Generalized proportional conflict redistribution rule applied to Sonar imagery and Radar targets classification

Generalized prop ortional conflict redistri bu tion rule applied to Sonar imagery and Radar targets classificat ion Arnaud Martin and Christophe Ossw ald August 17, 2021 Abstract In this chapter, w e present t w o applications in information fusion in order to ev al uate the generalized prop ortional conflict redistribution rule presented in the c hapter [5]. Most of the time the combination rules are ev al uated only on simple examples. W e study here different com b i nation rules and compare them in terms of decision on real d ata. Indeed, in real applications, we need a reliable decision and it is the final results that matter. Tw o applications are presented here: a fusion o f human ex perts opinions on the kind of underwater sediments dep i ct on sonar ima ge and a classifier fusion for radar targets recognition. Keywords: Exp erts fusion, classification, DST, DSmT, generalized PCR, S o nar, R adar. 1 In tro d uction W e hav e pres e n ted a nd discuss e d on s o me co m bina t ion rules in the chapter [5]. Our study was ess en tially o n the re dis t ribution of conflict rules. W e ha ve pro- po sed a ne w prop ortional conflict redis t ribution r ule . W e hav e seen that the decision can b e different following the rule. Mo st of the time the co m bination rules ar e ev aluated o nly on simple examples. In this chapter, we study different combination rules and compare them in terms of dec is ion on rea l data. In- deed, in real applications, w e need a reliable decis io n and it is the final results that matter. Hence, for a g iv en applicatio n, the b est combination rule is the rule given the b est r e sults. F or the decisio n step, different functions such a s credibility , plaus ibilit y and pig nis t ic probabilit y [9, 13, 2 ] are usua lly use d. In this chapter, w e present the adv an tages of the DSmT for the mo delization of real applications and also f or the com bination step. First, the principles of the DST and DSmT are recalled. W e present the fo rmalization of the b elief function mo dels, different r ules o f co m bination and decision. One the combination rule (PCR5) propo s ed by [12] for t w o exp erts is mathematically o ne of the best for the prop ortional re d istribution of the conflict applicable in the context of the DST and the DSmT. W e compare here an extension of this rule fo r mor e experts, the P CR6 rule pr e sen ted in the chapter [5]. Two applications a re presented here: a fusion o f hu man exp erts opinions on the kind of underw ater sediments depict on s o nar imag e and a classifier fusion for r adar ta rgets r ecognition. 1 The firs t a pplication relates the seab ed c haracteriza t ion, for instance in order to help the navigation o f Autonomous Underwater V ehicles or provide data to sedimentologists. The sonar images are obta ined with many imp erfections due to ins t rumentations measuring a huge n um ber o f physical da ta (geo metry of the device, co ordinates of the ship, mov emen ts of the sonar, etc.). In this kind of applications, the r ealit y is unknown. If hum an exp erts hav e to classify so nar images they can not pr o vide with certain t y the kind of sediment on the imag e. Thu s, for instance, in o rder to train a n automatic cla s sification algor ithm , w e m ust take in to account this difference and the uncertain t y of each expert. W e prop ose in this c hapter how to so lv e this human exp ert fusio n. The seco nd applica t ion a llows to really compar e the combination rules. W e present an applicatio n of classifier fusion in order to e xtract the infor mation for the auto matic target re cognition. The rea l data a r e provided b y measure s in the anechoic cham ber of E NSIET A (Bre s t, F rance) obtained illuminating 10 scale reduced (1 :48) targ ets of plane s . Hence, all the e x perimentations are controlled