral data, kernel function, area self-complementary filter

The pixel-wise classification of hyperspectral images with a reduced training set is addressed. The joint use of the spectral and the spatial information is investigated. The spectral information simply consists of the spectral value of each pixel. For the spatial information, we use an area filter to simplify the image and extract consistent connected components. These components are used to define an adaptive neighborhood for each pixel of the image. The vector median value of each component is defined as a spatial feature for the classification. Support Vector Machines are used for the classification and a composite kernel is used to combine both the spatial and the spectral information. Experiments are conducted on AVIRIS hyperspectral data. The proposed approach provides significant improvements in terms of classification accuracy when compared with a standard statistical method (maximum likelihood) and with a SVM classifier using the spectral information alone. Robustness with respect to the size of the training set is also investigated.

INTRODUCTION

The pixel-wise classification of hyperspectral data using both spectral and contextual information is investigated in this paper. It is well known that contextual information is useful for the classification of hyperspectral images [START_REF] Landgrebe | Signal Theory Methods in Multispectral Remote Sensing[END_REF]. Such information is usually modeled in a statistical framework using Markov Random Field theory (MRF). For instance, a recursive Bayesian classifier is proposed in [START_REF] Jackson | Adaptive bayesian contextual classification based on markov random field[END_REF], where the joint prior probabilities of the classes of each pixel and its spatial neighbors being modeled by MRF. Improved classification performances are obtained, the gain over standard methods being significant when the training set remains limited. Advance approaches can be found in [START_REF] Hegarat-Mascle | Ant colony optimization for image regularization based on a non-stationary markov modeling[END_REF], for the regularization of remote sensing images. These approaches are based on a statistical modelling of the interpixel dependency, where the neighborhood of every pixel is defined by a fixed set such as a 3 × 3 square. This strategy fails when considering pixels that are close to the border of a structure: The fixed shape neighborhood then includes pixels from different structures. For example, as shown in Fig. 1.(a), the classification of the marked pixel (roof) may be biased by the neighboring pixels that actually

The authors would like to thank Professor David A. Landgrebe for providing the hyperspectal data and the reference data used in these experiments. This research was supported in part by the Research Fund of the University of Iceland and the Jules Verne Program of the French and Icelandic governments (PAI EGIDE).

belong to the street. To overcome this problem, an adaptive neighborhood should be defined for each pixel. Furthermore, assuming that relevant structures have a sufficient area, this adaptive neighborhood should include a large number of pixels. In this paper, we thus propose to define the neighborhood of one given pixel as the connected zone resulting of a self-complementary area filtering. For each pixel, the spatial feature used for the classification is extracted from the neighborhood of pixels.

Another problem usually appears with MRF based methods and hyperspectral data: the problem of parameters estimation. When dealing with hyperspectral data, each pixel-vector x is composed of several components (dim(x) > 100). In such a high dimensional space, the statistical estimation becomes a difficult task [START_REF] Donoho | High-dimensional data analysis: the curses and blessing of dimensionality[END_REF]. For instance, the required number of training pixels for a reliable estimation is related to the square of the dimensionality for a quadratic classifier (e.g., the Gaussian Maximum Likelihood) [START_REF] Fukunaga | Introduction to statistical pattern recognition[END_REF]. In remote sensing applications, only limited ground-truth information is usually available. As a consequence, algorithms based on statistical modelling may not perform well in classification of hyperspectral data in full feature space [START_REF] Jimenez | Supervised classification in high dimensional space: geometrical, statistical and asymptotical properties of multivariate data[END_REF]. On the contrary, among the machine learning algorithms, Support Vectors Machine (SVM) [START_REF] Schölkopf | Learning with Kernels. Support vector machines, regularization optimization and beyond[END_REF] have demonstrated outstanding abilities to deal with hyperspectral data [START_REF] Melgani | Classification of hyperspectral remote sensing images with support vector machines[END_REF], even in the situation of very small training sets [START_REF] Fauvel | Evaluation of kernels for multiclass classification of hyperspectral remote sensing data[END_REF]. However, standard SVM are pixel-based classifiers and no spatial information is used in the classification process. Recent studies have investigated different approaches to deal conjointly with the spatial and spectral information using SVM. For instance, Camps-Valls proposes to estimate the local mean and variance for each pixel using a fixed square neighborhood. These statistical parameters are included in the classification process using a composite kernel [START_REF] Camps-Valls | Composite kernels for hyperspectral image classification[END_REF]. In [START_REF] Mercier | Partially supervised oil-slick detection by SAR imagery using kernel expansion[END_REF], Mercier uses the same kernel formulation but the spatial information is estimated by a wavelets decomposition of the image. In this paper, we have adopted the same kernel strategy: a linear combination of two kernels is used, one acting the spectral information and one acting on the spatial information.

