domain: spi phys math info

text:
Magali Barbier

Eric Bensana

David Doose

Florent Besson

Yann Le

Page Alseamar

Mikaël Leopoldoff
email: mleopoldoff@centralweb.fr

BARBIER, ONERA, A mission management system for a fleet of gliders A mission management system for a fleet of gliders

Keywords: underwater glider, fleet, mission management system, planning, supervision I

The objective of AGLIMMS project, whose acronym stands for Acoustic GLIders Mission Management System, is to efficiently coordinate a fleet of underwater gliders whose missions are to obtain physical, chemical, biological and/or acoustic measurements on a large 3D sea area. This paper describes planning and supervision functions under development and their integration in a global centralised architecture. A demonstration with three SeaExplorer from Alseamar is planned late 2019.

INTRODUCTION

Oceanographic missions performed by underwater gliders are various [START_REF] Rudnick | Ocean research enabled by underwater gliders[END_REF]: climate change monitoring, analysis of human activity (oil and mining industries), study of local flora and fauna, seabed mapping for its exploitation and protection… Whereas missions assigned to most Autonomous Underwater Vehicles last several days on several square dozen kilometres, underwater gliders can stay at sea for several months and scan thousands of kilometres. These lowcost vehicles move silently while interfering very little with their environment and their low weight allows light deployments and recoveries. Their drawbacks are a bad underwater positional accuracy in hundreds of meters and their limited power capacity, a few watts used for travelling (by changing buoyancy and using winged surfaces to produce a sawtooth-like trajectory) and for payload. When in mission, they surface every few hours for measurement data transmission and potential trajectory update. They are controlled by an experienced pilot who is alerted each time a glider surfaces and who can adapt the trajectory or change the mission if needed.

Using a fleet (sometimes called a network) of gliders either increases the spatial acquisition or reduces the processing time. The probability to detect sources of signals (gas leak, marine mammals, anthropogenic noise…) increases and positional accuracy is better thanks to a possible triangulation. Specific patterns such as comb allow for a better coverage of spatial areas as all synchronised gliders evolved in the same environment. Lastly, fleet missions can potentially proceed when a glider leaves the team, especially when reconfiguration issues are managed.

For an efficient coordination, the management of a fleet of gliders however requires more than individual experienced pilots. Many types of heterogeneous data must be assimilated: they stem from platforms, from the run of models (e.g. currents, sound propagation) and from external world (e.g. automatic ship identification systems AIS, bathymetry, weather). A centralised global Mission Management System (MMS) is thus required to help the pilots, by leaving them only high level tasks while algorithms perform the coordination activities. The objective of AGLIMMS is to build a centralised MMS to efficiently coordinate a fleet of gliders which needs are:

• reduced work for pilots who thus have rather a role of mission operators: need of a Man Machine Interface (MMI) allowing the definition of high level objectives and development of algorithms to compute a plan for the fleet of gliders and to supervise their sea execution;

• sea performance: optimised planning software for gliders coordination, online supervision of sea execution and online plan adaptation when surfacing;

• integration in the real world: use of environmental data and models, navigation models, consideration of data coming from real gliders via satellite links…

• Technology Readiness Level between 4 and 5: a demonstration with three SeaExplorer is planned late 2019.

This paper focuses on the planning and supervision functions of MMS.

III. PLANNING AND SUPERVISION SYSTEM (PSS)

AGLIMMS PSS aims at managing the coordination of all gliders actions for the achievement of missions in an optimised way. Fig. 2 shows interactions between planning and supervision functions and within the MMS. A fleet mission defined by an operator contains: the navigation area, an initial set of gliders (gliders can enter and leave an ongoing mission), several phases, and constraints between these phases. Five types of phases are available amongst them path following (a path being defined by a set of waypoints), 4D systematic coverage for data gathering, and 4D random coverage for signal detection. Fig. 3 shows an ongoing mission example that is used thereafter to illustrate the PSS functions. Glider g1 follows a path whereas gliders g2 and g3 are performing an acoustic barrier in midwater; g1 is underwater and its position is estimated, g2 and g3 are navigating around the mooring point and their position adequately known. Next phases are systematic coverages of two areas, one with two gliders in a comb formation amongst them g1, and one with one glider. In this example, the assignment of g1 to one coverage phase is imposed by the operator whereas g2 and g3 assignments to the coverage phases are not definitive: they can change each time a plan is computed (this matter is symbolized by a question mark on figures). Final step for all gliders are to join a recovery area then performing another acoustic barrier. Fig. 4 shows the related graph of phases.

IV. GLIDER ACTION COST MODEL

Optimised computation of plans requires the knowledge of the cost of one glider action when moving in its environment. Input data are the hydrodynamic model (reversed from real data), the maximal size of ballasts, their centring, electrical consumption model of the glider and of its payloads when used. Planning criteria relate to duration, time interval, consumption, and distance. Environment data are sea currents maps, bathymetry maps and potential forbidden areas.

A "glider action" (also further called "surfacing action") is composed of a sawtooth-like navigation profile between two surfacing points whose main variables are: vertical speed Vz, slope γ, bottom immersion Zb, top immersion Zt (as a glider does not surface after each ascent) and number of descent-ascent patterns Sr. Above side of Fig. 5 shows this profile without currents. As sea gliders are sensitive to currents that vary according to depth, the navigation profile may adapt to current layers; example on below side of Fig. 5 shows a 2D display but trajectories are 4D with X, Y, Z and time. In what follows, we distinguish a "surfacing action" (glider action given by its navigation profile as previously defined) from the "surfacing waypoint" (only its final position).

V. PLANNING FUNCTION

The planning function computes a "fleet plan" composed of all individual glider plans. A "glider plan" is one sequence of all glider actions from now up to end of mission or to a specific date. Planning algorithm can be called whenever the current situation evolves (see section VI on supervision), which generally leads to a regular update of the fleet plan.

Current fleet plan is used to make gliders dive. Each new fleet plan is displayed on the MMI. Offline, this displayed plan allows an operator to iteratively prepare new fleet missions; online, it allows them to potentially react to current situation.

