Fusion of Vegetation Indices Using Continuous Belief Functions and

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

1499

Fusion of Vegetation Indices Using Continuous Belief Functions and Cautious-Adaptive Combination Rule Abdelaziz Kallel, Sylvie Le Hégarat-Mascle, Laurence Hubert-Moy, and Catherine Ottlé

Abstract—The goal of this paper is to propose a methodology based on vegetation index fusion to provide an accurate estimation of the fraction of vegetation cover (fCover). Because of the partial and imprecise nature of remote-sensing data, we opt for the evidential framework that allows us to handle such kind of information. The defined fCover belief functions are continuous with the interval [0, 1] as a discernment space. Since the vegetation indices are not independent (e.g., perpendicular vegetation index and weighted difference vegetation index are linearly linked), we define a new combination rule called “cautious adaptive” to handle the partial “nondistinctness” between the sources (vegetation indices). In this rule, the “nondistinctness” is modeled by a factor
varying from zero (distinct sources) to one (totally correlated sources), and the fusion rule varies accordingly from the conjunctive rule to the cautious one. In terms of results, both in the cases of simulated data and actual data, we show the interest of the combination of two or three vegetation indices to improve either the accuracy of fCover estimation or its robustness. Index Terms—Belief functions, combination rule, evidence theory, fraction of vegetation cover (fCover), vegetation index fusion.

Fig. 1. Root-mean-square error of the fCover estimation versus the fCover using two vegetation indices (PVI and NDVI). The simulations are performed using the adding method (cf. Appendix II).

I. I NTRODUCTION

E

STIMATION of vegetation features from space is a great challenge for agronomist, hydrologist, and meteorologist communities. For example, land cover during winter in agricultural regions strongly influences soil erosion processes and water quality [1]. Therefore, the identification and monitoring of vegetation cover [the physical parameter being the fraction of vegetation cover (fCover)] constitutes a prior approach for the monitoring of water resources. Now, the use of vegetation indices [2] to estimate vegetation characteristics is very popular. They are empirical combinations between visible [generally Red (R)] and near-infrared (NIR) reflectances that show good correlation with plant growth, Manuscript received June 14, 2007; revised September 14, 2007. A. Kallel is with the Centre d’étude des Environnements Terrestre et Planétaires (CETP)/Institut Pierre-Simon Laplace (IPSL), 78140 Vélizy, France (e-mail: [email protected]). S. Le Hégarat-Mascle is with the Institut d’Electronique Fondamentale/ Architectures, Contrôle, Communication, Images, Systèmes (AXIS), Université de Paris-Sud, 91405 Orsay, France. L. Hubert-Moy is with the COSTEL UMR CNRS 6554 LETG/IFR 90 CAREN, Université de Rennes 2, 35 043 Rennes Cedex, France. C. Ottlé is with the Laboratoire des Sciences du Climat et de l’Environnement (LSCE)/Institut Pierre-Simon Laplace (IPSL), Centre d’Etudes de Saclay, 91191 Paris, France. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2008.916215

vegetation cover, and biomass amount. Besides these empirical indices, some methods based on the inversion of radiative transfer model have been proposed (e.g., [3]). Since these inversion methods provide an estimation of the vegetation fraction cover (among other parameters), they can also be interpreted as vegetation indices. However, even with such a large number of vegetation indices, it appears that none of them is sufficiently universal to replace all others. Fig. 1 shows the complementarity of vegetation indices in terms of fCover estimation. It plots the robustness of the fCover estimation versus the fCover using two different vegetation indices, namely, perpendicular vegetation index (PVI) and normalized difference VI (NDVI). For lower values of fCover (f < 0.4), the two vegetation indices show close performance. However, for f ∈ [0.4, 0.7] (respectively f ∈ [0.7, 1]), NDVI (respectively PVI) gives the more robust result. Then, the interest of combining several indices should be clear, provided that this combination would be “clever.” Then, in this paper, we propose to derive fCover by considering not only one but several vegetation indices using a scheme of fusion of these different vegetation indices. Each of the considered indices is assumed pertinent (to varying degrees) for our application—the fCover estimation. Now, for index combination, we chose to use the framework of the Dempster–Shafer (DS) evidence theory [4], which is interpreted and extended as

0196-2892/$25.00 © 2008 IEEE

1500

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

the theory of belief functions [5], [6]. This theory, which allows us to represent both imprecision and uncertainty, appears as a more flexible and general approach than the Bayesian one. Another of its advantages is its ability to handle not only single (or individual) hypothesis but also compound ones (unions of hypotheses). Applications were developed in medical imaging (e.g., [7]), object extraction (e.g., [8] and [9]), classification (e.g., [10]–[12]), and change detection (e.g., [13] and [14]). For our application, it allows us to model for some global ignorance, which is either present at the borders between the fCover classes or is due to the poor quality of some vegetation indices. However, before we can use the DS framework for our problem, two main assumptions need to be proven, which are not handled in classical applications of DS theory, such that those previously mentioned. 1) Dempster’s combination rule, or orthogonal sum, assumes the sources (or items of evidence to combine) to be distinct and independent (in terms of cognition). Clearly, this is not the case considering the vegetation indices that are (more or less) correlated. Now, Dubois et al. [15] propose a “cautious” rule of combination that applies in the case of nonindependent sources. However, we show that this rule may be too cautious, and we propose a parameterized rule that continuously goes from the DS orthogonal sum (for independent sources) or conjunctive rule (in the following, we do not consider the normalization associated to the closed-world assumption) to the cautious rule. 2) Generally, the considered frame of discernment is a discrete frame. However, in our case of fCover estimation, the frame of discernment is the interval [0, 1]. Then, rather than discretizing it into small subintervals, we consider the solutions proposed by Ristic et al. [16]–[18] to define belief functions on continuous frames of discernment. In such a case, the “mass” functions that are generalized into densities assigned to the real intervals and the “belief,” “plausibility,” and “commonality” functions become the integrals of mass densities. Finally, we underline that in this paper, we focus on the case where the focal elements (having nonnull mass) are consonant. This choice will be justified in Section III. It allows us to derive explicit solutions for the densities and integrals corresponding to belief functions and to show interesting properties as well. The remainder of this paper is organized as follows. Section II presents the evidence theory basic concepts, distinguishing the discrete and the continuous cases. Section III focuses on the consonant case and presents the proposed model of vegetation index fusion and the so-called “cautious-adaptive” proposed combination rule. Section IV gives the considered vegetation indices and shows the results obtained from their fusion. In particular, the results obtained using different combination rules (DS orthogonal sum or cautious-adaptive one) are compared. Section V concludes this paper. II. R EVIEW OF THE T RANSFERABLE B ELIEF M ODEL In this section, we summarize the main concepts of belief functions on discrete sets [4], [19] and the set of real numbers,

as proposed by Smets [17]. We also introduce, in the continuous case, a new function called the “canonical weight function (cwf)” that we relate to the basic belief density (bbd). One main interest of this function is the generalization of the discounting concept. A. Discrete Case 1) Basic Function Definition: Let Ω = {θ1 , . . . , θn } be a frame of discernment. The following four functions: basic belief assignment or “mass” (bba, m), “belief” (bel), “plausibility” (pl), and “commonality” (q) are then defined from 2Ω to [0, 1], such that [4], [19]


m(A) = 1

A⊆Ω

∀A ⊆ Ω, bel(A) =




m(B)

∅=B⊆A

∀A ⊆ Ω, pl(A) =




m(B)

B∩A=∅

∀A ⊆ Ω, q(A) =




m(B).

(1)

B⊇A

Since the relationships between the different functions are all bijective, the knowledge of only one of these functions allows the derivation of the others [20]. Now, by definition, we call the focal elements of Ω those elements with nonnull bba. A bba is called “normal” (respectively “nondogmatic”) if the ∅ is not a focal element (respectively Ω is a focal element). Finally, we note mI as the vacuous bba, the one such that mI (Ω) = 1 (Ω sole focal element). 2) BBA Ordering: Comparison of information sources may require an ordering relationship () between bba [21]. Among the possible ordering relationships [15], let us consider the q ordering (q ) and the s ordering (s ). For two bba’s m1 and m2 , m1 q m2 means that ∀A ∈ Ω, q1 (A) ≤ q2 (A). m1 s m2 means that there is a nonnegative  function S defined from 2Ω × 2Ω to [0, 1] satisfying ∀A ∈ Ω, B⊆A S(B, A) = 1 and that
m1 (A) = S(B, A)m2 (B). (2) B⊆A

S is called a specialization “function” [21], [22]. s implies q . According to an ordering, m1  m2 means that m2 is less informative than m1 . In the transferable belief model (TBM) [23], m2 is said to be “less committed” than m1 (m1 which is said to be “more committed” than m2 ). Then, we define, if they exist (that is not sure since all previous orderings are only partial), the least committed bba (LCbba ) of a set of bba and the most committed bba (MCbba ) of a set of bba, and we note S(m) [respectively G(m)] as the set of more committed (respectively less committed) bba than a given bba m. 3) Evidence Combination: The orthogonal sum (⊕) [4] combines two distinct evidences E1 and E2 (associated bba m1

KALLEL et al.: FUSION OF VEGETATION INDICES

1501

and m2 ) as follows: ∀A ⊆ Ω and A = ∅, m1 ⊕ m2 (A) =

1 K∅




m1 (X)m2 (Y )

where




X∩Y =A

(3)

K∅ = 1 −

m from q [20], one finds
log (w(X)) , i.e., log (q(A)) = − q(A) =

X⊇A



−1 =

w(X)

X⊇A

m1 (X)m2 (Y ).

