Abstract:
The invention relates to a method, a device and a system for merging information originating from several non-independent sensors. This invention makes it possible to prevent the same item of information from being reckoned twice during merging. The solution afforded consists of the creation of a new combination operator applying to latent belief structures. Said latent belief structures are obtained previously from conventional belief functions. These conventional belief functions are produced directly on the basis, for example, of the sensors of the system. The invention also proposes a means of transforming these latent belief structures into a probability distribution useful for decision taking.

Description:
PRIORITY CLAIM 
       [0001]    This application claims priority to PCT Patent Application Number PCT/EP2008/059660, entitled Procédé, Dispositif Et Système Pour La Fusion D&#39;informations Provenant De Plusieurs Capteurs, filed Jul. 23, 2008, also claiming priority to FR 07 05528, filed Jul. 27, 2007. 
     
    
     TECHNICAL FIELD 
       [0002]    The invention relates to the merging of information originating from several sensors and more particularly to the merging of imperfect information originating from non-independent sources. 
       BACKGROUND OF THE INVENTION 
       [0003]    Systems integrating several sensors are used in a great variety of fields such as site surveillance, maintenance, robotics, medical diagnosis or meteorological forecasting. Such systems for example carry out classification, identification and tracking functions in real time. 
         [0004]    To derive the best from multi-sensor systems, it is necessary to use an effective information merging scheme to combine the data originating from the various sensors of the system and generate a decision. 
         [0005]    According to the known art, certain information merging schemes rely on Dempster-Shafer theory (theory generalizing probability theory) and thus use belief functions. Belief functions are known for their ability to faithfully represent imperfect information. Information comprising an inaccuracy or an uncertainty or incomplete information is called imperfect information. The sensors of a system are considered to be sources of imperfect information, notably because of their inaccuracy. The term sensor is intended here in the broad sense. It includes physical devices for data acquisition (camera, radar, etc.) but also devices for processing these data. It is possible to establish a belief function on the basis of the data provided by most commercially available sensors. Belief combining schemes can be used. By their very nature these schemes are therefore particularly appropriate to the problem of the merging of imperfect information arising from sensors. 
         [0006]    A belief function can be represented with the aid of a function m called the mass distribution defined on a set of proposals Ω which is a space of possible worlds with finite cardinal. It associates degrees of belief, lying between 0 and 1, with parts A (groups of proposals, also called subsets) of Ω. These degrees of belief are determined on the basis of the available information. The set of parts of Ω is denoted 2 Ω .
       A mass distribution m satisfies the following two conditions:
           the mass ascribed to a subset A lies between 0 and 1: 0≦m(A)≦1, ∀ A ⊂ Ω.   the sum of the masses of all the subsets is equal to one:   
               
 
         [0000]    
       
         
           
             
               
                 ∑ 
                 
                   A 
                   ⊆ 
                   Ω 
                 
               
                
               
                 m 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
             
             = 
             1 
           
         
       
     
         [0010]    A multi-sensor system of classifier type used for the optical recognition of handwritten characters may be considered by way of nonlimiting example. It is assumed that the system is intended to determine whether the character formed on an image I is one of the letters ‘a’, ‘b’ or ‘c’. We therefore have a set of proposals Ω={a, b, c}. Each of the sensors of the system is a classifier which itself provides an item of information about the character to be recognized in the form of a belief function. It is assumed that there are two sensors of this type in our example. The following table gives an example of a belief function m 1  produced by a first sensor of the system and defined on 2 Ω . 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                   
               
               
                   
                 A 
                 m 1 (A) 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Ø 
                 0 
               
               
                   
                 {a} 
                 0 
               
               
                   
                 {b} 
                 0 
               
               
                   
                 {a, 
                 0.4 
               
               
                   
                 b} 
               
               
                   
                 {c} 
                 0 
               
               
                   
                 {a, c} 
                 0 
               
               
                   
                 {b, 
                 0.4 
               
               
                   
                 c} 
               
               
                   
                 Ω 
                 0.2 
               
               
                   
                   
               
             
          
         
       