Need for surfacing comes firstly from mission (surfacing needed for every heading changes in path), area coverage, specific operator constraints…) and secondly from navigation performing. The planning function uses then two algorithms, A1 calling A2 when necessary:

• A1 = fleet mission planning: selection of phases to perform, assignment of gliders to phases, choice of the sequence of phases visited by each glider, choice of the pattern and splitting for each phase and the computation of surfacing waypoints to answer mission constraints (further called "mission surfacing waypoint");

• A2 = glider navigation planning: computation of all surfacing actions to follow above mission surfacing waypoints while considering gliders characteristics and environmental constraints. This algorithm most of the time introduces new waypoints called "navigation surfacing waypoints". Note that mission surfacing waypoints are also navigation ones because the navigation profile has also to be computed from and towards them.

These two algorithms are illustrated for the example described in section III. Intermediate (because it does not represent a real plan) result of one fleet mission planning A1 is shown on Fig. 6 whereas result of several navigation planning A2 between each couple of mission surfacing waypoints is shown on Fig. 7. The final fleet plan includes all the surfacing actions. Current hypothesis for the definition of the fleet mission planning problem are: different types of gliders, temporal constraints between phases, phases that can be defined as optional, number of gliders needed for each phase or specific gliders already assigned to phases by the operator, wanted formation or not when more than one glider (if a formation is imposed we assume that the set of gliders is given), temporal information about starting and finishing times of each phase.

The fleet mission planning problem (A1) is modelled from the current mission definition and the current situation of the fleet as a Constraint Satisfaction Problem (CSP):

• The current mission definition (coming from the operator through the MMI) defines the current set of phases to be performed on a given horizon as well as logical temporal constraints between phases. This definition may also contain constraints on some aspects of the problem definition (intervals of values for start date as well as end date, phases pattern can be free or not, intermediate transit can be implicit or not…)

• The current fleet situation built from data collected from gliders and ongoing mission execution gather all information about the current status of each glider: true position if on surface or expected position and time of next surfacing, estimated status of battery level, current phase...

The global model embeds several sub-parts dedicated to specific aspects of the problem:

• Flow variables and constraints for gliders (as each glider visits a subset of the set of phases) and phases route definition;

• Assignment variables and constraints to select the set of gliders that will be used to perform each phase;

• Temporal variables and constraints bearing on starting time, duration and ending time for phases;

• Temporal and spatial constraints for each glider path definition (sequence of mission surfacing waypoints) taking into account pattern, formation and splitting aspects;

• Consumption variables and constraints for each glider involved in the mission.

These sub-problems are connected through the set on decision variables (mainly Booleans) defining the possible assignment of each glider to a phase and the possible flow of gliders between couples of phases.

Satisfaction algorithms can be used to solve this CSP. Moreover by defining a cost function based on some variables of the CSP, optimization can be performed too. The chosen framework for building and solving the CSP is the CHOCO tool [START_REF] Prud'homme | Choco Documentation[END_REF]. In a second phase we intend to test the possible use of the inCELL framework [START_REF] Pralet | Dynamic online planning and scheduling using a static invariant-based evaluation model[END_REF].

Inputs for the glider navigation planning problem A2 are: glider data, initial and final mission surfacing waypoints, glider action cost model introduced in section IV, 3D currents data (direction and speed by depth layers), bathymetry data (depth), and forbidden areas. The environment is meshed in cells. Output is the sequence of navigation surfacing waypoints between the two given mission surfacing waypoints.

Three algorithms have been implemented to solve A2. The first one gives quickly a solution then slowly the optimal one by browsing the tree structure of solutions to optimise each next waypoint looking for the global plan cost. It could be used offline during the preparation of a fleet mission. The second is a kind of A* algorithm with a traversal study of the tree structure of solutions; a first solution is obtained slowly but the optimal one quickly. It could be used when all goes well during mission. The third, a Monte Carlo Tree Search (MCTS), is efficient for the first waypoint but next waypoints have to be forced to obtain a global plan. It could be used when real and estimated trajectories are significantly different which means a quick next action is required.

An example of a result of the fist algorithm is shown on Fig. 8. These algorithms will be evaluated on real data and within the fleet mission planning A1.

VI. SUPERVISION FUNCTION

The supervision function triggers planned gliders actions at computed diving times and follows their execution. When a glider surfaces, it checks the current situation and selects next appropriate steps: triggering a glider dive, updating glider navigation plan, updating fleet mission plan. The supervision function can then call the planning function (see section V) to update/enhance a plan.

More broadly, asynchronous events received by supervision and which could trigger this replanning process (cf. Fig. 2) come from gliders but also from the operator and from the environment (via the database). Events can concern one or several gliders and one or several phases. For a given event, different strategies can be applied. Three types of events are identified:

• Extreme events leading to mission stop: stop order from the operator, end mission in a near future, and no planning solution found. This type of event turns off the supervision of all other events.

• Events which reaction concerns the fleet: leaving order to gliders from the operator (whatever the reason), entering order of new gliders from the operator, graph of phases modified by the operator, maximal duration reached for one phase, acoustic detection, new environment data available, and critical problem of a glider when surfacing. This type of events systematically leads to the triggering of a fleet mission planning.

• Events which reaction could concern only one glider: surfacing of a glider, availability of a glider for diving (after data loading), and anticipation of glider dive for safety reason. This type of events could trigger a fleet mission planning or lead to no reaction.

To react correctly to these asynchronous events, the supervision function must monitor mission situation:

• Executed, disrupted, or discarded phases are removed from the graph of phases.

• List of gliders contains those that the PSS can supervise.

• Current fleet plan is the last one computed by the planning function.

• Main glider state includes last diving data, next navigation waypoint and relative action (when under sea), next diving time (when in surface), ongoing phase, battery data.

• Main phase state includes ongoing or pending status, start date (real or planned), assigned gliders, and area (remaining area to be covered for coverage phases).

These situations, their updates and their reactions are managed through the formalism of Petri nets that allows the composition of elementary behaviours. A Petri net [START_REF] Murata | Petri nets: properties, analysis and applications[END_REF] is a bipartite graph with two types of nodes: places model the state of the system and transitions model its evolution. Petri net PSS_loop on Fig. 9 shows the call of planning function when an event occurs. Current state of this Petri net is given by a black token in the PSS_WAIT place. When an event occurs (triggering of event2assess transition), the current situation is assessed (token moves from PSS_WAIT place to SIT_ASSESS place); a decision is taken by supervision either to do nothing (assess2wait transition puts the net back in the PSS_WAIT state) or to call the planning function (assess2pln transition from SIT_ASSESS place to MISS_PLN place); when correctly computed (PLNok2react transition), the planned reaction is ran (place REACTION, behaviour is in other Petri nets).