and the reality is known. The results o f the fusion of thre e classifier s are studied in ter ms o f go od-cla s sification rates. This chapter is or g anized as follow: In the fir st section, we recall combination rules present ed in the chapter [5] and we compare in this chapter. The section 3 prop oses a mean to fuse human exp ert’s opinions in uncertain e nvironments such as the underwater milieu. This environmen t is descr ibed with so nar images the most appro pr iate in such environmen t. The last s e ction prese nts the r e s ult s of classifiers fusion in an a pplication of radar targets recognition. 2 Bac kgrounds on com bination rules W e reca ll here the combination r ules pr e sen ted and discuss e d in the chapter [5] and c ompared on t wo real applications in the forwards sec tio ns. F or more details o n the theory bases se e the c hapter [5 ]. In the co ntext of the DST, the no n-normalized co nj unctiv e rule is one o f the most used rule a nd is given by [13] for all X ∈ 2 Θ by: m c ( X ) = X Y 1 ∩ ... ∩ Y M = X M Y j =1 m j ( Y j ) , (1) where Y j ∈ 2 Θ is the resp onse of the exp ert j , and m j ( Y j ) the asso ciated basic belie f a ssignmen ts. In this c hapter, we focus o n rules where the conflict is redistributed. With the rule given in the Dub ois and P rade r ule [3], a mixed conjunctive and disjunctiv e rule, the conflict is redistr ibuted on partial ignor ance. This rule is g iv en fo r all X ∈ 2 Θ , X 6 = ∅ by: m DP ( X ) = X Y 1 ∩ ... ∩ Y M = X M Y j =1 m j ( Y j ) + X Y 1 ∪ ... ∪ Y M = X Y 1 ∩ ... ∩ Y M = ∅ M Y j =1 m j ( Y j ) , (2) where Y j ∈ 2 Θ is the resp onse of the exp ert j , and m j ( Y j ) the asso ciated basic belie f a ssignmen ts. 2 In the context of the DSmT, the non-no rmalized conjunctive rule ca n b e used for all X ∈ D Θ and Y ∈ D Θ . The mixed rule g iv en by the equation (2) has b een r ewrite in [10], and r ecalled DSmH, for all X ∈ D Θ , X 6≡ ∅ 1 by: m H ( X ) = X Y 1 ∩ ... ∩ Y M = X M Y j =1 m j ( Y j ) + X Y 1 ∪ ... ∪ Y M = X Y 1 ∩ ... ∩ Y M ≡∅ M Y j =1 m j ( Y j )+ X { u ( Y 1 ) ∪ ... ∪ u ( Y M )= X } Y 1 ,...,Y M ≡∅ M Y j =1 m j ( Y j ) + X { u ( Y 1 ) ∪ ... ∪ u ( Y M ) ≡∅ and X =Θ } Y 1 ,...,Y M ≡∅ M Y j =1 m j ( Y j ) , (3) where Y j ∈ D Θ is the resp onse o f the exp ert j , m j ( Y j ) the a ssociated bas ic belie f assignments, and u ( Y ) is the function g iving the union that comp ose Y [11]. F or example if Y = ( A ∩ B ) ∪ ( A ∩ C ), u ( Y ) = A ∪ B ∪ C . If we wan t to take the decisio n only o n the ele ments in Θ, some rules prop ose to redistribute the conflict prop ortionally on these elemen ts. The most accom- plished is the P CR5 given in [12]. The equation for M experts, fo r X ∈ D Θ , X 6≡ ∅ is g iv en in [1] by: m PCR5 ( X ) = m c ( X ) + M X i =1 m i ( X ) X ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ ( D Θ ) M − 1 M − 1 ∩ k =1 Y σ i ( k ) ∩ X ≡∅ M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) )1 l j >i ! Y Y σ i ( j ) = X m σ i ( j ) ( Y σ i ( j ) ) X Z ∈{ X,Y σ i (1) ,...,Y σ i ( M − 1) } Y Y σ i ( j ) = Z  m σ i ( j ) ( Y σ i ( j ) ) .T ( X = Z,m i ( X ) )  , (4) where σ i counts fr o m 1 to M av oiding i :  σ i ( j ) = j if j < i, σ i ( j ) = j + 1 if j ≥ i, (5) and:  T ( B , x ) = x if B is true , T ( B , x ) = 1 if B is false , (6) W e hav e pro posed ano t her prop ortional conflict re dis t ribution P CR6 rule in the chapter [5], for M exp erts, for X ∈ D Θ , X 6 = ∅ : m PCR6 ( X ) = m c ( X ) + (7) M X i =1 m i ( X ) 2 X M − 1 ∩ k =1 Y σ i ( k ) ∩ X ≡∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ ( D Θ ) M − 1        M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) m i ( X ) + M − 1 X j =1 m σ i ( j ) ( Y σ i ( j ) )        , 1 The notation X 6≡ ∅ means that X 6 = ∅ and following the chosen mo del in D Θ , X is not one of the element of D Θ defined as ∅ . F or example, i f Θ = { A, B , C } , we can define a model for which the expert can pro vide a mass on A ∩ B and not on A ∩ C , so A ∩ B 6 = ∅ and A ∩ B = ∅ 3 where σ is defined like in (5). m i ( X ) + M − 1 X j =1 m σ i ( j ) ( Y σ i ( j ) ) 6 = 0, m c is the conjunctiv e consensus rule giv en by the equation (1). The PCR6 and PCR5 rules are exa ctly the same for in the case of 2 exper ts. W e hav e also propo s ed tw o more generalized rules given by: m PCRf ( X ) = m c ( X ) + (8) M X i =1 m i ( X ) f ( m i ( X )) X M − 1 ∩ k =1 Y σ i ( k ) ∩ X ≡∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ ( D Θ ) M − 1        M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) f ( m i ( X )) + M − 1 X j =1 f ( m σ i ( j ) ( Y σ i ( j ) ))        , with the same notations that in the equation (7), and f a n incre a sing function defined b y the mapping of [0 , 1] o nto I R + . The seco nd gener alized rule is g iven by: m PCRg ( X ) = m c ( X ) + M X i =1 X M − 1 ∩ k =1 Y σ i ( k ) ∩ X ≡∅ ( Y σ i (1) ,...,Y σ i ( M − 1) ) ∈ ( D Θ ) M − 1 m i ( X ) M − 1 Y j =1 m σ i ( j ) ( Y σ i ( j ) ) ! Y Y σ i ( j ) = X 1 l j >i ! g m i ( X ) + X Y σ i ( j ) = X m σ i ( j ) ( Y σ i ( j ) ) ! X Z ∈{ X,Y σ i (1) ,...,Y σ i ( M − 1) } g   X Y σ i ( j ) = Z m σ i ( j ) ( Y σ i ( j ) ) + m i ( X )1 l X = Z   , (9) with the same notations that in the equation (7), and g an increasing function defined b y the mapping of [0 , 1] o nto I R + . In this c hapter, we choos e f ( x ) = g ( x ) = x α , with α ∈ I R + . 3 Exp erts fusion in Sonar imagery Seab ed c haracteriz a tion serves man y useful purp oses, e.g. help the navigation of Autono m ous Underwater V e hic le s or provide data to sedimentologists. In such sonar applica t ions, seab ed imag es are obtained with many imp erfections [4]. Indeed, in o r der to build imag e s , a huge num ber of physical data (geometry of the device, co ordinates of the s hip, movemen ts of the sonar , etc.) has to be taken in to acc oun t, but these data are p olluted with a large amount of noises caused by instrumentations. In addition, there a re some interferences due to the signa l traveling on multiple paths (re fle c t ion on the b ottom or surfa ce), due to speckle, and due to fa una and flo r a. Ther efore, sonar images have a lot of 4 imper fections s uch as imprecisio n and uncertaint y; thus sedimen t cla ssification on sonar images is a difficult problem. In this kind of applica tions, the reality is unknown a nd different exper ts can pro p ose different class ifications of the image. Figur e 1 ex hibit s the differences b et w een the int erpretation and the certaint y of tw o so nar exp erts trying to differentiate the type of s edimen t (ro c k, cobbles, s and, ripple, silt) or sha dow when the informa tion is invisible. Each color corresp onds to a kind of sedimen t a nd the asso ciated cer tain t y o f the exp ert for this sedimen t expr essed in term of sure, mo derately sure and no t sure. Thus, in order to tra in an automa t ic classification algor ithm, we m ust take in to a ccoun t this differ e nc e and the uncertaint y of each exp ert. Indeed, image classifica t ion is g enerally done on a lo cal par t o f the image (pixel, or most of the time on small tiles o f e.g. 16 × 16 or 32 × 32 pixels). F or example, how a tile of ro c k la beled as not sur e m ust b e ta ken into account in the learning step o f the classifier and how to take into acco un t this tile if a nother exp ert says that it is sand? Another problem is: how should we consider a tile with more than one sediment? Figure 1: Segmentation given by tw o exp erts. In this cas e , the space of discernment Θ repr esen ts the different kind of sediments on so nar images, such as ro ck, sand, s ilt, cobble, ripple o r shadow (that mea ns no sediment infor mation). The ex p erts g iv e their p erception and belie f a ccording to their certain t y . F or instance, the exp ert can be mo derately sure of his choice when he lab els one part of the image as b elonging to a certain class, a nd be totally doubtful on a no ther part