     
         [0011]    In this example, the sensor considers that it is as probable that the character to be recognized belongs to {a, b} as that it belongs to {b, c}. (m 1 ({a, b})=m 1 ({b, c})=0.4). m 1 (Ω)=0.2 represents the ignorance, that is to say the share of doubt, of the sensor. 
         [0012]    Dempster-Shafer theory makes it possible to combine the belief functions representing information arising from different sources, so as to obtain a belief function that takes into account the influences of each of the sources. The belief function thus obtained represents the combined knowledge of the various sources of imperfect information (the sensors). 
         [0013]    However, systems relying on this theory are based on the assumption that the merged information is independent. In practice two items of information can be considered to be independent if the sources associated with these items of knowledge are wholly unrelated. The concept of independence is fundamental since one of the constraints of information merging is to avoid counting the same item of information twice. It is obvious that this independence assumption is not satisfied by a certain number of multi-sensor systems. For example in the problem of the optical recognition of handwritten characters, the sensors may not be independent. Indeed, according to the known art, shape recognition schemes rely on automatic learning techniques using learning bases. If the sensors have been trained on the same learning bases, that is to say if the sensors have been set up using the same data, then the independence assumption required by the known art merging systems is not satisfied and therefore these systems may not be used. 
       SUMMARY OF THE INVENTION 
       [0014]    The invention is aimed notably at alleviating the problem cited previously by proposing a method, a device and a system for merging information originating from several non-independent sensors. This invention makes it possible to prevent the same item of information from being reckoned twice during merging. The solution afforded consists of the creation of a new combination operator applying to latent belief structures. Said latent belief structures are obtained previously from conventional belief functions. These conventional belief functions are produced directly on the basis, for example, of the sensors of the system. The invention also proposes a means of transforming these latent belief structures into a probability distribution useful for decision taking. 
         [0015]    For this purpose, the subject of the invention is a method for merging information originating from several sensors c 1 , c 2 , . . . , c n , n being the number of sensors, said method comprising the following steps:
       the acquisition of belief functions m 1 , m 2 , . . . , m n  arising from the sensors c 1 , c 2 , . . . , c n , said belief functions m 1 , m 2 , . . . , m n  being defined on a set of proposals Ω,   the calculation of latent belief structures LBS 1 , LBS 2 , . . . , LBS n  for each belief function m 1 , m 2 , . . . , m n , said latent belief structures LBS 1 , LBS 2 , . . . , LBS n  each comprising a pair of functions (w 1   c , w 1   d ), (w 2   c , w 2   d ), . . . , (w n   c , w n   d ),
 
characterized in that said sensors c 1 , c 2 , . . . , c n  are non-independent and in that it furthermore comprises the following steps:
   the calculation of a merged latent belief structure LBS m , said merged latent belief structure LBS m  comprising a pair of functions w m   c  and w m   d  calculated by applying a so-called weak-rule function to the functions (w 1   c , w 1   d ), (w 2   c , w 2   d ), . . . , (w n   c , w n   d ):       
 
         [0000]        w   m   c ( A )= w   1   c ( A )           w   2   c ( A )          . . .            w   n   c ( A ),  A⊂Ω 
 
         [0000]        w   m   d ( A )= w   1   d ( A )           w   2   d ( A )          . . .            w   n   d ( A ),  A⊂Ω 
        w m   c  and w m   d  being defined on all the subsets A of Ω,   the calculation of a probability distribution P m  on the basis of the functions w m   c  and w m   d .       
 
         [0020]    According to a variant of the invention, the acquisition of the belief functions is direct, said sensors c 1 , c 2 , . . . , c n  giving a belief function directly. 
         [0021]    According to another variant of the invention, characterized in that the acquisition of the belief functions is indirect, said belief functions m 1 , m 2 , . . . , m n  being calculated on the basis of the information arising from the sensors c 1 , c 2 , . . . , c n . 
         [0022]    According to a characteristic of the invention, the calculation of a probability distribution comprises the following steps:
       the calculation of commonality functions q m   c  and q m   d  on the basis of the functions w m   c  and w m   d ,   the calculation of functions pl m   c  and pl m   d  on the basis of the commonality functions q m   c  and q m   d ,   the calculation of probability distributions P m   c  and P m   d  on the basis of the functions pl m   c  and pl m   d ,   the calculation of a merged probability distribution P m  corresponding to the merged latent belief structure and calculated on the basis of the probability distributions P m   c  and P m   d .       
 
         [0027]    According to a characteristic of the invention, the calculation of the commonality functions q m   c  and q m   d  uses the following equations: 
         [0000]    
       
         
           
             
               
                 q 
                 m 
                 c 
               
                
               
                 ( 
                 B 
                 ) 
               
             
             = 
             
               
                 ∏ 
                 
                   B 
                   ⊄ 
                   A 
                   ⋐ 
                   Ω 
                 
               
                
               
                   
               
                
               
                 
                   w 
                   m 
                   c 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 q 
                 m 
                 d 
               
                
               
                 ( 
                 B 
                 ) 
               
             
             = 
             
               
                 ∏ 
                 
                   B 
                   ⊄ 
                   A 
                   ⋐ 
                   Ω 
                 
               
                
               
                   
               
                
               
                 
                   w 
                   m 
                   d 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
             
           
         
       
     
         [0028]    According to a characteristic of the invention, the calculation of the functions pl m   c  and pl m   d  uses the following equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       p 
                        
                       
                           
                       
                        
                       
                         1 
                         m 
                         c 
                       
                        
                       
                         ( 
                         A 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                              
                             ≠ 
                             B 
                           
                           ⊆ 
                           A 
                         
                       
                        
