A Support Vector Machine Binary Classification and ...

IJRIT International Journal of Research in Information Technology, Volume 3, Issue 5, May 2015, Pg.191-204

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ISSN 2001-5569

A Support Vector Machine Binary Classification and Image Segmentation of Remote Sensing Data of Chilika Lagloon 1

S.K.Sahu1, M.B.N.V. Prasad2, B.K.Tripathy3 P.G.Department of Statistics, Sambalpur University, Odisha, India, [email protected] 2 Dean, V.I.T.A.M., Berhampur, Odisha, India, [email protected] 3 Senior Professor, SCSE, VIT University, Vellore, Tamil Nadu, India, [email protected]

Support vector machine (SVM) and the inter-class reparability rule of hyper spectral data, binary tree SVM classifier based on separability measure among different classes of hyper spectral image classification. The binary tree SVM classifier is a approach to improve the accuracy of hyper spectral image classification and expand the possibilities for interpretation and application of hyperspectral remote sensing image. Segmentation of remote sensing images is a critical step in geographic object-based image analysis. Evaluating the performance of segmentation algorithms is essential to identify effective segmentation methods and optimize their parameters. In this study, we propose regionbased precision of evaluating segmentation of Chillika Lagoon, Odisha, India by using of ERDAS imagine-9.2 version software. We proposed to segment high-resolution remote sensing images. First, the over-segmented initial segmentation is produced by a region growing method by k mean clustering. By experiments using airborne operational modular imaging spectrometer II (OMIS II) data, satellite EO-1 Hyperion hyperspectral data and airborne AVIRIS data, the of Classification result image of novel binary tree SVM is obtained. 1. INTRODUCTION Support Vector Machines (SVM) for solving pattern recognition and nonlinear function estimation problems have been introduced . The idea of SVM is mapping the training data nonlinearly into a higher-dimensional feature space, then construct a separating hyperplane with maximum margin there. This yields a nonlinear decision boundary in input space. By the use of a kernel function, either polynomial, splines, radial basis function (RBF) or multilayer perceptron, it is possible to compute the separating hyperplane without explicitly carrying out the map into the feature space. While classical Neural Networks techniques suffer from the existence of many local minima, SVM solutions are obtained from quadratic programming problems possessing a global solution. Recently, least squares (LS) versions of SVM have been investigated for classification and function estimation. In these LS-SVM formulations one computes the solution by solving a linear system instead of quadratic programming. This is due to the use of equality instead of inequality constraints in the problem formulation. In such linear systems have been called Karush-Kuhn-Tucker (KKT) systems and their numerical stability has been investigated. This linear system can be efficiently solved by iterative methods such as conjugate gradient, and enables solving large scale classification problems. As an example we show the excellent performance on a multi two-spiral benchmark problem, which is known to be a difficult test case for neural network classifiers. In recent years, research has progressed in computer vision methods applied to remotely sensed images such as segmentation, object oriented and knowledge-based methods for classification of high-resolution imagery Argialas and Harlow (1990), Kanellopoulos et al. (1997). In Computer Vision,image analysis is considered in three levels: low, medium and high Argialas and Harlow (1990). Such approaches were usually implemented in separate software environments since low and medium level algorithms are procedural in nature, while high level is inferential and thus for the first one needs procedural languages while for the second an expert system environment is more appropriate. New approaches have been developed, recently in the field of Remote Sensing. Some of them were based on knowledge based techniques in order to take advantage of the expert knowledge derived from human photo-interpreters Argialas and Goudoula (2003), Yoo et al (2002), Yooa et al (2005). In particular within an Expert