of the ima ge. Mo reo ver, on a considered tile, more tha n one s edimen t can b e pr e sen t. Consequently we have to take in to ac c oun t all these aspects of the applica- tions. In or de r to s implify , we co nsider only tw o classes in the following: the ro c k referred as A , and the sand, referred as B . The prop osed mo dels can b e easily extended, but their study is easier to understand with only t w o class es. Hence, on certain tiles, A and B can b e pres e n t for one or mo re exp erts. The belief functions have to take int o acc o un t the certa int y given by the ex - per ts (referred resp ectiv ely as c A and c B , tw o n um bers in [0 , 1]) as well as the prop ortion of the kind of sedimen t in the tile X (referred as p A and p B , also t wo num bers in [0 , 1]). W e have tw o interpretations o f “the exp ert be liev es A ”: it can mea n that the exp ert thinks that there is A o n X and no t B , or it can mean that the exp ert thinks that ther e is A on X and it can also hav e B but he do es not say a nything about it. The first interpretation yields that hypotheses 5 A a nd B ar e exclusive and with the s econd they a re not e x clusiv e. W e only study the first case: A and B are exclusive. But on the tile X , the exp ert can also provide A and B , in this ca se the tw o pr opositions “the exp ert b eliev es A ” and “the exp ert b elieves A and B ” a re not exclusiv e. 3.1 Mo dels W e have prop osed five mo dels and studied thes e mo dels for the fusio n o f tw o exp erts [6]. W e present here the three last mo dels for tw o ex p erts and tw o classes. In this ca se the conjunctive r ule (1), the mixed rule (2) and the DSmH (3) ar e similar . W e give the obtained results o n a real database for the fusion of three exp erts in sonar. Mo del M 3 In our applica tion, A , B and C canno t b e co nsidered ex clusiv e on X . In order to prop ose a mo del following the DST, w e hav e to study exclusive classes only . Hence, in our application, w e can consider a spa c e of discernment of three exclusive clas ses Θ = { A ∩ B c , B ∩ A c , A ∩ B } = { A ′ , B ′ , C ′ } , following the no tations given on the figure 2. Figure 2: Nota t ion of the intersection of tw o cla s ses A a nd B . Hence, we ca n pro pose a new mo del M 3 given b y: if the exp ert s ays A :  m ( A ′ ∪ C ′ ) = c A , m ( A ′ ∪ B ′ ∪ C ′ ) = 1 − c A , if the exp ert s ays B :  m ( B ′ ∪ C ′ ) = c B , m ( A ′ ∪ B ′ ∪ C ′ ) = 1 − c B , if the exp ert s ays C ′ :  m ( C ′ ) = p A .c A + p B .c B , m ( A ′ ∪ B ′ ∪ C ′ ) = 1 − ( p A .c A + p B .c B ) . (10) 6 Note that A ′ ∪ B ′ ∪ C ′ = A ∪ B . On o ur n umerical example we o bt ain: A ′ ∪ C ′ B ′ ∪ C ′ C ′ A ′ ∪ B ′ ∪ C ′ m 1 0 . 6 0 0 0 . 4 m 2 0 0 0 . 5 0 . 5 Hence, the conjunctive r ule, the c r edibilit y , the plausibility and the pignistic probability are g iv en by: el ement m c bel pl betP ∅ 0 0 0 − A ′ = A ∩ B c 0 0 0 . 5 0 . 216 7 B ′ = B ∩ A c 0 0 0 . 2 0 . 066 7 A ′ ∪ B ′ = ( A ∩ B c ) ∪ ( B ∩ A c ) 0 0 0 . 5 0 . 2833 C ′ = A ∩ B 0 . 5 0 . 5 1 0 . 7167 A ′ ∪ C ′ = A 0 . 3 0 . 8 1 0 . 9333 B ′ ∪ C ′ = B 0 0 . 5 1 0 . 7833 A ′ ∪ B ′ ∪ C ′ = A ∪ B 0 . 2 1 1 1 where m c ( C ′ ) = m c ( A ∩ B ) = 0 . 2 + 0 . 3 = 0 . 5 . (11) In this example, with this mo del M 3 the decis ion will b e A with the maximum of the pignistic proba bilit y . But the decision could a priori be taken also on C ′ = A ∩ B b ecause m c ( C ′ ) is the highest. W e have seen that if we wan t to take the decision on A ∩ B , we must considere d the maximum of the ma sses b ecause of inclusion relations o f the credibilit y , pla usibilit y and pignistic proba bilit y . Mo del M 4 In the con text of the DSmT, we ca n write C = A ∩ B and easily prop ose a fourth mo del M 4 , without any considera tion on the exclusivity of the classes, g iv en by: if the exp ert s ays A :  m ( A ) = c A , m ( A ∪ B ) = 1 − c A , if the exp ert s ays B :  m ( B ) = c B , m ( A ∪ B ) = 1 − c B , if the exp ert s ays A ∩ B :  m ( A ∩ B ) = p A .c A + p B .c B , m ( A ∪ B ) = 1 − ( p A .c A + p B .c B ) . (12) This la st mo del M 4 allows to r epresen t our problem without adding an artificia l class C . Thus, the model M 4 based on the DSmT gives: A B A ∩ B A ∪ B m 1 0 . 