2018 International Conference on Advances in Sustainable Engineering and Applications (ICASEA), Wasit University, Kut, Iraq.
Classification of Remote Sensing Images via Fractal Discriptores Nadia M. G. Al-Saidi Hussam Yahya Abdul-Wahed
Depatment of Applied Sciences University of Technology Baghdad-Iraq
[email protected]
Depatment of Applied Sciences University of Technology Baghdad-Iraq
[email protected] and storing these data to be used in other applications. Remote sensing is considered as the main tool to recognize the world around us, due to its ability to recognize many natural phenomena. With the development of technology, the highresolution images need new tools to deal with the problems of analyzing and processing. Fractal techniques have been proved to be suitable for variety of pattern recognition applications based on FD as the main tool. We include here some previous works, which based on utilizing of FD in images analysis. The main applications are for image classification and feature extraction:
Abstract- In image analysis and recognition of natural scenes, fractal geometry takes an important place among other techniques by analyzing, coding and extracting of important features to represent the digital image. In this paper, we discussed the problem of classification of natural urban images based on some statistical approaches, such as co-occurrence matrix and the principle of variogram function, which provides a more descriptive way to analyze different textures. Moreover, a new fractal descriptor is proposed as one of the statistical approaches to serve as a feature extractor in classification of different land cover image types. The performance of this descriptor is investigated through hierarchal clustering that show encouraging results to some classical textural analysis method in terms of computing and accuracy.
In 2015, Al-Saidi et al. [6] proposed new generalization for the classical box counting dimension; they proved that, it has high robustness to noise. This generalized technique is then used in [7] to enhance the performance of fractal image compressions method, and in [8]. They combined this estimated measure with fractal image coding technique to enhance image retrieval methods.
Keywords— Image recognition; image classification; fractal dimension; co-occurrence matrix; variogram function.
I. INTRODUCTION Fractal dimension (FD) is defined as a factor used to improve the performance of image processing methods, and serves as an important feature of images [1]. It is an indicator of the complexity of the images and known as the meter of complexity. Complexity is a term used for describing the performance of a mathematical model or a system, in such a way that, the parts of the system work in multiple ways and follow special instructions. In the field of fractal theory, complexity is defined as a change in detail with change in scale [2]. Some of the current works use fractal geometry for analyzing of remote sensing images through textural features [3-5]. The texture of an image is a structure that preserves some statistical possessions in the color or brightness distribution and repeatedly makes patterns. The spatial relationship of texture patterns is organized randomly, or may be pairwise dependent. Nevertheless, there are regions that cannot be characterized by those elements only, because they have a significant pattern of brightness values. This problem defined a verity of changing that refers to the local irregularity in intensity from one point to another contained in a specific area. Different approaches utilized fractal dimension alone or with some statistical features in segmentation, classification, modelling and many other applications. The effectiveness of fractal complexity for explaining remote sensing problems had been established in order to make it suitable for image analysis. This work focuses on the problem of classification of image areas in remote sensing data based on new fractal descriptors. Remote sensing is the process of capturing data and avoiding physical interaction, then collecting
In 2002, Read and Lam [9] used the advantage of spatial autocorrelation, and fractal theory, especially fractal dimension for characterizing of unclassified remote sensing data. They concluded that, the fractal dimension is very useful for specifying different degrees of complexity in those data that represent different types of land covers. In 2004, Weng [10] analyzed three profiles from Landsat TM images by using fractal dimension approach to construct the spatial variability in the temperature of surface radiant in those images. His results showed that, the sequential changes in fractal dimension became valuable in identifying the increased textural complexity of the thermal surfaces as well as, the periodic dynamics of heat effects in images. In 2010, Feng and Chen [11] introduced the method of perimeter-area relationship to estimate the fractal dimension signifying the irregularity of urban images over certain years. Their results showed that, the self-similarity exists in both the built-up area and the urban area, and fractal properties have a tendency to become better defined with time. In 2017, Alonso et al. [12] used the mass exponent function to characterize the scaling property of multi-spectrum satellite images. They studied four band images that involve vegetation information, which are taken by different satellites but with similar geographic position. They have shown that, pixel size and the number of bites coding used for the image have a great influence in each bands of the image.
