with Spatial Information for Land-Cover Classification

(1)

with Spatial Information for Land-Cover Classification

Sinh Dinh Mai and Long Thanh Ngo⁽B⁾

Department of Information Systems, Le Quy Don Technical University, 236, Hoang Quoc Viet, Hanoi, Vietnam

{maidinhsinh,ngotlong}@gmail.com

Abstract. The paper proposes a method to use spatial information to interval type-2 fuzzy c-Means clustering (IT2-FCM) for problems of land cover classification from multi-spectral sattelite images. The spatial information between a pixel and its neighbors on individual band is used to calculate an interval of membership grades in IT2-FCM algorithm.

The proposed algorithm, called IIT2-FCM, is implemented on Landsat7 images in comparison with previous algorithms like k-Means, FCM, IT2- FCM to demonstrate the advantage of the approach in handling uncertainty or noise.

Keywords: Fuzzy clustering

·

Type-2 fuzzy sets

·

Land cover classification

·

Spatial information

1 Introduction

In image segmentation, the most important problem is to find a method to determine whether or not the considered pixel will belong to a certain cluster. The original algorithms like k-Means, fuzzy C-Means(FCM), interval type-2 fuzzy C- Means (IT2-FCM) exhibit the same strategy based on the Euclidean distance to compute the degree of similarity between objects and cluster centroids. Not only color based similarity of the pixels but the spatial relationship between pixels and their neighbors also certainly influence on the final clustering results.

Fuzzy clustering, especially type-2 fuzzy clustering, have exhibited the advantage in handing uncertainty of data. The fuzzy C-Means (FCM) and its variants have widely applied to various problems of image segmentation. FCMs with spatial information also proposed to enhance algorithms of image segmentation in the case of noise [9]-[12]. Z.Wang et al [11] proposed an adaptive spatial information theoretic fuzzy clustering to improve the robustness of the conventional FCM for image segmentation which handling the sensitivity to noisy data and the lack of spatial information. H. Liu et al [10] proposed a fuzzy spectral clustering with robust spatial information for image segmentation to overcome the noise sensitivity of the standard spectral clustering algorithm. K.S. Chuang et al

c Springer International Publishing Switzerland 2015

N.T. Nguyen et al. (Eds.): ACIIDS 2015, Part I, LNAI 9011, pp. 387–397, 2015.

DOI: 10.1007/978-3-319-15702-3 38

(2)

[12] introduced an algorithm of FCM with spatial information for image segmentation in which spatial function is the summation of the membership function in the neighborhood of each pixel under consideration.

Interval type-2 FCM (IT2-FCM) was proposed to handing the uncertainty [5] and have applied to various problems e.g. image segmentation, land cover classiﬁcation [2,3]. L.T.Ngo et al [13] also proposed an diﬀerent approach of type-2 fuzzy clustering by combining subtractive clustering and interval type-2 fuzzy sets and apply to image segmentation.

Land cover classification from multi-spectral satellite images is one of problems which have widely applied in real application and many approaches have been investigated, recently. W.Su et al [15] proposed method of object oriented information extraction for land cover classification of SPOT5 image, involves two steps: image segmentation and classification. Many other approaches based fuzzy logic also proposed in various manner such as using reformed fuzzy c-means clustering from color satellite image [14].

The paper deals with an approach to combine spatial information into IT2- FCM, called IIT2-FCM, to handle the uncertainty appearing from satellite images. The spatial information between a pixel and its neighbors on individual band is used to calculate an interval of membership grades in IT2-FCM algorithm. Experimental results on data study of Hanoi region with summarized data and validity indexes in comparison with other clustering, e.g k-Means, FCM, IT2- FCM. Especially, results is compared with survey data of the Vietnamese Center of Remote Sensing Technology (VCRST) to compare the accuracy between algorithms.

The paper is organized as follows: Section 2 is background of type-2 fuzzy sets and IT2-FCM; Section 3 introduces the IIT2-FCM algorithm; Section 4 shows experimental results of Hanoi region. Section 5 is conclusion and future works.

2 Background

2.1 Type-2 Fuzzy Sets

A type-2 fuzzy set in X is denoted ˜A, and its membership grade of x∈ X is μA˜(x, u), u ∈Jx ⊆[0,1], which is a type-1 fuzzy set in [0, 1]. The elements of domain ofμA˜(x, u) are called primary memberships ofxin ˜Aand memberships of primary memberships in μA˜(x, u) are called secondary memberships of xin A˜.