6 0 0 0 . 4 m 2 0 0 0 . 5 0 . 5 7 The obtained mass m c with the conjunctive yie lds : m c ( A ) = 0 . 30 , m c ( B ) = 0 , m c ( A ∩ B ) = m 1 ( A ) m 2 ( A ∩ B ) + m 1 ( A ∪ B ) m 2 ( A ∩ B ) = 0 . 3 0 + 0 . 2 0 = 0 . 5 , m c ( A ∪ B ) = 0 . 20 . (13) These re sults are exactly similar to the mo del M 3 . These tw o mo dels do not present ambiguit y and show that the mass on A ∩ B (rock and s and) is the highest. The generaliz ed credibility , the genera lized pla us ibilit y and the ge neralized pignistic probabilit y are giv en by: el ement m c Bel Pl GPT ∅ 0 0 0 − A 0 . 3 0 . 8 1 0 . 9333 B 0 0 . 5 0 . 7 0 . 783 3 A ∩ B 0 . 5 0 . 5 1 0 . 716 7 A ∪ B 0 . 2 1 1 1 Like the mo de l M 3 , on this example, the decision will be A with the max- im um of pignistic probability criteria. But here also the ma xim um of m c is reached for A ∩ B = C ′ . If we w an t to consider only the kind of p ossible sedimen ts A and B and do not allow their conjunction, we can use a pro portional co nflict redistr ibut ion rule such as the PCR rule: m P C R ( A ) = 0 . 30 + 0 . 5 = 0 . 8 , m P C R ( B ) = 0 , m P C R ( A ∪ B ) = 0 . 20 . (14) The credibility , the pla usibilit y and the pignistic pr obabilit y are given by: el ement m P C R bel pl betP ∅ 0 0 0 − A 0 . 8 0 . 8 1 0 . 9 B 0 0 0 . 2 0 . 1 A ∪ B 0 . 2 1 1 1 On this numerical ex ample, the decision will b e the same than the conjunctiv e rule, here the maximum of pignistic probability is reached for A (rock). In the next section we se e that is not a lw ays the case. Mo del M 5 Another mo del M 5 which can b e used in b oth the DST and the DSmT is giv en considering only one b elief function acco rding to the prop ortion by:    m ( A ) = p A .c A , m ( B ) = p B .c B , m ( A ∪ B ) = 1 − ( p A .c A + p B .c B ) . (15) 8 If for one exp ert, the tile contains only A , p A = 1, a nd m ( B ) = 0 . If for another exp ert, the tile c o n tains A and B , we tak e into accoun t the ce rtain t y and pro- po rtion of the tw o sediments but not o nly on one fo cal element. Consequently , we have simply: A B A ∪ B m 1 0 . 6 0 0 . 4 m 2 0 . 3 0 . 2 0 . 5 In the DST context, the conjunctive rule, the credibility , the plausibility and the pig nistic probability are giv en by: el ement m c bel pl b etP ∅ 0 . 12 0 0 − A 0 . 6 0 . 6 0 . 8 0 . 7 955 B 0 . 0 8 0 . 08 0 . 28 0 . 204 5 A ∪ B 0 . 2 0 . 88 0 . 88 1 In this ca se we do not hav e the pla usibilit y to decide o n A ∩ B , b e cause the conflict is on ∅ . In the DSmT co n tex t , the conjunctive rule, the gener alized credibilit y , the generalized pla usibilit y and the generaliz ed pignistic probability are given by: el ement m c Bel Pl GPT ∅ 0 0 0 − A 0 . 6 0 . 72 0 . 92 0 . 8933 B 0 . 0 8 0 . 2 0 . 4 0 . 63 33 A ∩ B 0 . 12 0 . 12 1 0 . 5267 A ∪ B 0 . 2 1 1 1 The decision with the maximum o f pignistic pr obabilit y criteria is still A . The PCR rule pr o vides: el ement m P C R bel pl betP ∅ 0 0 0 − A 0 . 69 0 . 69 0 . 89 0 . 79 B 0 . 11 0 . 11 0 . 31 0 . 2 1 A ∪ B 0 . 2 1 1 1 where m P C R ( A ) = 0 . 60 + 0 . 09 = 0 . 69 , m P C R ( B ) = 0 . 08 + 0 . 0 3 = 0 . 11 . With this mo del a nd example the PCR r ule, the decision w ill b e a lso A , and we do not hav e difference be tw een the conjunctiv e rules in the DST and DSmT. 3.2 Exp erimen tation Database Our databas e contains 42 sonar images provided b y the GESMA (Group e d’Etudes Sous-Marines de l’Atlan tique). These images were obtained 9 with a K lein 5 400 lateral so nar with a re solution of 2 0 to 30 cm in azimuth and 3 cm in r ange. The sea-b ottom depth was b et ween 15 m a n d 40 m. Three exper ts hav e manually s e gmen ted these images giving the kind of sediment (r o ck, cobble, sand, silt, ripple (horizontal, vertical or at 4 5 degr ees)), shadow or other (typically ships) parts o n images , help ed by the manual segmen- tation interface presen ted in figure 3. All sedimen ts are given with a certa in ty level (sure, mo derately sure or not sure). Hence, each pixel of every ima ge is lab eled as being either a certain type of sediment or a shadow o r other. Figure 3: Manual Segmentation Interface. The three e xperts pr ovide resp ectiv ely , 30 338, 3106 1, a nd 311 73 homoge- neous tiles, 806 9 , 7 5 27, and 7539 tiles with tw o sedimen ts, 57 5, 4 02, and 283 tiles with three sedimen ts, 14, 7, and 2 tiles with four, a nd 1, 0, and 0 tile for five sediments, and 0 for mor e. Results W e note A = r ock, B = cobble, C = sand, D = silt, E = ripple, F = shadow and G = other, hence we hav e seven cla sses and Θ = { A, B , C , D , E , F, G } . W e applied the genera lized mo del M 5 on tiles of size 32 × 3 2 g iven by:                        m ( A ) = p A 1 .c 1 + p A 2 .c 2 + p A 3 .c 3 , for ro c k, m ( B ) = p B 1 .c 1 + p B 2 .c 2 + p B 3 .c 3 , for cobble, m ( C ) = p C 1 .c 1 + p C 2 .c 2 + p C 3 .c 3 , for ripple, m ( D ) = p D 1 .c 1 + p D 2 .c 2 + p D 3 .c 3 , for sand, m ( E ) = p E 1 .c 1 + p E 2 .c 2 + p E 3 .c 3 , for silt, m ( F ) = p F 1 .c 1 + p F 2 .c 2 + p F 3 .c 3 , for shadow, m ( G ) = p G 1 .c 1 + p G 2 .c 2 + p G 3 .c 3 , for other, m (Θ) = 1 − ( m ( A ) + m ( B ) + m ( C ) + m ( D ) + m ( E ) + m ( F ) + m ( G )) , (16) where c 1 , c 2 and c 3 are the weigh ts asso ciated to the certitude r espectively: “sure”, “moder ately s ure” and “not sure”. The chosen w eights a r e here: c 1 = 2 / 3, c 2 = 1 / 2 and c 3 = 1 / 3. Indeed we hav e to consider the cases when the same kind of sediment (but with different certainties) is pr e sen t on the same tile. The prop ortion of eac h sedimen t in the tile asso ciated to these w eigh ts is noted, for instance for A : p A 1 , p A 2 and p A 3 . The total conflict b et w een the three ex perts is 0.2244. This conflict comes essentially from the difference of opinio n of the exp erts and not from the tiles 10 with mo r e than one sediment. Indeed, we hav e a weak auto-c onflict (conflict coming from the combination of the same exp ert three times). The v alues of the auto-conflict for the three exp erts are : 0.0 496, 0.0474, and 0.04 1 4. W e note a difference of decision b et ween the three com bination rules giving b y the equations (7) for the PCR6, (2) for the mixed rule and (1) fo r the conjunctive rule. The prop ortion of tiles with a different decis ion is 0 .11% b et w een the mixed rule and the co nj unctiv e rule, 0.66% b et w een the PCR6 and the mixed rule, a nd 0.73% b et w een the PCR6 and the conjunctive r ule. These res ult s s ho w that there is a difference of decision acco rding to the combination rules with the same mo del. How ev er, we c a n not know wha t is the b est decision, and s o what is the b est rule, b ecause on this application no ground truth is known. W e compare these same rules in another application, where the reality is co mpletely known. 4 Classifiers fusion in Radar target recognition Several t ypes o f classifiers have been develop ed in o rder to ex t ract the infor - mation for the automatic ta rget recognition (A TR). W e hav e no t ed that these per formances are different according to the cla ssifier and the radar target. W e hav e prop osed different approaches of infor mation fusion in o r der to o ut per form three ra dar target c la ssifiers [7]. W e pr e sen t here the res ults r eac hed by the fusion of three cla ssifiers with the conjunctiv e rule, the DSmH, the P CR5 and the P CR6. 4.1 Classifiers The three cla ssifiers used her e are the s a me than in [7 ]. The first one is a fuzzy K -nearest neighbo r classifier, the second o ne is a multila yer p erceptron (MLP ) that is a feed forward fully connected neura l netw ork. And the third one is the SAR T (Supervise d AR T) classifier [8] that uses the principle of pro tot ype generation like the AR T neural netw o rk, but unlike this one, the pro tot ypes are generated in a supervised manner. 