The originally of our proposed method is the definition of an adaptive neighborhood. This methodology is detailed in Section 2.1. Then, following [START_REF] Camps-Valls | Composite kernels for hyperspectral image classification[END_REF][START_REF] Mercier | Partially supervised oil-slick detection by SAR imagery using kernel expansion[END_REF], we present the classification using the spatial and the spectral information with the SVM based on a mixture of kernels, Section 3. Experimental results are presented and discussed in Section 4. Finally, conclusions are drawn in Section 5.

CONTEXTUAL INFORMATION

In this section, we present how neighbors-sets are defined and how the contextual information is estimated.

Self-complementary area filters

Self-complementarity is a very important property: the operator is its own complement (Ψ = CΨ), which is stronger than self-duality (Ψ = CΨC) [START_REF] Soille | Beyond self-duality in morphological image analysis[END_REF]. A self-complementarity filter treats structures in the image that have different local contrast equally, which is not possible with the self-duality property only. Thus, they analyze all the structures of an image at once, local extrema (be they minima or maxima) as well as regions with intermediate grey levels. Self-complementary area filters have been introduced to extend area opening and closing to all the structures of the image, not only its local extrema [START_REF] Soille | Beyond self-duality in morphological image analysis[END_REF]. This is of the utmost interest for the analysis of very high resolution remote sensing images: these connected filters enable to remove small meaningless structures (e.g. cars on the road) while preserving borders of interest with a very high accuracy and independently of the structures' contrast. P. Soille has proposed a two steps algorithm, which consists of:

1. Labelling all the flat zones that satisfy the area criterion λ, 2. Growing the labelled flat zones until a partition of the image is reached. A better preservation of the image structures is achieved by iterating the algorithm until the desired minimal size is obtained, e.g., let I be the image to process, then Ψ λ (I) = Ψ λ (Ψ λ-1 (. . . (Ψ2(I)))), where λ is the minimal size of the remaining flat structures.

As stated in the introduction, we define the neighborhood of each pixel as the connected zone resulting from the application of a self-complementary area filter. This is illustrated in Fig. 1 where The direct use of the area filter to hyperspectral remote sensing images is not possible, because of the lack of ordering relation. In order to overcome this shortcoming, several approaches can be considered. Using marginal ordering, one can apply the area filter on each band independently, but considering the high inter-band correlation, this is not appropriate [START_REF] Chanussot | Total ordering based on space filling curves for multivalued morphology[END_REF][START_REF] Lambert | Extending mathematical morphology to color image processing[END_REF]. In this article, we adopt the same strategy than in [START_REF] Benediktsson | Classification of hyperspectral data from urban areas based on extended morphological profiles[END_REF], where a Principal Component Analysis (PCA) [START_REF] Fukunaga | Introduction to statistical pattern recognition[END_REF] is applied to reduce the dimensionality and project data onto vector space where ordering relation exist. Then we compute the area filtering on the first principal component to extract the neighborhood of each pixel. The neighbors mask is applied on each band of the data and spatial information are extracted. The following scheme sum up the methodology:

x x,v 1 p / / x x Ψ / / Ωx / / ⊕ / / Ωx
where v 1 p is the first eigenvector corresponding to the largest eigenvalue.

Extracting spatial feature

Once the neighborhood of each pixel has been adaptively defined, spatial information is extracted. Considering the small average size of the neighbors set, a description using order statistics would not be reliable. Shape descriptors are not appropriate either as one given structure might be split into several consistent regions. (see Fig. 1

.(b):

The roof is divided into several triangles). As a conclusion, we propose to compute the vector median value of the neighbors set Ωx, for every pixel x:

Υx = med(Ωx) (1)
where dim(x) = dim(Υx) = n, the number of spectral bands. Unlike the mean vector, the median vector is a vector from the initial set, which ensures a certain spectral consistency. As a conclusion, every pixel now has two features: The spectral feature x which is the original value of each pixel and the spatial feature Υx which is the median value computed on each pixel's adaptive neighborhood. The easiest way to use both information would be to build a stack vector. In this paper, we propose to exploit the kernel trick [START_REF] Vapnik | Statistical Learning Theory[END_REF] of the SVM and design a composite kernel that allows to tune the relative influence of the extracted features. This is detailed in the next section.