Definition 1. [1] Atype−2 f uzzy set, denotedA, is characterized by a type-2˜ membership function μA˜(x, u)wherex∈X andu∈Jx⊆[0,1], i. e. ,

A˜={((x, u), μA˜(x, u))|∀x∈X,∀u∈Jx⊆[0,1]} (1) or

A˜=

x∈X

u∈Jx

μA˜(x, u))/(x, u), Jx⊆[0,1] (2) in which0≤μA˜(x, u)≤1.

(3)

At each value of x, sayx=x, the 2-D plane whose axes areuandμA˜(x, u) is called avertical sliceofμA˜(x, u). A secondary membership functionis a vertical slice of μA˜(x, u). It isμA˜(x=x, u) forx∈X and∀u∈Jx ⊆[0,1], i. e.

μA˜(x=x, u)≡μA˜(x) =

u∈Jx

fx(u)/u, Jx ⊆[0,1] (3) in which 0≤fx(u)≤1.

Type-2 fuzzy sets are called an interval type-2 fuzzy sets if the secondary membership function fx(u) = 1∀u∈Jx i. e. a type-2 fuzzy set are deﬁned as follows:

Definition 2. An interval type-2 fuzzy set A˜ is characterized by an interval type-2 membership function μA˜(x, u) = 1 wherex∈X andu∈Jx⊆[0,1], i. e.

, A˜={((x, u),1)|∀x∈X,∀u∈Jx⊆[0,1]} (4) Uncertainty of ˜A, denoted FOU, is union of primary functions i. e. F OU( ˜A) =

x∈XJx. Upper/lower bounds of membership function (UMF/LMF), denoted μA˜(x) andμA˜(x), of ˜Aare two type-1 membership function and bounds of FOU which is limited by two membership functions of an type-1 fuzzy set are UMF and LMF.

2.2 Interval Type-2 Fuzzy C-Means Clustering

In general, fuzzy memberships in interval type-2 fuzzy C means algorithm (IT2FCM) [5] is achieved by computing the relative distance among the patterns and cluster centroids. Hence, to define the interval of primary membership for a pattern, we define the lower and upper interval memberships using two different values ofm. In (5), (6) and (7),m1andm2are fuzzifiers which represent different fuzzy degrees.

IT2-FCM is extension of FCM clustering by using two fuzziness parameters m1, m2 to make FOU, corresponding to upper and lower values of fuzzy clustering. The use of fuzziﬁers gives diﬀerent objective functions to be minimized

as follows:

Jm1(U, v) =_N

k=1

_C

i=1(uik)^m¹d²_ik Jm2(U, v) =_N

k=1

_C

i=1(uik)^m²d²_ik (5) in which dik = xk −vi is Euclidean distance between the pattern xk and the centroid vi, C is the number of clusters and N is the number of patterns.

Upper/lower degrees of membership,uikandu_ikare determined as follows:

uik=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1 C j=1

dik

djk

_2/(m₁₋₁₎ if 1 C j=1

dik

djk

< 1 C 1

C j=1

dik

djk

_2/(m₂₋1) otherwise

(6)

(4)

u_ik=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1 C j=1

dik

djk

_2/(m₁₋1) if 1 C j=1

dik

djk

≥ 1 C 1

C j=1

dik

djk

_2/(m₂₋₁₎ otherwise

(7)

in whichi= 1, ..., C,k= 1, ..., N.

Because each pattern has membership interval as the upperuand the lower u, each centroid of cluster is represented by the interval between v^L and v^R. Cluster centroids are computed in the same way of FCM as follows:

vi = ( N k=1

(uik)^mxk)/( N k=1

(uik)^m) (8)

in which i = 1, ..., C. After obtaining v_i^R, v_i^L, type-reduction is applied to get centroid of clusters as follows:

vi= (v_i^R+v^L_i)/2 (9) For membership grades:

ui(xk) = (u^R_i (xk) +u^L_i(xk))/2, j= 1, ..., C (10) in which

u^Li = M l=1

uil/M, uil=

ui(xk) ifxil usesui(xk) forv_i^L

u_i(xk) otherwise (11)

u^R_i = M l=1

uil/M, uil=

ui(xk) ifxil usesui(xk) forv_i^R

ui(xk) otherwise (12) Next, defuzziﬁcation for IT2FCM is made as if ui(xk)> uj(xk) forj = 1, ..., C andi =j thenxk is assigned to clusteri.