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System environment, the classification step has been implemented through logic rules and heuristics, operating on classes and features, which were implemented by the user through an object-oriented representation De Moraes (2004), Moller-Jensen (1997). This object-oriented representation was mainly based on the image semantics and the explicit knowledge of the human expert. In order to classify each element of the image into the appropriate class, the knowledge based expert system represented the definitions of the classes through rules and heuristics, which an expert explicitly declares and develops within the system. As a result, more complex methods for image classification have been implemented and many more image features can be used for the classification step Smits and Annoni (1999).Very recently a new methodology called Object Oriented Image Analysis was introduced, integrating low-level, knowledge-free segmentation with high-level, knowledge-based fuzzy classification methods. Other fields of Artificial Intelligence have also been developed such as Computational Intelligence and Machine Learning involving Neural Networks, Fuzzy Systems, Genetic Algorithms, Intelligent Agents and Support Vector Machines Negnevitsky (2005). Machine learning is an integral part of Pattern Recognition, and in particular classification (Theodoridis and Koutroumbas (2003). Given that in the past,digital remote sensing used pattern recognition techniques for classification purposes, modern machine learning techniques have been also implemented for remote sensing applications and achieved very good classification results Binaghi et al (2003), Fang and Liang (2003), Theodoridis and Koutroumbas (2003), Huang et al (2002), Brown et al (2000), Foody and Mathur (2004).The Support Vector Machine (SVM) is a theoretically superior machine learning methodology with great results in the classification of high-dimensional datasets and has been found competitive with the best machine learning algorithms. In the past, SVMs were tested and evaluated only as pixel based image classifiers with very good results Huang et al (2002), Brown et al (2000), Foody and Mathur (2004), Gualtieri and Cromp (1999), Melgani and Bruzzone (2004). Furthermore, for remote sensing data it has been shown that Support Vector Machines have great potential, especially for hyperspectral data, due to their high-dimensionality Gualtieri and Cromp (1999), Melgani and Bruzzone (2004). In recent studies, Support Vector Machines were compared to other classification methods, such as Neural Networks, Nearest Neighbor, Maximum Likelihood and Decision Tree classifiers for remote sensing imagery and have surpassed all of them in robustness and accuracy Huang et al (2002), Foody and Mathur (2004). The various image segmentation methods, the region-based method is particularly suitable and thus widely used for segmentation of remote sensing images Schiewe et al.(2001), Carleer et al (2005) and Dey et al., (2010). The regionbased method can produce spatially contiguous segmented regions having inherent continuous boundaries and these regions can be viewed as image objects.

2. Support Vector Modeling SVM is based on two key elements: a general learning algorithm and a problem specific kernel that computes the inner product of input data points in a feature space. A SVM performs classification by constructing an Ndimensional hyper plane that optimally separates the data into two categories. SVM models are closely related to neural networks. In fact, a SVM model using a sigmoid kernel function is equivalent to a two-layer perception neural network. The input space is mapped by means of a non-linear transformation into a high dimensional feature space. The goal of SVM modeling is to find the optimal hyper plane that separates the data sets in such a way that the margin between the data sets is maximized. The vectors near the hyper plane are the support vectors (Figure). In other words the decision boundary should be as far away from the data of both categories as possible.

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Figure.1. The left picture separates the two categories with a small margin. The right picture has a maximized margin between the two categories, which is the goal of SVM modeling. The simplest way to divide two categories is with a straight line, flat plane or an N-dimensional hyper plane. This can unfortunately not been done with the two categories of Figure.2..2..

3

2

1

0 -1 (training) +1 (training) -1 (classified) +1 (classified) Support Vectors

-1

-2 -2

-1

0

1

2

3

4

5

Figure.2. An example of non-linear separation

To overcome this problem, the SVM uses a kernel function to map data into a different space where a hyper plane can be used to do the separation. The kernel function transforms the data into a higher dimension space to make it possible to perform the separation (Figure). There are a lot of different kernel function, used for a wide variety of applications.

Figure.3. Separation in a higher dimension.

