Analysis of Soil Properties and Characteristics Using Rescaled Range(R/S) Method
Soundararajan Ezekiel Robert A Hovis Rick A. Robbins
Computer Science Department Computer Science Department Ohio Department of
Ohio Northern University Ohio Northern University Natural Resources
Abstract
Fractal dimension (Hurst exponent) is a popular parameter for explaining certain phenomena and describing natural textures. In this paper, the classification of surficial soil images is studied by making use of some powerful texture features based on rescaled range (R/S) analysis. Our research demonstrated, first, that the Hurst exponent/fractal dimension reflected all the major aspects of soil data variability, and provided a unique quantitative characterization of the data spatial distributions and, second, that Hurst/fractal parameters might be useful for choosing an interpolation procedure for mapping soil data. Our method can be easily extended to a multiresolution feature vector which can serve as a texture descriptor.
1. Introduction
Fractal based texture analysis was first introduced by Pentland [1], where a correlation between texture coarseness and fractal dimensions of texture was demonstrated. A fractal is defined as a set for which the Hausdorff-Besicovich dimension is strictly greater than the topological dimension, there- fore fractal dimension is the defining property. Fractal models typically relate a metric property such as line length or surface area to the elementary length or area used as a basis for determining the metric property; measuring coastlines is a frequently used example. The relation between the ruler length and the measured coastline length can be considered a measure of the coastline’s geometric properties, e.g. its roughness. The functional relation between the ruler size r and the measured coastline L can be expressed as L= constant* r (1-FD) where FD is the fractal dimension. Fractal dimensions has been shown to correlate well with the function’s intuitive roughness. The topological dimension of an image is three- two spatial dimensions and a third dimension representing the image intensity. Considering the topological dimension E, the fractal dimension FD can be estimated from the Hurst exponent H as H=E-FD. The fractal concept developed by Mandelbrot[4] provides an excellent representation of the ruggedness of natural surfaces, it has been successfully applied to geographical simulation[6], texture analysis [1], and x-ray medical images. For images (E=3) and the Hurst exponent can be estimated by Rescaled Range Analysis Method. A small value of the fractal dimension FD (large value of the Hurst[2] exponent H) represents a fine texture, while the large FD (small H) corresponds to a coarse texture. Our method is a simple form of R/S analysis, used to study and correlate surficial soil properties and characteristics. Aerial photographic images on Pleistocene-age glacial deposits in west-central Ohio were used in the study. In this paper we are developing a method for classifying or categorizing pictorial data. From the experimental results it can be concluded that our approach is superior to the conventional methods
This paper is organized as follows. Section 2, considers the fractal dimension and the Hurst exponent. Section 3 introduces the Hurst based texture classification and segmentation. Section 4 presents the results and discussions. Finally, the conclusion and applications are given in Section 5.
2. Fractal dimension and Hurst exponent
Fractal dimension (FD) has been used to characterize data texture in a large number of fields. FD separates important classes of images and characterizes information which is not characterized by other texture features. A variety of procedures have been proposed for estimating the FD of images. These measurements are frequently referred to as dimension type- e.g., Cover Dimension (CD), Box-Counting Dimension (BCD), Hausdorff-Besicovitch Dimension (HD), Wavelet based FD, and etc. We will refer to these procedures as FD-estimators. In this section, we discuss the general concept of Rescaled Range (R/S) analysis for calculating the Hurst exponent.
