AN ANALYSIS OF THE FUZZY EXPERT SYSTEMS ARCHITECTURE FOR

International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm) Volume 2, Number 2, Pages 35–69, June 2004 Publisher Item Identifier S 1542-5908(04)10203-0/$20.00 Article electronically published on April 12, 2003 at http://www.YangSky.com/ijcc22.htm. Please cite this paper as: Ronei Marcos de Moraes, “An Analysis of the Fuzzy Expert Systems Architecture for Multispectral Image Classification Using Mathematical Morphology Operators(Invited Paper)”, International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm), Volume 2, Number 2, Pages 35–69, June 2004 . AN ANALYSIS OF THE FUZZY EXPERT SYSTEMS ARCHITECTURE FOR MULTISPECTRAL IMAGE CLASSIFICATION USING MATHEMATICAL MORPHOLOGY OPERATORS(INVITED PAPER) RONEI MARCOS DE MORAES Abstract. A fuzzy expert-system architecture for image classification was proposed by Moraes, Banon and Sandri (1998, 2000, 2002), whose rules are implemented through translation invariant mathematical morphology operators. In this paper, we analyze this architecture by an expert system that classifies an area of the Tapaj´s National o Forest, in Brazil. We compare this classifier with others classical classifiers and analyze also its performance in other areas with the same rules. We conclude that the expert systems for image classification are more accurate that others classical classifiers. However, the expert systems are very dependent of the knowledge of an expert and they are not adaptive for classification of others areas. c 2003 Yang’s Scientific Research Institute, LLC. All rights reserved. 1. Introduction The use of mathematical morphology has produced many applications in several areas of digital image processing, and also in what regards pattern recognition in binary images [32]. Work on the classification of gray-level images using mathematical morphology is still in its early stages. Some results can be found on shape and texture classification using granulometry [1, 10, 11]. Moraes [22] proposed a classifier based on an intersection of an elementary morphological operators: an erosion and an anti-dilation [3]: the sup-generating operator which it is the base of classification process [22]. Received by the editors April 4, 2003 / final version received April 12, 2003. Key words and phrases. Fuzzy expert systems, multispectral image classification, mathematical morphology, classification algorithms, performance analysis. The author would like to thanks to Gerald Jean Francis Banon, Sandra Aparecida Sandri and Michelle Pontes Seixas for valuable contributions to this work. c 2003-2004 Yang’s Scientific Research Institute, LLC. All rights reserved. 35 36 RONEI MARCOS DE MORAES Classifiers assisted by expert systems emerge as an alternative to reduce the computational cost of numerical classifiers. A great advantage of these classifiers is that the decision process can be sufficiently rich even when it involves only few pieces of information. Furthermore, knowledge stored as rules is explainable, reusable, and can be treated in a simple way [16]. A disadvantage is that they will not necessarily respond rightly when employed out of the context for which they were designed, contrary to what generally happens with pure numerical methods. In this paper, we can see the results when we apply the fuzzy classifier to classify a region out of context of rules done by an expert. Some practical implementations of such classifiers can be found in the literature: the ICARE system [9] uses a fuzzy expert system that employs a statistical pre-classifier, maps and old classifications of a region, to classify an image; another classifier [15] combines heuristic and numerical methods to classify ice on the sea using radar images; works derived from the ICARE system use neural networks in the pre-classification process [21, 34]. In those systems that use numerical pre-classifiers, the expert system can be regarded as a post-classifier. Moraes, Banon and Sandri propose the use of fuzzy sets and translation invariant operators of mathematical morphology to build expert systems for image classification [23, 24, 25]. The methodology consists in a fuzzy expert system architecture, whose rules are implemented through translation invariant mathematical morphology operators. Using such a methodology, it is possible to create simple and powerful expert systems that produce satisfactory classifications with only a small number of rules. In this paper we are interesting to evaluate the performance of this architecture by an expert system that classifies an area of the Tapaj´s National Forest, in Brazil. We o compare this classifier with others classical classifiers and analyze also its performance in others regions with the same rules. This work is organized as follows. Section 2 presents the basic concepts of fuzzy set theory, mathematical morphology and expert systems. Section 3 presents a general expert system architecture for image classification, using fuzzy set theory as underlying knowledge representation model. Section 4 presents an expert system implementation using mathematical morphology and a classifier developed by using this methodology, for an area in the National Forest of Tapaj´s, in Brazil. The first part of Section 5 presents a o short comparison of this classifier against classical classifiers in the literature. The second part analysis use of this classifier in others regions with the same rules. Finally, Section 6 brings the conclusion. FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 37 2. Basic Concepts Image classification consists of partitioning the image domain in classes. The classes can be derived using probability models [13, 16], fuzzy models [2, 19], expert systems [23], neural networks [26, 33] or combinations of them [9, 15, 21, 34]. In this work, we show a combination of three research areas to build architecture for multispectral image classification: fuzzy set theory, mathematical morphology and expert systems [25]. 