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Cartography and Geographic Information Science

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An Evaluation of Fractal Methods forCharacterizing Image Complexity

Nina Siu-Ngan Lam , Hong-lie Qiu , Dale A. Quattrochi & Charles W. Emerson

To cite this article: Nina Siu-Ngan Lam , Hong-lie Qiu , Dale A. Quattrochi & Charles W. Emerson(2002) An Evaluation of Fractal Methods for Characterizing Image Complexity, Cartography andGeographic Information Science, 29:1, 25-35, DOI: 10.1559/152304002782064600

To link to this article: https://doi.org/10.1559/152304002782064600

Published online: 14 Mar 2013.

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An Evaluation of Fractal Methods forCharacterizing Image Complexity

Nina Siu-Ngan Lam, Hong-lie Qiu, Dale A.Quattrochi and Charles W. Emerson

ABSTRACT: Previously, we developed an integrated software package called ICAMS (ImageCharactel-ization and Modeling System) to provide specialized spatial analytical functions for inter-preting remote sensing data. This paper evaluates three fractal dimension measurement methods thathave been implemented in ICAMS: isarithm, variogram, and a modified version of triangular prism.To provide insights into how the fractal methods compare with conventional spatial techniques inmeasuring landscape complexity, the performance of two spatial autocorrelation methods, Moran's Iand Geary's C, is also evaluated. Results from analyzing 25 simulated surfaces having known fractaldimensions show that both the isarithm and triangular prism methods can accurately measure a rangeof fractal surfaces. The triangular prism method is most accurate at estimating the fractal dimensionof surfaces having higher spatial complexity, but it is sensitive to contrast stretching. The variogrammethod is a comparatively poor estimator for all surfaces, particularly those with high fractal dimen-sions. As with the fractal techniques, spatial autocorrelation techniques have been found to be usefulfor measuring complex images, but not images with low dimensionality. Fractal measurement methods,as well as spatial autocorrelation techniques, can be applied directly to unclassified images and couldserve as a tool for change detection and data mining.

KEYWORDS: Fractal measurement, spatial autocorrelation, simulated surfaces, data mining

Introduction

The rapid increase in digital data volumesfrom new sensors, such as NASA'sEarthObserving System (EOS), Landsat-7, and

commercial satellites, raises a critical problem-howcan such an enormous amount of data be handledand analyzed efficiently? Efficient handling andanalysis of large spatial data sets is central toresearch that utilizes various layers of informationin different time periods. For example, long-termglobal change investigations require that new high-resolution data be combined with older, lower res-olution images. Analyses of natural disasters suchas hurricanes, fires, and earthquakes typically usetime series of satellite imagery for rapid change

Nina Lam is R. J. Russell Professor of Geography, LouisianaState University, Baton Rouge, LA 70803, USA. E-mail:< nlam@lsu.edu>. Hong-lie Diu is assistant professor ofGeography, California State University, Los Angeles. CA 90032,USA. E-mail: <hqiu@calstatela.edu>. Dale Ouattrochi isGeographer/Senior Research Scientist at NASA Global Hydrologyand Climate Center, Marshall Space Flight Center, AL 35812, USA.E-mail: <dale.quattrochi@msfc.nasa.gov>. Charles Emersonis Assistant Professor of geography, Western Michigan University,Kalamazoo, MI 49008, USA. E-mail: <charles.emerson@wmich

.edu>.

detection and continuous monitoring. Advancesin this type of large-scale spatial research requirenot only high-quality data sets, but also reliabletools to handle and analyze these data sets.

In a previous project, we developed a softwaremodule called ICAMS (Image Characterizationand Modeling System) to address the above needby focusing on the development of innovative spa-tial analytical tools and then bundling the toolstogether in a module so that it can be accessedby the broader research community. Detaileddescriptions of the theoretical background and thepractical need for developing ICAMS, as well as itssystem design and functionality, can be found inQuattrochi, et al. (1997) and Lam, et al. (1998). Inbrief, ICAMSis a software module designed to runon Intergraph-MGE and Arc/Info platforms thatprovides specialized spatial analytical functions forcharacterizing remote-sensing images. The mainfunctions of ICAMSare fractal analysis, variogramanalysis, spatial autocorrelation analysis, textureanalysis, land/water and vegetated/non-vegetatedboundary delineation, temperature calculation,and scale analysis.