2.1 THE SVM ALGORITHM CONSISTS OF TWO STAGES: Training stage: training samples containing labeled positive and negative input data to the SVM. This input data can consist of distance to border vectors, binary images, Zernike moments, and more. Each input data is represented by vector

xi

with label

yi = ±1, 1 ≤ i ≤ l , l is the number of samples. The decision boundary should classify all S.K.Sahu, IJRIT-193

IJRIT International Journal of Research in Information Technology, Volume 3, Issue 5, May 2015, Pg.191-204

points correctly, thus

(

)

yi w T x i + b ≥ 1, ∀i . The decision boundary can be found by solving the following 1 w2 . 2 y  w T x + b  ≥ 1 ∀i i i 

constrained optimization problem: Minimize Subject to

The Lagrangian of this optimization problem is:

L =

1 2

w

2

((

) )

− ∑ α i yi w T x i + b − 1 i

The optimization problem can be rewritten in terms of Lagrangian to zero: Maximize W (α ) = ∑ α −

i

α

i

α ≥ 0 ∀i . i

by setting the derivative of the

1 T ∑ α iα j y i y j x i x j 2 i, j

i subjectto α ≥ 0, ∑ α y = 0 ∀i i i i i


This quadratic programming problem is solver when:

αi


((

) )

y w Tx + b − 1 = 0 i i

∀i x

i

with

α >0 i

are support vectors. This is for a linear

separable problem, for more details about the non-linear problem Testing stage: the resulting classifier is applied to unlabeled images to decide whether they belong to the positive or the negative category. The label of x is simply obtained by computing:

w=

s ∑ α y x with t j j = 1, K , s t t t j =1 j j j

(

)

the indices of the s support vectors. Classify z as category 1 if the sum is positive, and category 2 otherwise.

2.2. LINEAR SVM AND NON LINEAR SVM The main idea of the SVM algorithm is that given a set of points which belong to one of the two classes, it is needed an optimal way to separate the two classes by a hyperplane as seen in the below figure. This is done by:



maximizing the distance (from closest points) of either class to the separating hyperplane minimizing the risk of misclassifying the training samples and the unseen test samples.

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Figure.4. Optimal Separating Hyperplane Depending on the way the given points are separated into the two available classes, the SVMs can be:
Linear SVM
Non-Linear SVM 2.5 versicolor (training) versicolor (classified) virginica (training) virginica (classified) Support Vectors 2

1.5

1

3

3.5

4

4.5

5

5.5

6

6.5

7

Figure.5. Linear Support vector machine

3

2

1

0 -1 (training) +1 (training) -1 (classified) +1 (classified) Support Vectors

-1

-2 -2

-1

0

1

2

3

4

5

Figure.6. Non-linear Support vector machine LINEAR SVMS

S be a set of points xi ∈ R d with i = 1,...., m . Each point xi belongs to either of two y i ∈ {− 1,+1}. The set S is linear separable if there are w ∈ R d and w0 ∈ R such that

Let

y i (w ⋅ xi + w0 ) ≥ 1,

classes, with label

i = 1,...., m

(w, w0 ) defines the hyperplane equation

w ⋅ x + w0 = 0 , named the separating hyperplane. The signed distance d i of a point xi to the separating hyperplane (w, w0 ) is given by: w ⋅ xi + w0 di = w The pair

it follows that:

yi d i ≥

1 w

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therefore

1 is the lower bound on the distance between points xi and the separating hyperplane (w, w0 ) .Given w

a linearly separable set S, the optimal separating hyperplane is the separating hyperplane for which the distance to the closest (either positive or negative) points in S is maximum, therefore it maximizes

1 . w

Figure.7. Optimal separating hyperplane NON-LINEAR SVMS The only way the data points appear in the dual form of the training problem is in the form of dot products xi ⋅ x j . Even if in the given space the points are non-linear separated, in a higher dimensional space, it is very likely that a linear separator can be constructed. So the solution is to map the data points from the input space

R d into some

R n (n > d) using a function Φ : R d → R n . Then the training algorithm will depend only on dot products of the form Φ( xi ) ⋅ Φ(x j ) .Constructing (via Φ ) a separating hyperplane with maximum space of higher dimension

margin in the higher-dimensional space yields a nonlinear decision boundary in the input space. Because the dot is computationally expensive kernel function are used. A kernel function K such that