2.1. Fuzzy Set Theory. In (classical) set theory, each subset A of an universe X can be expressed by means of a membership function µA : X → {0, 1}, where, for a given a ∈ X, µA (a) = 1 and µA (a) = 0 respectively express the presence and absence of a in A. A fuzzy set [35] or fuzzy subset is used to model an ill-known quantity. A fuzzy set A on X is characterized by its membership function µA : X → [0, 1], where [0,1] can be substituted by any bounded scale. We say that a fuzzy set A of X is “precise” when ∃c∗ ∈ X such that µA (c∗ ) = 1 and ∀c = c∗ , µA (c) = 0. A fuzzy set A will be said to be “crisp” when ∀c ∈ X, µA (c) ∈ {0, 1}. The intersection and union of two fuzzy sets are performed trough the use of t-norm and t-conorm operators respectively, which are commutative, associative and monotonic mappings from [0, 1] to [0, 1]. Moreover, a t-norm (resp. t-conorm ⊥) has 1 (resp. 0) as neutral element (e. g.: = min, ⊥ = max) [12]. Thus, we can define intersection and union of two fuzzy sets as: - The intersection of two fuzzy sets A and B, with membership functions µA (x) e µB (x) is a fuzzy set C with membership function given by: C = A ∩ B ⇔ µC (x) = {µA (x), µB (x)}, ∀x ∈ X. - The union of two fuzzy sets A and B, with membership functions µA (x) e µB (x) is a fuzzy set C with membership function given by: C = A ∪ B ⇔ µC (x) = ⊥{µA (x), µB (x)}, ∀x ∈ X. The complement of a fuzzy set A in X, denoted by ¬A is defined by: µ¬A (x) = n(µA (x)), ∀x ∈ X. where n : [0, 1] → [0, 1] is a negation operator which satisfies the following properties - n(0) = 1 e n(1) = 0 - n(a) ≤ n(b) se a > b - n(n(a)) = a, ∀x ∈ [0, 1] and a negation is a strict negation if it is continuous and satisfies - n(a) < n(b) se a > b. 38 RONEI MARCOS DE MORAES The main negation operator which satisfies these four conditions is n(a) = 1 − a. The implication function between two fuzzy sets A and B, with membership functions µA (x) e µB (x) is a fuzzy set C with membership function given by: C = A ⇒ B ⇔ µC (x, y) = {µA (x), µB (y)}, ∀x ∈ X, ∀y ∈ Y. where : [0, 1]2 → [0, 1] is an implication operator which obeys the following properties: ∀a, a , b, b ∈ [0, 1]: - If b ≤ b then I(a, b) ≤ I(a, b ); - I(0, b) = 1; - I(1, b) = b. The pure implications obeys too: - If a ≤ a then I(a, b) ≥ I(a , b); - I(a, I(b, c)) = I(b, I(a, c)). 2.2. Mathematical Morphology. Mathematical morphology studies mappings between complete lattices. It is useful in gray-level image processing since the set of gray-level images can be structured as a complete lattice. Bounded chains [3] are particular cases of lattice and are the appropriate models for gray levels. A gray-level image f is defined here as a mapping from a rectangle E ⊆ Z 2 to a bounded chain K. The set of gray-level images will be denoted by K E . Each ordered pair (p, f (p)) is called a pixel, where p ∈ E represents its position in the image domain E and f (p) is its gray level. Mappings between bounded chains [3] are very important because the elementary operators of mathematical morphology (dilation, erosion, antidilation and anti-erosion) on gray-level images can be constructed from elementary mappings between bounded chains, the so-called ELUT’s [3, 4, 5]. The following definition of ELUT’s is actually a characterization of operators which commute with sup and inf in the case of mappings between bounded chains. Definition 1. Let K1 and K2 be two bounded chains and let ψ be a mapping from K1 to K2 . Then: • ψ is a dilation ⇔ ψ is increasing and ψ(min K1 ) = min K2 . • ψ is a erosion ⇔ ψ is increasing and ψ(max K1 ) = max K2 . • ψ is a anti-dilation ⇔ ψ is decreasing and ψ(min K1 ) = max K2 . • ψ is a anti-erosion ⇔ ψ is decreasing and ψ(max K1 ) = min K2 . Using the elementary operators of mathematical morphology it is possible to construct new operators to solve image processing problems [6]. In Fig. 1, FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 39 Figure 1. Examples of elementary operators of mathematical morphology. we can visualize examples of elementary operators as dilation, erosion, antidilation and anti-erosion respectively, according to Definition 1, with K1 = K2 . We now recall some useful operators. Definition 2. Let K1 and K2 be two bounded chains, let (E, +, o) be an Abelian group and let B be a subset of E. A flat dilation by B is a mapping form K1 E to K2 E such that, for any f in K1 E and x in E, • δB (f )(x) = max{f (u + x) : u ∈ B}. Definition 3. Let K1 and K2 be two bounded chains, let (E, +, o) be an E Abelian group, let B be a subset of E and let f be an image in K1 . A E E conditional dilation by B given f is a mapping form K1 to K2 such that, for any g in K1 E , • δB,f (g) = δB (g) ∧ f , where ∧ is the image intersection. Definition 4. Let K1 and K2 be two bounded chains, let (E, +, o) be an E Abelian group, let B be a subset of E and let g be an image in K1 . An inf-reconstruction given the marker g and the subset B is a mapping form K1 E to K2 E such that, for any f in K1 E , • γB,g (f ) = δB,f ∞ (g), where the ∞ symbol means that the conditional dilation is applied repeatedly until stability. Definition 5. Let K1 and K2 be two bounded chains and let ψ be an ELUT from K1 to K2 . A fuzzy threshold operator given the ELUT ψ is a mapping form K1 E to K2 E such that, for any f in K1 E , • Φψ (f ) = ψ ◦ f . 40 RONEI MARCOS DE MORAES Since the ELUT’s are monotonous (increasing or decreasing) mappings, they can be interpreted as the fuzzy generalization of step function used in traditional threshold. That is, the output image Φψ (f ) is the composition of the ELUT with the input image. In this work the ELUT will represent a fuzzy set membership function. The fuzzy threshold operator generally induces a loss of information. It is interesting to note that important parts of the lost information may be recovered when applying an inf-reconstruction. Definition 6. Let K1 and K2 be two bounded chains and let d be a distance from E × E to K2 . A distance function given the distance d is a mapping form K1 E to K2 E such that, for any f in K1 E and x in E, • Ψd (f )(x) = d(x, {y ∈ E : f (y) = 0}), where, for any x ∈ E and X ⊂ E, K2 = {0, 1, ..., k2 }, with k2 = max{d(x, {y}) : x, y ∈ E} and d(x, X) = min{ d(x, y) + 1/2 : y ∈ X}, where d(x, y) is the Euclidean distance between x and y and α is the greater integer number lesser than α. 2.3. Expert Systems. Expert systems [28] use the knowledge of an expert in a given specific domain to answer non-trivial questions about that domain. For example, an expert system for image classification would use knowledge about the characteristics of the classes present in a given region to classify a pixel in an image of that region. This knowledge also includes the “how to do” methods used by the human expert. Usually, the knowledge in an expert system is represented by rules of the form: IF condition THEN conclusion . A simple rule of image processing could then be: IF the gray level of a pixel is between 0 and 13 in band #4 THEN the class of the pixel is River. Most rule-based expert systems allow the use of connectives AND or OR in the premise of a rule, and of connective AND in the conclusion. From rules and facts, new facts will be obtained through a inference process. In several cases, we do not have precise information about all classes. It is often the case in image processing that the knowledge in the rules cannot be expressed in a precise manner. In this case, it can be interesting to use a fuzzy rule-based expert system [36]. An example of simple fuzzy rule in image processing could then be: IF the gray level of a pixel is dark in band #4 THEN the class of the pixel is River. where ”dark in band #4” can be characterized by a fuzzy set. FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 41 3. A Fuzzy Expert System Architecture for Image Classification In this section, we present a general expert system architecture for image classification that uses fuzzy set theory as knowledge representation model, which was originally presented in [23] (see also [24, 25]). Let us suppose that we want to classify the pixels of an image f into m classes. Here we sometimes use the term information surface to refer to a mapping from E × K to K; such a mapping may represent a fuzzy set membership function. Each information surface can thus be seen as a L-fuzzy set [18], defined as a mapping µA : X → L, where X is the function domain and L is any set that is at least partially ordered. We propose here an image classifier having a fuzzy rule-based expert system architecture, in which the rule premises are translated in terms of combinations of mathematical morphology operators [23]. The firing of a rule on an image f results in the creation of a set of n information surfaces gk : E → K, k = 1, ..., n, where gk (p) corresponds to the degree of compatibility between the position p with class ck and K is a bounded chain. The system is implemented using two levels of abstraction: 1) the rules provided by experts are translated into combinations of mathematical morphology operators; and 2) the n information surfaces gk obtained by the application of these combinations are aggregated to yield the classification of each pixel. The expert system architecture proposed here is able to treat the whole image at the same time. However, for the sake of simplicity, we will detail the treatment at the pixel level. Let us suppose that all the rules in the knowledge base only employ the connective AND in the premise (the treatment for connective OR can be found elsewhere [23]): Rj : IF attr1 (p) = A1j AND . . . AND attrNj (p) = ANj j THEN class(p) = Bj where Aij and Bj , i = 1, ..., Nj , j = 1, ..., m are fuzzy sets, attri (p) is an attribute in the premise and class(p) is the attribute in the conclusion. The universe of discourse of Bj is C = {c1 , ..., cn }, the set of all possible classes. The universe of discourse of each Aij depends on attribute attri , and is not necessarily discrete. Given p, its classification is made in three stages: a) The compatibility of p in relation to the premise of each rule is verified. This yields a preliminary classification of p in relation to the class in the conclusion of each rule. b) The preliminary classifications yielded by the rules are aggregated into an imprecise global classification. Therefore, p can be classified as belonging to more than one class. 42 RONEI MARCOS DE MORAES c) A precise class is assigned to the pixel, i.e. a decision is reached about the classification of p. In stage (a), the classification of p in relation to rule Rj obeys the following scheme: a.1) The values of attributes attri in relation to p are compared to fuzzy sets Aij . This yields the compatibility degree of p in relation to each of the premises of a rule Rj and is calculated as hij (p) = µAij (attri (p)). a.2) The general compatibility degree of p in relation to rule Rj is then calculated as hj (p) = (h1j (p), ..., hNj j (p)), where is a t-norm e. g. operator min. a.3) The pixel classification is derived (according to rule Rj ) by applying an implication function between the general compatibility degree hj (p) and the rule conclusion, given by Bj . This value is denoted by fuzzy set Bj (p), with membership function on C given by: µBj (p) (ck ) = (hj (p), µBj (ck )), k = 1, ..., n, where is an implication function. Although t-norms are not implication functions, they are the usual choice for . The fuzzy classification of p in relation to rule Rj is thus represented by Bj (p). In stage (b), all fuzzy sets Bj (p) are aggregated into a single fuzzy set B , given by µB (p) (ck ) = ♦(µB1 (p) (ck ), ..., µBm (p) (ck )), where, the aggregation operator ♦ is usually given by a t-conorm, when is a t-norm [29]. When is a typical implication function, such as (a, b) = max(1 − a, b), then ♦ is usually a t-norm. Considering now the whole image, n information surfaces gk : E×K → K are derived, one for each ck in C: gk (p) = µB (p) (ck ) = ♦j=1,m µBj (p) (ck ) = ♦j=1,m Making mkj = µBj (ck ) and gkj (p) = (hj (p), µBj (ck )) (hj (p), mkj ), we get gk (p) = ♦j=1,m gkj (p). Let us suppose that the conclusion of each rule Rj classifies a pixel to a single class c∗ , i.e. Bj is a precise fuzzy set. Let us further suppose j that is a t-norm . Then, for ck = c∗ , mkj = µBj (p) (c∗ ) = 1 and j j FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 43 gkj (p) = (hj (p), mkj ) = (hj (p), 1) = hj (p). For ck = c∗ , we then j have gkj (p) = (hj (p), 0) = 0. In this case, for ck = c∗ in rules which use only connective AND, we get: j (1) and as consequence: (2) gk (p) = ♦j=1,m i=1,Nj gkj (p) = i=1,Nj µAij (attri (p)) µAij (attri (p)) In stage (c), a “defuzzification” is performed, i.e. only one class is assigned to each pixel. Here we choose for p the class for which p has the greatest membership degree in gk (p). We could also assign more than one class to each pixel, e.g., a pixel classified by rules as belonging to both c1 and c2 classes, could be classified as belonging to the “imprecise” class c{1,2} . In this case, one way to obtain the classification would be to apply a pre-fixed threshold lk to each information surface gk . 