This paper focuses on the use of the fractalmodule in ICAMS. Specifically, we evaluate theperformance of three fractal surface measure-ment algorithms that have been identified in the

Cartography and Geographic Information Science, 10l. 29, No.1, 2002, jJp.25-35

where:L = length of the curve;8 = step size;B = slope of the regression; andK = a constant.From the above equations, D is a function of

the regression slope B. The steeper the negativeslope (8 is a negative value), the higher the fractaldimension. The D value of a surface can be esti-mated in a similar fashion and is discussed in thenext section.

Applications of fractals generally fall into twogroups. The first uses the fractal model to simu-late real-world objects for both analytical anddisplay purposes. Fractals are commonly used incomputer graphics because they can simulate life-like landscapes, cities, or objects for video garnes,movies, and a variety of virtual reality applications(Peitgen and Saupe 1988; Batty and Longley 1994).Moreover, simulated fractal surfaces (fractionalBrownian motion (iBm) surfaces) or curves areregarded as ideal theoretical test data sets for test-ing a number of methods and models (Goodchild1980; Goodchild and Mark 1987). In cartographyand GIS, for example, fractal surfaces have been

fractal geometry, the fractal dimension (D), is a non-integer value that, in Mandelbrot's (1983) originaldefinition for fractals, exceeds the Euclidean topo-logical dimension. As the form of a point pattern,a line, or an area feature growsmore geometricallycomplex, the fractal dimension increases. Thefractal dimension of a point pattern can be anyvalue between zero and one, a curve between oneand two, and a surface between two and three.Increasing the geometrical complexity of a per-fectly flat two-dimensional surface (D=2.0) so thatthe surface begins to fill a volume, results in Dvalues approaching 3.0.

Self-similarityis the foundation for fractal analy-sis (Mandelbrot 1983),and is defined as a propertyof a curve or surface where each part is indistin-guishable from the whole, or where the form ofthe curve or surface is invariant with respect toscale. The degree of self-similarity,expressed as aself-similarity ratio, is used to define the theoreti-cal fractal dimension. In practice, a common wayto estimate the D value of a curve (e.g., a coastline)is to measure the length of the curve using vari-ous step sizes. The more irregular the curve, thegreater the increase in length as step size increases.D can be estimated by the following equations:

literature and implemented in ICAMS: isarithm,variogram, and triangular prism. The purposeof this study is to provide benchmark informationon the applicability and reliability of these fractalmethods for characterizing landscape complexityusing remote sensing imagery.

Fundamental research on the applicability andreliability of new spatial analytical techniques,such as fractals, is necessary. Although the fractaltechnique has been applied extensively, its useas a spatial technique for characterizing remotesensing images is rather uncommon and needsto be evaluated more thoroughly. In the past, itwas difficult to apply the fractal technique becausefractal algorithms were scattered throughout theliterature of diverse disciplines such as meteorol-ogy (Lovejoy and Schertzer 1985), astronomy(Barbanis et a1. 1999), and materials science (Luand Hellawell 1995), and they were not necessar-ily designed to work with remote sensing images.With ICAMS and its fractal module, where thethree m~or surface measurement methods iden-tified in the literature (isarithm, variogram, andtriangular prism) are modified and implemented,fractal measurement has become easier.

In this study, fractional Brownian motion (fBm)surfaces with known fractal dimensions are gen-erated to evaluate the performance of the threefractal surface measurement methods. A compari-son beuveen known and computed fractal dimen-sions provides an assessment of the reliabilityand effectiveness of the three fractal methods forcharacterizing and measuring landscape patterns.Furthermore, to provide insight into how the frac-tal methods compare with conventional spatialtechniques in measuring landscape complexity,this study also evaluates the performance of twospatial autocorrelation methods: Moran's I andGeary's C. Finally, since fractals are of particularuse in determining the effects of changes in pixelresolution as a result of image processing, a sen-sitivity analysis of the response of the fractal andspatial autocorrelation indices is conducted.

An Overview of FractalsFractal techniques were developed because mostspatial patterns of nature, such as curves and sur-faces, are so irregular and fragmented that classi-cal (Euclidean) geometry fails to provide tools forthe analysis of their forms. In Euclidean geometry,a point has an integer topological dimension ofzero, a line is one-dimensional, an area has twodimensions, and a volume three dimensions. In

Log L = K + B Log 8

D=l-B

(1)

(2)

26 CartograPhy and GeograPhic Information Science

used as test data sets to examine the performanceof various spatial interpolation methods (Lam1980) and the efficiency of a quad tree data struc-ture (Mark and Lauzon 1984).