VII. IMPLEMENTATION WITHIN MMS

MMS is centralised on a web-based platform which allows multiple users to prepare, supervise and analyse gliders missions. This web-based platform is described in [START_REF] Besson | A centralized webbased platform for combined glider and satellite observation analysis[END_REF]. Its actual version has been successfully tested in real conditions for the human piloting of gliders. Next step is to connect the PSS.

As supervision function is central in the PSS architecture, it thus carries out connections within MMS (see Fig. 2). To simulate the occurrence of events and evaluate PSS architecture, a client-server application has been developed. The server is the PSS that supervises the mission according to received events. The client is a especially created web page that allows sending asynchronous events towards the PSS server.

Running this simulation test-bed will complete standalone tests of the planning and supervision functions.

VIII. CONCLUSIONS AND FUTURE WORK

AGLIMMS project is in its final year. Planning and supervision functions have been specified, with parameters, constraints, variables and criteria for the first, and event and behavioural reactions for the second. Planning algorithms and supervision mechanisms are under development in parallel with standalone tests. Connection within the MMS is simulated through a client-server application.

Work is the next months prior to demonstration will be focused on the validation of the Planning and Supervision functions based on realistic test cases.

A demonstration with three SeaExplorer operating in a formation is planned late 2019 in the Mediterranean Sea. It will highlight the whole communication process, the PSS reaction to real events, and the ground replanning process to update the fleet mission plan. The scenario will be close to the example given in this paper with path following and systematic coverage phases.

Further work on this thematic could be to integrate the fleet of gliders in a global system that will include vessels and other types of autonomous vehicles and require for instance global planning capacities, to consider heterogeneous gliders and various payloads, to implement more generic team strategies and in relation to enhance communication strategies, to mathematically prove system robustness, and last but not least to onboard implement some decisional autonomy.

Four

  types of underwater gliders are on the market: SeaExplorer from Alseamar, Seaglider from University of Washington, Spray from Scripps Institution of Oceanography, and Slocum from Teledyne Webb Research. Several fleet experiments [2] [3] [4] [5] have been achieved with a fleet of above gliders, which were either individually operated by pilots discussing together or virtually moored. The structure of the paper is as follows. Section II presents AGLIMMS project in which this work falls. Section III introduces the Planning and Supervision System (PSS) and an illustrative example, then section IV the glider action cost model. Sections V and VI explains respectively planning and supervision problems then proposed solution algorithms. Section VII shows current PSS implementation in MMS before highlighting future work in section VIII. II. AGLIMMS PROJECT Three complementary partners are involved in the French AGLIMMS project (2016-2019): Alseamar (project lead) operate SeaExplorer (Fig. 1) since 2017; ONERA do extensive research on unmanned vehicles cooperation since 2007; CentralWeb develop big data management and webmapping systems.

Fig. 1 .

 1 Fig. 1. SeaExplorer underwater gliders from Alseamar.

Fig. 2 .

 2 Fig. 2. Planning and Supervision System.

Fig. 3 .

 3 Fig. 3. Example of current state of a fleet mission: areas of running phases are in green, pending in orange. Gliders estimated positions are in solid line, estimated in dashed line.

Fig. 4 .

 4 Fig. 4. Phases of the example mission. WPF = path following, MVF = virtual mooring, CSF = systematic coverage. Gliders are g1, g2 and g3.

Fig. 5 .

 5 Fig. 5. Navigation profile without (above) and with (below) currents (2D display).

Fig. 6 .

 6 Fig. 6. Fleet mission planning A1: mission surfacing waypoints and the computed plan are in red.

Fig. 7 .

 7 Fig. 7. Glider navigation planning A2: navigation surfacing waypoints and computed plans (between couples of mission surfacing waypoints) are in blue.

Fig. 8 .

 8 Fig. 8. Cartesian XY view of result of a glider navigation planning (A2) between two mission surfacing waypoints. Resolution is 3D.

Fig. 9 .

 9 Fig. 9. Example of Petri net modelling in the supervision function.

ACKNOWLEDGEMENT

This work is partially founded by the French defence procurement agency DGA.


text:
Imen Hammami
email: imen.hammammi@gmail.com

Jean Dezert
email: jean.dezert@onera.fr

Grégoire Mercier Cto

Kohonen-based Credal Fusion of Optical and Radar Images for Land Cover Classification

Keywords: Land cover classification, remote sensing, image fusion, Kohonen map, belief functions

This paper presents a Credal algorithm to perform land cover classification from a pair of optical and radar remote sensing images. SAR (Synthetic Aperture Radar) /optical multispectral information fusion is investigated in this study for making the joint classification. The approach consists of two main steps: 1) relevant features extraction applied to each sensor in order to model the sources of information and 2) a Kohonen mapbased estimation of Basic Belief Assignments (BBA) dedicated to heterogeneous data. This framework deals with co-registered images and is able to handle complete optical data as well as optical data affected by missing value due to the presence of clouds and shadows during observation. A pair of SPOT-5 and RADARSAT-2 real images is used in the evaluation, and the proposed experiment in a farming area shows very promising results in terms of classification accuracy and missing optical data reconstruction when some data are hidden by clouds.

I. INTRODUCTION

In this paper we consider the problem of the joint classification of farming landscape images from a multispectral optical image denoted I MS and a Synthetic Aperture Radar (SAR) image denoted I SAR . We assume the image registration problems resolved in a pre-processing step and we focus only on the classification problem. Classically, the classification is obtained from an information fusion approach which can be addressed at distinct levels [START_REF] Pohl | Multisensor image fusion in remote sensing: concepts, methods and applications[END_REF]:

• A pixel level (low level fusion) [START_REF] Jixian | Multi-source remote sensing data fusion: status and trends[END_REF]: Ideally, the images should be fused at this pixel information level, in which raw data extracted from each pixel, such as spectral or temporal information of the considered sources, are used. However, the design of efficient techniques for achieving this fusion is very difficult due to the complexity of joint processing of heterogeneous data, such as optical and radar data. This pixel-based heterogeneous fusion requires the use of accurate co-registered images that are often derived from a resampling process in a preprocessing step. The fusion techniques at this level require high computational resources and time. • At feature level (mid level fusion) [START_REF] Dromigny-Badin | Image fusion using evidence theory: applications to medical and industrial images[END_REF]: The feature (objet) level fusion is a mid (intermediary) level fusion approach.

