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Three New Implementations of the
Triangular Prism Method for Computing the
Fractal Dimension of Remote Sensing Images
Wanxiao Sun
Abstract
Based on Clarke’s (1986) triangular prism concept, this paper
proposes three new methods to compute the fractal dimension (D) of remote sensing images. Our first method involves
searching a pixel on each edge of a square whose digital
numbers (DN) value has the largest deviation from the central
pixel. Our second method uses a pixel on each edge of a
square whose DN deviation from the central pixel is closest to
the mean DN deviation from the central pixel to all pixels on
the same edge. In our third method, eight pixels on the four
edges of a square are used. Furthermore, common to the
three proposed methods is the use of actual DN of the central
pixel. The proposed computation methods have been tested
using both simulated fractal surfaces and real images. Results
show that the proposed methods appear to generally perform
better than Clarke’s 1986 method for synthetic images with
complex textures.
Introduction
Fractal-based texture analysis of remotely sensed images has
generated considerable interest in the remote sensing community in the past two decades (Pentland, 1984; De Cola, 1989;
Lam, 1990; De Jong and Burrough, 1995; Myint, 2003). A main
reason for this increased interest in the fractal model seems the
realization that incorporation of spatial information in digital
image analysis can help improve classification accuracies
(Haralick et al., 1973; Pratt et al., 1978; Gong and Howarth,
1990). As a relatively novel class of spatial technique, fractal
analysis appears to provide considerable potential for characterizing textures in remotely sensed images.
Fractal geometry was introduced and popularized by
Mandelbrot (1977) to model natural shapes (e.g., coastlines
and terrain) as well as other complex forms that traditional
Euclidean geometry fails to analyze. A major application of
fractal geometry in the geosciences has been the use of fractal
dimension to characterize the form of environmental phenomena at different scales (Goodchild, 1980; Mark and Aronson,
1984; Goodchild and Mark, 1987). The fractal dimension,
often denoted as D, is a central construct of fractal geometry.
It is called fractal dimension because it is a fractional (or
non-integer) number. A coastline’s fractal dimension, for
example, can take on any non-integer value between 1 and 2,
depending on the degree of irregularity of its form. The more
contorted a coastline is, the higher its fractal dimension.
Similarly, a terrain surface’s fractal dimension may be a non-
integer value between 2 and 3. As such, the fractal dimension
can be thought of as a parameter capable of capturing the
geometrical complexity of an object being analyzed.
A remotely sensed image can be viewed as a hilly terrain
surface whose “elevation” is proportional to the image gray
value. Technically, an image can be interpreted as a 3D space
where the x, y coordinates represent 2D position on the
image plane and the z coordinate represents the gray level
values or digital numbers (DN). Most remotely sensed images
are spatially and spectrally complex. As such, the fractal
dimension appears to be a useful parameter for measuring
the surface roughness (i.e., brightness differences) of remotely
sensed images. Several studies have used fractal techniques
to characterize textures and features in digital images. For
example, Pentland (1984) has shown that computed D values
are useful in edge detection, image segmentation, and other
image analysis applications. De Cola (1989) used fractal
techniques to examine the scaling characteristics of feature
classes constructed from Landsat TM images. Lam (1990)
demonstrated that different land-use types exhibit textures of
different D values. De Jong and Burrough (1995) presented a
“local D algorithm” to classify Mediterranean vegetation
types in remotely sensed images. Myint (2003) has recently
compared the accuracies of image classification using fractal
methods as well as other texture analysis techniques.
The utility of D as a texture measure depends to a large
extent on the availability and accuracy of the methods of
computing D. Several researchers (e.g., Tate, 1998; Lam et al.,
2002) have emphasized that developing efficient and reliable
algorithms is key to the application of fractal techniques to
digital image analysis. The triangular prism method was
proposed by Clarke (1986) to primarily compute the fractal
dimension of topographic surfaces. Despite this, Clarke’s
(1986) method has become one of the most often used methods for computing the fractal dimension of remotely sensed
images (Lam and De Cola, 1993; De Jong and Burrough, 1995;
Qiu et al., 1999; Lam et al., 2002; Myint, 2003). The purpose
of this study is to expand Clarke’s 1986 work. Specifically,
following Clarke’s (1986) triangular prism concept, we
first propose three new methods to construct the triangular
prisms used to compute the fractal dimension of digital
images. The performance of the proposed methods and
Clarke’s (1986) method is then tested on 45 synthetic surfaces and 72 real images.
Photogrammetric Engineering & Remote Sensing
Vol. 72, No. 4, April 2006, pp. 373–382.
Department of Geography and Environmental Resources,
Southern Illinois University-Carbondale, Carbondale, IL 62901
([email protected]).