The second group of applications utilizes thefractal dimension as an index for describing thecomplexity of cun'es and surfaces. Goodchild(1980) demonstrated that fractal dimension couldbe used to predict the effects of cartographic gen-eralization and spatial sampling, a result whichmay assist in determining the optimum resolutionof pixels and polygons used in remote sensingand GIS studies. Various applications of fractals ingeography and the geosciences, such as character-izing topography (Goodchild and Mark 1987) andurban landscapes (Batty and Longley 1994), havebeen documented in the literature. In particular,fractal analysis has been suggested as a useful tech-nique for characterizing remote sensing images aswell as identifying the effects of scale changes onthe properties of images (De Cola 1989, 1993;Lam 1990; Lam and Quattrochi 1992; Emersonet al. 1999).

While the fractal model is a fascinating tool forsimulation, its use as a technique for measuringand characterizing spatial phenomena has raisedcriticisms (Goodchild and Mark 1987; Lam 1990).At the theoretical level, the self-similarity propertyunderlying the fractal model assumes that theform or pattern of the spatial phenomena remainsunchanged throughout all scales, which furtherimplies that one cannot infer the scale of thespatial phenomena from its form or pattern. Thishas been considered unacceptable in principle bya number of researchers. Empirical studies haveshown that most spatial phenomena are not purefractals with a constant D, but instead D variesacross a range of scales. Rather than using D inthe strict sense as defined by Mandelbrot (1983),many researchers nowadays realize that strict self-similarity is rare in natural phenomena, and thatself-similarity must be estimated statistically andat certain scale ranges (Milne 1991). These find-ings, however, can be used positively. Lam andQuattrochi (1992) suggested that information onthe changes of D with scale could be used to sum-marize the scale changes of the phenomena andto interpret their underlying processes at specificranges.

At the technical level, a major impediment toapplying fractal measurements is that there arevery few algorithms readily available to research-ers for experimentation. For those who can accessthe fractal algorithms, the frustration is that theresults from applying differing algorithms often

Vbl. 29, No.1

contradict each other (Tate 1998). A thoroughevaluation of the various fractal measurementtechniques, such as the three fractal techniquesmentioned above, is necessary before they can beused to reliably characterize and compare changesin landscapes. Integrated software packages suchas ICAMS make it easier to carry out the evalua-tion tasks.

In a broader sense, the fractal technique maybe considered a textural measure for measuringimage complexity. As with other texture-basedanalytical techniques such as co-occurrence matri-ces (Haralick et al. 1973), localvariance (Woodcockand Strahler 1987), wavelets (Mallat 1989), andspatial autocorrelation statistics (Cliff and Ord1973), fractals have shown great potential in char-acterizing landscape patterns for global environ-mental studies. Unlike many spatial indices used inlandscape ecology such as contagion, dominance,and interspersion, the three fractal measurementmethods and the spatial autocorrelation statisticsimplemented in ICAMS can be applied directlyto unclassified images. This property makes thempotentially useful tools for summarizing the spatialcharacteristics of the image (i.e., metadata repre-sentation), data mining, and change detectionwithout the need for prior image classification.Ultimately, these methods can be used for environ-mental assessment and monitoring. A thoroughevaluation and comparison of these methods isneeded to identify their pros and cons in charac-terizing remote-sensing images.

Methods

Fractional Brownian SurfacesTo provide a benchmark for evaluating the threefractal surface measurement methods and theindices of spatial autocorrelation (Moran's I andGeary's C), fractional Brownian motion (fEm)surfaces with varying degrees of complexity (i.e.,fractal dimension) were generated using the sheardisplacement method (Peitgen and Saupe 1988;Goodchild 1980; Lam and De Cola 1993; Tate1998). The method starts with a surface of zeroaltitude represented by a matrix of square grids. Asuccession of random lines or cuts across the sur-face is then generated, and the surface is displacedvertically along each random line to form a cliff.This process is repeated until several cliffs are cre-ated between adjacent sample points. The set ofrandom lines generated has the property that theirpoints of intersection form a Poisson point process,while the angles of intersection are distributed

27

Dti + 1.The addition of" 1" isneeded to extendfr~~ma one-dimensional curve to a two-dimen-sional surface measurement. Analogous to thewalking-divider method for measuring lines, thealgorithm implemented in ICAMS evolved frommethods proposed in Goodchild (1980), Shelberget al. (1983), and Lam and De Cola (1993). Inorder to use the isarithm method in ICAMS, adata matrix of a given number of rows and col-umns must be specified, with the followingparam-eter input by the user: the number of step sizesto be used to measure the length of the isarithm;the isarithm interval (to determine which isarithmis going to be measured); and the direction fromwhich the operation proceeds (either row,column,or both).