(4)

X∩Y =∅

In the TBM, Smets and Kennes [19] propose not to do the normalization by the factor K∅ that leads to nonnormal bba ∩ ). Note that if q1 and called their rule the conjunctive rule ( and q2 are the commonality functions related to m1 and m2 , ∩ m2 is q12 (A) = q1 (A)q2 (A), the commonality related to m1  ∀A ∈ Ω. Some justifications following the principle of minimal commitment are given for both rules in [21], [22], [24], and [25]. More recently, and for two nondistinct evidences E1 and E2 , Smets [26] and Dubois et al. [15] propose a “cautious” rule dealing with the redundancy between the two evidences according to the s ordering m1  ∧ 2 = LCbba (Ss (m1 ) ∩ Ss (m2 )) .

(5)

This definition can be extended to any other ordering. Note that, since the ordering is not total, such a rule does not always exist. Note also that, in this paper, we only consider conjunctive rules (i.e., rules corresponding to a refinement of the information), but that disjunctive rules have also been proposed [27] (i.e., rules corresponding to a coarsening of the information), as well as fusion rules moving from the conjunctive to the disjunctive one [28], [29]. 4) BBA Decomposition: Shafer [4] defines simple support function (SSF) bba as follows. m is SFF if ∃A
Ω and w(A) ∈ [0, 1] such that   m(X) = w(A), = 1 − w(A),  = 0,

if X = Ω if X = A otherwise.

(6)

Such a bba is noted as Aw(A) . Now, m is a “separable” bba if [4]

m(Ω) . A⊆X
Ω w(X)

(9)

The analogy between the relationships linking m to q and − log w to log q will be used once again to extend the definition of w in the continuous case. ∩A⊆Ω Aw1 (A) and m2 =  ∩A⊆Ω Aw2 (A) , Now, writing m1 =  the conjunctive rule of combination writes ∩ Aw1 (A)w2 (A) .  m1 ∩2 =

(10)

A⊆Ω

Then, for separable bba, Denoeux [31] introduces the ∩A⊆Ω Aw1 (A) and w ordering (w ) as follows. Let m1 =  ∩A⊆Ω Aw2 (A) ; m1 w m2 means that ∀A ∈ Ωw1 (A) ≤ m2 =  w2 (A). Note that w implies s . Then, Denoeux [31] proposes to define the cautious rule for a separable bba as m1  ∧ 2 = LCbba (Sw (m1 ) ∩ Sw (m2 )) .

(11)

Such a rule exists and is unique, and m1  ∧ 2 is separable with w1  (A) = min{w (A), w (A)}, ∀A
Ω. 1 2 ∧ 2 B. Continuous Case 1) BBD and Associated Functions: The extending of the evidential theory to the continuous case [5], [16]–[18], [32] in the interval [0, 1] can be derived from a uniform dividing of [0, 1] into n subintervals θi = [ai , bi ] such that a1 = 0, bn = 1, and ∀i ∈ {2, . . . , n}, ai = bi−1 . Let Ωn = {θi , i ∈ {1, . . . , n}}, and let mn be the associated bba. Since the uncertainty in R can occur only between close values, the only focal elements are subintervals that write as union of adjoining singletons θi . For A ∈ Ωn , if mn (A) = 0, then ∃(i, j) ∈ {1, . . . , j} × {1, . . . , n}, such that A = k={i,...,j} θk . As in the probability case, tending n to infinity, mn is replaced by a density f defined on the subintervals of [0, 1] and by taking values in R+ , which is called bbd and checking [16], [17] y=1 x=1  

f (x, y)dxdy = 1.

(12)

x=0 y=x

m=

∩ Aw(A) .  A
Ω

(7)

Extending the w variation domain to [0, +∞[, Smets showed that there is a unique decomposition of any nondogmatic bba based on (7), and for any A
Ω [30] w(A) =



(−1)|X|−|A|+1

q(X)

bel, pl, and q are defined as (∀a, b/0 ≤ a ≤ b ≤ 1) [17] y=b x=b   bel ([a, b]) = f (x, y)dxdy x=a y=x x=b 

.

(8)

pl ([a, b]) =

X⊇A

By extension, we set w(Ω) = (1/m(Ω)) ≥ 1. Note that, using the similarity of log(w(A)) =  − X⊇A (−1)|X|−|A| log(q(X)) with the relationship deriving

y=1 

f (x, y)dxdy x=0 y=max{a,x} y=1 x=a  

q ([a, b]) =

f (x, y)dxdy. x=0 y=b

(13)

1502

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

By the derivation of bel and q, the following useful equations are obtained: ∂ 2 bel ([a, b]) ∂a∂b ∂ 2 q ([a, b]) f (a, b) = − . ∂a∂b

f (a, b) = −

(14) (15)

Finally, having combined all evidences, a decision (between the singleton elements of Ω) should be made. In the absence of information, in an analog way to the principle of maximal entropy, one assumes a uniform distribution of a given mass of a composed hypothesis over its singletons. Then, Smets [33] proposes the following transformation of a given bba m to a probability distribution called pignistic probability (Betf). In the continuous case, Betf is defined as follows [17]: y=1 x=a  

f (x, y) dxdy. y−x

∀a ∈ [0, 1[, Betf(a) = lim


→0 x=0 y=a+


where δ(u, ν) = δ(u)δ(ν), and δ is the Dirac function. The specialization operator s checks [17] ∀a, b, x, y ∈ [0, 1] such that a ≤ b and x ≤ y

(18)

Now, the specialization of a bbd f1 is a bbd f2 checking

The conjunctive combination of two bbds f1 and f2 is done as follows [17]: f2 (x, y)dydx

x=0 y=b y=b x=a  

f1 (x, y)dydx

x=0 y=b y=1 x=a  

f1 (x, b)f2 (a, y) x=0 y=b

+ f1 (a, y)f2 (x, b)dydx.

By using the analogy between the (m, q) relationship and the (− log w, log q) one, and by introducing ϕ as a density of − log w defined in [0, 1] × [0, 1]\(0, 1), from the relationship (15)

=

∂ 2 log (q ([a, b])) ∂a∂b

∂q([a,b]) ∂q([a,b]) ∂a ∂b

−

∂ 2 q([a,b]) ∂a∂b q ([a, b]) . 2

q ([a, b])

(22)

ϕ is called the cwf. It is a piecewise-continuous-dominated function over [0, 1] × [0, 1]\(0, 1), and we set ϕ(0, 1) = 0. q can be expressed versus ϕ as follows: ∀a, b/0 ≤ a ≤ b ≤ 1, and a > 0,   q ([a, b]) = q ([0, 1]) exp 



y=1 x=a  

 ϕ(x, y)dydx . (23)

Theorem 1: Let f be a bbd and ϕ be its cwf, and then, ∀a, b/0 ≤ a ≤ b ≤ 1, and a > 0   y=1 x=a     ϕ(x, y)dydx f (a, b) = q ([0, 1]) exp    × ϕ(a, b) +

x=a 

y=1 

ϕ(x, b)dx x=0



 ϕ(a, y)dy  . (24)

y=b

Proof: See Appendix I.
Now, we extend the definition of a separable bba to the continuous case. We say that f is separable if for each interval [a, b] of [0,1], ϕ(a, b) ≥ 0. Reciprocally, in [34], we show that if ϕ is a nonnegative function over each interval of [0, 1] different from [0, 1], then the function f defined as in Theorem 1 is a well-defined bbd. Theorem 2: Let f1 and f2 be the two bbds. Then, their conjunctive combination f1 ∩ 2 satisfies

y=1 x=a  

+

(21)

x=0 y=b

(19)

x=a y=x

+ f2 (a, b)

f0 (a, b) = 1.

a=0 b=a

x=0 y=b

a=x b=a

f1 ∩ 2 (a, b) = f1 (a, b)

α+

ϕ(a, b) = −

∀a, b ∈ [0, 1]×[a, 1], α f (a, b) = (1−α)f (a, b)+αδ(a, 1−b) (17)

y=b x=b   s(a, b|x, y)f1 (x, y)dbda. f2 (a, b) =

b=1 a=1  

(16)

2) BBD Discounting, Specialization, and Combination: In the continuous case, to combine nonreliable sources, Shafer [4] proposes the “discounting” process, which is later justified by Smets [23]. The degree of reliability of an evidence E is quantified by a factor (1 − α) with α ∈ [0, 1]. For a given bbd α f , the discounting gives a new bbd α f such that

s(a, b|x, y) = 0, if [a, b] ⊆ [x, y], a=y b=y   s(a, b|x, y)dbda = 1.

