The surface state of the Earth can be represented in remote sensing by three entities that are vegetation surfaces, aquatic surfaces and mineral surfaces. Any surface area observed, depending on the size of the area, may be a combination of these three entities.

Thus, it can be observed vegetation surface, aquatic surface, mineral surface, vegetation and aquatic surface, vegetation and mineral surface, mineral and aquatic surface, vegetation, mineral and aquatic surface.

The use of satellite images, for mapping purposes, has been the subject of several studies ( [1] [2] [3] [4] ). Researchers have used spectral indices to map vegetation surfaces, aquatic surfaces, and surfaces of bare soil and mansions.

These indices were used for image classification. Unfortunately, imperfections (uncertainties, inaccuracies, etc.) on the information produced by the images associated with said indices are observed ( [5] [6] ). Taking into account and good management of these imperfections are done by the theory of belief functions.

The Dempster-Shafer theory has been used successfully to handle cases of uncertainty, vagueness and ignorance in the classification of pixels to classes of vegetation surface, aquatic surface and mineral surface [4] .

In this article, the theory of plausible and paradoxical reasoning of Dezert- Smarandache (DSmT) is used to take into account the paradoxical character through the intersections of the elements vegetation surface, aquatic surface and mineral surface.

The general objective of the study is to develop a model of pixel classification by aggregating information using the DSmT theory, spectral indices NDVI (Normalized Difference Vegetation Index), MNDWI (Modification of Normalized Difference Water Index) and NDBaI (Normalized Difference Bare Index) and ASTER satellite images. It acts specifically first, to model the frameworks of discernment and reasoning and belief functions, then define the decision criteria and write algorithms and programming codes under the MATLAB software; finally realize and evaluate classified image.

The contribution of this study is to give an approach of unsupervised classification of mapping that takes into account the plausible and paradoxical characteristics related to the information of ground that is to say without needing to know the real spatial state of the ground concerned.

This paper, which proposes to report on the work carried out, presents successively the theory of plausible and paradoxical reasoning of Dézert-Smarandache, the material used, the methodological approach that guided the work and the results obtained.

Dezert-Smarandache theory (DSmT) is interpreted as a generalization of the theory of Dempster-Shafer (DST) ( [7] ). Its basic principle and formalism for aggregating information can be characterized by a four-stage structure of modeling, estimation, combination and decision.

Modeling consists in choosing the representation of the frameworks of discernment and reasoning and the models of the mass functions to be used.

The discernment framework Θ = { θ 1 , ⋯ , θ N } of DSmT is an exhaustive set of different hypotheses, not necessarily exclusive assumptions. The reasoning framework D^Θ associated, also called hyper-powerset, is the set of all possible propositions constructed from the elements of Θ, including the empty set (f), with the operators È and Ç. It is characterized by the following conditions ( [8] ):

1) ϕ , θ 1 , ⋯ , θ N ∈ D Θ ,

2) If A i , A j ∈ D Θ then A i ∩ A j ∈ D Θ and A i ∪ A j ∈ D Θ ,

3) There are no other elements belonging to D^Θ, except those obtained using rules 1 or 2.

The construction of the reasoning framework D^Θ can be obtained by a matrix product between the binary matrix of Dedekind and the coding vector of Smarandache ( [1] [7] ). Its cardinal increases according to the cardinal of the discernment framework on which it is based.

In order to fix ideas, it is considered the cardinal of Θ equal to 3. Thus, according to Dedekind ( [9] ), the cardinal of D^Θ is equal to 19. Then we get ( [1] ):

Θ = { θ 1 , θ 2 , θ 3 }

D Θ = { ϕ , θ 1 , θ 2 , θ 3 , θ 1 ∪ θ 2 , θ 2 ∪ θ 3 , θ 1 ∪ θ 3 , θ 1 ∪ θ 2 ∪ θ 3 , θ 1 ∩ θ 2 ,       θ 2 ∩ θ 3 , θ 1 ∩ θ 3 , θ 1 ∩ ( θ 2 ∪ θ 3 ) , θ 3 ∩ ( θ 1 ∪ θ 2 ) , θ 2 ∩ ( θ 1 ∪ θ 3 ) ,       θ 2 ∪ ( θ 1 ∩ θ 3 ) , θ 3 ∪ ( θ 1 ∩ θ 2 ) , θ 1 ∪ ( θ 2 ∩ θ 3 ) ,       ( θ 1 ∩ θ 3 ) ∪ ( θ 1 ∩ θ 2 ) ∪ ( θ 2 ∩ θ 3 ) , θ 1 ∩ θ 2 ∩ θ 3 }

In general, we define a generalized mass function m of D^Θ with values in [0,1] satisfying the following conditions of Equation (1):

{ ∑ A ∈ D Θ m ( A ) = 1 m ( ϕ ) = 0 (1)

where f is the empty set. The value m(A) quantifies the belief that the class sought belongs to the subset A of D^Θ (and to no other subset of A). The subsets A such that m(A) > 0 are called focal elements.