For each isarithm value and step size, thealgorithm classifies pixels below and above theisarithm value as white and black, respectively. Itthen compares each neighboring pixel along therows or columns and determines if the pairs areboth black or both white. If they are of differentcolors, it implies that an isarithm lies between twoneighboring pixels, which are then recorded asboundary pixels. The length of each isarithm lineis approximated by the total number of boundarypixels. It is possible that for a given step size thereare no boundary pixels. In this case, the missingisarithm line is excluded from the analysis. The

Figure 1. Three simulated surfaces and corresponding imagesfrom top to bottom, 0 = 2.1, 2.5, and 2.9.

(3)E[z. - Z .}2 = Idl 2/1l (rl+J

Fractal Measurement Methods

uniformly between 0 and 2rc. The heights ofthe cliffs are controlled by the parameter H,which determines how the variance betweentwo points relates to the separation distance. Inother words, H describes the persistence of thesurface and has values between 0 and 1. H isdefined as:

where E[Zi - Z(d+il is the expected variancebetween two points having a distance d. Thefractal dimensions of the simulated surfacesare equal to D = 3 - H (Mandelbrot 1975;Goodchild 1980; Lam 1980).

Figure 1 shows the effect of the H param-eter. A surface with D = 2.1 (H = 0.9) is quitesmooth with similar Z values occurring in adja-cent areas. Increasing D to 2.5 results in a lesssmooth, less uniform surface. The surface withD = 2.9 (H = 0.1) is quite rough, with high andlow values occurring at closely spaced intervals.The FORTRAN code for generating these frac-tal surfaces can be found in Lam and De Cola(1993).

To test which method (both fractals and spa-tial autocorrelation) measures image complexitymore accurately, five 512 x 512 surfaces for eachH level (0.1, 0.3, 0.5, 0.7, and 0.9) were generatedusing 3000 cuts. For each set of surfaces with vary-ing H, identical seed values were used in order togenerate the same sequence of random cuts and,therefore, visualize more easily the increase ofcomplexity as H decreases. The resulting fileswereconverted to 8-bit images normalized to digitalnumbers from 0 to 255. This generated five sets ofsimulated surfaces with fractal dimensions of 2.1,2.3,2.5,2.7, and 2.9.

The three fractal surface measurement methodsthat have been implemented in ICAMS-theisarithm, variogram, and triangular prismmethods-have been applied to real data and doc-umented in detail in various studies (e.g., Lam andDe Cola 1993; Jaggi et al. 1993; Xia and Clarke1997). Except in a preliminary study (Lam et al.1997), however, they have not been systematicallyevaluated using controlled, synthetic surfaces witha range of spatial complexity. The use of con-trolled surfaces provides a standard for compari-son, thereby illuminating the major characteristicsand differences among the methods.

The isarithm method uses the length of contoursof surface z-values as a means of determining thefractal dimension D of the surface, where D .., =

sUtJaa

28 CartograPhy and GeograPhic Information Science

~(~"U·"fI'···1DIr'IBf" .••rlllllp_1

Se1e4:t. br~olnU ttr.at.&1 IIIIJ.aIIP.n.rian:

rU'l!t lit: \n.u'lll\nlud8\n ••.•9.'l1tJ~::tO ••••. ,loU) ---;

••••if' ~ ~~ 2J7 ••••••.•• ~.'SC'.Lc:d pU.l.nt.: 5U l~.-J!JI01:l r- MGr :f~ •

•••• 11I'•. Cit 'Paupi ::lU 1.!J.J ~lDO

B:l:eakpo'Ulltl: O~ ..!.lIDOIIr_iIlJOIlIlI\.nt.'2: .,0 D~.J ~IDO

Figure 2. Sample output of a fractal analysis using the variogram method. The image on the left is a simulated fractionalBrownian surface of 512x512 pixels with a theoretical 0 = 2.5. The screen on the right shows that the computed 0 is 2.66with a regression fit r2 = 0.86. The parameters used in the calculation were 20 distance groups and a sampling interval ofevery 50th point.

total number of boundary pixels for each stepsize is plotted against step size in log-log form(forming the fractal plot), and a linear regressionis performed. The regression slope b is used todetermine the fractal dimension of the isarithmline, where D = 2-b. The final D of the surface isthe average of D values for those isarithms thathave an R2 greater than or equal to 0.9.