Note that the {q, s}-ordering definitions are like those in the discrete case. 3) CWF: Here, we propose to extend the canonical decomposition of a nondogmatic bbd (q([0, 1]) > 0) to the continuous case. We assume in this section that the bbd f is a sum of a nonnegative piecewise-continuous function (f0 ) over all intervals of [0, 1] and a Dirac function αδ(x, 1 − y) (where α ∈]0, 1]) such that

(20)

∀a, b/0 ≤ a ≤ b ≤ 1, ϕ1 ∩ 2 (a, b) = ϕ1 (a, b) + ϕ2 (a, b). (25)

KALLEL et al.: FUSION OF VEGETATION INDICES

1503




Proof: See Appendix I. Proposition 1: Let f1 and f2 be the two bbds such that ∀a, b/0 ≤ a ≤ b ≤ 1, ϕ1 (a, b) ≥ ϕ2 (a, b).

(26)

Then, there is f3 such that f1 = f2 ∩ 3. Proof: See Appendix I.
The w-ordering definition can be extended to the continuous case (ϕ ) by f1 ϕ f2 if ∀a, b/0 ≤ a ≤ b ≤ 1, ϕ1 (a, b) ≥ ϕ2 (a, b). As ∃f3 such that f1 = f2 ∩ 3 , then f1 s f2 . Therefore, ϕ implies s . III. P ROPOSED F USION M ODEL : C ONSONANT C ASE In this section, we deal with the consonant case that applies for our application. This case is convenient since supplementary properties occur. In particular, a consonant bba is separable, and there is an equivalence between the s and q orderings which allows the definition of the cautious rule and a new rule, taking into account the partial nondistinctness between different evidences. In the following section, first, the consonant case is justified, assuming that our knowledge is only partial, and represented rather by a set of possible values for the fCover (researched parameter) knowing the observation. Then, the properties specific to our model (consonant on real intervals) are presented.

on the real axis) is contained by the following one. In the following, we focus on the continuous case as it applies for our application, even if we mention the way the more interesting properties (and, particularly, the proposed combination rule) transpose in the discrete case. In the case of a unimodal pignistic density with mode µ = arg maxu Betf(u), the focal sets of the consonant least committed belief density are intervals [au , bu ] such that [16], [17]: Betf(au ) = Betf(bu ), where Betf is a bell-shaped density, au is uniquely defined as a function of bu , and f (a, bu ) = δ(a − au )(bu − au )(dBetf(bu )/dbu ), ∀bu ≥ µ. In our case, we assume univariate Gaussian pignistic functions. The parameters (µIVi , σIVi ) of these Gaussians are estimated from ground truth or Adding/PROSPECT simulations varying the vegetation/soil parameters. (µIVi , σIVi ) are empirically estimated from the obtained histograms of fCover values versus the vegetation index IVi ones. Now, the least committed bbd in the case of Gaussian pignistic density writes [17] as follows: f (a, bu ) = δ(a − au )2((bu − √ µ)2 / 2πσ 3 ) exp[−((bu − µ)2 /2σ 2 )], ∀bu ≥ µ, and au = 2µ − bu . Since for a given length u there is, at the maximum, one focal interval (Iu = [au , bu ]), and if u > u , Iu ⊂ Iu , let us introduce the notation h(bu ) = f (au , bu ): f (a, bu ) = δ(a − au )h(bu ) is a 1-D function that replaces f when the bbd is consonant. In the following, we assume that u ∈ [0, 1] and bu ∈ [µ, 1]. In this case, q simply writes [16] x=1 

A. Vegetation Index Mass Function Here, we present the way the mass functions (or bbd in the continuous domain) associated to the different vegetation indices are derived. Using the Adding model, we obtain simulations of satellite measurements (and then vegetation indices computed from) conditionally to fCover value and assuming vegetation/soil/acquisition parameters. Then, we can derive some distributions of fCover values versus vegetation index ones. However, the obtained knowledge is only partial and imprecise since the vegetation/soil parameters are unknown in our application, and therefore, the distributions of fCover values are spread by the necessary variability of vegetation/soil parameters in our simulations. Therefore, assuming that our knowledge corresponds rather to a potential betting behavior on the fCover value (in continuous domain), we model it by a pignistic probability density. Now, since the pignistic transform is a many-to-one transform, there is an infinite number of belief densities, which is said to be “isopignistic,” that induce the same pignistic probability density. Applying the least commitment principle [23], we choose, among all isopignistic belief densities, the least committed one, i.e., the one maximizing the commonality function q. This least committed belief density is a consonant density [16]–[18], [35], as defined in the next paragraph. In the discrete (respectively continuous) case, a bba (respectively bbf) is consonant if all the focal elements are nested: ∀A, B ∈ Ω, if m(A) = 0 and m(B) = 0, then A ⊆ B or B ⊆ A. Therefore, all the focal elements can be ordered in a socalled bba (respectively bbf) ordering set ∆ (respectively I), in such a way that each focal element (respectively focal interval

q(bu ) =

h(x)dx

(27)

x=bu

where q(bu ) is the commonality of the interval [au , bu ]. Moreover, by noting q  as the derivative of q, one has h(bu ) = −q  (bu ).

(28)

By doing a variable change in (23), we obtain 

x=1 

∀b ∈ [µ, 1[, q(bu ) = q(1) exp 

x=bu

 = exp−

x=1 

 ϕ(x)dx 

ϕ(x)dx exp

x=µ

 = exp −



x=b  u



x=1 

 ϕ(x)dx

x=bu

ϕ(x)dx .

(29)

x=µ

Therefore, the cwf expression is ϕ(bu ) = −

h(bu ) d log q(bu ) q  (bu ) = . =− dbu q(bu ) q(bu )

(30)

1504

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

Since h ≥ 0 and q ≥ 0, then ϕ ≥ 0, and therefore, h is separable. Moreover, the expression of h versus ϕ is  x=b   u ϕ(x)dx . (31) h(bu ) = ϕ(bu ) exp − x=µ

In the Gaussian Betf case, the LC bbd can be explained versus yu = (bu − µ)/σ  2 y yu2 σ h (yu ) = 2 √ exp − u 2 2π  2   yu yu yu σ q (yu ) = 2 √ exp − + erfc √ 2 2π 2 ϕσ (yu ) = yu +

π 2

yu2    2 yu exp y2u erfc √ 2

(32)

√  +∞ where erfc(x) = (2/ π) x exp[−t2 ]dt. B. New Combination Rule We already said that the vegetation indices to be combined are not necessarily independent (some of them are even strongly correlated). Therefore, the orthogonal sum ⊕ or the conjunctive ∩ are not suitable. On the other hand, the cautious rule rule  proposed by Smets et al. [15], [26], [31] may be “too cautious” in the sense that it estimates the “correlation” between two sources as the maximum of the possibly “redundant”1 information between the sources. Indeed, it is possible that two sources present an apparent correlation but are independent in a cognitive way (e.g., two experts can provide same diagnostic without having devised nor having the same formation, etc.). Therefore, we propose a parameterized combination rule that allows going from the conjunctive rule to the cautious one. We call this rule “cautious adaptive.” In the following section, we first present it in the discrete case and then its extension to the continuous case. 1) Discrete Case: Let ∆ be a set of n elements ωi (∆ = {ωi , i ∈ {1, . . . , n}}) such that the following conditions are achieved. 1) ∆ includes all focal elements. 2) ∀i ∈ {1, . . . , n}, |ωi | = i. 3) ∀(i, j) ∈ {1, . . . , j} × {1, . . . , n}, ωi ⊆ ωj . In the following, ∆ is called the “consonant ordering set.” Then, we define the ordering I of the Ω singleton elements θj (Ω = {θI(1) , θI(2) , . . . , θI(n) }) such that ∀i ∈ {1, . . . , n},

ωi = ij=1 θI(j) . In this section, we assume that all sources have the same consonant ordering set ∆. For consonant bba, the equivalence between s and q can be shown [34]. Moreover, for two consonant bba’s m1 and m2 , the bba m1  ∧ 2 defined by q1  ∧ 2 = min{q1 , q2 } is the unique optimal cautious solution following the q ordering [15] and, hence, the s ordering. 1 In

this sentence, the quotation marks mean that the word has to be understood in its common or popular sense instead of its mathematical definition.