978-1-5386-3540-7/18/31.00$©2018 IEEE
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2018 International Conference on Advances in Sustainable Engineering and Applications (ICASEA), Wasit University, Kut, Iraq. In the current work, new algorithms for fractal descriptors are proposed as one of the statistical approaches to serve as a feature extractor in classification of different land cover image types. This paper is designed as follows, in section 2; some technical background about fractal dimension is presented. The material and method of this work are discussed in section 3. Section 4 focuses on some results and its analysis, where the paper is concluded in section 5.
along with statistical methods. There are many techniques used to measure pattern distribution within a region of the image. The best-established computation on comparing values is the second-order statistics calculated from image [20]. It discriminates texture by the statistical properties of the grey levels with a surface. III. MATERIALS AND METHODS For remote sensing analysis, texture is a factor needed more developing. Most of the researchers are using fractal dimension alone to identify different textures that is often based on the boxcounting algorithm. Our method can be implemented through a couple of the texture factor, named variogram function besides Gray Level Co-Occurrence Matrix (GLCM).
II. TECHNICAL BACKGROUND In this section, a description of fractal geometry technique is briefly introduced. The reader can find a more extensive exposition in [1,13-15]. Mandelbrot [1] introduced the term “fractal” to describe various classes of sets that sharing the same properties, such as, self-similarity, having non integer dimensions, and it can be generated by iterator process. Many researchers defined fractal as an object with self-similarity. These properties are followed by an explanation in simple terms; it involves an introduction of fractal dimension and self-similarity, which was introduced by Grassberger in [16]. For this purpose, a geometric object is selfsimilar, if any small part of it looks similar or exactly the same as the original object. An object with these properties has an irregular shape, since it obtains small copies of itself. Therefore, the degree of this irregularity or (complexity) can be measured by fractal dimension (FD). An early formula to compute (FD) is the Hausdorff Besicovitch dimension, which can be derived from the Hausdorff measure [14]. The numeric computation of the Hausdorff dimension can be considered as complicate or sometimes difficult to estimate. Hence, numeric calculations are specified for a self-similar object that is composed of N selfsimilar parts scaled via some factor s. The relationship between their parts is represented by N=P-FD, when P is the reduction factor, and n is the number of pieces, such that: n
p i 1
s i
1
A. Variogram Function Statistical prediction may be based on the hypothesis that the set of the variables (v1,v2,…,vn) represents n realizations of a random variable v define on the set BR2. The regionalized variable is a variable realizes in a deterministic way within a spatial domain, but beyond that, it acts like a random variable [21]. The semi-variogram is a two-point statistical function that represents the increasing differences or decreasing in the relationship between regionalized variables, as the distance between them is increased. Semi-variogram function formula is written as: N
var(h )
(v (i , j ) v (i h , j )) ) 2
i 1
2H
…
(3)
where h is the lag distance between each pair of pixel values, v(i,j) is the pixel intensity at position (i,j), H is the total number of pairs of gray level values separated by a lag h . The semivariogram of an image is the average squared difference of pixel values separated by distance h. It is a two-point statistical method which represents the expanding differences or the decreasing in the relationship between pixels as the lag distance increasing. In order to preform pattern recognition, it is always a better way to partition the image into several areas and use this image as an initial set. For simplicity, suppose that the original image for the processing has equal number of rows M and columns N. The image is divided into fixed size square blocks, where the size of each block is 2n2n and nN. This practice is sufficient to process each square block separately in order to classify it. The procedure of feature extracting can be explained as follows: Let S be a square block of the image with size nn. Divide the image into equal size blocks to compute the semivariogram from its rows and columns as shown in Fig. 1. The procedure of computing for each row and columns as in Eq. (3) yields two vectors:
… (1)
The self-similarity dimension is calculated as follows:
log(N ) … (2) log( p ) The formula in Eq. (2) can be generalized to any self-similar set, where N is considered to be any self-similar measure and P be any scale parameter. This generalization gives many other estimations of fractal dimension, that can be applicable to different sets of object, which are not strictly self-similar, but they have a certain degree of self-similarity [13]. Several approaches in the literature were suggested [17-19] to estimate FD. These approaches can be classified through the box counting fractional Brownian motion (statistical method). The proper way of choosing a method to estimate FD depending on the nature of the image. Here, we are working with remote sensing images that have a lot of information about texture features with its gray level intensities. FD
𝑉𝑎𝑟1 𝑉𝑎𝑟1 𝑉𝑎𝑟2 𝑉𝑎𝑟2 𝑉𝑎𝑟3 𝑉𝑎𝑟3 . . . . . . 𝑉𝑎𝑟 𝑁𝛽 ] [𝑉𝑎𝑟𝑀𝛼 ] [
Texture analysis on images can be performed without addressing the fractal nature of the image. Most available researches, which used fractal techniques in texture analysis, had especially focused on fractal dimension. Certain computational methods based on texture features of brightness 100
2018 International Conference on Advances in Sustainable Engineering and Applications (ICASEA), Wasit University, Kut, Iraq. Homogeneity or inverse difference moment feature is used to measure image local homogeneity. The closeness distribution between the elements of the GLCM to the diagonal of GLCM is measured through this feature.