K (xi , x j ) = Φ(xi ) ⋅ Φ(x j )

is used in the training algorithm. All the previous derivations in the model of linear SVM are still viable by replacing the dot with the kernel function, since a linear separation is still done, but in a different space.The classes for Kernel Functions used in SVM are: (1) Polynomial:

(

) (

K x, x ' = x ⋅ x ' + c

)

q

(2) RBF (Radial Basis Function): x− x'

(

)

K x, x ' = e (3) Sigmoid:

(

)

−

2

2σ 2

(

K x, x ' = tanh αx ⋅ x ' − b

) (

)

x ∈ R d , therefore they are not difficult to compute. It remains to determine which kernel function K can be associated with a given (redescription space) function Φ . The kernel functions require calculations in

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Fig.8. Decision surface by a polynomial classifier

3. SVM CLASSIFIERS USING A GAUSSIAN KERNEL RBF This example shows how to generate a nonlinear classifier with Gaussian kernel function. First, generate one class of points inside the unit disk in two dimensions, and another class of points in the annulus from radius 1 to radius 2. Then, generates a classifier based on the data with the Gaussian radial basis function kernel. The default linear classifier is obviously unsuitable for this problem, since the model is circularly symmetric. Set the box constraint parameter to Inf to make a strict classification, meaning no misclassified training points. Other kernel functions might not work with this strict box constraint, since they might be unable to provide a strict classification. Even though the rbf classifier can separate the classes, the result can be over trained. Generate 100 points uniformly distributed in the unit disk. To do so, generate a radius r as the square root of a uniform random variable, generate an angle t uniformly in (0,Pi ), and put the point at (r cos( t ), r sin( t )).

1.5 1 0.5 0 -0.5 -1 r=1 -1.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-1 +1 Support Vectors

1.5 1 0.5 0 -0.5 -1 r=1 -1.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure.9.SVM Classifiers Using a Gaussian Kernel Using RBF

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3.1.SIGMOID KERNEL FUNCTION Sigmoid kernel, to train SVM classifiers, and adjust custom kernel function parameters. Scatter Diagram of Simulated Data 1 -1 1

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

-1

-0.8 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

Figure.10. SVM classifier using the sigmoid kernel function

4. IMAGE SEGMENTATION ALGORITHM

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Figure.11. Overview of the Region-based Image Segmentation Algorithm based on k-means clustering (RISA). 4.1. K-MEANS CLUSTERING OF REGION-BASED IMAGE SEGMENTATION A k-means clustering algorithm is used to create the image used for seed selection by assigning a cluster class to every pixel of a remote sensing image. A flow diagram of the k-means clustering algorithm used by RISA is shown in Fig.8.12. The algorithm starts by generating the spectral values for the predefined number of centroids (k). The centroids are generated by distributing them uniformly along the range of image multi-spectral values. The algorithm constructs a new image partition by associating each point with the nearest centroid. The centroids are then recalculated for the new clusters. This two-step procedure is repeated until it reaches the maximum number of iterations (m). Two parameters must be specified by the analyst: 1) the number of clusters (k), and 2) the maximum number of iterations (m). The default values are k = 20 and m = 10, and users can adjust them before running the algorithm. After clustering a single band or multi-spectral image, every pixel in the image is labeled based on the number of clusters (e.g., any number between 1 and 20). The clustered image is then used during the seed generation process.

Figure.12 . Flow diagram of the k-means clustering algorithm.