Smets [26] defines the “correlation” (m0 ) between two bba’s m1 and m2 as q0 = (q1  ∩ 2 /q ∧ 1  2 ). In our case q0 =

q1 q2 = max{q1 , q2 }. min{q1 , q2 }

(33)

m0 is the most committed bba in Gs (m1 ) ∩ Gs (m2 ). Now, the vacuous bba is the least committed bba in Gs (m1 ) ∩ Gs (m2 ). As said previously, when two distinct evidences give the same result, the use of the conjunctive combination is the right rule (confirming the assumptions supported by each evidence). Conversely, when the evidences are nondistinct, the cautious rule is suitable. In this paper, we propose to parameterize the nondistinctness by a parameter ∈ [0, 1] such that = 1 (respectively = 0) means that the evidences are nondistinct (respectively distinct). Then, we propose the discounting of the “correlation” m0 . The bba representing the “correlation” between the sources is now  m0 instead of m0 like in the cautious rule. More generally, we define the discounting w m0 as follows. The discounting of an SSF bba m = Aw results in α an increasing of w: α m = A w , such that α w = (1 − α)w + α. ∩A
nΩ Aw ). The Now, a consonant bba is separable (m =  proposed “generalized” discounting consists in a discounting of each evidence EA providing the SSF Aw (m being the α ∩A
Ω A w . Note combination of all Aw ’s) separately: αw m =  that in this notation, we have omitted to point out the eventual dependence of α with EA . In the most general case, the correlation between sources may depend on the considered assumption and so does the discounting of the source “correlation” m0 . With this general definition of discounting, if m1 and m2 are two nondogmatic consonant bba’s over the same ordering set, then if m1 w m2 , m2 can be viewed as a discounting of m1 (m2 =αw m1 ). Moreover, α m is an αw m of m [34]. ∧ is defined by Finally, the cautious-adaptive rule   q 1  ∧ 2 =

q1 ∩ 2 . w q0

(34)

2) Continuous Case: The extension of the cautious rule to the continuous case is restricted to the case when q1 (x) > q2 (x) for any x ∈]0, A] (or conversely), where x represents a reduced variable such as yu in (32). This restriction is motivated by the framework of our application. Now, following Kallel et al. [34], if h1 and h2 are two consonant bbds, such that h1 q h2 , where h1 and h2 are piecewisecontinuous functions over [0, A], h1 (0) = h2 (0), ∀x ∈]0, A], h2 (x) > 0, and q1 (x) < q2 (x), then h1 s h2 . Respecting the ∧ ) conditions of this theorem, we define the cautious rule ( = min{q , q }. The correlation bba (m ) is given as q1  1 2 0 ∧ 2 by q0 = max{q1 , q2 }. As in the discrete case and by analogy between w and ϕ, we propose the discounting of h0 , based on the degree of nondistinctness ( ∈ [0, 1]), and h0 is then replaced by ϕ h0 such that h0 ϕ ϕ h0 . The cautious-adaptive  ∧  rule can then be extended to the continuous case as follows: q 1  ∧ 2 =

q1 ∩2 . ϕ q0

(35)

KALLEL et al.: FUSION OF VEGETATION INDICES

1505

Fig. 2. Three vegetation index (IV1 , IV2 , and IV3 ) combinations. IV1 and IV2 are partially nondistinct and |µ2 − µ1 | <
, and then, they are combined using the cautious-adaptive rule ( is the level of nondistinctness). Their result bbd is combined with IV3 using the conjunctive rule.

C. Global Vegetation Index-Fusion Algorithm In the previous section, we propose a new fusion rule. For its practical application, we have to emphasize the three following points: 1) in which case the proposed combination rule can be used?; 2) how is the discounting done when the cautious-adaptive rule is required?; and 3) how is the decision taken? Concerning the first point, the cautious-adaptive rule is used only if the sources to be combined have the same consonant ordering set. Otherwise, they are considered as distinct, and the conjunctive rule applies. Practically, it means that after bbd estimation (as described in Section III-A), the values of µIVi (where IVi is the ith vegetation index to combine) are compared, and if their absolute difference is greater than a precision threshold , the indices are assumed distinct and combined with the conjunctive rule. Otherwise, the proposed cautiousadaptive rule is used. Fig. 2 shows an example of combination of three vegetation indices. We assume that IV1 and IV2 are partially nondistinct (according to the criterion |µ2 − µ1 | < ); therefore, they are combined using the cautious-adaptive rule. Conversely, IV3 is assumed independent from the bbd resulting from the combination (of IV1 and IV2 ), and thus, they are combined using the conjunctive rule. The generalized discounting is done as follows. Kallel et al. [34] show that ϕσ is a decreasing function of σ for any  value of yu (32). Therefore, if σ < σ  , then hσ ϕ hσ . In this ϕ σ paper, and for a given , we propose to define h by hσ/ . Fig. 3 shows an example of fusion using the cautious-adaptive rule, the resulted commonality curve ranges from the one corresponding to q1 , and the curve corresponding to the conjunctiverule fusion. Note that, when = 0.6, the result is close to the conjunctive one which means that a significant value of correlation should be at least about 0.8. Concerning the decision, it is classically taken using the pignistic probability density derived from the bbd of the combination of the considered vegetation indices. Note that, once one of the sources (vegetation indices) is distinct from the others, the combination is not consonant, and therefore, the considered intervals [au , bu ] may be any real intervals. Therefore, the pignistic probability density should be computed based on (16).

Fig. 3. Example of fusion between two bbds h1 and h2 using the new rule. h1 = h0.09 , and h2 = h0.1 . Three fusion results: conjunctive rule and cautious-adaptive rule with  = 0.6 and  = 0.8.

IV. A PPLICATION TO THE F USION OF V EGETATION I NDICES In this section, we analyze the performance of the proposed model through our application that is the combination of several vegetation indices in order to estimate the vegetation fraction cover. We first present the vegetation indices considered. Then, the performance of their fusion is evaluated in the case of simulated data: with or without noise. Finally, the case of actual data is considered. A. Considered Vegetation Indices During the last 20 years, works have been carried to follow continental surfaces from space. Numerous vegetation indices combining the reflectance measurements at two or several wavelengths have been proposed. The more popular index is probably the NDVI [36] that combines reflectance measurements in the Red and in the NIR domains without any other data. This index shows a good correlation with vegetation density. However, it suffers from some problems related to its dependence to acquisition geometry (sun position and looking angle) and spectral properties of the soil. Such dependences are at least partially taken into account by more recent indices. Indeed, more performing indices have been developed, particularly those that are taking into account the soil reflectance such as the soil-adjusted vegetation indices (SAVIs) [37]–[39] (for a review, one can refer to [2]). All these indices are highly correlated to the leaf area index (LAI) of the vegetation cover (e.g., [38] and [40]), particularly during the growing stage of the vegetation until a saturation level. The interests of these indices, particularly relative to the factors to which the indices are assumed robust, such as the acquisition conditions, the soil and vegetation spectral properties, the geometry of the vegetation cover, and the row effects, have been evaluated from robustness experiences using radiative transfer model simulations. Besides the empirical indices, some theoretical methods based on radiative transfer model inversion have also been proposed (e.g., [41]). However, since they handle a large number of parameters and estimate only few of them, the

1506

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

TABLE I VEGETATION INDICES. rNIR AND rR ARE THE NIR AND THE RED REFLECTANCES , RESPECTIVELY, AND a0 AND b0 A RE THE S OIL -L INE P ARAMETERS (S LOPE AND I NTERCEPT, R ESPECTIVELY )

other parameters (e.g., leaf orientation, pigment concentration, and mesophyll leaf structure that influence leaf transmittance and reflectance) have to be accurately calibrated, which may be difficult [42]. In [3], we propose to derive a semiempirical method allowing the inversion of the fCover. It is based on the inversion of a four-parameter model combining the reflectances in the Red and NIR domains. This model is semiempirical since it is based on radiative transfer modeling, but, owing to an ingenious parameterization of vegetation isoline set, it requires calibration of only four parameters that can be related to SAIL simulations [43]. To calibrate the model parameters, two optimization methods can be used: the Simplex (Sx) [44] and the Shuffled Complex Evolution algorithm (SCE-UA) [45], which are deterministic (local optimization) and heuristic methods (global optimization), respectively. SCE-UA is a priori more accurate, but it is also more complex and needs longer running time. Table I lists the used vegetation indices in this paper. The aim is to combine them to improve the fCover estimation performance both in terms of accuracy and robustness. B. Results in the Case of Simulated Data In this section, simulated data are considered in order to evaluate the fusion independent of the accuracy of the ground truth. These simulations have been obtained from the coupled Adding/PROSPECT. The adding method is proposed in [50] and [51] and briefly summarized in Appendix II. The PROSPECT model [42] provides the leaf albedo (leaf reflectance and transmittance) versus leaf features. In the Visible domain, the albedo is mainly due to pigment concentration (chlorophyll a + b) Ca+b . In the NIR domain, since the absorption is lower, the albedo is higher (it is related to the mesophyll