2.1 Rescaled Range Analysis: Methodology
Rescaled Range analysis is a simple process that is highly data intensive. Here are the sequential steps 1. Start with the whole observed data set that covers and calculate the mean Ā=(1/N)å ai 2. Next, sum the differences from the mean to get the cumulative total at each time point Xka from the beginning of the period up to any time: Xkm=å (ai – Ā), k=1,2,3,…n. 3. Calculate the range R(t)= max(Xka)-min(Xka) for k=1,2,3,…n. 4. Calculate the standard deviation S, of the values, ai of the observation over the period m, for which the local mean is Ā, S=[(1/N) å (ai – Ā)2]0.5 5. Calculate R/S=R(t)/S(t). 6. For the next stage, partition the time interval in to two blocks of size N/2=t and repeat the entire procedure , steps 1-5, and determined R/S for each segment of the data set of length N/2, then take averaged value. Repeat, using successively shorter t’s at each stage dividing the data set into non-overlapping segments and finding the mean R/S of these segments. 7. Plot the log-log plot, that is fit Linear Regression Y on X where Y=log (R/S) and X=log N. The exponent H is the slope of the regression line.
2.2 Example
As an example, R/S analysis has been applied to the signal[9] shown in Figure1. The log-log plot is shown in Figure 1. The signal produces an H value of 0.3569. Because the Hurst value is less than H=0.50, we say that signal exhibits the Hurst phenomena of antipersistance. This means that if the signal had been up in the previous period, it is more likely that it will be down in the next period and vice versa.
3.Hurst Based Texture Classification and Segmentation
In this section, we develop an efficient method for computing local Hurst exponents to measure the local roughness of an image by using the R/S technique. Image segmentation is one of the most important steps in our approach to image analysis and compression. Its main goal is to divide the image into parts that have the same roughness. We will discuss a simpler and more efficient version of the region based segmentation approach first, that is, Hurst based texture classification and segmentation.
Figure 1
The basic idea is to calculate the local Hurst exponents for an image. The local FD is then derived from the value of the Hurst exponent. A small value of FD represents a fine texture, while a large FD corresponds to a coarse texture. Based on this description, we can segment the image and find the edges with a simple thresholding method. Thresholding is the transformation of an input image I to an output binary image BI as follows: BI(i,j) =1 if I(i,j) ³T BI(i,j) = 0 if I(i,j)<T where T is the threshold
3.1Quincunx Neighborhood Q
Multifractal analysis is a new and promising approach to texture classification and image segmentation. In this method, an image I is segmented into a finite set of parts P1,P2,...,Ps which have different parameter FD such that I=ÈPi, PiÇPj=F, i¹j. One of the main problems is finding the local FD. To determine such a parameter FD, it is necessary to apply a mask at each pixel. Selection of such a mask is not an easy task. To obtain stable and useful results, masks of different size, shape, and position are considered. We use a special mask called Quincunx neighborhood to compute the local Hurst exponent.
Definition: A quincunx neighborhood is a set of the form:(1/2n)M-n([0,1]2) where M=[a,b]:
a=[1, 1]-1, b=[1,-1]-1 is the quincunx matrix for all odd values of n.
We represent the nodes by a distance vector d whose length is equal to the number of different distances from its origin. For example, the distance vector for a Quincunx neighborhood of size 4 to one decimal place approximation is d=[1,1.4,2,2.2,2.8,3,3.2,4].
3.2 Proposed Method
Our method is a simple form of R/S analysis[8]. Even though R/S analysis is defined for one dimensional time series, we have extended it to the case of two dimensional images. The range, Ri, for images is the difference between maximum and minimum pixel intensities along the linear traverse of pixels points in a distance vector d. That is, if Yi,1,Yi,2,...Yi,n is the set of pixel intensity values of points that lies within the distance d from the center, i=1,2,...m where m is length of distance vector d(i), then Ri=max(Yi,1,Yi,2,...Yi,n )-min(Yi,1,Yi,2,...Yi,n) Because the maximum and minimum values of Y will always be greater than or equal to zero, the range Ri will always be nonnegative. The general form of Einstein’s[10] T to the one half rule is Ri=c*iH. The subscript, i, for Ri refers to the rescaled range values for Y’s; c is a constant value, H is generally called the (local)Hurst exponent The local Hurst exponent H can be approximated by plotting the log (Ri) versus log(i) and solving for the slope through an ordinary least squares regression. Form the slope image S whose pixel values are the local Hurst values of each pixel in the image I. Since the values of d are fixed, we can store these values in a fixed vector which is computationally more efficient. The slope image S can be segmented by using thresholding techniques. Figure 2 a-b. shows the slope image and original image.