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0099-1112/06/7204–0373/$3.00/0
© 2006 American Society for Photogrammetry
and Remote Sensing
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The Proposed Methods
Clarke’s (1986) original triangular prism method makes use
of a discrete representation of the elevations of the Earth’s
surface such as in a digital elevation model (DEM). Based on
this data structure, the method takes elevation values (or the
equivalent of DN in an image) at the corners of squares,
interpolates a center value, divides the square into four
triangles, and then computes the top surface areas of the
prisms which result from raising the triangles to their given
elevations (Figure 1a). (Refer to Clarke, 1986 for a more
complete discussion.)
A critical step in the triangular prism method is the
calculation of the top surface area of the prisms. Clarke’s
(1986) method computes the top surface area by adding up
the areas of four triangles formed by the corner pixels of a
square and the interpolated center of the square. Obviously,
Figure 1. (a) 3D view of the triangular prism method
(after Clarke, 1986); (b) Top view of the corner pixels
(a, b, c, and d ) and the center point (e) used in Clarke’s
(1986) method (an example with step size 4).
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April 2006
the ways in which the prisms are constructed affect the
computed areas of the top surface, which in turn affect the
computed D values of the surface. Critical to the triangular
prism method is therefore the choice of the pixels used to
construct the prisms. The three methods proposed in this
study are inspired by approaches to construct the prisms
that are different from Clarke’s (1986) method.
Clarke (1986) used the average of the four elevations at
the corners of a square to represent the center of the square.
This approach clearly has the advantage of simple computation, but it may lead to error, because the interpolated value
assigned to the center of the square may not lie on the
surface being analyzed. This is especially the case where
sharp tonal variation exists in the neighboring pixels. For
example, suppose the DN values of the corner pixels are 1,
3, 2, 6, and the actual DN of the central pixel is 8. Using
Clarke’s (1986) method, the interpolated value assigned to
the central pixel would be 3, which is significantly different
from its true value. In the three methods developed in this
study, we propose to use the actual DN of the central pixel.
The advantage of this approach is that every point used to
calculate the area of the top surface can be assured to lie on
the surface.
In addition to using actual DN of the central pixel, the
three proposed methods focus on the choice of edge pixels
used to construct the triangular prisms. An edge pixel is
defined in this paper as a pixel that is located on the edges
of a square. Pixels a, b, c, and d in Figure 1b are examples
of edge pixels. For a square of side length (or step size) ,
there are edge pixels on each edge of the square, and
the total number of edge pixels is 4. As described above,
Clarke’s (1986) method uses the corner pixels of a square and
the interpolated center point to form the prisms. Two questions can be raised here. The first is how well the corner
pixels actually represent the surface being analyzed. In other
words, can approaches using other pixels better represent the
surface. Our first two methods are inspired by this consideration. The second question to be raised is: can use of more
edge pixels to approximate a surface yield more accurate
estimates of D. Our third method attempts to provide answers
to this question.
The Max-Difference Method
Our first method, which we call the Max-Difference method,
seeks to reduce potential errors introduced by the use of
corner pixels. It seems that in Clarke’s (1986) method, the
corner pixels are chosen simply because of their physical
locations. In other words, the actual DN values of edge pixels
have no effect on the selection of pixels used to construct the
prisms. The Max-Difference method represents a different
approach to construct the prisms by taking into account the
actual DN values of all edge pixels. Specifically, the MaxDifference method begins by searching a pixel on each edge
of a square whose DN has the largest deviation from the central pixel. This procedure is reiterated for a certain number of
geometrically increasing square sizes (e.g., 2, 4, 8, 16 . . . 2n)
specified by the user. The four edge pixels, one on each edge,
that meet this criterion are called max-difference edge pixels
and used to construct the prisms. It should be clear that the
max-difference edge pixels may or may not be corner pixels.
In cases where the max-difference edge pixels are not located
at the corners of a square, use of only corner pixels fails to
capture the details of the surface raised by the max-difference
edge pixels. By first searching the max-difference edge pixels,
the Max-Difference method can avoid this problem and hence
gives a better approximation to the surface.