The variogram method uses the variogram func-tion to estimate the fractal dimension. The vario-gram function describes how variance in surfaceheight between data points relates to their spatialdistance; and it is commonly used in many otherapplications such as kriging for characterizing thespatial structure of a surface. The only differencebetween the traditional variogram used in fractalestimation is that in the present approach, distanceand variance are portrayed in double-log form. Toderive the variogram function for a surface, vari-ance for all data pairs that fall into a specifieddistance interval, r(d), is calculated by:

n

y(d)= (1/2n) ~)Zi -Z(d+i}]2i=l (4)

where:n = total number of data pairs that fall in dis-

tance interval d, andz = surface value.

The slope of the linear regression performedbetween these two variables (in double log form)is then used to determine the fractal dimension D,where D = 3 - (b/2). Mark and Aronson (1984) pio-neered the use of the variogram method for fractalmeasurement. Detailed discussion of the methodcan also be found in Lam and De Cola (1993)and Jaggi et al. (1993). In ICAMS, the variogrammethod requires the following input parameters:the number of distance groups for computing thevariance, the sampling interval for determining thenumber of points used in the calculation, and thesampling method (regular or stratified random).Sampling only a subset of points for calculationis necessary, especially for large data sets such asremote sensing imagery, since the computationalintensity increases dramatically with an increasingnumber of data points.

Figure 2 shows a typical output from the vario-gram method in IeAMS on the Arc/Info platform.The window on the left is a simulated fractionalBrownian surface of 512 x 512 pixels with a theo-retical D = 2.5. The screen on the right showsthat the computed D is 2.66 with a regression fitr = 0.86. The parameters used in the calculationwere 20 distance groups and a sampling interval ofevery 50th point.

The triangular prism method (Clarke 1986;Jaggi et al. 1993) constructs triangles by connect-

11J1. 29, No. 1 29

Spatial Autocorrelation MethodsICAMS also contains modules for analyzing thespatial autocorrelation of images. Moran's I andGeary's C, two indices of spatial autocorrelation(Cliff and Ord 1973), have been commonly usedto measure the spatial autocorrelation of polygons,but they have seldom been applied to measureimages. Thus, their performance needs to beevaluated and compared with that of the fractalmethods. Moran's I is calculated from the follow-ing formula:

Unlike the isarithm method, the triangular prismmethod can be affected by the range of z valuesbecause the method compares area (unit squared)with grid cell length (unit). This is an importantproperty that was not pointed out in previous lit-erature. In order to make the fractal dimensions ofvarious surfacescalculatedfrom the triangular prismmethod comparable, normalization of the z valuesis necessary.To demonstrate this effect, a sensitivityanalysisof the three fractal dimension measurementmethods wasperformed, and the effectsof contraststretching on the resultant fractal dimensions aredemonstrated below.

ing the heights or z-values at the four corners of agrid cell to its center, with the center height beingthe average of the pixels at the four corners. Thesetriangular "facets" of the prism are then summedto represent the surface area. In the second step,the algorithm increases the step size from one?ixel to two pixels, thus forming 2 x 2 compositesof four adjacent pixels. The corners of each 2 x 2pixel composite are defined by the values of thefour adjacent 2 x 2 pixels, and the center value istheir average. The areas of the triangular prismsfor all 2x2 composites are then calculated andsummed. The algorithm continues until it reachesto the maximum step size specified by the user. Thelogarithm of the total of all the prism facet areas ateach step is plotted against the logarithm of stepsize (number of pixels), and the fractal dimensionis calculated by performing a linear regression onthe surface areas and step sizes.