𝑉𝑎𝑟(ℎ)𝛽= 𝑉𝑎𝑟(ℎ)𝛼 = where M stands for the size of rows entries, N stands for the size of columns entries. The vectors above signify the result of calculating the relationship between semi- variogram among pixel points in each row and column. The process above take place by constructing of 1D data point series out of 2D digital image by taking the variogram as a reduction of attribute order as shown in Fig.1. The elements in var and var are changed
M 1 N 1
HOM
a 0 b 0
Fig. 1.
M 1
(P
… (4) … (5)
couple of gray levels a and b, here d is the distance between them following direction θ:
... (9)
The database for remote sensing images was taken from UC Merced Land Use database [23]. This is a 21-class land use image dataset intended for research purposes. The size of each image is 512x512 pixels. The images were manually extracted from large images using the USGS National Map Urban Area Imagery collection for various urban areas around the country. The pixel resolution of this public domain imagery is one foot. In our research, we take several images, in each one, we try to partition its areas by using the proposed fractal descriptors.
The gray-level variation is measured through the contrast, which is known as a measure of intensity between two pixels (the reference and its neighbor).
F. Classification In this work, we used the hierarchical clustering method to classify the blocks of the remote sensing image. The agglomerative method was used with several merging procedures and distances. Hierarchical clustering starting by measuring the distances between each object in the data set and make a distance matrix or (proximity matrix). During each
M 1 N 1
(a, b )
(a, b )) 2
E. Data Base
… (6)
where (k , l ) (r d i , s d i ). There will be (M-1) neighbor resolution pairs for each row, and there are N rows, providing R (M 1)* N nearest horizontal pairs. Haralick [22] proposed 14 texture features that are useful for describing texture properties. In this study, only three statistical features are extracted from GLCM, which are contrast, homogeneity and energy they are defined as follows:
,d
,d
D. The Fractal Descriptors Algorithm Three texture features were computed from VLCM; contrast, energy and homogeneity. They are determined for all blocks. In each block, the descriptors are computed for different offsets resulting in three features for each offset. The next process is the estimation of fractal descriptors from the change of scaling of the offsets. Finally, we compute the slop of linear regression of the log-log plot that resulting in new features called fractal descriptors. Algorithm 2 shows the necessary steps and functions to calculate the fractal descriptors.
GLCM of size G *G denoted by P ,d for a shifted vector d = (di, dj) is constructed as follows: The non-normalized frequencies are built as a function of angle θ and distance d such that, the element of P ,d represents the sum of occurrence for a
P
… (8)
C. The proposed algorithm: Variogram Co-Occurrence Matrix(VLCM) In this step, the semi- variogram values in Var1 and Var2 were computed for different lag distances from the block image. The number of distances h was computed for half of data set. Nevertheless, for remote sensing images is typically calculated for h=20. The mean value of semi- variogram can be computed for all elements to each h. For each Var1 and Var2, a new modified method is constructed to compute the occurrences of the elements of the semi- variogram values inside each vector. This proposed matrix is called variogram level Co-occurrence Matrix (VLCM). Each entry in VLCM represents the number of pairs of row or column with certain distance that identifies variogram value. The necessary steps and functions to compute VLCM matrix for each block of the image are listed in Algorithm 1.
B. Gray Level Co-Occurrence Matrix (GLCM) The GLCM statistical technique describes the distribution of gray levels and their texture relationship. Haralick [10] was the first who introduced the concept of a gray level co-occurrence matrix. GLCM works through specifying the relationship of neighboring pixels. Take an image u (i , j ) of size M*N which is represented as the form u (i , j ):0 i M 1,0 j N 1 , with G gray levels. Then the
CON | a b |2
P ,d (a, b )
a 0
Extracting variogram from rows or columns
P ,d |[(r , s ),(k , l )] G *G : u (r , s ) a,u (k , l ) b |
2
Energy is the sum of the squares of the co-occurrences. This feature is useful for roughness estimation. Energy is small for regions with large pixel-intensity variations, while it is large for uniform regions.
to the nearest integer values in order to form a new matrix holding the frequencies for each variogram by the following formulas:
var1 Round (var (i ) min(var ) *10)) min(var ) var2 Round (var (i ) min(var ) *10)) min(var )
1
1 | a b |
… (7)
a 0 b 0
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2018 International Conference on Advances in Sustainable Engineering and Applications (ICASEA), Wasit University, Kut, Iraq. iteration of this classification, the smallest entry in the proximity matrix is determined. Then, the two clusters, which are separated by the distance given in the proximity matrix, are merged. Finally, the row in which the distance lies is deleted according to some selection criteria. The agglomerative algorithm is started by distributing the data set objects in an individual cluster, then merging them gradually. Algorithm 3 shows all the necessary steps and function for computing the classification process.