The following special mass functions are defined ( [10] ):

ü a mass function m is said to be normal when m(f) = 0;

ü a mass function m is said to be categorical when it has a single focal element A such that m(A) = 1. In the case where A is a set, knowledge is certain but imprecise.

When A = {𝜃_k}, knowledge is certain and precise;

ü a mass function m is said to be empty (or total ignorance) when the function m is categorical in Θ: m(Θ) = 1;

ü a mass function m is said to be Bayesian if all the focal elements are singletons of Θ in Equation (2):

∑ θ k ∈ Θ m ( θ k ) = 1 ; (2)

ü a mass function m is said to be dogmatic if m(Θ) = 0;

ü a mass function m is said to be consonant if all the focal elements are nested;

ü a mass function m is said to have a simple support when it has 2 focal elements, one of which is Θ (Equation (3)):

{ m ( A ) = 1 − ω ,   A ∈ D θ m ( Θ ) = ω m ( B ) = 0 ,   ∀ B ∈ D θ \ A (3)

In this case, the function m can also be denoted A^ω where ω represents the weight of the ignorance of the mass function m.

The estimation consists in determining all the parameters of the mass functions selected at the modeling stage. This is a difficult problem that does not have a universal solution. The difficulty is further increased if we want to assign masses to compound hypotheses involving intersections and/or unions ( [11] ).

The combination is the grouping phase of the information, from the mass functions of the different information sources, using an operator adapted to the formalism of the modeling. The DSmT has two types of combination ( [12] ): the classic version and the hybrid version. Consider n initial mass functions m 1 , m 2 , ⋯ , m n representing the respective information of n different sources, which can be combined according to the DSmT.

The classic combination of DSmT is (Equation (4)):

m ( C ) = ∑ A 1 ∩ A 2 ∩ ⋯ ∩ A n = C m 1 ( A 1 ) ∗ m 2 ( A 2 ) ∗ ⋯ ∗ m n ( A n )   A i ( i = 1 , ⋯ , n ) ∈ D θ ,   ∀ C ∈ D θ (4)

The hybrid combination of DSmT is used in the presence of integrity constraints applied to D^Θ. An integrity constraint of a set U is an impossibility of considering a mass assignment to this set ( [12] ). The mass of the set U is then assigned to the empty set f. Thus, the hybrid combination is defined ( [13] ) by Equations (5)-(8):

m ( A ) = ϕ ( A ) ∗ [ H 1 ( A ) + H 2 ( A ) + H 3 ( A ) ] ,   ∀ A ∈ D θ (5)

H 1 ( A ) = ∑ X i ( i = 1 , ⋯ , n ) ∈ D θ X 1 ∩ X 2 ∩ ⋯ ∩ X n = A m 1 ( X 1 ) ∗ m 2 ( X 2 ) ∗ ⋯ ∗ m n ( X n ) (6)

H 2 ( A ) = ∑ X i ( i = 1 , ⋯ , n ) ∈ ϕ [ ∪ i = 1 n u ( X i ) ) = A ] ∨ [ ( ∪ i = 1 n u ( X i ) ∈ ϕ ) ∧ ( A = Θ ) ] m 1 ( X 1 ) ∗ m 2 ( X 2 ) ∗ ⋯ ∗ m n ( X n ) (7)

H 3 ( A ) = ∑ X i ( i = 1 , ⋯ , n ) ∈ D θ X 1 ∪ X 2 ∪ ⋯ ∪ X n = A X 1 ∩ X 2 ∩ ⋯ ∩ X n = ϕ m 1 ( X 1 ) ∗ m 2 ( X 2 ) ∗ ⋯ ∗ m n ( X n ) (8)

The function f(A) of Equation (5) is a binary function equal to zero for empty or impossible sets A and is worth the unit for the others. In Equation (7), the set u(X_i) represents the union of all the objects of the set X_i. Thus, this equation indicates that if the union of objects is also a constraint, then the mass is either assigned to the union of all the singletons that form the objects, or to total ignorance. Equation (8) indicates that the mass is attributed to the union of all the objects of the sets X_i, if the intersection is a constraint.

Several rules exist to achieve a hybrid combination. The 5th version of the family of rules of combination with Proportional Redistribution of Conflict (PCR5) is presented in the following, because it gives better results ( [14] ).