Computer implementation of the Max-Difference method
is slightly more complex than that of Clarke’s (1986) method
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due to the fact that the max-difference edge pixels can be any
pixels on the edges of a square and, as a result, more base
distances need to be calculated. The term “base distance” is
used in this paper to mean the distance between two pixels
on the base plane of the image (see Figure 1a). Furthermore,
we use Tc to denote the base distance between the central
pixel and an edge pixel and Te, the base distance between two
edge pixels. In Figure 2 ea, eb, ec, and ed are examples of Tc,
and ab, bc, cd, and da are examples of Te. In Clarke’s (1986)
method, the base distances Tc between the center of a square
and each of the corner pixels are the same and equal to
d ( 12/2), and the base distances Te between each pair of the
corner pixels are also the same and equal to , where is step
size (Figure 1b). In other words, in Clarke’s (1986) method,
only two sets of base distances need to be calculated. In our
method, on the other hand, the base distances Tc between the
central pixel and each of the max-difference edge pixels can
be different. This is also true of the base distances Te between
each pair of the max-difference edge pixels. As a result of this,
each of these eight base distances needs to be computed.
The Max-Difference method is implemented in six steps:
Step 1. For the upper edge of a square, find the max-difference
edge pixel (e.g., a in Figure 2) and record its location as
La, La [1,], step size, Using the same procedure,
find the max-difference edge pixels on the other three
edges (e.g., b, c, d in Figure 2) and record their locations
as Lb, Lc, and Ld, respectively, Lb, Lc, and Ld [1,],
step size.
Step 2. Calculate the base distances Tc between the central
pixel and each of the four max-difference edge pixels.
Equation 1 shows an example of how to compute Tc
between e and a in Figure 2:
Tc(ea) 1(La 1 d/2)2 (d/2)2
where La is the location of the max-difference edge
pixel a, and is step size.
Figure 2. Labels of edge pixels used in searching the
max-difference edge pixels (a, b, c, and d) and computing the base distances (Tc) between the central pixel
(e) and the four max-difference edge pixels, ea, eb, ec,
and ed (an example with step size 4).
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
(1)
Step 3. Calculate the base distances Te between each pair of
the max-difference edge pixels. To calculate Te, we first
define two arrays x(k) and y(k), where k [1,4] and
step size, to record the x, y coordinates of the
edge pixels in a raster coordinate system (see Figure 3).
Equations 2-1 through 2-4 are used to compute the x, y
coordinates of the pixels along the upper, right, lower,
and left edges.
upper
x(i) i
y(i) 1
lower
right
(2-1)
x(i d) d 1
y(i d) i
(2-2)
left
x(i 2d) d i 2
x(i 3d) 1
y(i 2d) d 1
(2-3) y(i 3d) d i 2 (2-4)
where: i [1,] and is step size.
Once the x, y coordinates of each edge pixel are
determined, Te between each pair of the max-difference
edge pixels can be calculated using the Pythagorean
Theorem.
Step 4. Calculate the top surface area of the prisms circumscribed by a square. The base distances Tc and Te
obtained in steps 2-3, together with the DN values
of the central pixel, the max-difference edge pixels
and the corner pixels, are used to solve the lengths of
sides of the triangles at the top of the prisms by the
Pythagorean Theorem. The top surface area of each
prism then can be computed using Heron’s formula
(see Clarke, 1986 for the formulas used in this step).
Note that the number of prisms used in the computation may vary, depending on the location of the maxdifference pixels. Adding up the top surface areas
of all prisms gives the top surface area of the prisms
circumscribed by the square.
Figure 3. Labels of edge pixels used in computing the
base distances (Te) between pairs of max-difference edge
pixels, ab, bc, cd, and da, in a raster coordinate system
(an example with step size 4).
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Step 5. Calculate the total area of the top surface by summing
up the top surface areas of the prisms circumscribed by
all squares of a given step size needed to cover the
surface (Figure 4).
Step 6. Compute the fractal dimension (D) value. The total area
of the top surface can be calculated repeatedly for
increasing step sizes by iterating Steps 1 through 5.
To obtain equally-spaced observations on the independent variable for the regression, step size is increased by
powers of 2 (i.e., 2, 4, 8, 16 . . . 2n). As the size of
the squares used to cover the surface increases, more
details in the top surface get lost and, as a result, the
estimated total area of the top surface decreases (Figure 4). Once the total areas of the top surface are
calculated for a certain number of step sizes, plot
log (total top surface area) against log (step size), fit
a least-squares regression line through the data points,
and calculate the slope (b) of the regression line. The
fractal dimension (D) of the surface then is computed
as D 2 b, where b is the slope of the log-log
regression line.
The procedure described above can be illustrated with
the following example (Figure 4). Suppose we use four step
sizes (i.e., 2, 4, 8, and 16) to estimate the D of an image.
We first compute the total area of the top surface for step
size 2, resulting in an estimate of the top surface area
(e.g., 3259.6). We repeat this calculation for step sizes 4,
8, and 16, and obtain another three sets of estimated area
of the top surface (e.g., 3196.1, 3158.1, and 3110.2). Plot
log (3259.6) versus log (2), log (3196.1) versus log (4), and
so forth; fit a least-squares regression line through the four
data points and calculate the slope (b) of the regression
line (e.g., b 0.0296). The fractal dimension (D) of the
image then is computed as D 2 b 2 (0.0296)
2.0296.