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             
                                
                               B 
                                
                             
                             + 
                             1 
                           
                         
                          
                         
                           
                             q 
                             m 
                             c 
                           
                            
                           
                             ( 
                             B 
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   
                     A 
                     ∈ 
                     
                       2 
                       Ω 
                     
                   
                   , 
                   
                     
                       p 
                        
                       
                           
                       
                        
                       
                         1 
                         m 
                         c 
                       
                        
                       
                         ( 
                          
                         ) 
                       
                     
                     = 
                     0 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       p 
                        
                       
                           
                       
                        
                       
                         1 
                         m 
                         d 
                       
                        
                       
                         ( 
                         A 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                              
                             ≠ 
                             B 
                           
                           ⊆ 
                           A 
                         
                       
                        
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             
                                
                               B 
                                
                             
                             + 
                             1 
                           
                         
                          
                         
                           q 
                           m 
                           d 
                         
                          
                         
                           ( 
                           B 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   
                     A 
                     ∈ 
                     
                       2 
                       Ω 
                     
                   
                   , 
                   
                     
                       p 
                        
                       
                           
                       
                        
                       
                         1 
                         m 
                         d 
                       
                        
                       
                         ( 
                          
                         ) 
                       
                     
                     = 
                     0 
                   
                 
               
             
           
         
       
     
         [0029]    According to a characteristic of the invention, the calculation of the probability distributions P m   c  and P m   d  uses the following equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         P 
                         m 
                         c 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             ω 
                             k 
                           
                           } 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         κ 
                         
                           - 
                           1 
                         
                       
                        
                       p 
                        
                       
                           
                       
                        
                       
                         1 
                         m 
                         c 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             ω 
                             k 
                           
                           } 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ∀ 
                   
                     
                       ω 
                       k 
                     
                     ∈ 
                     Ω 
                   
                 
               
             
           
         
       
       
         
           
             
               with 
                
               
                   
               
                
               κ 
             
             = 
             
               
                 ∑ 
                 
                   
                     ω 
                     k 
                   
                   ∈ 
                   Ω 
                 
               
                
               
                 p 
                  
                 
                     
                 
                  
                 
                   1 
                   m 
                   c 
                 
                  
                 
                   ( 
                   
                     { 
                     
                       ω 
                       k 
                     
                     } 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         P 
                         m 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             ω 
                             k 
                           
                           } 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         κ 
                         
                           - 
                           1 
                         
                       
                        
                       p 
                        
                       
                           
                       
                        
                       
                         1 
                         m 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             ω 
                             k 
                           
                           } 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ∀ 
                   
                     
                       ω 
                       k 
                     
                     ∈ 
                     Ω 
                   
                 
               
             
           
         
       
       
         
           
             
               with 
                
               
                   
               
                
               κ 
             
             = 
             
               
                 ∑ 
                 
                   
                     ω 
                     k 
                   
                   ∈ 
                   Ω 
                 
               
                
               
                 p 
                  
                 
                     
                 
                  
                 
                   1 
                   m 
                   d 
                 
                  
                 
                   ( 
                   
                     { 
                     
                       ω 
                       k 
                     
                     } 
                   
                   ) 
                 
               
             
           
         
       
     
         [0030]    According to a characteristic of the invention, the calculation of the merged probability distribution P m  uses the following equation: 
         [0000]    
       
         
           
             
               
                 P 
                 m 
               
                
               
                 ( 
                 
                   { 
                   
                     ω 
                     k 
                   
                   } 
                 
                 ) 
               
             
             = 
             
               
                 κ 
                 
                   - 
                   1 
                 
               
                
               
                 
                   
                     P 
                     m 
                     c 
                   
                    
                   
                     ( 
                     
                       { 
                       
                         ω 
                         k 
                       
                       } 
                     
                     ) 
                   
                 
                 / 
                 
                   
                     P 
                     m 
                     d 
                   
                    
                   
                     ( 
                     
                       { 
                       
                         ω 
                         k 
                       
                       } 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               with 
                
               
                   
               
                
               κ 
             
             = 
             
               
                 ∑ 
                 
                   
                     ω 
                     k 
                   
                   ∈ 
                   Ω 
                 
               
                
               
                 
                   
                     P 
                     m 
                     c 
                   
                    
                   
                     ( 
                     
                       { 
                       
                         ω 
                         k 
                       
                       } 
                     
                     ) 
                   
                 
                 / 
                 
                   
                     P 
                     m 
                     d 
                   
                    
                   
                     ( 
                     
                       { 
                       
                         ω 
                         k 
                       
                       } 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0031]    The subject of the invention is also a device for merging information comprising:
       means for the acquisition of belief functions m 1 , m 2 , . . . , m n  on the basis of the information arising from sensors,   means for merging the belief functions originating from the means for the acquisition,   characterized in that the means for merging information implement the method according to the invention.       
 