The SVM classifier is surely one of the most popular kernel learning algorithms. It performs robust non-linear classification using the kernel trick. The idea is to find a separating hyperplane in some feature space induced by the kernel function while all the computations are done in the original space [START_REF] Vapnik | Statistical Learning Theory[END_REF]. Given a training set S = {(x 1 , y1), . . . , (x ℓ , y ℓ )} ∈ R n × {-1; 1}, the decision function is found by solving the convex optimization problem:

max α g(α) = ℓ X i=1 αi - 1 2 ℓ X i,j=1 αiαjyiyjk(x i , x j ) subject to 0 ≤ αi ≤ C and P ℓ i=1 αiyi = 0 (2)
where α are the Lagrange coefficients, C a positive constant that is used to penalize the training errors, and k the kernel. To be an acceptable kernel, k should be a positive semi-definite function [START_REF] Aronszajn | Theory of reprodusing kernel[END_REF]. A short comparison of kernels for remotely sensed image classification can be found in [START_REF] Fauvel | Evaluation of kernels for multiclass classification of hyperspectral remote sensing data[END_REF]. When the optimal solution of ( 2) is found, i.e. the αi, the classification of a sample x is achieved by looking to which side of the hyperplane it belongs:

y = sgn ℓ X i=1 αiyik(x i , x) + b ! . ( 3
)
To construct the kernel that works with both the spatial and the spectral information at the same time, we use the positive linear combination of two kernels, namely the spectro-spatial kernel:

K λ σ,µ (x i , x j ) = µk spect σ (x i , x j ) + (1 -µ)k spat σ (x i , x j ) ( 4
)
where λ is the area filtering parameter and µ a weighting parameter which controls the relative influence of each kernel. The latter parameter µ has to be tuned during the training process. The spectral kernel k spect σ is chosen as the standard Gaussian kernel:

k spect σ (x i , x j ) = exp " - x i -x j 2 2σ 2 « ( 5
)
where the norm is the Euclidean-norm and σ ∈ R + tunes the flexibility of the kernel. The spatial kernel k spat σ is also Gaussian but uses the extracted spatial features Υx:

k spat σ (x i , x j ) = exp " - Υ x i -Υ x j 2 2σ 2 « . ( 6
)
The one versus all strategy was chosen for the multi-class classification problem [START_REF] Schölkopf | Learning with Kernels. Support vector machines, regularization optimization and beyond[END_REF]. It allows us to tune the parameter µ for each class independently.

EXPERIMENTAL RESULTS

Hyperspectral data from the optical AVIRIS sensor were used in the experiments. The data contains 220 data channels and is 145×145 pixels. It was collected in June 1992 over the Indian Pine Test site. Sixteen classes of interest are defined (see Table 1). The reference data consist of 10366 pixels that are not uniformly distributed; class 9 has only 20 labelled pixels while class 11 has 2468 labelled pixels. All the spectral bands were used, even though some of them were known to be noisy. No pre-processing were done.

Different training sets were randomly constructed from the reference data with 5, 10, 15, and 20 pixels by class, respectively. Each experiment is repeated five times and the averaged results are reported. The validation was done using the whole reference data.

The accuracy is assessed using the Kappa coefficient of agreement κ, i.e., the percentage of agreement corrected by the amount of agreement that could be expected due to chance alone [START_REF] Liua | Comparative assessment of the measures of thematic classification accuracy[END_REF].

Several sizes of the area filtering parameter λ were tested. According to the low spatial resolution, we have used relatively low spatial parameter: from 3 to 20 pixels. The SVM parameters were tuned for each binary sub-problem by cross-validation: σ 2 = {0.5, 1, 2, 4}, µ ∈ {0.1, 0.2, . . . , 0.9} and C = 200. The optimization problem was solved using a modified version of the LIBSVM library (http://www.csie.ntu.edu.tw/ ˜cjlin/libsvm).

Experimental results are given in Fig. 2. With 5 pixels by classes, corresponding to less than 1% of the available reference data, the κ is as low as 31.4 using the spectral information only. On the contrary, a significant increase of κ is obtained when using the additional spatial information with the proposed method (typically around 50% increase), for instance κ = 47.30 for λ = 17. Naturally, better classification accuracies are obtained using the larger training sets. For instance, when 20 training samples are used for each class, κ reaches over 70 whatever the value of λ, while using only the spectral bands leads to κ = 46.6.