The triangular prism method implemented inICAMS is different from the original algorithmdiscussed in Clarke (1986) and Jaggi et al. (1993).We show below that step size (0) instead of step sizesquared (0 2) should be used to regress with theprism areas to derive the correct fractal dimen-sion. Consider the basic equations for definingthe dimension of a fractal curve such as the KochCurve (Mandelbrot 1967; Feder 1988):

N(o) = KOD (5)

n n

nL LWijZiZjI(d) = i j

wI,zi(10)

Rearranging the above equation into logarith-mic form becomes:

where:o = step size;N(o) = number of steps;L(o) = length of the curve; andK = a constant.To extend the curve definition to area, we can

deduce the following expression of fractal dimen-siol'. for an object whose area isA and step size 0:

L(o) = N(o)o = Ko I-D

A(o) = N(o)o 2 = Ko 2-D

Log A = K + (2-D) Logo

(6)

(7)

(8)

where:w = weight at distance d, so thatJij = 1 if point j is within distance d of point i,

otherwise w .. = 0;yz's = deviations (i.e., z.= x - x for variable x);

and 1 I mean

W = the sum of all the weights where i :f- j.Moran's I varies from + 1.0 for perfect positive

autocorrelation (a clumped pattern) to -1.0 forperfect negative autocorrelation (a checkerboardpattern).

Geary's C contiguity ratio, another index ofspatial autocorrelation that is similar to Moran's I,uses the formula:

where:

B = (2-D) is the slope of the regression;K = a constant, and therefore:

(n-I)I,I,Wij( Xi-Xj fC( d)= i j

2WI,Zi2

i

(11)

Hence, the original algorithm was modified andimplemented in ICAMS(Zhao 2001).

D = 2-B (9)with the same terms listed above. Geary's C nor-mally ranges from 0.0 to 3.0, with 0.0 indicatingpositive autocorrelation, 1.0 indicating no auto-correlation, and values greater than 1.0 indicatingnegative autocorrelation.

30 CartograPhy and GeograPhic Information Science

Surface Triangul;:>rFractal Isarithm Prism Variogram Mean Standard Moran's I Geary's C

Dimension D D D Deviation

2.9 2.9989 2.9114 3.0518 132.0440 23.4960 0.1838 0.8172

2.7 2.8525 2.7233 2.9752 127.6700 28.6640 0.8390 0.1610

2.5 2.5192 2.3703 2.7007 123.3600 42.5840 0.9949 0.0051

2.3 2.1661 2.0718 2.1853 128.6760 53.0580 0.9999 0.0001

2.1 2.0424 2.0390 2.0140 129.6000 57.3720 1.0000 0.0000

Table 1. Summary statistics for the simulated surfaces.

Image Resampling and ContrastStretchingImage resampling is used when images are recti-fied, in analyses of scale effects, or when older,low-resolution imagery is combined with newerhigh-resolution images. Bilinear resamplinguses the four closest neigh!:ors to perform a two-dimensional linear interpolation to obtain theoutput cell value (the output cell can be of any sizeor orientation). This averaging process reducesthe dynamic range of the output brightness values,which is frequently overcome by contrast stretch-ing. The effects of this resampling procedure onthe performance of different fractal and spatialautocorrelation methods vary. To investigate theseeffects, a 1024 x 1024 pixel fractional Browniansurface having a known fractal dimension of 2.9was generated. This produced a complex surfacein which the brightness values change rapidlywithin a short distance. Bilinear resampling wasused to reduce the resolution of the image, result-ing in resampled images having 512 x 512, 256 x256, 128x 128, and 64 x 64 pixels. The pixel sizesranged from one pixel width in the original sur-face, to resampled pixels that were two, four, eight,and sixteen pixels wide.

Results and DiscussionTable 1 lists the descriptive statistics for the 512x 512 simulated surfaces, which show that as Dincreases, the standard deviation of the surfacegenerally decreases. Standard deviation does notmeasure the spatial arrangement of the bright anddark pixels, just their relationship to the meanbrightness value. In other words, standard devia-tion, which is a non-spatial measure of variation,bears no relationship with D, a spatial measureof variation. A moderate inverse relationshipbetween fractal dimension and standard devia-tion has also been reported in previous studieswhere real remote-sensing images were measured

l1J1.29, No.1

(Lam et al. 1998). In that study, it was suggestedthat both spatial and non-spatial indices, such asfractal dimension and standard deviation, couldbe used together to form a broad impression ofan image even without viewing it. Therefore, theyshould be considered as part of the metadata (i.e.,key descriptors) for the image. For example, whenan image has a high standard deviation but arelatively lowfractal dimension (such as the D=2.1surface in Figure 1), the surface would most likelyexhibit a spatially homogeneous pattern with adetectable trend. On the contrary, if an image hasa low standard deviation but a high fractal dimen-sion (such as the D=2.9 surface), the surface ismuch more fragmented and spatially varying.