3 Interval Type-2 Fuzzy C-Means Clustering with Spatial Information

3.1 Spatial Information

In fact, the image information is stored as numeric values so the problem of image partitions is usually based on the degree of similarity among these values to decide whether an object belongs to any region in the image. Therefore the key to determine a pixel will belong to certain area is based on the similarity in these colours, which is calculated through a function of the distance in the color

(5)

space dik = xk −vi i.e Euclidean distance between the pattern xk and the centroidvi.

We use a mask of sizenxnto slip on the image, the center pixel of mask is the considered pixel. The number of neighboring pixels is determined corresponding to the selected type of mask i.e., 8 pixels for the 8-directional mask or 4 pixels for the 4-directional mask.

To determine the degree of inﬂuence of a neighboring pixels for the center pixels, a measure spatial informationSIik is deﬁned on the basic of the degree ukiand the attraction distancedki as follows:

SIik= _N

j=1uijd⁻_kj¹ _N

j=1d⁻¹_kj (13)

in whichuij is the membership degree of the neighboring elementxj to the clusteri. The distance attractiondkj is the squared Euclidean distance between elements (xk, yk) and (xj, yj). According to this formula, the value of spatial information is at greater pixel on the mask while many neighboring pixels they have similar color them and the opposite. We use the inverse distance d⁻¹_kj, because the closer neighbors xj is to the centerxk the more inﬂuence it has on the result.

A new distance measure is deﬁned as follows:

Rik=xk−vi²(1−αe^−SI^ik) (14) whereSijis spatial relationship information between elementsxkand clusters i, α ∈ 0,1 is the parameter that controls the relative factor of neighboring pixels. Ifα= 0,Rikis the squared Euclidean distance and the algorithm becomes the original standard FCM.

The idea behinds the use of this spatial relationship information is: Consider the local nxn neighborhood with the center xk has large intensity diﬀerences with the closest neighboring pixelsxk, which has similar intensity as the cluster centroid vi. If the neighborhood attraction SIij, takes a large value then the expression (1−αe^−SI^ik) will take a small value forα = 0. After each iteration of the algorithm, the central element xk will be attracted to the clusteri. If the neighborhood attraction SIik continuously take a large value till the algorithm terminate, the central elementxk will be assigned to the clusteri.

3.2 Interval Type-2 Fuzzy C-Means Clustering with Spatial Information

The main idea of the IIT2-FCM algorithm is extended from IT2-FCM by adding the spatial information to calculate distance between clusters and pixels. The steps are described as follows:

Firstly, Initialization of matrix centroid V.

Secondly, the primary membershipsuikandu_ikfor a pattern are corespond- ing to two fuzziﬁers m1 andm2which were chosen by heuristic.

(6)

Then value of spatial information of each pixelsSIik is computed. Because each pattern has membership interval u, u with the upper bound u and the lower boundu, soSIij will be an interval with two bounds which are computed corresponding to the upper and lower membership grades as follows:

SIik= _N

j=1uij(dkj)⁻¹ _N

j=1(dkj)⁻¹ (15)

SI_ik= _N

j=1u_ij(dkj)⁻¹ _N

j=1(dkj)⁻¹ (16)

Then the value of spatial information is defuzziﬁed as:

SIik= (SIik+SI_ik)/2 (17) Compute the distance as the following formula:

Rik=xk−vi²(1−αe^−SI^ik) (18) We deﬁneIk ={i|1 ≤i ≤C, Rik = 0} in which k = 1, N and xk−vi is the Euclide distance between data samples k and cluster i in d-dimensional space. In case ofIk=∅,uikandu_ikare determined in the same way of formulas (5) and (6) by replacing the distancedij by the new distance Rij as follows:

uik=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1 C

j=1

(Rik/Rjk)^2/(m¹⁻¹⁾

if 1

C j=1

(Rik/Rjk)

< 1 C 1

C j=1

(Rik/Rjk)^2/(m²⁻¹⁾

if 1

C j=1

(Rik/Rjk)

≥ 1 C

(19)

u_ik=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1 C

j=1

(Rik/Rjk)^2/(m¹⁻¹⁾

if 1

C j=1

(Rik/Rjk)

≥ 1 C 1

C j=1

(Rik/Rjk)^2/(m²⁻¹⁾

if 1

C j=1

(Rik/Rjk)

< 1 C

(20)

Otherwise, ifIk =∅, uik anduik are determined:

uik=

0 ifi /∈Ik i∈Ik

uik= 1ifi∈Ik (21)

u_ik=

0 ifi /∈Ik i∈Ik

u_ik= 1ifi∈Ik (22)

(7)

in whichi= 1, C,k= 1, N.