It is mainly based on the extraction of one, or more, feature maps by computing the relevant descriptors of objects from each input image under analysis. Then, a fusion strategy must be applied to combine attributes from different images related to a same spatial area. This approach induces a loss of information in the processing due to the replacement of original raw data by the attributes extracted which usually are not 100% reliable. • At decision level (high level fusion) [START_REF] Dong | Survey of multispectral image fusion techniques in remote sensing applications[END_REF]: The decision level operates directly on individual decisions found by applying a proper processing for each image analyzed separately. Although this level is considered the most robust among the three, the fusion solution solution is not globally optimal since it seeks to optimize each source individually at first. There is no joint image processing for making the classification.

Due to the complexity of development of efficient low level fusion methods and their high computational resource requirement, most of users working in remote sensing prefer to work at mid level fusion or high level fusion. In this paper, we propose a new method for joint classification from optical and radar images developed at the mid fusion level exploiting features information being a good compromise between low level and high level fusion techniques. Our main contribution lies in the joint use of images to extract the features in each image before making their fusion for the classification. For this, Kohonen's unsupervised map classification framework [START_REF] Kohonen | The self-organizing map[END_REF], [START_REF] Kraaijveld | A nonlinear projection method based on Kohonen's topology preserving maps[END_REF] is used to provide an effective way for dealing with the heterogeneity of the data sources. Our technique is based on the belief functions because they offer a mathematical framework for modeling epistemic or subjective uncertainty (i.e., uncertainty resulting from imperfect knowledge), as well as stochastic uncertainty (i.e., uncertainty resulting from data diversity). In the context of this work, the multispectral optical image I MS is considered as reliable if no clouds or shadows affect the data. The SAR image I SAR is known to be noisy [START_REF] Oliver | Understanding synthetic aperture radar images[END_REF] due to unavoidable speckle noise effects. Moreover, with no polarimetric capability (which is the case in this study), the information that can be extracted from the SAR image is much less reliable than the one from the optical data. Different features are extracted from I SAR and I MS . For I SAR , we use six local texture descriptors [START_REF] Materka | Texture analysis methods -a review[END_REF], [START_REF] Haralick | Texture features for image classification[END_REF] to take into account the information computed from the neighborhoods surrounding the SAR pixel under concern. At the feature level, each pixel (i, j) of the SAR image is characterized by a 6-tuple 1 denoted κ(i, j). The featured-SAR image is denoted I κSAR . The surface reflectance of the different bands of the multispectral optical image I MS are used directly as features. To guarantee that the classes are defined in a homogeneous fashion both from the optical and from the SAR observations, a first coarse joint classification is performed to link the spectral signatures of I MS and the SAR texture descriptors of I κSAR . For this, a simple K-means classifier is used with an appropriate distance that accounts for the heterogeneity of the joint observation and cross-calibration factor, which in turn accounts for the relative dynamics between the two observations. Once this coarse joint classification is done, at each pixel (i, j) of both registered images the fusion is achieved thanks to belief functions. This of course requires the construction of BBAs m MS (•) and m κMS (•) defined with respect to a common classification frame of discernment, and a decision-making strategy for making the classification. This paper is organized as follows. Sections II and III recall respectively the basics of belief functions, and Kohonen's map technique. Section IV proposes a technique of estimation of belief functions from heterogeneous data in the case of complete and missing optical data. Section IV. Some results obtained from real remote sensing images are presented in Section V, followed by a conclusion with perspectives in the last section.

II. BASICS OF BELIEF FUNCTIONS

Shafer introduced Belief Functions (BF) [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] to model the epistemic uncertainty, to reason under uncertainty and to combine distinct sources of evidence. BF are based on preliminary Dempster's works [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF], [START_REF] Dempster | A generalization of Bayesian inference[END_REF]. We consider a finite discrete frame of discernement (FoD) Θ = {θ 1 , θ 2 , . . . , θ n }, with n > 1, and where all exhaustive and exclusive elements of Θ represent the set of the potential solutions of the problem under concern. The set of all subsets of Θ is the power-set of Θ denoted by 2 Θ . The number of elements (i.e. the cardinality) of 2 Θ is 2 |Θ| . A basic belief assignment (BBA) associated with a given source of evidence is defined as the mapping m(•) : 2 Θ → [0, 1] satisfying the conditions m(∅) = 0 and

A∈2 Θ m(A) = 1. The quantity m(A) is the mass of belief of subset A committed by the source of evidence (SoE). A focal element X of a BBA m(•) is an element of 2 Θ such that m(X) > 0. The set of all focal elements of m(•) is denoted F Θ (m) {X ⊆ Θ|m(X) > 0} = {X ∈ 2 Θ |m(X) > 0}.
The belief and plausibility functions are defined by

Bel(A) = X∈2 Θ X⊆A m(X) and P l(A) = X∈2 Θ X∩A =∅ m(X). (1)
The width P l(A) -Bel(A) of the belief interval [Bel(A), P l(A)] is called the uncertainty on A committed by the SoE. It represents the imprecision on the (subjective) 1 More precisely, κ(i, j) (µ, σ, β 1 , β 2 , f 6 , f 5 ), i.e the mean, standard deviation, skewness, kurtosis, sum-average and inverse different momentsee [START_REF] Haralick | Texture features for image classification[END_REF] for details and their definitions in appendix.

probability of A granted by the SoE which provides the BBA m(•). When all elements of F Θ (m) are only singletons, m(•) is called a Bayesian BBA [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] and its corresponding Bel(•) and P l(•) functions are homogeneous to a same (subjective) probability measure P (•). The functions m(•), Bel(•) and P l(•) are one-to-one [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. If a source of evidence is known reliable at some degree with a reliability factor2 α ∈ [0, 1], then its BBA m(•) must be discounted (modified) to work with a new BBA defined by m (X) = α • m(X) for X = Θ, and by m (Θ) = α • m(Θ) + (1 -α). This is the classical Shafer's discounting method [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. A more refined reliability discounting technique called contextual discounting can be found in [START_REF] Mercier | Contextual discounting of belief functions[END_REF]. In the combination of BBAs, one can also take into account (if one wants) the importance factor of each source. The importance level of the sources enters in many multi-criteria decision-making problems involving BF. For this, one commits to each source a priority factor3 β ∈ [0, 1] and we apply the importance discounting method proposed in [START_REF] Smarandache | Fusion of sources of evidence with different importances and reliabilities[END_REF]. More precisely, we have to work with a new (non normalized) BBA defined by m (X) = β • m(X) for X = ∅, and by

m (∅) = β • m(∅) + (1 -β) = 1 -β.
Of course, after the combination of BBAs a normalization step is necessary before making the final decision.