Fractal Measurement MethodsThe isarithm, triangular prism, and variogramfractal measurement methods were used on eachof the 25 simulated surfaces. tor the isarithmmethod, the input parameters were number ofstep sizes = 5, isarithm interval = 10, and direc-tion of operation = both row and column. Theinput parameters for the variogram method were20 distance groups and sampling every 20th pixelfrom both rowsand columns of a regular grid. Theonly input parameter required by the triangularprism method is number of step sizes; as with theisarithm method, it was fixed to five step sizes. Theslope of the regression based on these five stepswas used to compute fractal dimension in the isa-rithm and triangular prism methods.

Figure 3 shows the average fractal dimensionD for each of the five sets of fractional Browniansurfaces. The triangular prism method is thebest estimator for the rougher surfaces with frac-tal dimensions of 2.9 and 2.7, although it is lessaccurate at D = 2.5 and 2.3. For fractal dimen-sions around 2.5, the isarithm method is themost accurate. All the methods underestimatethe fractal dimension for the D = 2.3 and 2.1surfaces. Root mean squared errors (RMSE) foreach set of surfaces by each fractal method were

31

Figure 3. Comparison between mean estimated fractal dimension andtheoretical fractal dimension.

1heoretical Fractal Dimension

computed by taking the square root of theaverage of the sum of squared differencesbetween the estimated and the theoreticalfractal dimensions. The results are shownin Table 2. Both the isarithm and thetriangular prism methods yield similarRMSE (0.1068 and 0.1082), and they aregenerally lower than that of the variogrammethod. The triangular prism method hasthe worst RMSE for D = 2.3 surfaces, butbest RMSE for D = 2.7 and 2.9 surfaces.The isarithm method is generally the bestestimator for the three smoothest surfaces,while the variogram method is a compara-tively poor estimator for all of the surfaces,particularly those with higher fractaldimensions.

3.1

5 2.9'iiic

CI1.§ 2.7omt:i 2~ .5LL.

uiWcrnCI1:2 2.1

1.92.1 2.3 2.5 2.7 2.9

I-+- Isarithm I____Tri.Prism

-+- Variogram

Moran's I and Geary's C

Figure 4. Relationship between theoretical fractal dimension and spatialautocorrelation indices.

0.2

-+- Moran's I___ Geary's C

2.92.72.52.3Theoretical Fractal Dimension

spatial autocorrelation techniques are ineffectiveand do not reflect accurately the complexity of thesimulated surfaces.

Resampling and Contrast StretchingThe triangular prism and isarithm methods formeasuring fractal dimension and Moran's I andGeary's C indices of spatial autocorrelation wereexamined for their sensitivity to resampling andcontrast stretching. The variogram method formeasuring fractal dimensions was not evaluatedin this analysis due to its instability when used onsmall images having relatively few pixels. A spa-tially complex, 1024 x 1024 fractional Browniansurface having a specified H value of 0.1 (D2.9) was used in this analysis. Figure 5 shows that

o2.1

1.2 -

,IllC

~ 0.4::!E

,Ill 0.8r:-IOQ)

C)- 0.6

As expected, there is an inverse relation-ship between Moran's I and Geary's C, asshown in Table 1 and displayed in Figure4. The more complex surfaces with higherfractal dimension are more random intheir spatial configurations and haveMoran's I values close to 0.0 and Geary'sC values close to 1.0 (negatively autocor-related). The smoother surfaces with lowfractal dimensions have Moran's I valuesof 1.0 and Geary's C values of 0.0 (posi-tively autocorrelated).

Unlike the fractal methods, no RMSEwere computed for the spatial autocorrela-tion statistics, as there were no theoreticalspatial autocorrelation values for each setof D surfaces. However, the spatial auto-correlation values computed for each setof surfaces were very similar, with varianceclose to zero; therefore, the mean Moran'sI and Geary's C values could be used to comparewith the theoretical fractal dimension. In compar-ing the spatial autocorrelation statistics with thetheoretical fractal statistics, the results presentedin Figure 4 show that Moran's I (and by extension,Geary's C) could be used to measure the spatialcomplexity for surfaces with D > 2.5. Moran'sI value decreases (Geary's C increases) as Dincreases, with a noticeble drop Uump for Geary'sC) from D = 2.7 to D = 2.9. Although the rangeof Moran's I values is small, recent studies haveshown that these small differences can be used todiscriminate surface features as accurately as thefractal techniques (Myint 2001). For surfaces withlower fractal dimensions, this study shows that the