5. SVM WITH A BINARY TREE CLASSIFICATION STRATEGY The first analyzed approach proposes a face detection system using linear support vector machines with a binary tree classification strategy. The result of this technique are very good as the authors conclude: “The experimental results show that the SVMs are a better learning algorithm than the nearest center approach for face recognition.” General

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information subsection describes the basic theory of SVM for two class classification. A multi-class pattern recognition system can be obtained by combining two class SVMs. Usually there are two schemes for this purpose. One is the one-against-all strategy to classify between each class and all the remaining; The other is the one-againstone strategy to classify between each pair. While the former often leads to ambiguous classification, the latter one was used for the presented face recognition system. A bottom-up binary tree for classification is proposed to be constructed as follows: suppose there are eight classes in the data set, the decision tree is shown in the figure below where the numbers 1-8 encode the classes. By comparison between each pair, one class number is chosen representing the “winner” of the current two classes. The selected classes (from the lowest level of the binary tree) will come to the upper level for another round of tests. Finally, the unique class will appear on the top of the tree.

Figure.13. The bottom-up binary tree used for classification Denote the number of classes as c, the SVMs learn carry out comparisons of can decompose

c

as

c(c − 1) discrimination functions in the training stage, and 2

c − 1 times under the fixed binary tree structure. If c n1

c=2 +2

n2

+2

n3

+ ... + 2

ni

where

does not equal to the power of 2, we

n1 > n 2 > .... > ni . Because any natural

number (even or odd) can be decomposed into finite positive integers which are the power of 2. If , and if

c

is even

c

is odd,

ni = 0

n i > 0 . It can be noticed that the decomposition is not unique, but the number of comparisons in

the test stage is always

c − 1.

6. LEAST SQUARES SUPPORT VECTOR MACHINES Given a training set of N data points

{ y k , x k }kN=1 , where x k ∈ R n is the k-th input pattern and y k ∈ R is the k-th

output pattern, the classifier can be constructed using the support vector method in the form

N  y ( x) = sign ∑ α k y k K ( x, x k ) + b   k =1  where

αk

are called support values and b is a constant. The

K (⋅,⋅) is the kernel, which can be either

K ( x, xk ) = x x (linear SVM); K ( x, xk ) = ( x x + 1) (polynomial SVM of degree d); T k

T k

d

K (x, xk ) = tanh[κ xkT x + θ ] (multilayer perceptron SVM), or K (x, xk ) = exp{− x − xk

2 2

/ σ 2 } (RBF

SVM), where κ θ , and σ are constants. For instance, the problem of classifying two classes is defined as

wT ϕ ( x k ) + b ≥ +1 if y k = +1  T w ϕ ( x k ) + b ≤ −1 if y k = −1 This can also be written as

y k [ wT ϕ ( xk ) + b] ≥ 1, k = 1,..., N S.K.Sahu, IJRIT-200

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where

ϕ (⋅) is a nonlinear function mapping of the input space to a higher dimensional space. LS-SVM classifiers N

min J LS ( w, b, e) = 12 wT w + γ w, b , e

1 2

∑e

2 k

k =1

subjects to the equality constraints

yk [wT ϕ ( xk ) + b] = 1 − ek , k = 1,..., N The Lagrangian is defined as N

{

L( w, b, e;α ) = J LS − ∑α k y k [ wT ϕ ( x k ) + b] − 1 + ek

}

k =1

with Lagrange multipliers

αk ∈ R

(called support values).

The conditions for optimality are given by

 ∂L = 0  ∂w  ∂∂Lb = 0  ∂L  ∂ek = 0  ∂L = 0  ∂α k for

→ → → →

w = ∑k =1 α k y k ϕ ( xk ) N

∑

N

α k yk = 0 α k = γ ek k =1

y k [ w T ϕ ( x k ) + b ] − 1 + ek = 0

k = 1,..., N . After elimination of w and e one obtains the solution 0  b   0  YT =   T −1    α Y ZZ I + γ     1v 

with

Z = [ϕ ( x1 )T y1 ;...;ϕ ( x N )T y N ], Y = [ y1 ;...; y N ], 1v = [1;...;1], e = [e1 ;...; eN ] and α = [α1 ;...; α N ] .