1.
u (i , j ) of
2. 3.
data of the given image Define h, number of distance pixels Define k, number of pixels in a single row For each row(r) of u (i , j )
4. 5.
2. 3.
coni n , j 1,coni n , j 2, ...coni n , j n 4. 5. 6. 7.
the
Compute the proximity matrix T of the data in CON as in Algorithm 2. Assign each element as a cluster on itself. Set the number of the desired clusters to be cl. Specify the small entry in the distance matrix, T
T (coni ,con j ) min(Tij ) 8.
Merge the two clusters coni and conj to form a new cluster
9.
Update the distance matrix T by assigning distance value of
In this work, the measure of cluster improvement or superiority is discussed. One type of measure allows us to compare different sets of clusters, known as an internal controller. It measures the total similarity, which is established on the corresponding similarity for pairwise fractal descriptors. Consider two objects ci,cj, the internal quality can be measured by calculating the cophenetic correlation coefficient such that,
7.
Compute the matrix w(h,r) contains var(r) is calculated for different values of h 8. Compute MW=mean (w (h, r)) over h. 9. Compute the occurrences of each element in MW to be Stored the in VLCM matrix. Algorithm 2. Fractal Descriptors Algorithm
u (i , j ) of
2. 3. 4.
data of the given image Define d , number of offsets Compute VLCM of each block For each d , compute
size M*N, where
by:
IV. RESULT AND ANALYSIS
Round the values in each row
Read
conij
T (conij ) min(T (coni ,con j ))
z 2( k 1)
1.
conij .
10. Repeat (9). 11. Stop when the determined cluster is greater than c1.
var(r ) Round (var(r ) min(var(r ) *10)) min(var(r ))
5. 6.
Calculate fractal descriptor for each block as in Algorithm 2. Store the values of step (2) in a one dimensional vector that represent the data set in a form of the location
CON coni , j , coni 1, j ...con i n , j
For each h while i+hM Compute z=v(i,j)-v(i+h,j) Compute variogram of each row var(r)=
6.
size M*N, where
u (i , j ) represents
Read
u (i , j ) of size M*N, where u (i , j ) represents the data
of the given image
Algorithm 1. VLCM Matrix Algorithm 1.
Read
u (i , j ) represents
the
(T (c ,c i
coh
((T (c ,c i
CON Contrast (d ) HOM Homogeneity (d ) ENG Energy (d )
j
) T ave )(t (ci , c j ) t ave )
…(10)
ij
j
2
2
) T ave ) (t (ci , c j ) t ave ) )
ij
where T (ci ,c j ) represents the distance between the elements in the proximity matrix and Tave is the average of these distances. While t (ci ,c j ) represents the height of the dendrogram plot, that is the distance of clusters which contain the two objects ci ,c j . The average of t (ci ,c j ) is denoted by
End for Compute the slop as a linear regression of
log(CON (d ) log(d ) log(HOM ) Fh log(d ) log(ENG ) Fe log(d )
Fc
t ave . The best value of coh for different linkages methods were calculated in this section. The closer the value of coh to1, the more effective the clustering result will be as reflected the original data clustering nature. The quantity coh represents a dendrogram that conserves the joint distances between the original object data, which consider as the fractal descriptors.
The procedure in Algorithm 3 can be done in the same way with the remaining fractal descriptors; energy and homogeneity. The algorithm starts with defining an image u(i,j) then divide it into certain number of blocks. The fractal descriptor is computed for each block and stored in a one dimension vector. The clustering algorithm is aimed to classify image blocks by the proposed features in order to show the performance of each one of them, and how they affect the clustering results.