In DST framework, Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] did propose to combine s ≥ 2 distinct sources of evidence represented by BBAs m 1 (.), . . . , m s (.) over the same FoD with Dempster's rule (i.e. the normalized conjunctive rule) which is commutative and associative. The justification and behavior of Dempster's rule have been strongly disputed from both theoretical and practical standpoints as reported in [START_REF] Dezert | Why Dempster's fusion rule is not a generalization of Bayes fusion rule[END_REF]- [START_REF] Dezert | On the validity of Dempster's fusion rule and its interpretation as a generalization of Bayesian fusion rule[END_REF]. In particular, the high (or even very low) conflict level between the sources can be totally ignored by Dempster's rule which is a very serious problem [START_REF] Dezert | On the validity of Dempster-Shafer theory[END_REF]. Also, Shafer's conditioning (based on Dempster's rule) is inconsistent with the probabilistic conditioning and Generalized Bayes' Theorem [START_REF] Dezert | Total Belief Theorem and Generalized Bayes' Theorem[END_REF]. We don't assume, nor claim, that there must be a single optimal belief fusion method that fits all situations, but it is reasonable and it makes sense to test and evaluate other combination techniques in applications. In this work, we focus on the Proportional Conflict Redistribution rule #6 (PCR6) developed in DSmT (Dezert-Smarandache Theory [START_REF] Smarandache | Advances and applications of DSmT for information fusion[END_REF]) because it preserves the specificity of information through its conflict redistribution principle and it usually provides better performances than Dempster's rule as reported by many researchers working in different fields of applications. Of course, the price to pay for using PCR6 is its higher computational complexity with respect to Dempster's rule. The decision-making from a BBA can be done in many ways (max of belief, max of plausibility, max of some approximate probability measure built from the BBA, etc) [START_REF] Dezert | Decision-Making with Belief Interval Distance[END_REF].

III. KOHONEN'S MAP TO MODEL BF

Kohonen's map, performs an unsupervised neural networkbased competitive learning [START_REF] Kohonen | The self-organizing map[END_REF]. It defines a non-linear mapping from an input space (say R p ) onto a regular array of N × M nodes as illustrated in Fig. 1. A reference vector, also called a weighting vector, w(i, j) ∈ R p is associated with the node at each position (i, j) with 1 ≤ i ≤ N and 1 ≤ j ≤ M . During the projection of x ∈ R p to the map, the input vector is compared to each w(i, j) by using an appropriate metric. The best match w x (the node that minimizes this metric) is defined as the output of the Self-Organizing Map (SOM): thus, the input data x is mapped onto the SOM at location (i x , j x ), where w(i x , j x ) is the most similar neuron to x (also called the winning neuron).

Thanks to its topology-preserving property, samples in R p which are not too far from each other are projected onto the map in the same area. Thus, the neurons w are located on the map according to their similarity. Moreover, it can be considered that the SOM achieves a non-uniform quantization that transforms x to w x by minimizing the given metric.

We present briefly the method for the construction of BBA from Kohonen's SOM. This method has been presented in details in [START_REF] Hammami | The Kohonen map for Credal Classification of Large Multispectral Images[END_REF], [START_REF] Hammami | Kohonen's map approach for the belief mass modeling[END_REF], and it has already been applied successfully for SPOT images classification. In this paper we apply it for joint classification of optical and SAR images.

We consider a given FoD Θ = {θ 1 , θ 2 , . . . , θ K } which correspond to the set of classes for the pixels in the images under analysis. The aim is to classify each pixel into one of these classes. Typically, θ 1 may represent the cereal class, θ 2 the bare soil class, etc. The BBA construction requires a SOM and an initial classification that defines the classes θ i , i = 1, . . . , K. The class centers in R p are denoted as C 1 , . . . , C K while their projection onto the map are associated with the weighting vectors {w C1 , . . . , w C K }. The mass of the class θ i for i = 1, . . . , K is defined directly on the map by

       m(x ∈ θ i ) ∼ 1, if w x =w Ci m(x ∈ θ i ) ∼ d map (w x , w Ci ) -1 K =1 d map (w x , w C ) -1 , otherwise (2)
where d map (•, •) stands for the distance along Kohonen's map.

It is mainly based on Euclidean norm, and it uses the index that locates the two vectors on the map. That is

d map (w 1 , w 2 ) = (i w1 -i w2 ) 2 + (j w1 -j w2 ) 2 (3)
if w 1 (resp. w 2 ) is located at position (i w1 , j w1 ) (resp.

(i w2 , j w2 )) on the map. The mass of compound hypotheses, say m(x ∈ θ k ∪ θ ), is defined directly in the feature space because our definition of disjunctions of hypotheses expresses the lack of discrimination. We relate this mass value to a scaling effect between the sample x under concern and the disjunction of classes θ k and θ as follows

m(x ∈ θ k ∪ θ ) ∼ 1 -tanh(β • z) (4)
with

z = d R p (C k , C ) d R p (x, C k ) + d R p (x, C ) 0 < k, K, k = .
and where the β parameter stands for the level of ambiguity.

d R p (•, •) denotes a distance metric in R p . It can be defined through Euclidean norm L 2 (R p ), but also through a spectral perspective, such as the spectral angle mapper or the spectral information divergence. It may also be based on Kullback-Leibler divergence or mutual information, when dealing with SAR [START_REF] Inglada | A New Statistical Similarity Measure for Change Detection in Multitemporal SAR Images and its Extension to Multiscale Change Analysis[END_REF].

The justification for modeling m(x ∈ θ k ∪ θ ) by ( 4) is as follows. If a sample x is very close to the class center C k in comparison with any other class center C , then there is no ambiguity in believing that x belongs to class θ k , whereas if the distances between x and the class centers C k and C are of the same scale then it becomes uncertain to commit x with the class θ k or with the class θ , and it is more prudent in this case to commit x with the disjunction θ k ∪ θ .

The mass committed to the total ignorance Θ is based on the distance of a sample x to the map. It therefore requires a comparison of the distances in the feature space, as well as the distances in Kohonen's space. It can be expressed as follows:

m(x ∈ Θ) ∼ 1 -min d R p (x, w x ) d R p (C x , w Cx ) , d R p (C x , w Cx ) d R p (x, w x ) (5)
where C x is the class center of x, w Cx is its projection on the map. The formula (5) can be used as an outlier detector. In order to work with normalized BBAs in the fusion process, we apply a classical normalization step.