         [0035]    The subject of the invention is also a system for merging information comprising:
       at least two sensors, collecting data and providing information,   means for merging information originating from said sensors, said merging means being linked to the sensors,   characterized in that said sensors are non-independent and in that the means for merging information comprise the device for merging information according to the invention.       
 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0039]    The invention will be better understood and other advantages will become apparent on reading the detailed description given by way of nonlimiting example and with the aid of the figures among which: 
           [0040]      FIG. 1  represents an exemplary embodiment of a system for merging information according to the invention. 
           [0041]      FIG. 2  represents a chart of the steps implemented in the method for merging information according to the invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0042]    It is proposed to illustrate the method according to the invention, by way of nonlimiting example, with the previously mentioned multi-sensor system of classifier type used for the optical recognition of handwritten characters. It is assumed that the system is intended to determine whether the character formed on an image I is one of the letters ‘a’, ‘b’ or ‘c’. 
         [0043]    The term sensor is intended here in the broad sense. It includes physical devices for data acquisition (camera, micro, etc.) but also devices for processing these data, in the example: a classifier. 
         [0044]    An exemplary embodiment of a system for merging information according to the invention is illustrated by  FIG. 1 . Such a system comprises a first classification system C 1    101  and a second classification system C 2    102 . It is assumed that each of the two classification systems comprises a device allowing the acquisition of the image I  104  such as a video camera for example, and means for processing the signal comprising character recognition on the basis of the captured image I and the creation of a belief function m(A) indicating the degree of belief of the sensor in the alternatives ({a}, {b} and {c}) and in the sets of alternatives ({a,b}, {a,c}, and {b,c}). 
         [0045]    According to the known art, shape recognition schemes rely on automatic learning techniques using predetermined learning bases. In this example, we assume that the two classification systems C 1    101  and C 2    102  are not independent since they have been trained on the same learning base. The independence assumption made in the merging systems according to the known art is therefore not satisfied. 
         [0046]    The system furthermore comprises a device  103  for merging the information originating from the sensors C 1  and C 2 . The device  103  for merging the information comprises:
       means for the acquisition  105  of belief functions m 1 , m 2 , . . . , m n  arising from the sensors  101 ,  102 ,   means  106  for merging the belief functions m 1 , m 2 , . . . , m n  originating from said sensors  101 ,  102 .       
 
         [0049]    The acquisition can be done in a direct manner when a sensor produces a belief function directly or in an indirect manner when the belief function is calculated on the basis of the information provided by the sensor. 
         [0050]    These means  106  are noteworthy in that they implement the method according to the invention and in that they make it possible to calculate a probability distribution on the basis of the belief functions provided by the two non-independent sensors C 1  and C 2 . The merging means  106  can be a computer or an integrated circuit implementing the method according to the invention. 
         [0051]    The system must recognize in the acquired image one of the three letters ‘a’, ‘b’ or ‘c’. We therefore have a set of proposals Ω={a, b, c}. 
         [0052]    The table below presents the various intermediate calculations performed by applying the method according to the invention. 
         [0053]    The first column contains the subsets of Ω. The second and third columns contain the values of the functions m 1 (A), m 2 (A) provided by the two sensors. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 A 
                 m 1  (A) 
                 m 2  (A) 
                 w c   1  (A) 
                 w d   1  (A) 
                 w c   2  (A) 
                 w d   2  (A) 
                 w c   m   
                 w d   m   
                 P m   
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Ø 
                 0 
                 0 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                   
               
               
                 {a} 
                 0 
                 0 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 9/19 
               
               
                 {b} 
                 0 
                 0 
                 1 
                 5/9 
                 1 
                 1 
                 1 
                 5/9 
                 5/19 
               
               
                 {a, b} 
                 0.4 
                 0 
                 1/3 
                 1 
                 1 
                 1 
                 1/3 
                 1 
                   
               
               
                 {c} 
                 0 
                 0 
                 1 
                 1 
                 1 
                 5/9 
                 1 
                 5/9 
                 5/19 
               
               
                 {a, c} 
                 0 
                 0.4 
                 1 
                 1 
                 1/3 
                 1 
                 1/3 
                 1 
                   
               
               
                 {b, c} 
                 0.4 
                 0.4 
                 1/3 
                 1 
                 1/3 
                 1 
                 1/3 
                 1 
                   
               
               
                 Ω 
                 0.2 
                 0.2 
               
               
                   
               
             
          
         
       
     