4. Implementation and Application The Tapaj´s National Forest (Fig. 2) is a federal forest reservation and o it is geographically inside located of an imaginary rectangle of coordinates, whose vertexes are: 02◦ 43 27 S and 55◦ 26 01 W and 04◦ 07 39 S and 54◦ 45 00 W, in the Brazilian state of Par´. We must to note the small City of Aveiro a inside the National Forest, to Southwest of the Brazilian city of Santar´m, e also in the state of Par´. The City of Aveiro is an autonomous municia pal district with 1714 inhabitants (data of the 1991 Brazilian Demographic Census) and with an area of 17157,9 km2 [14]. The area used in that experiment is a small part of the park in the area of the City of Aveiro. In what follows, we present rule examples based on an application using bands #3, #4, #5 and #7 from Landsat TM images obtained on August 7th, 1995, from this Tapaj´s region in Brazil. Figure 3 o shows the RGB composition image of the region and the image has 500x500 pixels. Mathematical morphology operators treat the whole image at once, and can be seen as first order logic functions [23]. In our work, we use these operators in rule premises mostly to extract image attributes for the expert system and to implement fuzzy characteristics such as “near”, “very near”, “distant”, “very distant”, etc. We will use the following conventions to name some membership functions: gA denotes a mapping from E → {0, ..., 255}; µA denotes a mapping e from {0, ..., 255} → {0, ..., 255}; µbA denotes a mapping from {yes, no} → f 44 RONEI MARCOS DE MORAES Figure 2. Localization of the Tapaj´s National Forest, in o Par´ state, Brazil [27]. a {0, 255}; and µcA denotes a mapping from C → {0, 255}, where C is the set f of classes in a given application. Let µdark for band #4 and µvery dark for band #5 be defined as trapezoidal membership functions, respectively as FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 45 Figure 3. RGB composition image of an area of the Tapaj´s National Forest. o  if s ≤ 10  255,  255 ∗ (17 − s) µdark (s) = , if 10 < s ≤ 17 ;  7  0, otherwise  if s ≤ 7  255,  255 ∗ (12 − s) µvery dark (s) = , if 7 < s ≤ 12 .  5  0, otherwise 46 RONEI MARCOS DE MORAES Figure 4. Example of application of rules to an area of Tapaj´s National Forest Landsat image in Brazil. A o Tapaj´s National Forest band #4 image; B River extraco tion; C Fuzzy Threshold extract region of interest and others regions; D Fuzzy Distance image; E Minimum of the images C and D; F Minimum of the image E and Fuzzy Negation of the image B; G Extraction of components at right of River; H Minimum of the images F and G; I Final result of rules application. A set of rules to extract the class Contact Areas at Right Margin of River (see Fig. 4) could then be {R1 , R2 , R3 , R4 , R5 }, are described bellow. R1 : IF radiometry of band #4= dark AND radiometry of band #5= very dark THEN position ∈ briver . FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 47 Membership functions µdark and µvery dark are ELUTs of fuzzy threshold operators that are applied to a gray-level image (in this case, to bands #4 and #5). These functions are dilations and erosions at same time, in conformity to the definitions of Section 2.2. Let us suppose that we have = = min and ♦ = max. Considering that the conclusion of R1 represents a precise fuzzy set, using equations (1) and (2) we obtain the information surface gbriver (p) from R1 : gbriver (p) = min{µdark (f4 (p)), µvery dark (f5 (p))} The above expression can be rewritten introducing the fuzzy threshold operator of Section 2.2 (Definition 5): gbriver = inf{Φµdark (f4 ), Φµvery dark (f5 )}. Rule R2 is given by: R2 : IF radiometry of band #3= bright AND position ∈ near a river AND position ∈ ¬briver THEN position ∈ bpcamr . where µbright is an ELUT for a fuzzy threshold operator (as in this case of rule R1 ); near a river is a fuzzy set defined in terms of the distance function Ψd , according to Definition 6 of Section 2.2. µnear a river = Φµnear (Ψd (gbriver )), river ¬briver is the complement of fuzzy set briver , calculated as µ¬b (p) = 255 − µb (p), and bpcamr are the ”possible contact areas at margins of river river” defined by 255, if y = yes 0, if y = no With the same specifications used above for gbriver , we obtain for gbpcamr : µb pcamr (y) = gpcamr (p) = min{µbright (f3 (p)), µnear a river (p), f255 (p) − µbriver (p)} where f255 is a white image, i.e. ∀p ∈ E, f255 (p) = 255. In terms of morphological operators, membership function µnear a river is a fuzzy threshold operator on the distance function. We then have 48 RONEI MARCOS DE MORAES gpcamr = inf{ Rule R3 is given by: R3 : Φµbright (f3 ), Φµnear (Ψd (gbriver ))(p), f255 − µbriver } IF position ∈ bpcamr AND position ∈ right of river THEN position ∈ bpcarmr . where bpcamr are the ”possible contact areas at margins of river”; bpcarmr are the ”possible contact areas at right margin of river” and right of river is a fuzzy set defined in terms of the inf-reconstruction operator γg , according to Definition 4 of Section 2.2. µright of river = γgEastmark (g¬b river ) where γg (f ) reconstruct the connected component of g¬b , given a marker river defined by gEastmark (Eastmark is a marker defined by a vertical column of one pixel width at the image right border)[7]. We obtain then an implementation of rule R3 : gpcarmr = inf{gbpcamr , γgEastmark (g¬b river } Rule R4 bellow is used to filter noises and is given by: R4 : IF position ∈ bpcarmr AND position ∈ conected component THEN position ∈ bcarmr . where bpcarmr are the ”possible contact areas at right margin of river”; bpcarmr are the ”contact areas at right margin of river” and connected component is a fuzzy set given by pixels positions presents at possible contact areas at right margin of river given a marker by thinning [3] of possible contact areas at right margin of river by a line 1 × 3. We obtain then a implementation of rule R4 : gcarmr = inf{gbpcarmr , γT hinning line(gpcarmr ) (gbpcarmr )} Let ccarmr be a class ”Contact Areas at right margin of River”. Rule R5 relates fuzzy set ccarmr with fuzzy set bcarmr : R5 : IF position ∈ bcarmr THEN class = ccarmr FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 49 255, if c = ccarmr 0, otherwise. The implementation of rule R5 by equations (1) and (2), yields image gccarmr as result, where gccarmr (p) = µccarmr (p) (ccarmr ). These rules are applied to an image and the results of each step of this process are shown in Fig. 4. Image A is the Tapaj´s National Forest band o #4. Image B is the implementation of rule R1 for the River extraction. Images