Ideally, if we consider that the both sources are totally reliable, this BBA construction could be applied separately with I MS (using a SOM MS and Euclidean distance metric) to build m MS (•) for each pixel of I MS . Similarly, one could apply this method with I κMS (using a SOM SAR and Euclidean distance metric) to build m SAR (•) for each pixel of I κMS . In practice however, the things are no so perfect and SAR images appear as non-reliable source of information because of speckle noise which makes their extracted features less reliable. The optical images can also appear as partially unreliable in the areas where there are clouds or cloud shadows because the optical information can be missing at these pixel locations. To overcome this serious problem a joint SOM is defined through a hybrid neuron definition that takes into account the p spectral signatures of the optical data, and the six texture descriptors of the SAR data.

A. Joint and enslaved Kohonen's SOM for BBA construction

In a general context, let x = {x 1 , x 2 , . . . , x p } ∈ R p and y = {y 1 , y 2 , . . . , y q } ∈ R q be the two heterogeneous observations provided by two heterogeneous sensors. The input samples of the proposed hybrid SOM are done through the co-located observations z = (x, y) with which a distance must be associated. This (hybrid-joint) distance is a linear combination of two metrics defined respectively in R p and in R q , that is

d(z, z ) d R p (x, x ) + α • d R q (y, y ), (6)
with z = (x, y) and z = (x , y ) belong to R p+q . The parameter α is a cross-calibration factor, that accounts for the relative dynamics between x and y. Since d(z, z ) is defined though a cartesian product metric space and two distances d R p (•, •) and d R q (•, •), it remains a distance.

According to this definition of a joint feature space and its related distance metrics, it is possible to perform a training of a joint SOM where the weighting vectors are defined with w z = (w x , w y ) ∈ R p+q . Nevertheless, this joint processing of our heterogeneous data does not take into account, at this stage, the reliability of the sources. Optical and SAR data interfere in the same manner in the location of each class center on the map (the class center of winning neurons w C k , k ∈ {1, 2, . . . , K}), while SAR data is much less reliable than optical data in the land cover classification accuracy (but in case of missing optical information due to cloud occultation of course). Then, instead of a joint processing, an enslaved processing is set up to perform SOM training and a SOMbased BBA of the SAR data only.

Enslaved SOM training starts with a classical SOM training of the optical data only, which generates a SOM MS . Then, the neurons of SOM MS are concatenated by the q = 6 texture-based SAR feature components to fit the R p+q of the joint processing. The training of this hybrid map begins, but only the last q-components (dedicated to the SAR data) are modified. In this case, the optical part is preserved, while the SAR part follows the optical part in the location of classes on the map (locations of the winning neurons w C k ). This defines SOM κSAR|MS from which the BBA m SAR (•) is built for each pixel of the SAR image. The scheme of fusion of reliable 4 multispectral optical and SAR image is given in Fig. 2.

B. Dealing with missing optical data

When the optical sensor acquires a scene in the presence of clouds, two kinds of missing data must be considered: the parts of the data that are hidden by the clouds themselves, and the 4 Here we assume that the information at a given pixel in the optical image is not missing, i.e. there is no cloud occultation effect at this pixel. parts that are affected by the shadow of the clouds. In this case, a binary mask allows the training of Kohonen's map using valid (reliable) optical data only. The BBA of pixels located in occulted areas could be simply modeled by a vacuous BBA, that is by m MS (Θ) = 1. However this standpoint ignores the joint observation of optical and radar sensors. In fact, the optical pixel may be recovered by using the joint Kohonen's map SOM κSAR|MS which reflects the links between optical and radar parts in the observation. Based on this remark, when a pixel x MS is considered missing in the optical image due to the presence of clouds or shadow, the co-located radar observation y SAR is considered. Its winning neuron in the radar restriction of SOM κSAR|MS allows us to consider the optical part of Kohonen's map. This spectral signature is substituted for x MS to recover the missing information.

IV. OPTICAL AND RADAR JOINT CLASSIFICATION

For each pixel of the fused image, the fusion process consists in combining the BBAs m MS (•) and m SAR (•) built by the method presented in the previous section. In order to take into account the reliability of the optical and radar sources in the fusion process, we apply a different discounting on the BBA m MS (•) and on m SAR (•) before their combination. We apply a contextual discounting [START_REF] Mercier | Contextual discounting of belief functions[END_REF] of m MS (•) and an importance discounting [START_REF] Smarandache | Fusion of sources of evidence with different importances and reliabilities[END_REF] of m SAR (•) to better fuse the two heterogeneous sources of information that do not yield the same kind of physical information with a different level of discrimination efficientcy. More precisely, before making the combination of the two BBAs, we discount them as follows:

• Contextual discounting of m MS (•) m MS (X) = Y1,Y2∈2 Θ |Y1∪Y2=X m MS (Y 1 )m λi θi (Y 2 ) (7)
where m λi θi (•) represents the BBA attributed to θ i ∈ Θ which enters in the contextual discounting method. λ i ∈ [0, 1] is a tuning parameter associated with the class θ i . This BBA is defined by m λi θi (∅) = λ i and m λi θi

(θ i ) = 1 -λ i . • Importance discounting of m SAR (•) m SAR (∅) = 1 -β m SAR (X) = β • m SAR (X), if X ∈ 2 Θ \ {∅} (8)
where β is a chosen importance factor in [0, 1]. The fusion is achieved by combining the discounted BBAs m MS (•) and m SAR (•) with PCR6 rule [START_REF] Smarandache | Advances and applications of DSmT for information fusion[END_REF], followed by a normalization step as explained in [START_REF] Smarandache | Fusion of sources of evidence with different importances and reliabilities[END_REF]. The justification of PCR6 rule comes from the fact this rule makes a rational redistribution of partial conflict proportionally only to the masses of propositions that are involved in the conflict. Hence, PCR6 rule offers the preservation of the specificity of the information in the fusion process. PCR6 is preferred to Dempster's rule because Dempster's rule provides counterintuitive results [START_REF] Dezert | On the validity of Dempster-Shafer theory[END_REF], and it does not respond to importance discounting as shown in [START_REF] Smarandache | Fusion of sources of evidence with different importances and reliabilities[END_REF]. The result of the combination of m MS (•) and m SAR (•) with PCR6 is denoted symbolically as m P CR6 (•) = [m MS ⊕ m SAR ](•). The normalization step after applying PCR6 is necessary in our application because m MS (∅) > 0 implies m P CR6 (∅) > 0. Therefore, after applying PCR6 rule we normalize all masses values m P CR6 (X) for X = ∅ by m P CR6 (X)/(1 -m P CR6 (∅)) → m P CR6 (X), and we set m P CR6 (∅) = 0 in order to get a fused and normalized BBA from which the final decision will be made. In this work, we adopt the classical maximum of betting probability, denoted BetP (•), decision-making strategy introduced in [START_REF] Smets | The transferable belief model[END_REF]. We use this decision-making strategy because it is simple to implement, and it offers a compromise between the pessimistic (max of belief) and the optimistic (max of plausibility) decisional attitudes. More precisely, we classify the pixel under analysis in the class θ i ∈ Θ, if BetP (θ i ) = max j {BetP (θ j )}.