Because each pattern has membership intervaluand u, so each centroid of cluster is represented by the interval betweenv^L andv^R.

Algorithm 1:Algorithm ﬁndv^L andv^R Step 1: Finduij, u_ij, by the equations (19)-(20).

Step 2: Setmis a constant satisﬁed m≥1;

Computev_j = (v_j1, ..., v_jM) by the equation (8) withuij= ^(u^ij^+u₂ ^ij⁾. Sort N patterns on each of M features in ascending order.

Step 3: Find index k such that: xkl ≤ v_jl ≤ x(k+1)l with k = 1, .., N and l= 1, .., M.

Step 4: Calculatev^”by following equation: In case v^” is used for ﬁndingv^L

v^”= k i=1

xiμA(xi) + ^N

i=k+1

xiμA(xi) k

i=1μA(xi) + ^N

i=k+1μA(xi)

(23)

In case v^” is used for ﬁndingv^R

v^”= k i=1

xiμA(xi) + ^N

i=k+1

xiμA(xi) k

i=1μA(xi) + ^N

i=k+1μA(xi)

(24)

Step 5: Ifv =v^”go to Step 6 else set v =v^” then back to Step 3.

Step 6: Setv^L=v or v^R=v.

After obtainingv^R,v^L, to get centroid of clusters by (9).

For membership gradesui(xk) base on the formula (10), (11) and (12).

Next, defuzziﬁcation for algorithm IIT2-FCM: if ui(xk) > uj(xk) for j = 1, ..., C andi! =j thenxk is assigned to clusteri.

Algorithm 2:IIT2-FCM algorithm Step 1:Initialization

1.1 The two parameters of fuzzym1,m2 (1< m1, m2), errore. 1.2 Initialization centroidV = [vi], vi ∈R^d.

Step 2:Compute the fuzzy partition matrixU,U and update centroid V:

2.1. Calculate the value of spatial informationSIikby formulas (15), (17).

2.2. Calculate matrix of membership gradesUikby formulas (18), (20).

2.3. Assign a pattern to a cluster.

2.4. Update the centroid of clustersV^j = [v^j1, v2^j, ..., v_c^j] by using the algorithm of ﬁndingv^L andv^R and the formula (8).

Step 3:Verify the stop condition:

IfM ax(|J^(j+1)−J^(j)|), go to step 4, otherwise go to step 2.

Step 4:Report the clustering results.

(8)

4 Land-Cover Classification Using IIT2-FCM

The IIT2-FCM based algorithm for land cover classification is implemented on Landsat7 images of Hanoi region, Vietnam ( 21ô54’23.11”N, 105ô03’06.47”E to 20ô55’14.25”N, 106ô02’58’.57”E) with area is 1128.km²and resolution is 30m× 30m. We use 6 bands of 1, 2, 3, 4, 5 and 7 as inputs of clustering algorithms and the image is classified into 6-classes corresponding to 6 types of land covers as in Figure1.

Class1: Rivers, ponds, lakes Class2: Rocks, bare soil

Class3: Fields, sparse tree Class4: Planted forests, low woods

Class5: Perennial tree crops Class6: Jungles.

Fig. 1.Six types of land covers

Fig. 2.The bands of Hanoi region

(9)

Table 1.Land cover classification of IIT2-FCM, IT2-FCM, FCM, k-Means algorithms

Bands of Landsat7 image of Hanoi region are shown in Fig.2. The experimental results are demonstrated in Fig.3 in which (a), (b), (c), (d) images are classiﬁcation result of IIT2-FCM, IT2-FCM, FCM, k-Means algorithms, respec- tively. Fig.3shows that the classiﬁcation of IIT2-FCM gives the better clusters.

Beside, summarized data of algorithms is compared with the survey data of VCRST which is demonstrated in Table1in which IIT2-FCM exhibits the best quality of clusters with the lowest standard deviation of diﬀerence, i.e. SD is 4.4.

Fig. 3.Land cover classification. a) IIT2-FCM; b) IT2-FCM; c) FCM; d) k-Means.

(10)

To evaluate the quality of clusters, several validity indexes are computed from results of the mentioned algorithms. We considered various validity indexes such as the Bezdeks partition coeﬃcient (PC-I) [8], the Dunns separation index (DI), the Davies-Bouldins index (DB-I), and the Separation index (S-I), Xie and Benis index (XB-I), Classiﬁcation Entropy index (CE-I) [7]. The validity indexes are shown in the Table2.