C, D, E are intermediate results of rule R2 and image F is its final result. Image G is a intermediate result of rule R3 and image H is its final result. Image I is the final result of rule R4 . Note that in Fig. 4, we do not show the implementation of rule R5 and that zoom boxes show details of region of interest in each image. It was used the Khoros system, version 2.1 [17], to make digital processing of these images. An existing visual classification with eight classes for the area was used as reference map (see Fig. 5-Left) to allow a comparison with the results obtained by the expert system [23] (see Fig. 5-Right). For a better comparison, the colors are the same in both classifications. where ccarmr is given by: µccarmr (c) = Figure 5. Left - Visual classification results; Right - Classification by the expert system. According to the visual examination by an expert, the classification results obtained by the system can be considered as quite satisfactory. The statistical comparison between the two classifications can been performed using Percentual of Correct Classification, Kappa or Tau coefficient. 50 RONEI MARCOS DE MORAES The Percentual of Correct Classification uses the reason of the total number of coincident pixels for the two classifiers for the total number of pixels present in the image. Is is an intuitive relation of agreement. Let M be the number of classes presents in the image, let N be the total number of pixels and nij the elements of classification matrix of two classifications (0 < i, j ≤ M ). The Percentual of Correct Classification is given by: M nii G= 2 with variance σG given by: i=1 N , G(1 − G) . N The most severe critic to that coefficient is its optimism, since it only analyzes the main diagonal of the classification matrix. The coefficient Kappa K tries to ponder the agreements in the classification matrix, taking in consideration the classes in the image. It is given by: 2 σG = M M P0 − Pc ; Pc = ; K= , N N2 1 − Pc where ni+ is the sum at line i in the classification matrix and n+i is the 2 sum at column i in the same matrix. The variance σK is given by: P0 = i=1 i=1 2 σK = 2 P0 (1 − P0 ) 2(1 − P0 ) + 2P0 Pc − θ3 (1 − P0 )2 θ4 − 4Pc + + , N (1 − Pc )2 N (1 − Pc )3 N (1 − Pc )4 M M nii ni+ n+i where: nii (ni+ + n+i ) θ3 = i=1 nii (ni+ + n+i )2 ; θ4 = i=1 N2 N3 For effect of calculations, a form just approached by the first portion of 2 the σK can be used. But, in the calculations accomplished in this article, the complete form of the variance was used. The main critic to that coefficient is its pessimistic form, because it incorporates the information of the main diagonal besides the remaining of the matrix. Recently, the coefficient Tau (T ) was proposed [20], with the objective of improving the test Kappa. This coefficient is called coefficient Tau and it is FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 51 given for: M nii P0 = and the variance 2 σT i=1 N ; T = P0 − 1/M , 1 − 1/M is given by: 2 σT = P0 (1 − P0 ) . N (1 − 1/M )2 In a comparative analysis, Brites et al. [8] conclude that Tau coefficient should be the coefficient that best reflects the accuracy of a classification processes. A numerical comparison between the two classifications has been performed using the Percentual of Correct Classification, Kappa and Tau coefficients. The Percentual of Correct Classification results were 86.09% with variance 4.79 ×10−7 . The Kappa results were 81.62% with variance 6.26 ×10−7 . The Tau results were 84.11% with variance 7.86 ×10−7 , and thus the classification can also be considered satisfactory in statistical terms. It is important to notice that the expert system classification was made using only the expert knowledge modelled by the rules and the original image, without pre-classification or post-classification. 5. Performance Evaluation 5.1. A Short Comparison Against Classical Classifiers in the Literature. There are several image classifiers in the literature. However, we use some classifiers in this evaluation and all these classifiers are supervised classifiers. In this class of classifiers, we have to provide some information to classifier, before the beginning of classification process. Although they do not possess an exact theoretical correspondence to the proposed classifier, they are very used in the practice and they possess vast available bibliography. In the following sections, we will present in a summarized plenty way those classifiers and to follow the results of its classifications in comparison to the proposed classifier. We must to note that for all the presented classifications, the samples of training for classifiers were always the same ones [31]. 5.1.1. The Maximum Likelihood Classifier. The classifier of Maximum Likelihood it is one of the most applied methods in the area of Remote Sensing [30]. He uses information about the distribution of the samples to compute a distance measure that is used to compute the most probable class for a 52 RONEI MARCOS DE MORAES pixel. Its surface of decision can take the form of parables, circles and ellipses, as shown in the illustration 6 [30]. In the illustration, the unknown point will be classified in agreement not only to the center of the samples of the classes, but also with regard to distribution of the points of those classes. Figure 6. Maximum Likelihood classification for a pixel in 2-D space [30]. The classifier of Maximum Likelihood assumes that the classes are unimodals and they follow a gaussian statistical distribution. Its discriminant function is given by: gi (x) = ln p( i) − 1 1 ln |Σi | − (x − µi )t Σ−1 (x − µi ) i 2 2 where p( i ) is a priori probability for the class i, µi is a mean vector and Σi is a covariance matrix for the data of class i and |Σi | is the mathematical determinant of covariance matrix. The classification is given by the choice of maximum g(x)for all the classes i ∈ N , where N is the total number of possible classes. FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 53 The a priori probabilities are not always available or possible of they be certain and they can be considered the same for all the N classes. If all the covariance matrix Σi are the same, the equation above is reduced to: 1 (x − µi )t Σ−1 (x − µi ) i 2 which is known as classifier by the distance of Mahalanobis. gi (x) = Figure 7. Maximum Likelihood image classification. Being known that the classifier of Maximum Likelihood assumes the