V. EXPERIMENTAL RESULTS

We present the results of joint classification obtained with this new fusion method applied on real SPOT-5 and RADARSAT-2 images taken at almost same period of time (April 2015). Our study covers a 11.5 × 9km 2 area of the flat agricultural Beauce region, located in the south-west of Paris, France which is characterized by very large fields dominated by rape and cereal (wheat, barley, corn) crops. In this experience, the multispectral image I MS is acquired by the SPOT-5 French satellite, and the SAR iamage I SAR is acquired by the RADARSAT-2 Canadian satellite in Ultra-Fine mode. The SPOT-5 image is characterized by a size of 1145 × 903 pixels, with a spatial resolution of 10m, and has four bands (Green (G), Red (R), Near InfraRed (NIR) and Medium InfraRed (MIR)). The RADARSAT-2 image is composed of 3850 × 3010 pixels, with a spatial resolution of 3m. Both HH and HV polarization channels are available in the SAR image, but we did use only HH polarization because it interacts more efficiently than HV-polarization with agricultural crops. Fig. 3-(a) and Fig. 3-(b) show, respectively, the false color composite of the SPOT image and its registered SAR image. We distinguish five different land cover types (classes) in Fig. 3(a):

A. Preprocessing of SAR image

The fusion process is achieved at the coarser resolution of both images, that is at the resolution of SPOT5 image (i.e. 10m). For this, the RADARSAT-2 image is first processed in order to extract the local statistical parameters (µ, σ, β 1 , β 2 , f 6 , f 5 )); the processing is done through a sliding window of 51 × 51 for (µ, σ, β 1 , β 2 ) and 15 × 15 for the Haralick texture parameters (f 6 , f 5 ) estimation. In order to prevent bordering effects between parcels, a naive map extracted from the multispectral image serves as a mask in the local parameter estimation of radar data. This guarantees a parameter estimation on effective homogeneous areas and preserves the borders of each parcel. The choice of analysis window size is based on our object dimensions of interest. Therefore, our analysis windows size is proportional to those field dimensions. The 3m-resolution SAR feature image is then downsampled to a 10m-resolution SAR feature image, and then registered to the SPOT geometry. Fig. 4 shows a false color composition of the radar information at a 10m resolution. The color composition is shown with RGB=(µ, σ, β 1 ). In comparing visually Fig.

B. Fusion & joint classification results without missing data

Here we consider the simplest case where there is no missing data in the optical image when there is no cloud nor shadow effect. In order to merge the belief degrees associated with each pixel from the two input images, a unified FoD is required. The simple classes of this FoD are defined using the K-means unsupervised classifier [START_REF] Hartigan | Algorithm AS 136: A K-Means Clustering Algorithm[END_REF], where the parameter K is set to 5. One applies the K-means classifier to a stack image collecting the spectral information of the SPOT image I MS and the texture information of the (downsampled and registered) SAR image I κSAR . We have taken the cross-calibration factor α = 5.10 -3 in (6) because it corresponds to the average ratio between the mean value of the optical and radar data. The ground truth was collected in July 2015, while the data were acquired in April, and as a result, any ambiguity between different kinds of crops could not be resolved, as many fields were still in the seedling state. Hence, it was decided that only five classes could be discriminated. The results of the joint classification are shown in Fig. 5. This classification is used as reference image (the ground truth of classification result) for the performance analysis of our fusion method.

Fig. 6 presents the Kohonen's maps SOM MS and SOM κSAR|MS trained with the optical and the SAR information. SOM MS in Fig. 6-(a), shows essentially the distribution of spectral signatures representing cover soils (in red) and bare soils (in cyan) in this farming area. In the marginal dark zone between these two types of spectral signatures, there is a location dedicated to man-made structures (buildings and roads) and a forest area (darker, at the bottom left area of SOM MS ). The enslaved map SOM κSAR|MS of Fig. 6-(b), shows the same kind of neurons at the same location on the map, viewed by the textural parameters extracted from SAR data.