The Mean-Difference Method
Our second method, which we call the Mean-Difference
method, is similar to the Max-Difference method in that, in
choosing the pixels used to construct the prisms, it takes
into account the actual DN values of all edge pixels. The
difference is in the criterion used. Specifically, the MeanDifference method uses a pixel on each edge of a square
whose DN deviation from the central pixel is closest to the
mean of DN deviations from the central pixel to all pixels
on the same edge. The pixels that meet this criterion are
called mean-difference edge pixels and used to construct
the prisms.
Similarly to the Max-Difference method, computer
implementation of the Mean-Difference method also consists
of six steps. The procedures and equations used in Steps 2
through 6 are the same as in the Max-Difference method
described above. The only difference is in the first step.
Specifically, the first step in the Mean-Difference method
involves searching a mean-difference pixel on each edge of a
square using the following procedure.
For each edge of a square (e.g., the upper edge), find
the mean-difference edge pixel (e.g., a in Figure 2) using
Equations 3-1 through 3-3 and record its location as La, La
[1,], step size.
h(i) ABS [h(e) h(i)]
(3-1)
h MEAN [h(i)]
(3-2)
MeanPixel MIN {ABS[ h h(i)]}
(3-3)
where: i ith pixel on an edge, i [1,], is step size; h(i)
absolute DN difference between the central pixel and ith
edge pixel; h(e) DN of the central pixel; h(i) DN of ith
edge pixel; h mean of DN deviations from the central
pixel to all pixels on the same edge; ABS, MEAN, MIN
the absolute, mean, and minimum function, respectively;
and MeanPixel the mean-difference pixel.
Using the same procedure, finding the mean-difference
edge pixels on the other three edges (e.g., b, c, and d in
Figure 2) and recording their locations as Lb, Lc, and Ld,
respectively, Lb, Lc, and Ld [1,], step size.
Figure 4. Illustration of the reiteration procedure to
establish the relationship between the top surface area
of the prisms and the step size (or square size) used in
the triangular prism method: (a) step size 2, top
surface area (64 squares); (b) step size 4, top
surface area (16 squares); (c) step size 8, top
surface area (4 squares); (d) step size 16, top
surface area (1 square); (e) fitting a least-squares
regression line to derive fractal dimension (D).
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April 2006
The Eight-Pixel Method
Our third method, which we call the Eight-Pixel method,
is inspired by the question whether use of more pixels
can better approximate an image surface. The idea is that,
except for step size 1, there are more than four pixels
on the edges of a square available to represent the surface.
More importantly, as the size of the square used to cover the
surface increases, it may well be the case that the geometry
of the top surface circumscribed by the square becomes
increasingly “complicated,” and use of only four pixels may
not represent the surface so well. As such, it seems reasonable to argue that using more pixels to represent a surface
should be able to capture more details of the geometry of
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used. Several researchers (e.g., Tate, 1998; Lam et al., 2002)
have suggested that a good strategy for evaluating fractal
computation methods is to test them using synthetic data
with known generated D values. In this approach, simulated data can serve as a benchmark for assessing the performance of the computation methods used. A comparison between computed and generated D values provides
an indication of the performance of the methods being
examined.
Fractional Brownian motion (FBM) is a widely used
method of generating fractal surfaces with known D values.
The variogram of an FBM surface has the form (Goodchild,
1980; Goodchild and Mark, 1987):
E[z(x) z(x d)]2 k(|d|)2H
Figure 5. Top view of the eight edge pixels (a, b, c, d,
e, f, g, and h) and the central pixel (o) used in the
Eight-Pixel method (an example with step size 4).
the surface. This is particularly true when there is sharp
gray-level variation in the neighboring pixels. Based on this
line of reasoning, the Eight-Pixel method attempts to better
approximate a surface by using eight edge pixels. The eight
pixels used to construct the prisms are the four corner pixels
as in Clarke’s (1986) method, and four middle pixels on the
edges of a square. A middle pixel is defined as a pixel that is
halfway between a pair of corner pixels (Figure 5).
The computation of the top surface area of the eight
prisms in the Eight-Pixel method is similar to the procedures
described in Clarke (1986). In terms of computer implementation, three differences exist between these two methods.
•
•
•
First, since the Eight-Pixel method uses eight edge pixels,
two sets of base distances between the central pixel and
the edge pixels (Tc) need to be calculated. The first is
the base distances between the central pixel and each
of the corner pixels (e.g., oa, oc, oe, and og in Figure 5).