TABLE II SET OF SIMULATIONS USED TO EVALUATE THE PERFORMANCE OF THE METHOD. WHEN “Sdv = 0” (BY DEFAULT FOR THE FOUR LAST PARAMETERS), THE PARAMETER VALUE IS GIVEN IN FIRST (OR UNIQUE) COLUMN; ELSE , IT IS A RANDOM VALUE GENERATED FOLLOWING A N ORMAL D ISTRIBUTION N (M, Sdv).

leaf structure, N). Adding inputs are the leaf reflectance and transmittance simulated by PROSPECT, the vegetation density (fCover), the leaf area distribution, and the sun and observation geometries: the sun zenithal angle (θs ), the observation zenithal and azimuthal angles (θo , ϕo ), the hot-spot parameters (hs) [52], and the soil reflectances (Rsoil,R , Rsoil,NIR ) that are entirely determined by Red soil reflectance variation [minRso,R , maxRso,R ] and the soil-line equation slope and intercept (a0 , b0 ). These values are then used to simulate remotesensing measurements, from which the vegetation indices listed in the previous section are computed. The fCover is sampled from 0 to 0.99 with a step equal to 10−2 . The soil-line parameters are fixed to (1.6,0), and the reflectance variation in the Red domain ranges from 0.025 to 0.325. Numerous simulations have been done. Here, we only present the two following typical cases, respectively: without noise and with noise (“noise” refers to the fact that the simulation parameters are random variables instead of having a deterministic value). Table II shows the set of parameters which

KALLEL et al.: FUSION OF VEGETATION INDICES

1507

Fig. 4. Fusion performance (robustness versus accuracy) evaluation from the Adding/PROSPECT-coupled model simulated data. (a) Comparison of the results obtained using each vegetation index individually or combining several vegetation indices. (b) Difference between the performance of the multiindex fusion and the best performance obtained from any vegetation index involved in the fusion.

have been used for simulations. The first simulation set is without noise; it presents a green vegetation with high pigment concentration and low mesophyll structure value. The second set introduces spatial vegetation parameter variations through a nonnull standard deviation corresponding to different kinds of cultures and the existence of senescent vegetation. Finally, in this paper, the “correlation” between vegetation indices was set as an a priori (supervised) parameter that can be defined from general knowledge about the indices. We assumed the independence between vegetation indices (i.e.,

= 0) except between the following: 1) PVI and WDVI that are linearly linked and for which we assume therefore a correlation of 1.0; 2) RVI and NDVI that are unlinearly linked (NDVI = (RVI − 1)/(RVI + 1)) for which we assume a correlation of 0.9; 3) Sx and SCE-UA which only differ by the kind of optimization (local or global) for which we assume a correlation of 0.8. The performance of the method has been evaluated through two parameters: 1) the method accuracy represented by the mean L1 error that is the average of the absolute difference between the estimated fCover value and the actual one (i.e., the one used for the considered Adding/PROPSPECT simulation) and 2) the method robustness represented by the standard deviation of L1 error. In the following figures, the x- and y-axes deal with these two parameters, respectively. Fig. 4 shows the results obtained in the case of data without noise (Table II, first scenario). In Fig. 4(a), method robustness versus method accuracy is shown, respectively, considering each vegetation index individually (before or after correction of its bias evaluated during the learning step of pignistic probability parameters) or the combination of two or three vegetation indices. When three indices are combined, we compare the

result using the classical conjunctive rule (assuming source cognitive independence) and the proposed cautious-adaptive combination rule. In particular, Fig. 4(a) allows the comparison of these two combination rules through the results called “PVI + WDVI + 1IVEG” and “PVI, WDVI, 1IVEG” that correspond to the conjunctive rule applied to three indices, respectively (among the nine ones considered), two of them being PVI and WDVI, and cautious-adaptive rule applied to the same indices (or through the results called “RVI + NDVI + 1IVEG” and “RVI, NDVI, 1IVEG,” and through the results called “SCE − UA + Sx + 1IVEG” and “SCE − UA, Sx, 1IVEG,” whose interpretation is the same as for “PVI + WDVI + 1IVEG” and “PVI, WDVI, 1IVEG” but replacing PVI and WDVI by RVI and NDVI or by SCE-UA and Sx). First, we note that the result of a combination of two indices is generally better than the use of only one index and that the combination of three indices is generally better than the combination of two. Then, we compare the fusion with the conjunctive rule to the one with the cautious-adaptive rule. The results are plotted with empty and full symbols, respectively. We note that they are rather close (that is not surprising since they use the same set of vegetation indices and the same evidential framework), but there is a quasi-systematic slight improvement using the cautious-adaptive rule. Now, in order to evaluate precisely the performance of the fusion relative to the individual vegetation indices, Fig. 4(b) compares the evidential fusion with the better (for each of the two criteria: accuracy and robustness of the fCover estimation) indices involved in the considered evidential fusion. Since Fig. 4(b) shows the signed difference between fusion performance and individual index one, negative values mean that fusion performs better than any involved vegetation index. From Fig. 4(b), we note that there are two clusters of points. The first one corresponds to points (i.e., combinations) with definitely negative values, i.e., corresponding to what we call “optimal” fusion since it induces an improvement of both

1508

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

Fig. 5. Fusion performance (robustness versus accuracy) evaluation from the Adding/PROSPECT-coupled model noisy simulated data. (a) Comparison of the results obtained using each vegetation index individually or combining several vegetation indices. (b) Difference between the performance of the multiindex fusion and the best performance obtained from any vegetation index involved in the fusion.

criteria (accuracy and robustness). The second set of points is around zero with most values slightly positive, i.e., corresponding to a fusion either improving only one of the two criteria or none relatively to the “best” vegetation index. In this latter case, the fusion induces an intermediate performance relative to those of the vegetation indices involved in the considered fusion. In particular, the combinations involving the TSAVI or the SCE-UA are better than the “best” index only about one time out two, which is not very surprising since these indices are already performing well. Note that once more, we can check that the (empty symbols) conjunctive rule performs worse than the (full symbols) cautious-adaptive one. Fig. 5(a) and (b) is essentially the same as Fig. 4(a) and (b) except that it corresponds to noisy simulations (Table II, second scenario). Comparing Figs. 4 and 5, we note that, in the nonoisy case, good performance was obtained from combinations involving the RVI or NDVI, whereas the PVI and WDVI lead to rather poor performance, which is the contrary in the noisy case. Poor performance was obtained from combinations involving the RVI or NDVI, whereas the PVI or WDVI lead to good performance. Indeed, in the absence of noise, the vegetation isolines have an increasing slope (versus fCover) as modeled by RVI or NDVI, whereas when noise increases, the vegetation isolines become more parallel as modeled by PVI or WDVI. We also note that in the absence of noise, TSAVI performs much better than SAVI or MSAVI, whereas when noise becomes important, better performance is obtained by SAVI or MSAVI (rather than TSAVI). These comments show the difficulty of choosing a universally “best” vegetation index (practically, noise level is unknown so that we cannot use this information for vegetation index choice). The vegetation indices called “SCE-UA” and “Sx” are interesting since they are adaptive and, therefore, more robust (we check on the two noise-level simulations that, in both cases, lead to the second best performance). Another solution to improve individual index performance is

to consider vegetation index combination. An interesting point put forward by Fig. 5(b) is that, in a combination of several indices, when the “correlated” indices present good performance, the combination result is robust (and performing) versus the combination rule (conjunctive or cautious adaptive)—see the case of combinations involving PVI and WDVI or SCE-UA and “Sx”—and when the “correlated” indices present poor performance, the use of the cautious-adaptive rule allows the improvement of the results in a significant way—see the case of combinations involving RVI and NDVI. Fig. 6 compares the evidential fusion with the estimation given by the arithmetic mean of the fCover estimated from each index involved in the considered evidential fusion. Indeed, in the absence of any a priori information about the accuracy of the considered indices, the best estimator is the arithmetic mean. Negative values correspond to a better performance (smaller L1 error) of the evidential fusion in comparison to the arithmetic-mean estimator. First, one can point again the previous conclusions concerning the interest of the cautiousadaptive combination rule relative to the conjunctive one. Then, from Fig. 6, the interest of the evidential framework is clear since almost all considered combinations exhibit negative values, involving an improvement of both performance accuracy and robustness, whatever is the case: with or without noise. C. Results in the Case of Actual Data Now, in order to test the fusion in the case of actual data, we consider both remote-sensed data and reference field data (ground truth) measurements acquired simultaneously over the Yar watershed (61 km2 ). This watershed, which is located in a fairly intensive farming area (field size is about 1 ha) in northern Brittany, France, has been carried out for several years. In 2006, over 81 fields, ground-truth measurements of

KALLEL et al.: FUSION OF VEGETATION INDICES

1509

Fig. 6. Comparison (difference of the first- and second-order moments on the errors) between evidential fusion and average estimation. (a) Simulated data without noise. (b) Noisy simulated data.