gaussian distribution for the classes (univariate gaussian in the case of only one band or multivariate gaussian in the case of more than one band), it can 54 RONEI MARCOS DE MORAES have intersections among the distributions. We say that given three classes A, B and C, the tails of the distributions A and B have a small intersection and for a pixel x in that intersection, the probabilities will be very small that that pixel x it belongs to any one of the classes in subject. For that reason, we could abstain from classifying that point, attributing a minimum limit of probability, below which our doubt that that point belongs to any class it is big. That is the call rejection option that would attribute to the point a generic class D, non present among the available classes A, B and C. There is another option, that comes being still studied recently, that is to attribute to the point x both classes A and B. however, the rejection option is the more classic treatment form. There is still a third option that is the one of classifying all the points by the largest probability of belonging to a class (even if the probability is small). As our classifier does not possess rejection option and option for classification in more than one class, we will adopt that last treatment form when we use Maximum Likelihood classifier. For Maximum Likelihood method we use no rejection option and with rejection option with the following parameters: 0.1%, 0.5%, 1.0%, 2.5%, 5.0%, 10.0% and 50.0% of rejected pixels. The better result is obtained when we use no rejection option and the percentual of correct classification is equal to 58.85%. The image classification is showed in Fig. 7. For classifier by the distance of Mahalanobis, we use the same options and parameters and the better result is obtained when we use no rejection option too and the percentual of correct classification is equal to 53.93%. Figure 8 shows the image classification of the area of the Tapaj´s National Forest, in Brazil, o by classifier by the distance of Mahalanobis. 5.1.2. K-NN Classifier. The classifier K-NN (K-Nearest Neighbor) it is a non-parametric supervised classification method in which a pixel is associated to classes depending on the largest number of neighboring points that belong to that class. Usually, the classification algorithm selects neighboring K with closer well-known classes to the point x that is being investigated and it selects the class that has more points in the neighborhood [30]. Differently of parametric classifiers, where we want to estimate p(x| i ), we can estimate the probability a posteriori p( i |x) by examining the classes of the well-known points that are close to an unknown point. Figure 9 shows a classification example K-NN in the bi-dimensional space of attributes [30]. We have 3 classes with its samples and we want to determine the classes of the points unknown 1 and 2. The central idea of the classification is simple and intuitive: to classify a point, first we took its closer neighboring K and inside of that set, we found the most representative class. In Fig. 9, the 7 nearest neighbors of each unknown point are FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 55 Figure 8. Image classification by classifier by the distance of Mahalanobis. shown by the lines that connect them. In that example, the unknown point 1 will be classified as class B and the unknown point 2 will be as class A. The algorithm of the classification K-NN should calculate the distance of all the spectral signatures of the neighboring points to the point to be classified, to determine the minimum distance that includes K samples and to count the points for each class which are inside of that size group K. For a great amount of points, that is expensive in computational terms. A variation of that algorithm, that is a little faster, it consists of to select points that are inside of a ray hyper-sphere R and to count the points for 56 RONEI MARCOS DE MORAES Figure 9. K-NN classification for a pixel in 2-D space [30]. each class, which are inside of that hyper-sphere. The problem is that the user should decide R in an empiric way and they can happen cases of hyperspheres without any point. Figure 10 [30] shows the same example of classification of Fig. 9 with the modified algorithm. In that case the points that will be considered as neighbors are inside the points of circle (due to reduction of the dimension for 2) centered in the point to be classified. For the unknown point 1, there are 5 points inside or partially inside of the circle and therefore he will be classified as class B. For the unknown point 2 only a point exists of the class A and therefore it will be classified as class A. This approach could also work as a rejection option: if does not exist any sample inside of a ray hyper-sphere R, we can reject any class attribution for the pixel. In the case of the unknown point 2, if the ray of the circle were a little smaller, we could not attribute class to it. For K-NN method we use a set of parameters of test, i.e. K = {4, 5, 6, 10}. The better result is obtained when we use K = 10 and the percentual of correct classification is equal to 32.47%. Figure 11 shows the image FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 57 Figure 10. Variation of K-NN classification for a pixel in 2-D space [30]. classification of the area of the Tapaj´s National Forest, in Brazil, by K-NN o classifier. 5.1.3. Minimum Distance Classifier. The classifier of Minimum Distance is a simple classifier that just measures the Euclidean distance from the medium center of the samples to the point x unknown in the n-dimensional space [30]. The point x it is attributed to a class whose center is closer. Figure 12 illustrates a simple classification in the 2-D space [30]. In that figure, three different classes and two unknown points exist (point 1 and point 2) for which one wants to attribute a class. In the case of the point 1, the attributed class is the class B and of the point 2 it is the class A. The main advantages of this method are: simplicity and speed, by using only one measure of the samples of the classes. A classic case of mistake of that classifier happens when the medium center of a class A is more distant than the one of a class B, but the group of the class B as a whole is not distant. The pixel will be classified as being belonging the class A, when in fact it belongs the class B. 58 RONEI MARCOS DE MORAES Figure 11. K-NN image classification. For Minimum Distance method we use no rejection option and with rejection option with the following parameters: 0.5, 1.0, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 15.0 50.0, and 70.0 standard deviation. The better result is obtained when we use no rejection option and the percentual of correct classification is equal to 58.85%. Figure 13 shows the image classification of the area of the Tapaj´s National Forest, in Brazil, by Minimum o Distance Classifier. 