This set of base distances are calculated in the same way
as in Clarke’s (1986) method, and they are all equal to
d ( 12/2), where is step size. Another set of base distances needed to be calculated is the distances between
the central pixel and each of the four middle pixels
(e.g., ob, od, of, and oh in Figure 5), which have the same
value of 0.5.
Second, in the Eight-Pixel method, the base distances
between two neighboring edge pixels (Te) (e.g., ab, bc, cd,
and de in Figure 5) are the same and equal to 0.5, whereas
this value is in Clarke’s (1986) method.
Third, eight triangles are used in the Eight-Pixel method to
compute the top surface area of the prisms, whereas four
triangles are used in Clarke’s (1986) method.
(4)
Where E[ ] denotes the statistical expectation, z(x) is the
elevation of the surface at coordinates denoted by the vector x,
d is a displacement vector, k is a constant, |d| is the magnitude (length) of the displacement vector, and H is a parameter
in the range 0 to 1. The fractal dimensions (D) of simulated
surfaces are equal to D 3 H (Goodchild, 1980).
The FBM surfaces used in this study were generated
using the shear displacement method (Saupe, 1988). Detailed
description of this method can be found in Goodchild (1980)
and Lam and De Cola (1993). Since synthetic surfaces are
realizations of a stochastic process, it is necessary to use
multiple sample data sets at each D value to adequately
compare the performance of different methods of computing
fractal dimensions. In this study, five samples of synthetic
surfaces were generated for each H level (or each D value)
from 0.1 through 0.9 in steps of 0.1 (i.e., H 0.1, 0.2, . . .
0.9 or D 2.9, 2.8, . . . 2.1). In other words, a total of
45 synthetic surfaces were used in this study. Following
the approach presented in Tate (1998) and Lam et al. (2002),
an identical seed value of the random number generator
was used to generate each set of the surfaces with varying
H. For each surface generated, a total of 3,000 fault lines
were applied. Each of the synthetic surfaces was generated using a 513 513 grid. The generated surfaces were
stretched to 8-bit images with DN ranging from 0 to 255.
Three sample surfaces and their corresponding images
representing generated D 2.1, 2.5, and 2.9 are displayed
in Figure 6.
To demonstrate the utility of fractal techniques in the
analysis of real remote sensing images, we also tested the
four computation methods on 72 real images. The real
images used for this purpose were extracted from a digital
aerial photograph of Carbondale, Illinois acquired on 25
January 2001. The digital aerial photograph has a spatial
resolution of 0.9843 ft 0.9843 ft and was acquired in three
visible bands: red, green, and blue. Six land-use/land-cover
types with different textural appearance, i.e., crop field,
forest, parking lot, pasture, residential area, and lake surface,
were selected for this analysis. For each of the six landuse/land-cover types, 12 samples of 129 by 129 pixels in the
red band were used in the computation. The 72 sample
images used in this analysis are displayed in Figure 7. Note
that each column in Figure 7 contains a different sample
of each land-use/land-cover type (labeled crop field, forest,
parking lot, pasture, residential, and lake surface) while each
row consists of 12 different samples (labeled S1 to S12) of
the same land-use/land-cover type.
Test Data
The estimates of D computed from real data such as DEMs
and remote sensing images are of unknown accuracy. As
such, computed D values of real data may not explain
precisely the effectiveness of the computation methods
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Results and Discussion
The input parameters used in the implementation of the
proposed methods and Clarke’s (1986) method are specified
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Figure 6. Samples of synthetic surfaces and their corresponding images representing
generated D 2.1 (a), 2.5 (b), and 2.9 (c). The upper-left corner of the images (right
column) corresponds to the origin (0,0) of the synthetic surfaces (left column).
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Figure 7. Samples of real images: each column contains a different sample of each land-use/land-cover
type (labeled crop field, forest, parking lot, pasture, residential, and lake surface); each row consists of
12 different samples (labeled S1 to S12) of the same land-use/land-cover type.
as follows. Because the proposed methods use actual DN of
the central pixel, the starting step size used in these methods was 2, followed by step sizes of 4, 8, 16 . . . , and up
to 512. In other words, nine step sizes were used in the
proposed methods. Clarke’s (1986) method does not require
the use of actual DN of the central pixel. Therefore, the
starting step size used in this method was 1 and the number
of step sizes used was ten. To compute the D values of the
real images, seven step sizes were used in the proposed
methods and eight in Clarke’s (1986) method. Following the
work of Lam et al. (2002), all the methods tested in this
study, including Clarke’s (1986) method, used step size ()
instead of step size squared (2) in the regression to derive
the slope of the regression line. As such, the term “Clarke’s
method” used in the following text should be interpreted as
a modified version of Clarke’s 1986 method.