4.Texture Based Image Segmentation
This section describes the classification of surficial images from the slope image. First, we start with a soil image to be analyzed and then apply our method explained in section 3.2 with respect to Quincunx mask of size five and form the Slope image S. We segment this slope image into three portions by a simple thresholding
Figure 2.a
Figure 2.b
method. The second segment is shown in Figure 3 below. This gives the texture with identical fractal dimensions. This means that it divides the image into parts that have the same roughness.
Figure 3
4. 1.Result and Discussion
In this paper, all the aerial photographic images were captured and digitized at 100 dpi using a Microtek Scanmaker X6 scanner. The imagery is 1981 infra-red aerial photography that is provided by the United States Cartographic and Geospatial Center in Fort Worth, Texas for use in soil survey work. The scanned color imagery was then converted to grayscale for analytical purposes. Each photographic image is classified into the four major land cover classes (cultural areas, hydrographic areas, vegetative areas, and bare soil surface areas, using our techniques described in section 2 and 3. Currently, soil scientists for the Ohio Department of Natural Resources-Division of Soil and Water Conservation and the United States Department of Agriculture-Natural Resources Conservation Service compile these maps manually by delineating soil boundaries using pens and/or pencils. The delineation boundaries are drawn in the field and then transferred onto a mylar photobase and overlays[7]. Our method would allow for the future application of electronic delineation of soil boundaries. See Figure 4
Figure 4
5. Conclusion
In this paper, we have proposed a new texture based Rescaled Range (R/S) Analysis model. The use of this fractal model helps us to measure the degrees of roughness of the soil images and classify into different classes. That is, segmenting the image into different features including soils. Further, the experimental result shows that this model would be much faster than the traditional manual method to classify the soil images. This evaluation suggests that the model is an excellent tool in analyzing the surficial soil images. However, additional refinement of the fractal analysis capabilities needs to be explored. This would be performed by electronically scripting a program that would group the pixels with similar qualities and to delineate these areas of similar pixel qualities. These delineations would then be tested and correlated in the field or by overlaying an electronic map of the field-verified soil delineation boundaries for comparison purposes. In addition, this method has the potential for soil scientists to utilize historical aerial photographs as additional analytical tools. Further applications of R/S based models are being studied.
6. References
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[2]. H.Hurst., “Long-term storage capacity of reservoirs,” Trans. Amer.Soc. Civil. Engrs.,116:770-808 1951
[3]. B. B. Mandelbrot., “How Long is the Coast of Great Britain, Statistical Self Similarity and Fractional Dimension,” Science, 1967.
[4] B.B Mandelbrot, “Fractal Geometry of Nature,” San Francisco, CA Freeman 1982.
[5]. J.Feder., “Fractals,” New York, London Plenum Press, 1989.
[6] A.Fournier et.al; “Computer rendering of stochastic models,” ACM Commun, vol 25, pp 371-384,1982.
[7]K.E. Miller, R.A Robbins, “Soil Survey of Hardin County, Ohio.,“USDA-Soil Conservation Service . Govt. Printing Office,1994
[8] Robert A Hovis, Soundararajan E. “Texture Based Image Compression by Using Qiuncunx Mask,” IASTED International Conference on Applied Informatics , Innsbruck, Austria, February 19 - 22, 2001.
[9] Robert A Hovis, Soundararajan E. “Seismic Signal Processing by Using Rescaled Range Analysis (R/S) based Fractal Dimension,“ IASTED International Conference on Applied Informatics (AI2001), Innsbruck, Austria, February 19 - 22, 2001.
[10]AEinstein.,”Uberdievondermolekularkineticschen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten teilchen,”AnnalsofPhysics,322,1908