5.1.4. Classification by Parallelepiped Method. This classifier belongs the class of the parametric supervised classification methods [30]. It assumes FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 59 Figure 12. Minimum Distance classification for a pixel in 2-D space [30]. that all the points that belong to a class assume values inside of the limits of a parallelepiped in the n-dimensional space which is defined by the samples of the class. Figure 14 shows a simple example of classification with that method in the 2-D space. The inherent disadvantages of that method can already be visualized: a pixel x can stranger be classified in 4 ways (A, B, C or ”?”), even so there is intersections in the attribute 1 for the classes A and C. This problem can be solved by the attribute 2, where there are not intersections among the classes. Even so it is evident that if for the attribute 2 there was also intersection there would be difficulty or even it would disable to attribute a class to that pixel. Is there also the problem of the rigidity of the method: a pixel x unknown with attributes in any of the areas described for ”? ” will it be placed under doubt. This is an inherent characteristic to that method: it have a definition of rejection option in its definition. These two points against the application of that method would already place out it of subject in terms of an use well happened without the due cares. However, we placed it in that study due to another fact: for being 60 RONEI MARCOS DE MORAES Figure 13. Minimum Distance image classification. a classifier that places very defined limits to its classes, he can be seen under the optics of the Mathematical Morphology as being an union of supgenerators (intersections of erosions and anti-dilations). Ally to this fact, we also have the advantages of the simplicity and speed of that method for the cases us which he is applied. It is also useful as a pre-classifier, in the sense that we can more thoroughly study the points rejected by him, to verify the need or not of to sub-divide classes or to verify the existence of new classes in an image. For Parallelepiped method we use no rejection option and with rejection option with the following parameters: 0.5, 0.7, 0.9, 1.0, 1.1, 1.2, 1.3, 1.5, 1.7, FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 61 Figure 14. Classification by Parallelepiped Method for a pixel in 2-D space [30]. 2.0, 2.5, 3.0, 4.0, 4.9, 4.95 and 5.0 standard deviation. The better result is obtained when we use 4.0 standard deviation and the percentual of correct classification is equal to 27.68%. Figure 15 shows the classification of the area of the Tapaj´s National Forest, in Brazil, by Parallelepiped Method. o Although that classifier to have a correspondence with the Mathematical Morphology, its results were very poor in relation to the specialist system. It can be noticed that it classified River (in blue) with relative precision, even so it made a mistake completely in the forest classes, City of Aveiro and Antropized Area, placing great part of those areas in doubt (it corresponds to the gray tone). It was considered the weakest result among the presented. 5.2. Performance in Other Regions with the Same Rules. In the next sections, we presented an application of those rules to two different images. In the first case, we classified an image that includes a part of the previous area. In the second case, we totally classified an image different from the previous area. Unfortunately, because we do not have reference 62 RONEI MARCOS DE MORAES Figure 15. Image classification by Parallelepiped Method. maps for those cases, we can not make statistical comparisons between classifications. However, we can observe the behavior of the rules of the fuzzy expert system when it is applied to different domains. 5.2.1. Classification of an Area of 730 × 350 Points Including a Part of the Previous Area with the Same Rules and One Class More. The images were captured by the satellite Landsat TM in the same date of the image of 500 × 500 pixels and through the same bands. The difference is the displacement to East of the image, that does not show the area in the opposed margin of the river, and this image has more pixels in the axis FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 63 East-West and less pixels in the axis North-South. It presents in a clear way an additional area to East (that contains a new non defined class) and it also contains already a part of the image classified previously. Figure 16 shows the band 5 of that image captured by the satellite Landsat TM. They were necessary some small fittings for the system classified that new image. It was of to hope the system already recognized the areas similar to those found with success in the original image and it obtained a classification without the need of corrections. In the case of the texture, the classifier recognized the whole texture, even of the area not worked previously. However, marker construction is not adapted to any size of image and it resulted in a too big marker for the class Dense Forest (in darker green), although the texture went well recognized. That did with that the new class was marked in some areas that they didn’t correspond to the reality. Finally, the final result was an image classified with the new class being recognized as Secondary Forest (in clearer green) and the contours are not the same to the true contours of the class, although it approaches enough in some few areas. Therefore, the result was below what it was waited (see Fig. 17). Figure 16. Tapaj´s National Forest area including a part o of the previous area captured by sensor number 5 of Landsat TM satellite, treated by histogram modification. Probably if the rules were reformulated, we could have a closer form of being automated to accomplish that classification in larger areas than the 64 RONEI MARCOS DE MORAES original area, although there was the restriction that new classes didn’t exist and that the area didn’t suffer abrupt alterations in elapsing of the time. Figure 17. Image classification by expert system with some small fittings in the rules. 