Analysis of Soil Properties and Characteristics Using Rescaled Range(R/S) Method
Soundararajan Ezekiel
Robert A Hovis
Computer Science Department Computer Science Department
Ohio Northern University
Ohio Northern University
s-ezekiel@onu.edu
r-hovis@onu.edu
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, therefore 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
Rick A. Robbins
Ohio Department of
Natural Resources
rfrobb@wcoil.com
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 FDestimators. 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) 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= (ai – Ā), k=1,2,3,…n. 3.
Calculate the range R()= 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)  (ai – Ā)2]0.5
5. Calculate
R/S=R()/S(). 6. For the next stage, partition the
time interval in to two blocks of size N/2= 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 ’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) T 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=Pi, PiPj=, ij. 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.
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.
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 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 fieldverified 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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Natural Scenes,” IEEE Trans on Pattern Analysis
and Machine Intelligence,1984 ,666-674.
[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.,”Uberdievondermolekularkinetics
chen Theorie der Warme geforderte Bewegung
von in ruhenden Flussigkeiten suspendierten
teilchen,”AnnalsofPhysics,322,1908