         [0054]    The method for merging information originating from several sensors according to the invention is applied hereinafter to the system comprising two sensors, said sensors being non-independent. However the method according to the invention can be applied to a system comprising a larger number of sensors. 
         [0055]      FIG. 2  represents a chart of the steps implemented in the method for merging information according to the invention. 
         [0056]    The first step  201  of the method according to the invention is the acquisition of the belief functions m 1 , m 2  arising from the sensors. In the example, two functions m 1 , m 2  are acquired, corresponding respectively to the two sensors C 1  and C 2 . In this example, the first sensor C 1  considers that it is as probable that the character to be recognized belongs to {a, b} as that it belongs to {b, c}. (m 1 ({a, b})=m 1 ({b, c})=0.4). m 1 (Ω)=0.2 represents the ignorance, that is to say the share of doubt of the sensor. The second sensor C 2  considers that it is as probable that the character to be recognized belongs to {a, c} as that it belongs to {b, c}. 
         [0057]    The second step  202  of the method according to the invention is the calculation of latent belief structures LBS 1  and LBS 2  for each belief function m 1 , m 2  obtained on the basis of each of the sensors. Each of the latent belief structures LBS 1  and LBS 2  is calculated by using the canonical decomposition of a belief function. The canonical decomposition is a function denoted w and calculated on the basis of the belief function m by way of a function q called the commonality function. The canonical decomposition w 1  of the belief function m 1  is therefore calculated  206  on the basis of the function q 1 . The function q 1  is calculated  205  with the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         q 
                         1 
                       
                        
                       
                         ( 
                         A 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           B 
                           ⊇ 
                           A 
                         
                       
                        
                       
                         
                           m 
                           1 
                         
                          
                         
                           ( 
                           B 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   A 
                   ∈ 
                   
                     2 
                     Ω 
                   
                 
               
             
           
         
       
     
         [0058]    We have, for example, q 1 ({a,b})=m 1 ({a,b})+m 1 (Ω)=0.4+0.2=0.6; q 1 (Ω)=m 1 (Ω)=0.2. 
         [0059]    The function is w 1  calculated  206  with the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         w 
                         1 
                       
                        
                       
                         ( 
                         A 
                         ) 
                       
                     
                     = 
                     
                       
                         ∏ 
                         
                           B 
                           ⊇ 
                           A 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             q 
                             1 
                           
                            
                           
                             ( 
                             B 
                             ) 
                           
                         
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             ( 
                             
                               
                                  
                                 B 
                                  
                               
                               - 
                               
                                  
                                 A 
                                  
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   A 
                   ∈ 
                   
                     
                       2 
                       Ω 
                     
                      
                     \ 
                      
                     
                       { 
                       Ω 
                       } 
                     
                   
                 
               
             
           
         
       
       
         
           
             where the operator |A| represents the cardinal of the set A. 
           
         
       
     
         [0061]    We have for example: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       w 
                       1 
                     
                      
                     
                       ( 
                       
                         { 
                         
                           a 
                           , 
                           b 
                         
                         } 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         
                           q 
                           1 
                         
                          
                         
                           ( 
                           
                             { 
                             
                               a 
                               , 
                               b 
                             
                             } 
                           
                           ) 
                         
                       
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ( 
                           
                             
                                
                               
                                 { 
                                 
                                   a 
                                   , 
                                   b 
                                 
                                 } 
                               
                                
                             
                             - 
                             
                                
                               
                                 { 
                                 
                                   a 
                                   , 
                                   b 
                                 
                                 } 
                               
                                
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     × 
                     
                       
                         
                           q 
                           1 
                         
                          
                         
                           ( 
                           Ω 
                           ) 
                         
                       
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ( 
                           
                             
                                
                               Ω 
                                
                             
                             - 
                             
                                
                               
                                 { 
                                 
                                   a 
                                   , 
                                   b 
                                 
                                 } 
                               
                                
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       0.6 
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ( 
                           
                             2 
                             - 
                             2 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     × 
                     
                       0.2 
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ( 
                           
                             3 
                             - 
                             2 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     0.2 
                     0.6 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     / 
                     3 
                   
                 
               
             
           
         
       
     
         [0062]    The canonical decomposition w 2  of the belief function m 2  is calculated in a similar manner. 
         [0063]    A latent belief structure LBS 1  is a pair of functions w 1   c  and w 1   d  representing respectively the confidence and the distrust for a given set A. The calculation  207  of the functions w 1   c  and w 1   d , whose values appear in the fourth and fifth columns of the above table, uses the following equations: 
         [0000]    
       
         
           
             
               
                 
                   w 
                   1 
                   c 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
               = 
               
                 1 
                 ⋀ 
                 
                   
                     w 
                     1 
                   
                    
                   
                     ( 
                     A 
                     ) 
                   
                 
               
             
             , 
             
               A 
               ∈ 
               
                 
                   2 
                   Ω 
                 
                  
                 \ 
                  
                 
                   { 
                   Ω 
                   } 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   w 
                   1 
                   d 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
               = 
               
                 1 
                 ⋀ 
                 
                   1 
                   
                     
                       w 
                       1 
                     
                      
                     
                       ( 
                       A 
                       ) 
                     
                   
                 
               
             
             , 
             
               A 
               ∈ 
               
                 
                   2 
                   Ω 
                 
                  
                 \ 
                  
                 
                   { 
                   Ω 
                   } 
                 
               
             
           
         
       
       
         
           
             where the operator           is the minimum between two values. 
           