5.2.2. Classification of an Area of 400 × 400 Pixels Totally Different From the Previous Area with the Same Rules and Different Classes. Again, the images were taken by the satellite Landsat TM in the same date of the original image and by the same bands. The difference of that time, is the displacement to the north of the image, for an area that does not have anything in common with the previous areas, in the area of the small city of Itapai´na (see Fig. 2). The objective of that new experience, is to analyze u the behavior of the rules in an area totally different from the original area for which the classifier was done. Figure 18 shows the band 5 of that image captured by the satellite Landsat TM. Although some attributes of the image are recognized, the construction of the marker for the class Dense Forest is incompatible with the image and that it causes an error. To solve this, some rules were disabled and the classifier worked with a summarized group of rules. It can be observed in Fig. 19 that the result is regarding the constant classes in the original area, that is to say, the classes Tapaj´s River (in blue), o Antropized Area (in yellow) and Secondary Forest (in clearer green). It is obvious that those classes do not correspond everybody to the reality. In that case, the classes Tapaj´s River and Antropized Area really correspond, o but the Secondary Forest does not correspond. The small city of Itapai´na u FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 65 Figure 18. Tapaj´s National Forest area totally different o from the previous area captured by sensor number 5 of Landsat TM satellite, treated by histogram modification. that consists in the map (Fig. 2) was not located. The reason is that she is smaller than the City of Aveiro (in red in Fig. 5-right) and it does not possess the same form. That checks what the theory observes, that is to say, an applied specialist system to a differentiated domain can take to mistakes (the class Secondary Forest is not present in that area and the city was not detected). However, nevertheless two classes were detected with relative success: Tapaj´s River and Antropized Area. o 66 RONEI MARCOS DE MORAES Figure 19. Image classification by expert system with a summarized group of rules. 6. Conclusions In this work, we analyze a general expert systems architecture for image classification that uses fuzzy set theory as knowledge representation model, presented by Moraes, Banon, and Sandri [24]. In this architecture, the rules are implemented using mathematical morphology operators. The main contribution of this approach is the homogeneous application of mathematical morphology and fuzzy set theory for image classification [25]. FUZZY EXPERT SYSTEMS FOR MULTISPECTRAL IMAGE CLASSIFICATION 67 The translation invariant property of the morphological operators makes it easy to extract the attributes from the image, either by characteristics related to shape or by radiometry, independently of their localization in the image. This important characteristic of translation invariant operators allows the application of the same combinations of operators, with small adjustments, to another image in the same area. As any other knowledge based systems, this approach has the disadvantage of requiring an expert to furnish the rules, in this case, someone with a good knowledge about the region to be classified. Furthermore, a minimum knowledge about mathematical morphology operators is necessary to the application builder. We construct an application of this architecture to classify an area of Tapaj´s National Forest in Brazil. The application has shown that the imo plemented classifier is quite accurate, even with a small number of rules, when we compare with a reference map, using statistical comparisons. We compare this classifier with others classical classifiers presented in the literature and it presented results significantly superiors. However, we must to note that the classical classifiers analyze the pixels exclusively for pixels spectral reflection, without taking in its bill position in the image or other knowledge variables. Among the presented classifiers, the classifier of Maximum Likelihood was what supplied less distorted results and the classifier by Parallelepiped Method obtained the worst performance in that experiment. When this expert system was applied to images in different domains, we verified also that, using the same rules, it is not possible to classify another image, even if this has similar characteristics or an image covering a larger area than the original. It would be necessary to add more rules, to modify or same to exclude some of the existent rules to reach the wanted precision, because the alteration of the domain of the system demands the adaptation of the system of rules. A larger number of rules could possibly improve the accuracy of the presented classifier, but with higher computational cost. The system also proved to be efficient; an application such as the one shown here would only take a few minutes in an ordinary personal computer. In the present version of the system, we do not have performed preclassification or post-classification. A post-classification step could however improve even further the overall quality of the classification, specially in what regards noisy images. In future works, we intend to include incorporating pre-classification and post-classification techniques and the aggregation of other sources of information, such as geographical information systems. 68 RONEI MARCOS DE MORAES References [1] Aubert, A.; Jeulin, D.; Hashimoto, R. Surface texture classification from morphological transformations. In: Mathematical Morphology and its Applications to Image and Signal Processing. Eds. Goutsias, J.; Vincent, L.; Bloomberg, D. S. 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Fuzzy Sets, Information and Control, v. 8, p.338-353, 1965. [36] Zadeh, L. A. Fuzzy Logic, Computer, v. 1, p.83-93, 1988. Department of Statistics, Federal University of Para´ ıba (UFPB), Brazil, ´ ˜ Cidade Universitaria s/n, Joao Pessoa, Para´ 58051-900, Brazil ıba E-mail address: ronei@de.ufpb.br, (http://www.de.ufpb.br/∼ronei/)