Fig. 7. Performance (robustness versus accuracy) evaluation from actual data. (a) Comparison of the results obtained using each vegetation index individually or combining several vegetation indices. (b) Difference between the performance of the multiindex fusion and the best performance obtained from the vegetation index involved in the fusion.

fCover values have been acquired with a 5% precision for low fCover values (fCover ≤ 25%) and a 10% accuracy for high fCover values (fCover > 25%). The distribution of the observed fCover values presents two modes: one mode around one since the majority of fields are covered in winter, and one mode around 0.1. Indeed, fields with mean vegetation are few since the leaseholders are either respecting the law (winter coverage) or not respecting it, but rarely respecting it only partially. Concomitant to the ground-truth campaign, a very high resolution Quickbird image has been acquired over the Yar basin on March 22, 2006: pixel 2.8 × 2.8 m and four frequency bands, namely, Blue (450–520 nm), Green (520–600 nm), Red (630–690 nm), and NIR (760–900 nm). Preprocessing of the data in order to correct the atmospheric effects and to calibrate

the data to perform the Sx and SCE-UA fCover estimations is described in [3]. Fig. 7 shows the obtained results, as previously presented, in terms of performance accuracy and robustness. In this part of this paper, the whole set of two index combinations has been divided as follows: 1) the combinations involving either the Sx or the SCE-UA fCover estimation; 2) among the remainder combinations, those involving either the SAVI or the TSAVI or the MSAVI; and 3) the remainder combinations (involving only the basic indices: PVI, WDVI, RVI, and NDVI). From Fig. 7(a), the combinations involving either the Sx or the SCE-UA estimations are better than others. However, they do not improve the results obtained using only one of these two indices (except for the combination with PVI, but the

1510

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

Fig. 8. Performance of the fCover estimation versus the fCover: Comparison of monoindex estimation and their fusion in the case of the three vegetation indices PVI, NDVI, and TSAVI: (a) Mean L1 error per fCover value and (b) per fCover, mean L1 error value multiplied by the number of fields.

improvement is very small). Indeed, SCE-UA and, at a lower degree, Sx estimations already provide very good results, and therefore, their improvement seems difficult. Now, these indices are rather complicated in terms of soft development, and one may prefer to perform the fusion of several “simpler” indices. In Fig. 7(b), we show the improvement (relative to the “best” vegetation index among those involved in a combination) due to fusion. We focus on the case of the classical indices (excluding Sx and SCE-UA). The indices involved in a considered combination are indicated besides the combination-resulting point, omitting the extension “Vegetation Index” (“VI”): “PVI” becomes “P,” “SAVI” becomes “SA,” etc.). The more interesting combinations are those involving complementary indices such as PVI and NDVI and one of the ∗SAVIs (SAVI, TSAVI, or MSAVI). A more surprising observation is the closeness of the results. On our data set, there is an important “correlation” between the vegetation indices in the sense that the fCover estimations they provide are very close. We explain the absence of soil-line sensitivity by the fact that in winter, the soil is very wet (almost saturated) due to the abundant rains; therefore, the surface reflectance special variability is too low to let the soil-line dependence appear. Then, fusion of several indices seems less informative than in the case of the simulated data where vegetation indices appear more complementary. Finally, we note that the combination of three indices does not really improve the result obtained in combining two indices. Finally, Fig. 8 shows the L1 error versus the fCover. It considers the case of vegetation indices PVI, NDVI, and MSAVI, and their successive combinations PVI + NDVI, and PVI + NDVI + MSAVI. However, similar results may be observed with other combinations [considered on Fig. 7(b)]. Fig. 8(a) shows the mean error variation. Improvements due to fusion occur successively for fCover = 1 when combining PVI and NDVI and for fCover = 0.075 when adding TSAVI to the previous PVI + NDVI combination. For other fCover values, the combination does not lead systematically to improvements. However, as already noted, the fCover histogram is bimodal with the two modes corresponding to values where the estima-

tion is improved by fusion. Fig. 8(b) shows the contribution to the L1 error global mean (for a given fCover = p, the mean error value times the number of fields having fCover = p). We clearly see the effect of the large number of fields completely covered. V. C ONCLUSION In this paper, we consider the problem of fusion of different sources that are not necessarily independent and in a continuous discernment space. This problem was raised in the framework of the fCover estimation from remote-sensing data. However, the defined solutions may be applied to similar problems, particularly data fusion of different sensor measurements to estimate other real parameters. Then, the contributions of this paper are of two kinds: theoretical, in the domain of fusion, and thematic, in the domain of vegetation monitoring. Concerning the fusion contributions, the results are the following. 1) First, to handle the continuous case, we define the “cwf” that can be viewed as a density of − log(w), where w is the mass of Ω (discernment frame) of the SSFs involved in the decomposition of a nondogmatic bba. The interest of this “cwf” is for redundance modeling in the case of source combination according to the proposed rule recalled just previously. 2) Second, we propose a new combination rule called cautious adaptive that allows taking into account the partial redundancy between sources through a supervised parameter. We show that, when this parameter varies from zero to one, the proposed rule varies from the classical conjunctive combination to the cautious rule defined by Smets. The main advantage of the proposed rule is then to control the level of a priori redundancy between the sources since two sources may be apparently correlated (if they provide the same analysis) but independent in a cognitive way. Concerning the vegetation monitoring, we show that combining several vegetation indices can improve either the accuracy

KALLEL et al.: FUSION OF VEGETATION INDICES

1511

of the fCover estimation or its robustness. In some optimal cases, it can even lead to improvement of both accuracy and robustness. These conclusions were drawn from the analysis of simulated data using the coupled Adding/PROSPECT radiative transfer model, either with or without noise simulating the natural variability of soil/vegetation parameters. In the case of actual data, results seem less clear which is mainly due to the specific distribution of the ground fCover (mainly bimodal and with poor fCover accuracy). In the case of the classical vegetation indices, the combination of two or three of them may improve the estimation relatively to the best estimator among the indices involved. However, in the case of more sophisticated inversion methods (such as the Sx or the SCE-UA), there is no real improvement. Perspectives are also twofold. First, future studies will deal with automatic learning of the “correlation” or “redundancy” between sources. This learning can only be based on the analysis of the conjoint statistics of the considered sources. Another problem that will occur is the modeling of the partial redundancy between three or more sources. Indeed, to preserve the associativity of the combination, this notion of redundancy has to be carefully defined. The second perspective is related to our specific application. A more discerning model of the vegetation index “correlation” function could be defined by introducing the fCover dependence. It will allow modeling, for example, a higher “correlation” of the indices, taking into account the soil line at lower fCover values. More precise a priori information about the reliability of each vegetation index versus the fCover inversion domain can also be taken into account. A PPENDIX I P ROOFS OF T HEOREMS Proof: Theorem 1: ∂2 ∂ 2 q ([a, b]) =−  ∂a∂b ∂a∂b  y=1 x=a         × q ([0, 1]) exp  ϕ(x, y)dydx    

f (a, b) = −

Proof: Theorem 2: ϕ1 ∩ 2 (a, b) = − =−

∂ 2 log (q1 ([a, b]) q2 ([a, b])) ∂a∂b

=−

∂ 2 (log (q1 ([a, b])) + log (q2 ([a, b]))) ∂a∂b

∂ 2 log (q1 ([a, b])) ∂ 2 log (q2 ([a, b])) − ∂a∂b ∂a∂b = ϕ1 (a, b) + ϕ2 (a, b). (37)