         
       
     
         [0065]    We have, for example, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       w 
                       1 
                       c 
                     
                      
                     
                       ( 
                       
                         { 
                         
                           a 
                           , 
                           b 
                         
                         } 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     1 
                     ⋀ 
                     
                       
                         w 
                         1 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             a 
                             , 
                             b 
                           
                           } 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     ⋀ 
                     
                       1 
                       / 
                       3 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       1 
                       / 
                       3 
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                       
                           
                       
                        
                       
                           
                       
                     
                      
                     
                       
                         w 
                         1 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             a 
                             , 
                             b 
                           
                           } 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     ⋀ 
                     
                       1 
                       
                         
                           w 
                           1 
                         
                          
                         
                           ( 
                           
                             { 
                             
                               a 
                               , 
                               b 
                             
                             } 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     ⋀ 
                     
                       1 
                       
                         1 
                         / 
                         3 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   1 
                 
               
             
           
         
       
     
         [0066]    We proceed in a similar manner to calculate the functions w 2   c  and w 2   d  whose values appear in the sixth and seventh columns of the above table. 
         [0067]    The method according to the invention is noteworthy in that it furthermore comprises the following steps. 
         [0068]    The third step  203  of the method according to the invention is the calculation of a merged latent belief structure LBS m  by applying a rule called a weak rule to the values calculated in the previous step  202 . The merged latent belief structure LBS m  takes the form of a pair of functions w m   c (A) and w m   d (A). The weak rule is defined by: 
         [0000]        w   m   c ( A )= w   1   c ( A )           w   2   c ( A ),  Aε 2 Ω \{Ω}
 
         [0000]        w   m   d ( A )= w   1   d ( A )           w   2   d ( A ),  Aε 2 Ω \{Ω}
 
         [0069]    The fourth step  204  of the method according to the invention is the calculation of a probability distribution P m  on the basis of the merged latent belief structure LBS m . 
         [0070]    According to the known art, it is possible to obtain a probability distribution on the basis of a mass distribution. The LBS obtained in the previous step  203  can be converted into a mass distribution, however this mass distribution is potentially signed, that is to say it can take its values in the set of reals IR rather than in the interval [0,1]. Now, the scheme according to the known art for obtaining a probability distribution on the basis of a mass distribution is not applicable to signed mass distributions. The method according to the invention is noteworthy in that it makes it possible to calculate a probability distribution directly on the basis of the merged LBS, that is to say without needing to calculate an intermediate mass distribution. 
         [0071]    The calculation  204  of a probability distribution P m  on the basis of the merged latent belief structure comprises the following sub-steps. 
         [0072]    The first sub-step  208  is the calculation of the commonality functions corresponding to the functions w m   c  and w m   d  calculated previously and complying with the following equations: 
         [0000]    
       
         
           
             
               
                 q 
                 m 
                 c 
               
                
               
                 ( 
                 B 
                 ) 
               
             
             = 
             
               
                 ∏ 
                 
                   B 
                   ⊄ 
                   A 
                   ⋐ 
                   Ω 
                 
               
                
               
                 
                   w 
                   m 
                   c 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 q 
                 m 
                 d 
               
                
               
                 ( 
                 B 
                 ) 
               
             
             = 
             
               
                 ∏ 
                 
                   B 
                   ⊄ 
                   A 
                   ⋐ 
                   Ω 
                 
               
                
               
                 
                   w 
                   m 
                   d 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
             
           
         
       
     
         [0073]    We have for example: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       q 
                       m 
                       c 
                     
                      
                     
                       ( 
                       
                         { 
                         
                           a 
                           , 
                           b 
                         
                         } 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         w 
                         m 
                         c 
                       
                        
                       
                         ( 
                          
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         c 
                       
                        
                       
                         ( 
                         
                           { 
                           a 
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         c 
                       
                        
                       
                         ( 
                         
                           { 
                           b 
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         c 
                       
                        
                       
                         ( 
                         
                           { 
                           c 
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         c 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             a 
                             , 
                             c 
                           
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         c 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             b 
                             , 
                             c 
                           
                           } 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     * 
                     1 
                     * 
                     1 
                     * 
                     1 
                     * 
                     
                       1 
                       3 
                     
                     * 
                     
                       1 
                       3 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     9 
                   
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 
                   
                     
                       q 
                       m 
                       d 
                     
                      
                     
                       ( 
                       
                         { 
                         
                           a 
                           , 
                           b 
                         
                         } 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         w 
                         m 
                         d 
                       
                        
                       
                         ( 
                          
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           a 
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           b 
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           c 
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             a 
                             , 
                             c 
                           
                           } 
                         
                         ) 
                       
                     
                      
                     
                       
                         w 
                         m 
                         d 
                       
                        
                       