For each generated D value, the three proposed methods
and Clarke’s (1986) method were used to compute five
estimates of D using the synthetic images described above.
The mean of the computed D values obtained by each of the
four methods at each generated D level then was calculated
and the results are presented in Figure 8. The deviations of
the computed D values from generated D values at each D
level are plotted in Figure 9. The standard deviation and
root mean squared error (RMSE) of the computed D values
are reported in Table 1. RMSE were calculated using the
n
following formula:
RMSE
e a [Dest(i) Dgen(i)]2 fnn, where
B i1
Dest is computed D, Dgen is generated D, and n is the number
of synthetic surfaces used.
It can be seen from Figures 8 and 9 that the four computation methods returned smaller estimated D values than
the generated D values for the synthetic images over the
entire range of fractal dimension. Measured by the extent
of the deviations of the estimated D values from the generated D values, the performance of the four methods varies
with respect to the smoothness or roughness of the images
used. Specifically, the four methods produced D values
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Figure 8. Plot of mean computed fractal dimensions
obtained using Clarke’s (1986), Max-Difference, MeanDifference, and Eight-Pixel methods versus generated
fractal dimensions.
closer to those generated ones for images with smooth to
quite “rough” surfaces (e.g., generated D 2.1–2.2 and
2.5–2.7), but they resulted in larger deviations for extremely
“rough” images (e.g., generated D 2.9).
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Figure 9. Deviations of computed fractal dimensions
from generated fractal dimensions resulted from Clarke’s
(1986), Max-Difference, Mean-Difference, and Eight-Pixel
methods.
Another pattern displayed in Figures 8 and 9 is that,
in terms of the deviations of computed D values from generated D values, the three proposed methods appear to
generally perform better than Clarke’s (1986) method. This
also is reflected in the generally smaller RMSE yielded by the
three proposed methods compared with those generated by
Clarke’s (1986) method (Table 1). Furthermore, the results
from the test for equality of means show that the differences
between the mean computed D values obtained by the three
proposed methods and those obtained by Clarke’s (1986)
method are all statistically significant where generated D
2.4 (see Table 1). Among the three proposed method, the
Max-Difference method returned improved estimates of D
over the entire range of fractal dimension and the improvements were statistically significant. This result seems to
suggest that use of max-difference edge pixels appears able
to capture more details of the geometry of image surfaces.
TABLE 1.
COMPARISON
Generated
D
Clarke
stdv
OF THE
PERFORMANCE
2.1
0.0019
0.0850
Max-Difference
stdv
0.0033
0.0751
RMSE
delta mean 0.0099***
t
(6.087)
Mean-Difference stdv
0.0018
RMSE
0.0789
delta mean 0.0060***
t
(4.979)
Eight-Pixel
stdv
0.0020
0.0788
RMSE
delta mean 0.0061***
t
(5.049)
RMSE
OF
The improved results achieved by the Mean-Difference
method for images with complex textures suggest that use of
mean-difference edge pixels also appears to be a sound
approach to approximate image surfaces.
The Eight-Pixel method was outperformed by the MaxDifference method over the range of D 2.1–2.5. Nevertheless, compared to Clarke’s (1986) method, the Eight-Pixel
method also delivered improved estimates of D and the
improvements were statistically significant for relatively
“rough” images with generated D 2.4. These results seem
to suggest that the strategy of using more edge pixels to
approximate an image surface appears generally effective.
This seems especially true of the images with greater textural complexity. This finding makes sense because using
more pixels that lie on the image surface should be able to
capture more details of the geometry of the surface.
Another factor that may have contributed to the improved
accuracies of computed D values using the proposed methods is their use of actual DN values of the central pixel.
The advantage of using actual DN values of the central pixel
is that it ensures that all the pixels used to construct the
prisms lie on the image surface being analyzed and therefore
can better approximate the surface. A disadvantage of this
approach is the loss of one data point for the generation of
the regression line.
The results from the analyses of the 72 real images are
shown in Figure 10. Three observations can be made here.
First, the three proposed methods and Clarke’s (1986) method
appear generally effective in differentiating the six landuse/land-cover types examined in this study. The four
methods did a particularly good job in distinguishing those
land-cover features with apparently different textures such as
parking lot, residential area, lake surface, forest, and crop
field. This is reflected in the relatively large “distances”
between the computed D values of these land-use/land-cover
types. Second, for the same data set, that is, the same image
samples of a land-use/land-cover type, the computed D values
obtained by the four methods vary. Take the land-use type
“residential area” for example. The mean computed D values
for the “residential” image data set obtained using Clarke’s
(1986), Max-Difference, Mean-Difference, and Eight-Pixel
methods are 2.506, 2.497, 2.566, and 2.494, respectively.