Note that when considering larger area thresholds (λ > 20), the classification accuracy decreases since the area filter is too strong and structures of interest start to disappear. Consequently, one should set λ according to the spatial resolution of the data and the minimal size of the structures one is interested in.

Table 1 sums up the kernel parameters found during the training process for each binary sub-problem. The value of µ differs for the different classes. This tends to prove that the amount of each kind of information actually needs to be tuned carefully during the training process and should not be fixed a priori to the same value for every classe.

The obtained results are very promising. They clearly outper- 1. Also, substantial improvement over the classical SVM is obtained, as can be seen from Fig. 2.

CONCLUSION

The classification of hyperspectral data with a spectro-spatial SVM classifier is investigated. Influence of a very small training set is studied in an empirical way. In order to include spatial information in the analysis, an adaptive definition of every pixel's neighborhood is proposed. Using an area self-complementary filter, the image is simplified to remove irrelevant structures. The remaining flat zones are considered as sets of neighbors. For each flat zone, the vector median value is extracted to take spatial neighborhood into account.

The classification is performed using support vector machines and a spectro-spatial kernel. Comparisons with state of the art algorithms, a Gaussian ML and a SVM, showed a clear increased of the classification accuracy. Some open questions remain to be addressed. The parameter λ in the area filtering has to be tuned. It does not seem critical, as there is a large range of values for which the classification accuracy remains almost equal. Furthermore, prior knowledge on the data resolution and the size of the structures of interest can be used. The kernel parameter has also to be tuned properly, some a priori information on the physical nature of the class should help to reduce the range of tested values and thus reduce the global training time. Our ongoing works include the definition of additional spatial information, such as textural information.

Fig. 1 .

 1 Fig. 1. Inter-pixel dependency estimation. (a) Original image and fixed square neighborhood. (b) Filtered image and neighbors-set defined using flat zones. (c) Original image with the defined neighbors-set Ωx.

Fig. 1 .

 1 (b) is the area filtering of Fig. 1.(a). The image is partitioned into flat zones. Each flat zone is consistent and hence belongs to one single structure in the original image. Furthermore, the smallest structures have been removed and only the main structure of interest remain. All the pixels belonging to one given flat zone are considered as neighbors. The neighborhoods defined in this way are applied on the original image. Fig. 1.(c) shows the morphological neighborhood Ωx associated with the spotted pixel x. It is obviously more homogeneous and spectrally consistent than the fixed square featured on Fig. 1.(a).

Fig. 2 .

 2 Fig. 2. Kappa coefficient of agreement. The horizontal axis represents the number of training samples use for each class and the vertical axis represents the κ values in percentage (+ : Classical SVM using the spectral information only; * : proposed approach with λ = 5; × : Proposed approach with λ = 9; • : Proposed approach with λ = 13; △ : Proposed approach with λ = 17).

Table 1 .

 1 Kernel parameters found for λ = 17 and 20 training samples by classes. κ= 75.80, Overall Accuracy: 78.78%, Average Accuracy: 85.57%.

	Classes	1	2	3	4	5	6	7	8
	Name	Alfalfa	Corn-notill	Corn-min	Corn	Grass/Pasture Grass/Tress	Grass/Pasture-mowed	Hay-windrowed
	µ	0.6	0.4	0.5	0.4	0.4	0.1	0.9	0.4
	σ 2	0.5	0.5	0.1	4	0.5	2	4	2
	Classes	9	10	11	12	13	14	15	16
	Name	Oats	Soybeans-notill Soybeans-min	Soybeans-clean	Wheats	Woods	Bldg-Grass-Tree-Drives Stone-steel Towers
	µ	0.7	0.1	0.1	0.2	0.7	0.3	0.3	0.8
	σ 2	4	2	2	1	0.5	0.5	4	0.5
	form results obtained on the same data set and presented in [1], pages
	334-346. With supervised feature reduction (Decision Boundary
	Feature Extraction [1]) and a Gaussian Maximum Likelihood classi-
	fier, the overall accuracy was only 63.3% for a training set made of
	688 samples and 72.4% for a training set made of 1490 samples. Us-
	ing the same reference set, we achieved 78.78% of overall accuracy,
	with only 320 training samples see Table