4.2 Database The databas e is the same than in [7]. The real data were obtained in the anechoic cham ber of ENSIET A (Brest, F ra nce) using the exp erimen tal s e t up shown on figure 4. W e hav e consider ed 10 scale reduced (1:48) targ ets (Mirag e, F14, Ra f ale, T orna do, Harr ie r , Apache, DC3, F16, Jagua r and F117). Each targ e t is illuminated in the a cquisition phase with a freq uency stepp ed signal. The data snapshot contains 32 frequency steps, uniformly distr ibut ed ov er the band B = [11650 , 178 50]MHz, which res ults in a frequency increment of ∆ f = 200MHz. Cons equen tly , the sla nt range r esolution and ambiguit y window are giv en by: ∆ R s = c/ (2 B ) ≃ 2 . 4 m, W s = c/ (2∆ f ) = 0 . 7 5 m. (17) The complex sig na ture obtained from a backscattered snaps ho t is coher en tly int egrated via FFT in order to achiev e the slant ra nge pr ofile corresp onding to a g iv en asp ect of a given target. F or each of the 10 targets 15 0 range profiles 11 Figure 4: E xperimental setup. are th us genera t ed cor responding to 150 angular p ositions, from -50 degrees to 69.50 degrees, with a n angular incremen t of 0.5 0 de g rees. The data ba se is rando m ly divided in a tr a ining set (for the three sup ervised classifiers ) and test set (for the ev aluation). When all the r ange pro files are av ailable, the training set is formed by ra ndomly selecting 2/3 o f them, the others being c o nsidered as the test set. 4.3 Mo del The numerical outputs of the class ifiers for eac h tar get and eac h classifier , no r- malized b et ween 0 and 1, define the masses . In or der to keep only the most credible classes we consider the t wo highest v a lues of these outputs referred as o ij for the j th classifier and the targe t i . Hence, we obtain only three fo cal elements (tw o ta rgets a nd the ignorance Θ). The classifier does not pro vide equiv alent belief in mean. F or example, the fuzzy K -near est neighbors cla ssifier pr o vide ea sily a belief of 1 for a targe t , whereas the t wo o th er classifier s provide alwa ys belief no t null on the second target and igno rance. In o r der to give the s ame w eight to each classifier, we weigh t ea c h b elief by an adaptive thresho ld given by: f j = 0 . 8 mean ( o ij ) . 0 . 8 mean ( b ij ) , (18) where mean ( o ij ) is the mean of the b elief o f the t w o targets on all the previous considered sig nals fo r the classifier j , mean ( b ij ) is the similar mea n on b ij = f j .o ij . f j is initialized to 1. Hence, w e exp ect the mean of b elief o n the targets tends to w ard 0.8 for each class ifier, and 0.2 on Θ. Moreov er, if the b elief mass o n Θ for a given signal and classifier is le ss than 0.001, w e keep the maximum o f the mass and force the other in or der to r e a c h 0.001 on the ignorance and s o avoid total conflict with the co nj unctive rule. 4.4 Results W e have conducted the divis ion of the databa se into tr aining data base and tes t database, 8 00 times in or der to estimate b etter the go od- classification ra tes . 12 Rule Conj. DP PCR f √ x PCR g √ x PCR6 PCR g x 2 PCR f x 2 PCR5 Conj. 0 0.68 1.5 3 1.60 2.02 2.53 2.77 2.83 DP 0.68 0 0.94 1.04 1.47 2.01 2.27 2.37 PCR f √ x 1.53 0.9 4 0 0.23 0.61 1.15 1 .49 1.67 PCR g √ x 1.60 1.0 4 0.2 3 0 0.44 0.99 1.29 1.46 PCR6 2.0 4 1.4 7 0.61 0.4 4 0 0 .55 0.88 1.08 PCR g x 2 2.53 2.0 1 1.1 5 0.99 0.55 0 0.39 0.71 PCR f x 2 2.77 2.2 7 1.4 9 1.29 0.88 0 .39 0 0.51 PCR5 2.8 3 2.3 7 1.67 1.4 6 1.08 0.71 0.51 0 T a ble 1: Pro p ortion of targets with a different decision (%) W e hav e o bt ained a total conflict of 0.41 76. The auto-conflict, reached b y the combination o f the sa me class ifie r thr e e times, is 0 .1570 for the fuzzy K -nearest neighbor, 0.4055 for the SAR T and 0.3613 for the multila y er per ceptron. The