CLARKE’ (1986), MAX-DIFFERENCE, MEAN-DIFFERENCE,
USING THE SYNTHETIC SURFACES
2.2
2.3
2.4
0.0056
0.1693
0.0076
0.1599
0.0095*
(2.208)
0.0059
0.1624
0.0069*
(1.887)
0.0059
0.1626
0.0067
(1.845)
0.0155
0.2082
0.0177
0.1889
0.0196*
(1.873)
0.0155
0.1945
0.0137
(1.392)
0.0170
0.1946
0.0138
(1.345)
0.0230
0.1994
0.0230
0.1614
0.0382**
(2.638)
0.0236
0.1713
0.0284*
(1.883)
0.0249
0.1704
0.0294*
(1.940)
2.5
2.6
2.7
AND
EIGHT-PIXEL METHODS
2.8
2.9
0.0284
0.0251
0.0131
0.0076
0.0086
0.1842
0.1767
0.1862
0.2344
0.3150
0.0223
0.0171
0.0071
0.0103
0.0063
0.1310
0.1121
0.1112
0.1427
0.2142
0.0529** 0.0643***
0.0749***
0.0919***
0.1008***
(3.287)
(4.749)
(11.209)
(16.386)
(21.248)
0.0286
0.0250
0.0136
0.0089
0.0092
0.1396
0.1155
0.1096
0.1454
0.2191
0.0453** 0.0621***
0.0769***
0.0890***
0.0959***
(2.515)
(3.894)
(9.282)
(17.411)
(17.202)
0.0293
0.0253
0.0137
0.0074
0.0091
0.137
0.1109
0.1033
0.1388
0.2121
0.0480** 0.0668***
0.0833***
0.0956***
0.1030***
(2.631)
(4.188)
(9.816)
(20.088)
(18.364)
All
Images
0.2036
0.1495
0.1541
0.1511
stdv standard deviation of computed D values.
RMSE root mean sqaured error of computed D values.
delta mean (mean computed D values obtained by the proposed method) minus (mean computed D values obtained by Clarke’s method).
t t values from the test for equality of mean computed D values obtained by the proposed method and Clarke’s method.
***p 0.01; **p 0.05; *p 0.1; two tailed tests.
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Figure 10. Computed fractal dimensions of the real
images representing six land-use/land-cover types using
Clarke’s (1986), Max-Difference, Mean-Difference, and
Eight-Pixel methods.
Third, when applied to the images representing relatively
homogeneous land-cover features such as pasture and crop
fields, the four methods returned consistent estimates of D,
reflected in the relatively small variations in the estimated D
values for these land-cover features. When applied to the
images with less homogeneous land-use/land-cover features
such as parking lots, residential areas and, to a lesser extent,
forest, on the other hand, the variations in the estimated D
values for the same land-use/land-cover type appear relatively
large. This result may suggest that the spatial arrangements of
individual objects in these images, such as the orientation, size
and spacing of houses, cars, and trees, may all have had an
effect on the computed D values. The apparently large variations in the computed D values for the images representing
“lake surface” seemed counter-intuitive since “lake surface” is
usually perceived “smooth.” A possible explanation to this
seemingly unusual result is that the aerial photograph used in
this study was taken in January, a time when parts of the lakes
in the study area were frozen. As such, factors such as the size,
depth, and distribution of ice in the lake were highly variable
from one image to another. All these factors appear to have
affected the spatial patterns of tonal variations (i.e., brightness
or darkness) in the images used, which in turn resulted in
large variations in the computed D values.
It should be noted that, because the proposed methods
involve comparing the DN values of all edge pixels and using
more edge pixels, they are computationally more complex
than Clarke’s (1986) method. For example, the computing time
needed to calculate the D value of a real image of 129 129
pixels with complex textures (estimated D 2.61 and 2.69)
was about eight seconds using the Max-difference method,
compared to about two seconds using Clarke’s (1986) method.
The findings of the present study have implications for
the application of fractal techniques to remote sensing image
analysis. First, our results from the analyses of the 72 real
images suggest that fractal techniques appear to be a promising tool for characterizing the textural complexity of remotely
sensed images. The computed D values provide a potentially
useful source of textural information that could be used
to supplement digital image classification problems. Since
the main purpose of the present study was to develop new
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
algorithms and compare the performance of different computation methods based on the triangular prism concept, we
only calculated a single global D value, that is, an average
(or mean) D, for each synthetic and real image. This approach
has been used in several studies focusing on algorithm evaluation, such as in Tate (1998) and Lam et al. (2002). It should
be noted that when fractal dimensions are to be used as
textural information to segment remote sensing images, computing global D values does not serve the purpose. What is
needed in these applications is the “local” D values reflecting
local tonal variations of different land-use/land-cover types.