Mercer’s condition is applied to the matrix

Ω kl

Ω = ZZ T with = y k y l ϕ ( x k ) T ϕ ( xl ) = y k y l K ( x k , xl )

The kernel parameters, i.e. σ for RBF kernel, can be optimally chosen by optimizing an upper bound on the VC dimension. The support values αk are proportional to the errors at the data points in the LS-SVM case, while in the standard SVM case many support values are typically equal to zero. When solving large linear systems, it becomes needed to apply iterative methods.

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Figure.14. Flow chart of Least square Support vevtor machine

7. EXPERIMENT AND ANALYSIS The study area (Geological setup of the area), covering the Chillika Lagoon and its surroundings in Odisha, India, and lying between latitudes 19° 0’ 0’’N and 20°N, and longitudes 84° 50’ 0’’E and 85°40’0’’E has been extensively surveyed using ground-based geological techniques of Image segmentations by using of ERDAS imagine-9.2 version software. By experiments using airborne operational modular imaging spectrometer II (OMIS II) data, satellite EO-1 Hyperion hyperspectral data and airborne AVIRIS data, the Classification result image of novel binary tree SVM is obtained.

(a) (b) Fig.15.(a) False Color Composition (FCC) Image of Chillika Lagoon, Odisha Fig.15(b). Image Segmentation of Chillika Lagoon After the entire data set is normalized (the value of each pixel is between zero and one), samples are selected. Taking into account all spectral and texture features, the pixel purity index (PPI) of the image is computed. One thousand pure pixels are obtained for all classes and endmembers are chosen as samples. In this experiment, all pure pixels together with ground truth are taken as training samples at last. The classification problem involves the identification of eight land cover types ( crop land (230 pixels), inhabited area (111pixels), inhabited area1 B (67

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pixels), crop land B (78 pixels), water (103 pixels), road (65 pixels), bare soil (91 pixels), plant (88 pixels)) for the OMIS data set.

(a)

(b)

Fig.16. (a) False color composite of OMIS II hyperspectral (image of Changpin, Beijing, China) remote sensing image. Fig.16.(b). Classification result image of novel binary tree SVM. TABLE .1. ADAPTIVE BINARY TREE SVM CLASSIFICATION CONFUSION-MATRIX. Crop Land

Inhabited Inhabited area area1

Crop land

Water

Bare soil

Road

Plant

User accuracy

Crop Land

134

0

0

2

0

0

0

0

98.53%

Inhabited area

0

34

26

0

0

36

0

0

35.42%

0

26

0

0

67

0

0

27.96%

0

0

164

0

0

0

0

100.00%

Inhabited 0 area1 0 Crop land Water

0

0

0

0

135

0

0

0

100.00%

Road

0

0

0

0

0

103

0

0

100.00%

Bare soil

0

0

0

0

0

0

103

2

97.22%

Plant

0

0

0

0

0

0

3

105

97.22%

Producer 100.00% 100.00% 50.00% accuracy 8. CONCLUSIONS

98.80% 100.00%

50.00% 97.17% 98.13% 96.52%

Support vector machine (SVM) and the inter-class separability rule of hyperspectral data, binary tree SVM classifier based on separability measure among different classes of hyperspectral image classification.The binary tree SVM

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 5, May 2015, Pg.191-204

classifier is a approach to improve the accuracy of hyperspectral image classification and expand the possibilities for interpretation and application of hyperspectral remote sensing image. Segmentation of remote sensing images is a critical step in geographic object-based image analysis. Evaluating the performance of segmentation algorithms is essential to identify effective segmentation methods and optimize their parameters. In this study, we propose region-based precision of evaluating segmentation of Chillika Lagoon, Odisha by using of ERDAS imagine-9.2 version software.We proposed to segment high-resolution remote sensing images. First, the over-segmented initial segmentation is produced by a region growing method by k mean clustering. By experiments using airborne operational modular imaging spectrometer II (OMIS II) data, satellite EO1 Hyperion hyperspectral data and airborne AVIRIS data, the Classification result image of novel binary tree SVM is obtained.

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S.K.Sahu, IJRIT-204