=−


Proof: Proposition 1: Let ϕ = ϕ1 − ϕ2 . ϕ ≥ 0, ∀a, b/ 0 ≤ a ≤ b ≤ 1; thus, there is f such that ϕ is its following cwf. ∩ f. Therefore, according to Theorem 2, f1 = f2  A PPENDIX II A DDING M ETHOD The adding method [50], [51], [53], [54] is based on the assumption that a vegetation layer receiving a radiation flux, from bottom or top, partially absorbs it and partially scatters it upward or downward, independent of the other layers. Thus, the relationships between fluxes are given by operators providing the output flux density distribution versus the input flux density distribution. The generalization of the adding operators presented in [53] and [54] in the continuous case dealing with radiance hemispherical distribution is as follows. With Le being the radiance in the observation solid angle Ωe = (θe , ϕe ) (θe is the zenithal angle, and ϕe the azimuthal angle in the observation direction) provided by the scattering of an incident source flux by the medium dEi (Ωi ), within a cone of solid angle dΩi = sin(θi )dθi dϕi (θi is the zenithal angle, and ϕi is the azimuthal angle in the source direction), the bidirectional reflectance distribution function (BRDF) is

x=0 y=b

∂ = q ([0, 1]) ∂b   y=1      × − ϕ(a, y)dy    y=b   y=1 x=a       ϕ(x, y)dxdy  × exp    x=0 y=b   y=1 x=a     ϕ(x, b)dx ϕ(a, y)dy  = q ([0, 1]) ϕ(a, b) +   × exp 

x=0 y=1 x=a  



 ϕ(x, y)dydx .

y=b

(36)

∂ 2 log (q1 ∩ 2 ([a, b])) ∂a∂b

r(Ωi → Ωe ) =

dLe (Ωi , Ωe ) dLe (Ωi , Ωe ) = dEi (Ωe ) Li (Ωi ) cos(θi )dΩi

(38)

where Li is the radiance provided by the source. Passing through a medium, the source radiation flux produces a radiance in the Ωe direction and the bidirectional transmittance t(Ωi → Ωe ) =

dLe (Ωi , Ωe ) . dEi (Ωi )

(39)

For both cases, Le is obtained by integrating the source flux over the hemisphere  Le (Ωe ) = {t, r}(Ωi → Ωe )Li (Ωi ) cos(θi )dΩi . Π

x=0 y=b




 over hemisphere (40)

1512

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008

The scattering operators R and T , which give the outward radiance Li from an incident radiance defined over the whole hemisphere Le , are as follows:  R[Li ](.) =

r(Ωi → .)Li (Ωi ) cos(θi )dΩi

(41)

t(Ωi → .)Li (Ωi ) cos(θi )dΩi .

(42)

Π



T [Li ](.) = Π

The reflectance operator of the combination of two layers 1 and 2 (below layer 1) is Rt = Rt,1 + Tu,1 ◦ (I − Rt,2 ◦ Rb,1 )−1 ◦ Rt,2 ◦ Td,1 (43) where Rt , Rb , Tu , and Td represent the reflectance operators of the top and bottom of the layer and the transmittance operators upward and downward, respectively. The second subscripts refer to the layer number. To compute the canopy BRDF, the operators are discretized. The zenithal and azimuthal angles are sampled, the operators are replaced by matrices, and “◦” are replaced by the matrix multiplication. The derivation of the adding method operator is as follows. For one vegetation layer, the top and bottom reflectance operators and the downward and upward transmittance operators require the estimation of top and bottom bidirectional reflectances, and the downward and upward bidirectional transmittances are derived based on the SAIL formalism [43], [55]. Since the SAIL model assumes that the diffuse fluxes are semiisotropic, the operator derivation is only correct for a thin layer (LAI < 10−2 ) where the diffuse-flux contribution is small. Therefore, in the adding method, the considered layer is divided into thin sublayers. The whole-layer reflectance operator is derived by using the adding method (43). R EFERENCES [1] S. Dabney, J. Delgado, and D. Reeves, “Using winter cover crops to improve soil and water quality,” Commun. Soil Sci. Plant Ann., vol. 32, no. 7/8, pp. 1221–1250, 2001. [2] G. Rondeaux, M. Steven, and F. Baret, “Optimization of soil-adjusted vegetation indices,” Remote Sens. Environ., vol. 55, no. 2, pp. 95–107, Feb. 1996. [3] A. Kallel, S. L. Hégarat-Mascle, C. Ottlé, and L. Hubert-Moy, “Determination of vegetation cover fraction by inversion of a four-parameter model based on isoline parametrization,” Remote Sens. Environ., vol. 111, no. 4, pp. 553–566, Dec. 2007. [4] G. Shafer, A Mathematical Theory of Evidence. Princeton, NJ: Princeton Univ. Press, 1976. [5] P. Smets, “Un modéle mathémati co-statistique simulant le processus du diagnostic médical,” Ph.D. dissertation, Univ. Libre de Bruxelles, Bruxelles, Belgium, 1978. [6] P. Smets, “The combination of evidence in the transferable belief model,” IEEE Pattern Anal. Mach. Intell., vol. 12, no. 5, pp. 447–458, May 1990. [7] I. Bloch, “Some aspects of Dempster–Shafer evidence theory for classification of multi-modality medical images taking partial volume effect into account,” Pattern Recognit. Lett., vol. 17, no. 8, pp. 905–916, Jul. 1996. [8] J. Van Cleynenbreugel, S. Osinga, F. Fierens, P. Suetens, and A. Oosterlinck, “Road extraction from multi-temporal satellite images by an evidential reasoning approach,” Pattern Recognit. Lett., vol. 12, no. 6, pp. 371–380, Jun. 1991.

[9] F. Tupin, I. Bloch, and H. Matre, “A first step toward automatic interpretation of SAR images using evidential fusion of several structure detectors,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 3, pp. 1327– 1343, May 1999. [10] T. Denoeux, “A k-nearest neighbor classification rule based on Dempster–Shafer theory,” IEEE Trans. Syst., Man, Cybern., vol. 25, no. 5, pp. 804–813, May 1995. [11] S. Le Hégarat-Mascle, I. Bloch, and D. Vidal-Madjar, “Application of Dempster–Shafer evidence theory to unsupervised classification in multisource remote sensing,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 4, pp. 1018–1031, Jul. 1997. [12] S. Le Hégarat-Mascle, I. Bloch, and D. Vidal-Madjar, “Introduction of neighborhood information in evidence theory and application to data fusion between radar and optical images with partial cloud cover,” Pattern Recognit., vol. 31, no. 11, pp. 1811–1823, Nov. 1998. [13] S. Le Hégarat-Mascle and R. Seltz, “Automatic change detection by evidential fusion of change indices,” Remote Sens. Environ., vol. 91, no. 3/4, pp. 390–404, Jun. 2004. [14] S. Le Hégarat-Mascle, R. Seltz, L. Hubert-Moy, S. Corgne, and N. Stach, “Performance of change detection using remotely sensed data and evidential fusion: Comparison of three cases of application,” Int. J. Remote Sens., vol. 27, no. 16, pp. 3515–3532, Aug. 2006. [15] D. Dubois, H. Prade, and P. Smets, “New semantics for quantitative possibility theory,” in Proc. 6th ECSQARU, 2001, pp. 410–421. [16] B. Ristic and P. Smets, “Belief function theory on the continuous space with an application to model based classification,” in Proc. IPMU, Perugia, Italy, 2004, pp. 1119–1126. [17] P. Smets, “Belief functions on real numbers,” Int. J. Approx. Reason., vol. 40, no. 3, pp. 181–223, Nov. 2005. [18] F. Caron, E. B. Ristic, Duflos, and P. Vanheeghe, “Least committed basic belief density induced by a multivariate Gaussian: Formulation with applications,” Int. J. Approx. Reason., 2007. to be published. [19] P. Smets and R. Kennes, “The transferable belief model,” Artif. Intell., vol. 66, no. 2, pp. 191–234, Apr. 1994. [20] P. Smets, “The application of the matrix calculus to belief functions,” Int. J. Approx. Reason., vol. 31, no. 1/2, pp. 1–30, Oct. 2002. [21] R. R. Yager, “The entailment principle for Dempster–Shafer granules,” Int. J. Intell. Syst., vol. 1, no. 4, pp. 247–262, 1986. [22] F. Klawonn and P. Smets, “The dynamic of belief in the transferable belief model and specialization-generalization matrices,” in Proc. 8th Annu. Conf. UAI, 1992, pp. 130–137. [23] P. Smets, “An axiomatic justification for the use of belief function to quantify beliefs,” in Proc. Int. Joint Conf. AI, Chambery, France, 1993, pp. 598–603. [24] D. Dubois and H. Prade, “On the unicity of Dempster’s rule of combination,” Int. J. Intell. Syst., vol. 1, no. 2, pp. 133–142, 1986. [25] D. Dubois and H. Prade, “A set theoretical view of belief functions,” Int. J. Gen. Syst., vol. 12, pp. 193–226, 1986. [26] P. Smets, “The concept of distinct evidence,” in Proc. IPMU, Palma de Mallorca, Spain, 1992, pp. 789–794. [27] D. Dubois and H. Prade, Possibility Theory: An Approach to Computerized Processing of Uncertainty. New York: Plenum, 1988. [28] M. Fauvel, J. Chanussot, and J. A. Benediktsson, “Decision fusion for the classification of urban remote sensing images,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 10, pp. 2828–2838, Oct. 2006. [29] J. Chanussot, G. Mauris, and P. Lambert, “Fuzzy fusion techniques for linear features detection in multitemporal SAR images,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 3, pp. 1292–1305, May 1999. [30] P. Smets, “The canonical decomposition of a weighted belief,” in Proc. IJCAI, Montreal, PQ, CA, 1995, pp. 1896–1901. [31] T. Denoeux, “The cautious rule of combination for belief functions and some extensions,” in Proc. FUSION, Florence, Italy, 2006, pp. 1–8. [32] T. M. Strat, “Continuous belief functions for evidential reasoning,” in Proc. AAAI, Austin, TX, 1984, pp. 308–313. [33] P. Smets, “Decision making in the TBM: The necessity of the pignistic transformation,” Int. J. Approx. Reason., vol. 38, no. 2, pp. 133–147, Feb. 2005. [34] A. Kallel, S. L. Hégarat-Mascle, and L. Hubert-Moy, “Combination of partially non-distinct consonant beliefs: The cautious-adaptive rule,” Int. J. Approx. Reason., 2007, submitted for publication. [35] P. Smets, “Quantified epistemic possibility theory seen as an hyper cautious transferable belief model,” in Proc. Rencontres Francophones sur la Logique Floue et ses Appl. (LFA), La Rochelle, France, 2000, pp. 343–353. [36] J. W. Rouse, R. H. Haas, J. A. Schell, D. W. Deering, and J. C. Harlan, “Monitoring the vernal advancement retrogradation of natural vegetation,” NASA/GSFC, Greenbelt, MD, 1974. Final Report, Type III.