Such local D values can be computed using local “moving
window” techniques. Although a thorough discussion of
the implementation of the triangular prism method at local
window levels is beyond the scope of this study, we direct
readers interested in this issue to the work presented by
De Jong and Burrough (1995).
Second, our results indicate that the triangular prism
technique, as implemented in our proposed methods and
Clarke’s (1986) method, appears more suitable for computing
the fractal dimension of remote sensing images with smooth
to quite rough surfaces, but it tends to perform poorly on
extremely rough images. This means that, to obtain reliable
results using a fractal technique such as the triangular prism,
one would need some knowledge about the textural complexity of the image to be analyzed. Fortunately, a large number
of other techniques, such as the co-occurrence matrices
(Haralick et al., 1973), local variance (Woodcock and Strahler,
1987), wavelets (Mallat, 1989), and spatial autocorrelation
statistics (Cliff and Ord, 1973), are available for characterizing textural features in digital images. As such, using other
texture analysis techniques to develop a “feel” for the image
data set to be analyzed appears desirable before a fractal
dimension computation method is applied.
Third, our study also shows that even using the same
approach for computing fractal dimension, such as the
triangular prism, different estimation procedures, such as
the four implementations of the triangular prism approach
examined in this study, may yield different computed D
values for the same data set. Studies comparing different
approaches for computing fractal dimension (e.g., the
variogram, triangular prism, and isarithm) have also reported
that variation in computed D values can be introduced by
the choice of general approaches (Tate, 1998; Lam et al.,
2002; Myint, 2003). Taken together, these results seem to
suggest that the choice of computation methods is an
important issue in the application of fractal techniques to
digital image analysis. Researchers need to be aware of the
comparative performance of different computation methods
proposed in the literature and the biases that are associated
with a particular method or a particular implementation
procedure.
Conclusions
How to extract the often complex and erratic textures in digital
images and use such information to improve image classification has been a major research issue for remote sensing
scholars for decades. In this context, fractal geometry appears
appealing because it offers us powerful tools for analyzing
fragmented and irregular forms, which seem characteristic of
textural features in remotely sensed images. There is a large
and growing literature examining the potential of the fractal
model in digital image processing. A review of this literature
suggests that fractal techniques appear capable of capturing
certain aspect of the spatial variations of gray-levels in digital
images that may not be captured by other texture analysis
techniques (Mandelbrot, 1977; Pentland, 1984). As such,
incorporating fractal parameters such as the fractal dimension
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into a digital image processing procedure appears to be a
promising approach to enhancing image analysis results. It
should be noted, however, that the fractal dimension describes
only one aspect (i.e., the shape) of the textures in digital
images. This means that use of fractal dimension alone may
not be sufficient to completely characterize image textures. It
appears that the utility of fractal dimension may be explored
to a fuller extent when it is used in conjunction with other
texture measures and perhaps spectral information as well.
Developing efficient and reliable algorithms for computing fractal dimension is key to fully exploring the potential
of fractal techniques in remote sensing applications. In this
study, we proposed three new alternatives based on Clarke’s
(1986) triangular prism concept to compute the fractal dimension of remotely sensed images. Our results indicate that
in comparison with Clarke’s (1986) method, the proposed
methods appear able to generate improved estimates of fractal dimensions for synthetic images with complex textures.
Results from the analysis of real remote sensing images show
that the proposed methods appear effective in differentiating
different land-use/land-cover types examined in this study.
The performance of the three proposed methods and
Clarke’s (1986) method reported in this study was based on
the analyses of 45 synthetic surfaces and 72 real images.
More thorough evaluation of the proposed methods clearly is
needed to establish the merits and drawbacks of these new
methods in comparison with other existing methods. In this
context, testing the proposed methods on a variety of real
image data and developing efficient algorithms to compute
local D values of real images, analogous to the work of
De Jong and Burrough (1995), are especially desirable because
the utility of any computation methods is ultimately measured by their performance in the analysis of real remote
sensing images. It is our hope that the results presented in
this paper could spark further interest in the evaluation of
various fractal computation methods and, thereby, contribute
to the development of more effective techniques and efficient
algorithms for image texture analysis.
Acknowledgments
I am grateful to the office of Research Development and
Administration (ORDA) at Southern Illinois University
carbondale for providing support for this research. I would
like to thank the three anonymous reviewers for their very
helpful comments on an earlier draft of this paper.
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(Received 02 November 2004; accepted 22 February 2005; revised
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