In a simplifying process, Djiknavorian ( [12] ) proposes a dynamic procedure which can be put in the form of the following algorithm (Dj):

Beginning s 1 = m 1 For   i = 1   to   n − 1   do s i + 1 ( C ) = ∑ A , B ∈ D θ A ∩ B = C s i ( A ) ∗ m i + 1 ( B ) + ∑ D ∈ D θ D ∩ C = Φ [ ( s i ( C ) ) 2 ∗ m i + 1 ( D ) s i ( C ) + m i + 1 ( D ) + ( m i + 1 ( C ) ) 2 ∗ s i ( D ) m i + 1 ( C ) + s i ( D ) ] ,   ∀ C ∈ D θ End m ( C ) = s n ( C ) ,   ∀ C ∈ D θ End

where m_i (i = 1 to n) represent n mass functions corresponding to n different information sources and m the combined mass function obtained by the PCR5 rule.

The most commonly used decision rules for DSmT are based on combined mass functions, credibility functions, plausibility functions or pignistic probability functions.

The functions of credibility (Bel), of plausibility (Pls) and of pignistic probability (BetP) are defined from D^Θ in [ 0 , 1 ] and are given respectively by Equations (9)-(12):

B e l ( A ) = ∑ B ∈ D θ ,   B ⊆ A m ( B )   ,   ∀ A ∈ D θ (9)

P l s ( A ) =   ∑ B ∈ D θ ,   B ∩ A ≠ ∅ m ( B )   ,   ∀ A ∈ D θ (10)

B e t P ( A ) = ∑ θ i ∈ A B e t P ( θ i ) ,     ∀ A ∈ D θ (11)

where

B e t P ( θ i ) = ∑ A ∈ D θ , θ i ∈ A m ( A ) | A |   ,     ∀ θ i ∈ Θ       i = 1 , ⋯ , n (12)

| A | is the cardinal of the set A

Moreover, the interval [ B e l ( A ) , P l s ( A ) ] , known as the confidence interval, which quantifies the ignorance of the source on the hypothesis A, can be used for a decision.

Following this presentation of the DSmT, the material used in this study is presented in the next section.

The tools used are software and data.

With regard to the software, it was first used ENVI 4.7 to preprocessing ASTER images, then MATLAB to develop a model based on the use of the spectral indices NDVI, MNDWI and NDBa, and the DSmT to characterize states of the vegetation, aquatic and mineral surfaces. The data are of two types: field data and remote sensing data.

Field data consist of geographical coordinates of fixed points and outcrops. They are collected in the Toumodi-Yamoussoukro-Tiébissou area in the V Baoulé in Côte d’Ivoire. The remote sensing data used are derived from the ASTER sensor and are rectified satellite images of the scene AST_L1A_00301102004105832, covering this area. This sensor has 14 bands with a broad spectral region covering the visible and near infrared (VNIR-Visible and Near Infrared), the medium infrared (SWIR-Short-Wave Infrared: Tape 4, Band 5, Band 6, Band 7, Band 8 and Band 9) and Thermal Infrared (TIR-Thermal Infrared: Band 10, Band 11, Band 12, Band 13 and Band 14).

The spatial resolution associated with the said images is 15 m in the visible and the near infrared, 30 m in the medium infrared and 90 m in the thermal infrared.

The approach used consisted first of a preprocessing on the ASTER satellite images under ENVI, and then it was developed a classification model based on the calculation of spectral indices (NDVI, MNDWI and NDBaI) and the use of DSmT. Concretely, it was a question of modeling the discernment and reasoning frameworks, the mass functions as well as the functions of measuring the evidence, and defining the decision criteria. In addition, algorithms and programming codes in language were realized under Matlab software and the classified image was generated and evaluated.

The results presented derive from the classifications made with the hybrid DSmT based on the PCR5 rule for the above four models generated by an integrity constraint and simplifications.

In this article, in order to map the surface state of the Toumodi-Yamoussoukro- Tiébissou zone in the “V Baoule” of Côte d’Ivoire, the theory of plausible and paradoxical reasoning of Dézert-Smarandache was proposed with the aim of taking into account the paradoxical nature of the intersections of the vegetation surface, the aquatic surface and the mineral surface from Aster satellite images and the NDVI, MNDWI and NDBaI spectral indices. After a preprocessing on the ASTER satellite images, a model of surface state characterization based on the calculation of spectral indices (NDVI, MNDWI and NDBaI) and the DSmT with the computation PCR5 rule, was developed. It gives an average rate of 93.34% for well classified pixels and a field compliance rate of 96.37%. However, the results could be improved by a supervised approach to mass functions, in particular through the extensive use of learning elements. This model could be used, with appropriate adjustments, for other mapping purposes.

Okaingni, J.-C., Ouattara, S., Kouassi, A., Vangah, W.J., Koffi, A.K. and Clement, A. (2017) Modeling and Characterization of Vegetation, Aquatic and Mineral Surfaces Using the Theory of Plausible and Paradoxical Reasoning from Satellite Images : Case of the Toumodi-Yamoussoukro-Tiébissou Zone in V Baoulé (Côte d’Ivoire). Open Journal of Applied Sciences, 7, 520-536. https://doi.org/10.4236/ojapps.2017.710038