                         ( 
                         
                           { 
                           
                             b 
                             , 
                             c 
                           
                           } 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     * 
                     1 
                     * 
                     
                       5 
                       9 
                     
                     * 
                     
                       5 
                       9 
                     
                     * 
                     1 
                     * 
                     1 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     25 
                     81 
                   
                 
               
             
           
         
       
     
         [0074]    The second sub-step  209  is the calculation of the plausibility functions pl m   c  and pl m   d  on the basis of the commonality functions q m   c  and q m   d  calculated previously and complying with the following equations: 
         [0000]    
       
         
           
             
               
                 p 
                  
                 
                     
                 
                  
                 
                   1 
                   m 
                   c 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     
                        
                       ≠ 
                       B 
                     
                     ⊆ 
                     A 
                   
                 
                  
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       
                          
                         B 
                          
                       
                       + 
                       1 
                     
                   
                    
                   
                     
                       q 
                       m 
                       c 
                     
                      
                     
                       ( 
                       B 
                       ) 
                     
                   
                 
               
             
             , 
             
               A 
               ∈ 
               
                 2 
                 Ω 
               
             
             , 
             
               
                 p 
                  
                 
                     
                 
                  
                 
                   1 
                   m 
                   c 
                 
                  
                 
                   ( 
                    
                   ) 
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
               
                 p 
                  
                 
                     
                 
                  
                 
                   1 
                   m 
                   d 
                 
                  
                 
                   ( 
                   A 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     
                        
                       ≠ 
                       B 
                     
                     ⊆ 
                     A 
                   
                 
                  
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       
                          
                         B 
                          
                       
                       + 
                       1 
                     
                   
                    
                   
                     
                       q 
                       m 
                       d 
                     
                      
                     
                       ( 
                       B 
                       ) 
                     
                   
                 
               
             
             , 
             
               A 
               ∈ 
               
                 2 
                 Ω 
               
             
             , 
             
               
                 p 
                  
                 
                     
                 
                  
                 
                   1 
                   m 
                   d 
                 
                  
                 
                   ( 
                    
                   ) 
                 
               
               = 
               0 
             
           
         
       
     
         [0075]    We have for example: 
         [0000]    
       
         
           
             
               p 
                
               
                   
               
                
               
                 1 
                 m 
                 c 
               
                
               
                 ( 
                 
                   { 
                   a 
                   } 
                 
                 ) 
               
             
             = 
             
               
                 
                   q 
                   m 
                   c 
                 
                  
                 
                   ( 
                   
                     { 
                     a 
                     } 
                   
                   ) 
                 
               
               = 
               
                 1 
                 3 
               
             
           
         
       
     
         [0076]    The function pl m   d  is then calculated in a similar manner. 
         [0077]    The third sub-step  210  is the calculation of the probability distributions P m   c  and P m   d  on the basis of the plausibility functions pl m   c  and pl m   d  calculated previously. This is done via the following equations: 
         [0000]    
       
         
           
             
               
                 
                   P 
                   m 
                   c 
                 
                  
                 
                   ( 
                   
                     { 
                     
                       ω 
                       k 
                     
                     } 
                   
                   ) 
                 
               
               = 
               
                 
                   κ 
                   
                     - 
                     1 
                   
                 
                  
                 p 
                  
                 
                     
                 
                  
                 
                   1 
                   m 
                   c 
                 
                  
                 
                   ( 
                   
                     { 
                     
                       ω 
                       k 
                     
                     } 
                   
                   ) 
                 
               
             
             , 
             
               
                 ∀ 
                 
                   
                     ω 
                     k 
                   
                   ∈ 
                   
                     Ω 
                      
                     
                         
                     
                      
                     with 
                      
                     
                       
                           
                       
                        
                       
                           
                       
                     
                      
                     κ 
                   
                 
               
               = 
               
                 
                   ∑ 
                   
                     
                       ω 
                       k 
                     
                     ∈ 
                     Ω 
                   
                 
                  
                 
                   p 
                    
                   
                       
                   
                    
                   
                     1 
                     m 
                     c 
                   
                    
                   
                     ( 
                     
                       { 
                       
                         ω 
                         k 
                       
                       } 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0078]    We have for example: 
         [0000]    
       
         
           
             
               
                 P 
                 m 
                 c 
               
                
               
                 ( 
                 
                   { 
                   a 
                   } 
                 
                 ) 
               
             
             = 
             
               
                 
                   p 
                    
                   
                       
                   
                    
                   
                     1 
                     m 
                     c 
                   
                    
                   
                     ( 
                     
                       { 
                       a 
                       } 
                     
                     ) 
                   
                 
                 
                   
                     p 
                      
                     
                         
                     
                      
                     
                       1 
                       m 
                       c 
                     
                      
                     
                       ( 
                       
                         { 
                         a 
                         } 
                       
                       ) 
                     