Table 2.The various validity indexes from classification of Hanoi region Validity index k-Means FCM IT2-FCM IIT2-FCM

DB-I 4.531 3.4983 2.3981 1.1246 XB-I 1.761 1.1784 0.6823 0.1382 S-I 0.9821 0.6287 0.3834 0.0917 CE-I 1.323 0.9869 0.5872 0.1972 PC-I 0.6543 0.6982 0.7282 0.8628

Note that the validity indexes are proposed to evaluate the quality of clustering. The better algorithms exhibit smaller values of T-I, DB-I, XB-I, S-I, CE-I and the larger value of PC-I. The results in Table2show that the IIT2FCM have better quality clustering than the other individual algorithm such as IT2-FCM, FCM and k-Means.

In summary, classification results can be explained that, the boundary of water and soil classes are usually quite clearly, while the vegetation classes are often confused in which both grasses and trees. With the resolution of 30m×30m, the difference of classification results can be acceptable to assess land cover on a large area.

5 Conclusion

This paper presents a method of clustering algorithm based on combining spatial information into IT2-FCM to handle uncertainty better. The experiments were carried out based on Landsat7 image of Hanoi region to assess the advantage of the proposed algorithm.

The next goal is to implement further research on hyperspectral satellite imagery for environmental classiﬁcation, assessment of land surface temperature changes.

References

1. Mendel, J.M., John, R.I.: Type-2 fuzzy sets made simple. IEEE Trans. on Fuzzy Systems10(2), 117–127 (2002)

2. Ngo, L.T., Nguyen, D.D.: Land cover classification using interval type-2 fuzzy clustering for multi-spectral satellite imagery. In: IEEE-SMC, pp. 2371–2376 (2012)

(11)

3. Nguyen, D.D., Ngo, L.T.: Multiple kernel interval type-2 fuzzy c-means clustering.

In: IEEE Int’ Conference on Fuzzy Systems, pp. 1–8 (2013)

4. Bezdek, J.C., Ehrlich, R., Full, W.: FCM: The Fuzzy C-Means clustering algorithm.

Computers and Geosciences10(2), 191–203 (1984)

5. Hwang, C., Rhee, F.C.H.: Uncertain Fuzzy Clustering: Interval Type-2 Fuzzy App- roach to C-Means. IEEE Trans. on Fuzzy Systems15(1), 107–120 (2007)

6. Genitha, C.H., Vani, K.: Classification of satellite images using new fuzzy cluster centroid for unsupervised classification algorithm. In: 2013 IEEE Conf’ on ICT, pp. 203–207 (2013)

7. Wang, W., Zhang, Y.: On fuzzy cluster validity indices. Fuzzy Sets and Systems 158, 2095–2117 (2007)

8. Bezdek, J., Pal, N.: Some new indexes of cluster validity. IEEE Transactions on Systems, Man and Cybernetics28(3), 301315 (1998)

9. Zhao, F., Fan, J., Liu, H.: Optimal-selection-based suppressed fuzzy c-means clustering algorithm with self-tuning non local spatial information for image segmentation. Expert Systems with Applications41, 4083–4093 (2014)

10. Liua, H., Zhao, F., Jiao, L.: Fuzzy spectral clustering with robust spatial information for image segmentation. Applied Soft Computing12, 3636–3647 (2012) 11. Wang, Z., Song, Q., Soh, Y.C., Sim, K.: An adaptive spatial information-theoretic

fuzzy clustering algorithm for image segmentation. Computer Vision and Image Understanding117, 1412–1420 (2013)

12. Chuang, K.S., Tzeng, H.L., Chen, S., Wu, J., Chen, T.J.: Fuzzy c-means clustering with spatial information for image segmentation. Computerized Medical Imaging and Graphics30, 9–15 (2006)

13. Ngo, L.T., Pham, B.H.: Approach to image segmentation based on interval type- 2 fuzzy subtractive clustering. In: Pan, J.-S., Chen, S.-M., Nguyen, N.T. (eds.) ACIIDS 2012, Part II. LNCS, vol. 7197, pp. 1–10. Springer, Heidelberg (2012) 14. Sowmya, B., Sheelarani, B.: Land cover classification using reformed fuzzy C-

means. Journal of Sadhana36(2), 153–165 (2011). Springer

15. Su, W., Zhang, C., Zhu, X., Li, D.: A Hierarchical Object Oriented Method for Land Cover Classification of SPOT 5 Imagery. WSEAS Trans. on Information Science and Applications6(3), 437–446 (2009)