auto-conflict for the fuzzy K -nearest neighbor is weak b ecause it happ ens many times that the mas s is only on one class (and ignora nce), wherea s there a re tw o classes with a non-null mass for the SAR T and multila y er p erceptron. Hence, the fuzzy K -nearest neig h b or r educe the total conflict during the combination. The total conflict is here higher than in the previous application, but it c omes here from the mo delization essentially and not from a difference of opinion giving by the classifiers. The pr oportion of ta rgets with a different decision is g iving in per cen tage, in the table 1. These p ercentages are more imp ortant for this a pplication than the previous applica t ion on sona r images . Hence the conjunctive r ule a nd the mixed rule are very similar. In terms o f similarity , we ca n give this o rder: conjunctive rule, the mixed rule (DP), PCR6f a nd PCR6g with a concave mapping, PCR6, PCR6f a nd PCR6g with a convex mapping, and PCR5 . The final decision is taken with the maxim um of the pignistic pr obabilities. Hence, the results reached by the genera liz ed PCR ar e significa n tly b etter than the c o nju nctive rule and the PCR5, a nd b ett er than the mixed rule (DP). The conjunctive rule and the PCR5 give the worth classifica tio n rates on these data (there is no sig nifican tly differ ence), wher eas they have a high pro portion of targets with a diff erent dec is ion. The be st classification rate (see table 2) is o btained with PCR f √ x , but is not s ignifican tly b etter than the results obtained with the other versions PCR f , using a differen t co nca v e ma ppin g. 5 Conclusion In this chapter, we have prop osed a study of the combination rules compar ed in terms of decision. The generalized propo rtional conflict redis t ribution (PCR6) rule (presented in the c hapter [5]) have been e v aluated. W e hav e shown on re a l data that there is a difference of dec is ion following the choice of the combination rule. This difference can b e very small in p ercen tage but allows significantly difference in go o d-classification rates. Mor eo v er, hig h prop ortion with a different decision do es not lead to a high differenc e in terms of go od-clas sification rates. 13 Rule % confiance In terv al Conjunctive 89 .83 [89.7 5 : 89.91 ] DP 89.99 [89.9 0 : 90.0 8 ] PCR f x 0 . 3 90.100 [9 0.001 : 90 .2 00] PCR f √ x 90.114 [9 0.015 : 90 .2 13] PCR f x 0 . 7 90.105 [9 0.006 : 90 .2 04] PCR g √ x 90.08 [89.9 8 : 90.1 8 ] PCR6 90.05 [89.9 7 : 90.1 3 ] PCR g x 2 90.00 [89.9 1 : 90.1 0 ] PCR f x 2 89.94 [89.8 3 : 90.0 4 ] PCR5 89.85 [89.7 5 : 89.8 5 ] T a ble 2: Go od-cla ssification r ates (%) The last application sho ws tha t we can ac hiev e b etter go od-cla ssification r ates with the generalized PCR6 than with the conjunctive rule, the DSmH, o r PCR5. The first presented application shows that the mo delization on D Θ can re- solve easily some problems . If the applicatio n need a decisio n step and if we wan t to consider the conjunctions of the elements of the discernment spac e, we hav e to take the decision directly on the masses (and not on the credibilities, plausibilities or pig nistic pr o babilities). Indeed, these functions are increasing and can no t give a decision on the conjunctions o f elemen ts. In real applica- tions, most of the time, ther e is no ambiguit y and we can take the decision, else we have to prop ose a new decisio n function that ca n r eac h a decision on conjunctions a nd also on singletons. The conjunctions of elemen ts can be consider ed (and so D Θ ) in many ap- plications, especially in image pro cessing, where an ex p ert can provide element with more than one cla sses. In estimation applica tions, where int erv a ls ar e con- sidered, encroaching interv als (with no empt y intersection) can provide better mo delization. References [1] J. Dezert and F. Sma randac he. 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