KALLEL et al.: FUSION OF VEGETATION INDICES

[37] A. R. Huete, “A soil-adjusted vegetation index (SAVI),” Remote Sens. Environ., vol. 25, no. 3, pp. 295–309, Aug. 1988. [38] F. Baret and G. Guyot, “Potentials and limits of vegetation indices for LAI and APAR assessment,” Remote Sens. Environ., vol. 35, no. 2/3, pp. 161–173, Feb./Mar. 1991. [39] J. Qi, A. Chehbouni, A. R. Huete, and Y. H. Kerr, “A modified soil adjusted vegetation index,” Remote Sens. Environ., vol. 48, no. 2, pp. 119– 126, May 1994. [40] J. G. P. W. Clevers and W. Verhoef, “LAI estimation by means of the WDVI: A sensitivity analysis with a combined PROSPECT–SAIL model,” Remote Sens. Environ., vol. 7, pp. 43–64, 1993. [41] F. Baret and S. Buis, “Estimating canopy characteristics from remote sensing observations,” in Review of Methods and Associated Problems. New York: Springer-Verlag, 2007. [42] S. Jacquemoud and F. Baret, “Prospect: A model of leaf optical properties spectra,” Remote Sens. Environ., vol. 34, no. 2, pp. 75–91, Nov. 1990. [43] W. Verhoef, “Earth observation modeling based on layer scattering matrices,” Remote Sens. Environ., vol. 17, no. 2, pp. 165–178, Apr. 1985. [44] J. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J., vol. 7, pp. 308–313, 1965. [45] Q. Duan, S. Sorooshian, and V. K. Gupta, “Effective and efficient global optimization for conceptual rainfall-runoff models,” Water Resour. Res., vol. 28, no. 4, pp. 1015–1031, 1992. [46] A. J. Richardson and C. L. Wiegand, “Distinguishing vegetation from soil background information,” Photogramm. Eng. Remote Sens., vol. 43, pp. 1541–1552, 1977. [47] J. G. P. W. Clevers, “The application of a weighted infrared-red vegetation index for estimating leaf area index by correcting for soil moisture,” Remote Sens. Environ., vol. 25, no. 1, pp. 53–69, Jul. 1989. [48] R. L. Pearson and L. D. Miller, “Remote mapping of standing crop biomass for estimation of the productivity of the short grass prairie,” in Proc. 8th Int. Symp. Remote Sens. Environment, M. Erim and A. Arbor, Eds., 1972, pp. 1357–1381. [49] F. Baret, G. Guyot, and D. Major, “TSAVI: A vegetation index which minimizes soil brightness effects on LAI and APAR estimation,” in Proc. Joint 12th Can. Symp. Remote Sensing and IGARSS, 1989, vol. 4, pp. 1355–1359. [50] A. Kallel, S. L. Hégarat-Mascle, C. Ottlé, and L. Hubert-Moy, “Canopy bidirectional reflectance calculation based on adding method and sail formalism,” in Proc. IGARSS, Barcelona, Spain, 2007, pp. 2554–2557. [51] A. Kallel, W. Verhoef, S. L. Hégarat-Mascle, C. Ottlé, and L. HubertMoy, “Canopy bidirectional reflectance calculation based on adding method and SAIL formalism,” Remote Sens. Environ., 2007, submitted for publication. [52] A. Kuusk, “Photon–vegetation interactions,” in Applications in Optical Remote Sensing and Plant Ecology. Berlin, Germany: Springer-Verlag, 1991, ch. The Hot Spot Elect in Plant Canopy Reflectance, pp. 139–159. [53] H. C. van de Hulst, Light Scattering by Small Particles. New York: Dover, 1981. [54] K. Cooper, J. A. Smith, and D. Pitts, “Reflectance of a vegetation canopy using the adding method,” Appl. Opt., vol. 21, no. 22, pp. 4112–4118, 1982. [55] W. Verhoef, “Light scattering by leaf layers with application to canopy reflectance modelling: The SAIL model,” Remote Sens. Environ., vol. 16, no. 2, pp. 125–141, Oct. 1984.

Abdelaziz Kallel received the M.S. degree in telecommunications from the Ecole Supérieure des Communications, Tunis, Tunisia, in 2003. He is currently working toward the Ph.D. degree at the Centre d’étude des Environnements Terrestre et Planétaires/ Institut Pierre-Simon Laplace, Vélizy, France. His research interest concerns canopy modeling based on radiative transfer theory, data fusion, and image processing. His current works deal with vegetation monitoring using remotely sensed visible/ infrared data based on radiative transfer modeling, data fusion, and image classification.

1513

Sylvie Le Hégarat-Mascle received the Ph.D. degree in signal and image processing from the École Nationale Supérieure des Télécommunications, Paris, France, in 1996, and the “Habilitation à Diriger des Recherches” from Versailles University, Versailles, France, in 2006. She was an Associate Professor with Versailles University from 1998 to 2005. Since 2006, she has been a Professor with Université de Paris-Sud, Orsay, France. Her research interests include statistical pattern recognition, image analysis, and data fusion. She works with Bayes theory and Dempster–Shafer evidence theory. In the remotesensing domain (SAR and optical sensor), concerned applications include landcover/land-use classification and vegetation and soil moisture monitoring.

Laurence Hubert-Moy received the Ph.D. degree in geography from the Université de Rennes II, Rennes, France. She is currently a Professor with the Université de Rennes II. Her research focuses in the area of agriculture applications of remote-sensing data. Her work is mostly dedicated to land use and cover changes for environmental management from optical images. Specifically, she focuses her efforts in various critical areas such as local and regional land cover and land-cover change mapping in intensive farming areas, wetland characterization, and spatial modeling as applied to image understanding and use. Dr. Hubert-Moy received the Bronze Medal of the French National Center for Scientific Research in 2003.

Catherine Ottlé received the M.S. and Ph.D. degrees in physics from the Université de Paris, Paris, France, in 1981 and 1983, respectively. In 1985, she was with the Centre National de la Recherche Scientifique, Saclay, as a Research Scientist. She was also with the Centre d’étude des Environnements Terrestre et Planétaires, Paris. She has been with the Laboratoire des Sciences du Climat et de l’Environnement/Institut Pierre-Simon Laplace, Centre d’Etudes de Saclay, Paris, since 2006. Her research interests include the applications of remote sensing to the study of the land surface processes for hydrology and vegetation monitoring with an emphasis on inversion and assimilation techniques.