Many cluster analysis procedures such as: single linkage, complete linkage, weighted average, unweighted average distance, median distance, weighted centroid distance, and ward’s method were discussed. The mean values Mave and t (ci ,c j ) were listed in Table 1 for each of linkage values with the cophenetic correlation coefficient for each of them. The lag distance (h) of the variogram function takes the initial step of calculation for extracting fractal descriptors. In the initial step, the variogram for each row of the image is calculated for specified number of h, the mean value of all distances was taken for each row. The fractal behavior of the descriptors shows
Algorithm(3): Clustering by Fractal Descriptors
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2018 International Conference on Advances in Sustainable Engineering and Applications (ICASEA), Wasit University, Kut, Iraq. better performance as it approaches to the linear fitting line. The following resultant figures summarize this issue. Starting with dividing the image in Fig. 2 as an initial stage.
Fig. 2.
Fig. 6.
Tested image
Table.1 shows the performance of the proposed algorithm with different measures of linkages and distances. In Table.1, the performance of the proposed algorithm with different measures of linkages and distances is illustrated. Fig.7 shows the dendrogram stages of merging the clusters throw each steep for city block measure, noticed that for city block distance measure, the initial stages for merging the clusters iterate in a very close way, by choosing the closest member to unite with it.
In Fig. 3, the linear fitting model is presented for the contrast descriptor in log-log plot
Fig. 3.
Log-log plot of contrast descriptors
The same procedure is applied for the homogeneity of the fractal descriptor; Fig. 4 and Fig. 5 show the linear fitting model:
Fig. 4.
Clustering by homogeneity fractal descriptor with five max clusters
Fig. 7.
The dendrogram result for city block distance
Based on the relationships observed in equation (10), the cophenetic coefficients were obtained, considering that the relationship between two clusters assigned in the same cluster can be signified by the average distance within this cluster, and the association between two clusters that assigned in different clusters can be represented by the average distance between them. For example in Fig.7, based on City block distance, the intera-distance between the blocks 24 and 25 is simply the average distance within cluster 2, which is t (c24 ,c 25 ) = 1.93.
Log-log plot of contrast descriptors
However, the distance between the block cluster 12 and 13 is the average distance between clusters 2 and 4, which is t (c12 ,c13 ) =
4.15.This procedure is done for every cluster block that contains sub-clusters in order to calculate the required intra-distances. TABLE I: THE VALUE OF COH FOR DIFFERENT DISTANCE AND LINKAGE MEASURE
Distance Measure
Fig. 5.
Log-log plot of homogeneity descriptors with h=7
A much better presentation for the homogeneity descriptor through the fitted model is shown in Fig. 5, one can see most of the points for h=7 are effectively matching. Since the used clustering method in this work is unsupervised, the user can choose a number of clusters on prior knowledge. The number of clusters can be determined by cutting the dendrogram at a specific stage of merging elements to choose the maximum cluster. The result of this procedure is illustrated in Fig. 6:
City block
Euclidian
103
Linkage method single linkage Complete linkage weighted average linkage Unweighted average linkage
0.6055 0.7443 0.6870 0.7430
Median linkage Centroid linkage
0.7430 0.7430
Ward linkage single linkage Complete linkage weighted average linkage Unweighted average linkage
0.738 0.6549 0.7430 0.7415 0.7415
2018 International Conference on Advances in Sustainable Engineering and Applications (ICASEA), Wasit University, Kut, Iraq. Distance Measure
Mahalanobis
Minkowski
Linkage method Median linkage
0.7456
Centroid linkage Ward linkage single linkage Complete linkage weighted average Unweighted average Median linkage
0.7418 0.7368 0.6112 0.7330 0.7496 0.7647 0.7434
Centroid linkage Ward linkage single linkage Complete linkage weighted average linkage Unweighted average linkage Median linkage Centroid linkage Ward linkage
0.7634 0.6654 0.6873 0.7431 0.7374 0.7233 0.7456 0.7223 0.7195
[7]
[8]
[9]
[10]
[11]
V. CONCLUSION
[12]
The new proposed method of feature extraction that is called fractal descriptors, made a satisfactory result for the problem of image recognition and classification. The variogram function of each row shows a great result of descriptors like fractal behavior shape. It was found that, a different number of lag distances for each land cover would satisfy the fractal behavior. Taking the mean value of variogram function for different lag distances by each row is an empirical process for finding descriptors that take fractal shape. Contrast descriptor and homogeneity descriptor of the fractal descriptors had been extracted. For the purpose of block classification, hierarchical clustering was chosen for its simplicity. The fractal descriptors show better recognition results when using small block size in the dividing process. In addition, for the sake of distance and linkages, city block distance measure shows a better result than other distance measures by using the cophenetic correlation coefficient.
[13] [14] [15] [16] [17]
[18]
[19]
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