One sees that the purple area in the middle of SOM κSAR|MS map, appears quite homogeneous, while we have two very different areas in SOM MS map. This illustrates the fact that the optical sensor is mainly sensitive to the presence of chlorophyll in this farming area, while the radar is sensitive to the surface roughness. However, surface roughness may appear similar, from a radar observation, in bare soil and also in cover fields, depending on the plantation. Nevertheless, the wooded area is clearly discriminated from SAR sensor. The wooded area appears in brown-yellow in Fig. 4 and at the bottom left of SOM κSAR|MS in Fig. 6-(b). The SAR sensor does not help in the discrimination between bare soil and cover soil; nevertheless, it easily discriminates the 2 kinds of cover fields that appear in red from a SPOT point of view (e.g., Fig. 5), as an area at the top right of the SOM κSAR|MS map appears in brown. It helps to do discrimine between corn seed in winter and in early spring. The use of the enslaved radar part of SOM κSAR|MS allows the Credal classification to tackle this ambiguity in order to improve the final joint classification. From Kohonen's maps SOM MS and SOM κSAR|MS , the estimation of BBAs is then performed. The contextual discounting factors λ i involved in [START_REF] Oliver | Understanding synthetic aperture radar images[END_REF] were calculated using results given by the confusion matrix [START_REF] Elouedi | Discountings of a belief function using a confusion matrix[END_REF] derived from the cross decisions given by the optical source and the decisions of the reference data only. Each λ i is estimated using the percentage of correct classifications of the target class θ i . In our application, the contextual reliability factors are: λ 1 = 0.85, λ 2 = 0.86, λ 3 = 0.52, λ 4 = 0.9, and λ 5 = 0.87. Based on the subjective attribute of SAR source, we have taken β = 0.4 as importance discounting factor for discounting the BBA m SAR (•) with ( 8). It maximizes class discrimination of the overall process. Fig. 7 shows the classification result drawn from the fusion of optical and SAR information based on PCR6 rule and max of BetP (•) decision-making strategy. The measure of performance (i.e. confusion matrix) is established by comparing the result of Fig. 7 with the classification ground truth of Fig. [START_REF] Kohonen | The self-organizing map[END_REF]. A Correct Classification Rate (CCR) of 77.25% has been evaluated from Table I, and a Kappa Index of 0.74 which is considered to a good agreement in [START_REF] Cohen | A coefficient of agreement for nominal scales[END_REF]. The accounting for source reliability for each class allows the decision making to mitigate ambiguous fields such as those which are slightly covered, but may still be considered as bare soils, or the seedling fields which may not have the same surface roughness. This impacts mainly the class C 5 (fields mainly barley), which can be classified as C 2 or C 3 , depending on the roughness signature and density of chlorophyll. The same analysis has been done with missing data in the optical image. For this, we have applied artificial occultation masks on SPOT5 image to simulate the presence of clouds or shadows. Two masked regions, namely, zone 1 and zone 2, were selected in order to hide different kinds of ground cover. The mask of zone 1 mainly covers a non-agricultural area composed of 18,056 pixels, while the mask of zone 2 covers some covered and some bare soils composed of 6,300 pixels, as shown in Fig. 8(a). The joint classification result is shown in Fig. 8(b). A quantitative analysis gives an overall classification accuracy of CCR = 73.94% which is very close to CCR = 77.25% obtained when processing the cloud-free SPOT5 image with SAR image.

VI. CONCLUSION

A new method has been proposed for combining complementary information drawn from optical and radar remote sensing data for land cover classification in agricultural areas. The proposed approach refers to the direct combination of heterogeneous data at the pixel level in the features space, and considers the image registration problems as resolved. This new approach is based on hybrid training of Kohonen's map using heterogeneous data for mass functions estimation. This step helps to deal with the heterogeneity of data sources by representing those in the same semantic meaning through co-located observations. The methodology benefits from this joint training of heterogeneous data to restore missing parts of optical data. It is worth noting that the enslaved processing is relevant when one of two sources of information is considered to be more accurate than the other. When the cloud coverage is too significant, the related source of information can no longer be considered as sufficiently accurate. In that case, a joint processing should be preferred in order to recover missing data. Our experimental tests on real images showed the benefit of heterogeneous joint processing in the analysis of a complex farming area, even in the case of missing data. A deeper analysis should be performed to better link the heterogeneous data fusion process with Credal theory and the hybrid training of Kohonen map when joint training data is available.

APPENDIX -SAR TEXTURE DESCRIPTORS

Let p(i), i = 0, 1, ..., N g -1 be the pmf of the intensity levels occurrence, where N g is the number of gray levels. p(i) is calculated by dividing the histogram value h(i) by the total number of pixels of the odd sliding window of size w × w, that is p(i) h(i)/w.w. We use the following SAR texture descriptors:

• The mean µ: It is the average level of intensity of the region of texture

µ = E[i] = Ng-1 i=0 i × p(i)
• The standard deviation σ: it describes the variation of intensity around the mean where p(i, j) is the value of the Gray Level Cooccurrence Matrix (GLCM) at the cell (i, j) [START_REF] Haralick | Texture features for image classification[END_REF]. • Haralick's descriptor f 6 : It is the sum average descriptor defined as

f 6 = 2Ng i=2 i × Ng-1 k=0 Ng-1 =0 k+ =i p(k, )

Figure 1 .

 1 Figure 1. Schematic view of Kohonen's Self-Organizing Map [22].

Figure 2 .

 2 Figure 2. Fusion framework between a reliable optical multispectral image and a SAR image.

• C 1 :

 1 The brown class C 1 corresponds to wooded areas. C 1 appears in light grey on SAR image of Fig. 3(b). • C 2 : The dark red field class C 2 corresponds mainly to durum wheat (planted in winter). This class has no explicit signature in the SAR image. (a) SPOT5/Take5 data acquired on April 20 th , 2015. False color composite: RGB = (NIR, R, G) c CNES (b) Registered RADARSAT-2 HH F5 mode Ascending acquired on April 23 rd , 2015. RADARSAT-2 Data ans Products c MacDONALD, DETTWILER and ASSOCIATES LTD -All Rights Reserved

Figure 3 .

 3 Figure 3. Multispectal (a) and radar (b) used in the experiment (Beauce region, France).

Figure 4 .• C 3 :

 43 Figure 4. Iκ SAR image corresponding to the SAR texture information with false color composition: RGB=(µ, σ, β 1 )

  (4) with (3)(a), one sees that C 1 class (wooded area) appears in yellow in Fig. (4), C 3 class (rape) appears in light brown and C 4 appears in dark magenta.

Figure 5 .

 5 Figure 5. Reference image built from Unsupervised K-means classification (with K = 5 classes) applied jointly on multispectral and SAR information

Figure 6 .

 6 Figure 6. SOM maps of size 65 × 65 neurons. The map (a) has been trained with the optical data only, while the map (b) has been enslaved to map (a) and trained with radar data. Hence, co-located neurons bring the same ground information between the 2 maps

Figure 7 .

 7 Figure 7. Joint classification result with PCR6 and max of BetP strategy.

  (a) The original SPOT image with missing data at the location of the mask. (b) Classification of the heterogeneous data with missing data.

Figure 8 .

 8 Figure 8. Results for joint classification with simulated cloud cover. The decision is performed by the maximum of pignistic probability over all simples hypotheses

σ 2 •-µ σ 3 =• 2 = E i -µ σ 4 =- 1 i=0(i -µ) 4 σ 4 × p(i) - 3 •

 2324143 = E[(i -µ) 2 ] = [ µ) 2 × p(i)] -1/The skewness β 1 : It describes how symmetric intensity distribution is around the mean β 1 = E i The normalized kurtosis β 2 : It measures the flatness excess of the intensity distribution β NgHaralick's descriptor f 5 : It is the inverse difference moment which measures local similarities in the image i -j) 2 × p(i, j)

α = 1 means that the source is 100% reliable, whereas α = 0 means that the source is totally unreliable.

β = 1 means that the source is considered as 100% important for the analyst, whereas β = 0 means that the source is considered as not important at all.