                   
                   + 
                   
                     p 
                      
                     
                         
                     
                      
                     
                       1 
                       m 
                       c 
                     
                      
                     
                       ( 
                       
                         { 
                         b 
                         } 
                       
                       ) 
                     
                   
                   + 
                   
                     p 
                      
                     
                         
                     
                      
                     
                       1 
                       m 
                       c 
                     
                      
                     
                       ( 
                       
                         { 
                         c 
                         } 
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   
                     1 
                     3 
                   
                   
                     
                       1 
                       3 
                     
                     + 
                     
                       1 
                       3 
                     
                     + 
                     
                       1 
                       3 
                     
                   
                 
                 = 
                 
                   1 
                   3 
                 
               
             
           
         
       
       
         
           
             The function P m   d  is then calculated in a similar manner. 
           
         
       
     
         [0080]    The fourth sub-step  211  is the calculation of a probability distribution P m  corresponding to the merged latent belief structure LBS m  and calculated on the basis of the probability distributions P m   c  and P m   d : 
         [0000]    
       
         
           
             
               
                 
                   P 
                   m 
                 
                  
                 
                   ( 
                   
                     { 
                     
                       ω 
                       k 
                     
                     } 
                   
                   ) 
                 
               
               = 
               
                 
                   κ 
                   
                     - 
                     1 
                   
                 
                  
                 
                   
                     
                       P 
                       m 
                       c 
                     
                      
                     
                       ( 
                       
                         { 
                         
                           ω 
                           k 
                         
                         } 
                       
                       ) 
                     
                   
                   / 
                   
                     
                       P 
                       m 
                       d 
                     
                      
                     
                       ( 
                       
                         { 
                         
                           ω 
                           k 
                         
                         } 
                       
                       ) 
                     
                   
                 
               
             
              
             
                 
             
           
         
       
       
         
           
             
               with 
                
               
                   
               
                
               κ 
             
             = 
             
               
                 ∑ 
                 
                   
                     ω 
                     k 
                   
                   ∈ 
                   Ω 
                 
               
                
               
                 
                   
                     P 
                     m 
                     c 
                   
                    
                   
                     ( 
                     
                       { 
                       
                         ω 
                         k 
                       
                       } 
                     
                     ) 
                   
                 
                 / 
                 
                   
                     P 
                     m 
                     d 
                   
                    
                   
                     ( 
                     
                       { 
                       
                         ω 
                         k 
                       
                       } 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0081]    We therefore have: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       m 
                     
                      
                     
                       ( 
                       
                         { 
                         a 
                         } 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         
                           P 
                           m 
                           c 
                         
                          
                         
                           ( 
                           
                             { 
                             a 
                             } 
                           
                           ) 
                         
                       
                       / 
                       
                         
                           P 
                           m 
                           d 
                         
                          
                         
                           ( 
                           
                             { 
                             a 
                             } 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             P 
                             m 
                             c 
                           
                            
                           
                             ( 
                             
                               { 
                               a 
                               } 
                             
                             ) 
                           
                         
                         / 
                         
                           
                             P 
                             m 
                             d 
                           
                            
                           
                             ( 
                             
                               { 
                               a 
                               } 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             P 
                             m 
                             c 
                           
                            
                           
                             ( 
                             
                               { 
                               b 
                               } 
                             
                             ) 
                           
                         
                         / 
                         
                           
                             P 
                             m 
                             d 
                           
                            
                           
                             ( 
                             
                               { 
                               b 
                               } 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             P 
                             m 
                             c 
                           
                            
                           
                             ( 
                             
                               { 
                               c 
                               } 
                             
                             ) 
                           
                         
                         / 
                         
                           
                             P 
                             m 
                             d 
                           
                            
                           
                             ( 
                             
                               { 
                               c 
                               } 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         ( 
                         
                           1 
                           3 
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           5 
                           23 
                         
                         ) 
                       
                     
                     
                       
                         
                           ( 
                           
                             1 
                             3 
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             5 
                             23 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             1 
                             3 
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             9 
                             23 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             1 
                             3 
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             9 
                             23 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       23 
                       15 
                     
                     
                       437 
                       135 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     9 
                     19 
                   
                 
               
             
           
         
       
     
         [0082]    Likewise, we calculate P m ({b}) and P m ({c}): 
         [0000]    
       
         
           
             
               
                 P 
                 m 
               
                
               
                 ( 
                 
                   { 
                   b 
                   } 
                 
                 ) 
               
             
             = 
             
               
                 
                   P 
                   m 
                 
                  
                 
                   ( 
                   
                     { 
                     c 
                     } 
                   
                   ) 
                 
               
               = 
               
                 5 
                 19 
               
             
           
         
       
     
         [0083]    The probability distribution P m  corresponding to the example is given in the tenth column of the above table. The conclusion of the multi-sensor system according to the invention is that the most probable character on the image  1  is ‘a’.