32 CartograPhy and Geographic Information Science

ConclusionsThe three fractal surface measurements methodsimplemented in ICAMS, including the isarithm,variogram, and modified triangular prism meth-

Surface Fractal TriangularDimension Isarithm Prism Variogram

2.9 0.1011 0.0463 0.15442.7 0.1532 0.0656 0.28182.5 0.0236 0.0845 0.30082.3 0.1364 0.2066 0.15752.1 0.0647 0.0478 0.0868

All Surfaces 0.1068 0.1082 0.2126

the triangular prism method is strongly affectedby reductions in image contrast. Smoothing animage by averaging larger and larger groups ofpixels into a coarser resolution image results in asharp drop off in D from slightly higher than thesimulated value of 2.9 to a D value close to 2.5.Stretching the range of pixel values back to theoriginal 0 to 255 range reduces the drop in mea-sured fractal dimension.

The isarithm method is not as strongly affectedby contrast stretching. In Figure 5, the isarithmmethod overestimates the simulated surface frac-tal dimension of 2.9, regardless of whether thedynamic range is restored by stretching or not.Since the isarithm interval is fixed, the fractaldimension is estimated from fewer isarithms in theunstretched images (when there is a smaller rangeof values) than it would be in an image stretchedto the full dynamic range. l<orboth the triangularprism and isarithm methods, the fractal dimen-sion measurement is relatively stable as the imageis resampled to coarser resolutions, as indicatedby the generally horizontal trend in these plots(except in the case of un stretched triangular prismmeasurements). This is a characteristic of the idealfractal nature of the fractional Brownian surfacesused in this analysis. The coarser versions of theoriginal surface are similar in complexity to theoriginal, particularly if they are normalized to theentire 0 to 255 dynamic range of pixel values. Realimages of the earth depart from this fractal ideal,and often exhibit complex responses to changes inresolution (Emerson et al. 1999).

Figure 6 shows measurements of Moran's I andGeary's C on the same 1024 x 1024 images. Inthis case, contrast stretching has little effect oneither index, although there is a tendency for thetwo indices to converge to a value of 0.5 as manysmaller pixels are averaged into fewer larger ones.

ods, were evaluated using25 simulated surfacesof varying degrees ofcomplexity. Based on theresults from this analysis,we conclude that boththe isarithm and triangu-lar prism methods can beused to accurately ana-

Table 2. Root mean squared error for estimated fractal dimension of the simulated surfaces. Iyze images over a rangeof fractal dimensions,

although the triangular prism method is best forhighly complex surfaces that are characteristic ofremotely sensed imagery. The triangular prismmethod, however, is sensitive to the actual rangesof values. To ensure comparability and accuracyof measurement, the range of values should benormalized before computation by the triangularprism algorithm. The variogram method may notbe appropriate for complex images, as it tends toyield higher fractal dimension estimates than theother methods. The variogram method, however,would be useful for measuring fractal surfaces oflow dimensions. The spatial autocorrelation tech-niques are useful for measuring complex images,but not images with low dimensionality.

These results are useful for research on develop-ing and improving spatial analytical techniquesand our continuous updating and new imple-mentations in ICAMS.These techniques could beused to examine a host of related research ques-tions. For example, fractal measurement could beused to determine if different environmental andecological landscapes and processes (e.g., coast-lines, vegetation boundaries, and wetlands) havecharacteristic fractal dimensions. With accuratemeasurement methods, these fractal techniquescould be used to identify regions of varyingdegrees of complexity, and ultimately be used asa part of metadata or as a tool for data miningor change detection without the need to classifyimages beforehand, thereby increasing the effi-ciency of continuous environmental monitoring. Itis also possible to apply fractal techniques locallyto achieve more accurate feature identificationand change detection. Research is underway toexplore these latest applications. For example, weare currently adding modules to ICAMS for localfractal and wavelet analysis. We also plan to fur-ther evaluate the applicability of these fractal andrelated spatial measurement methods for monitor-ing environmental changes in a number of locali-ties, such as Xinjiang, China, and Atlanta, USA,where field investigations are currently underway.Furthermore, we will utilize the multi-resolution

VOl. 29, No.1 33

Figure 5. Effect of resampling and contrast stretching on computed fractaldimensions.

3.20 -property of different typesof imagery and spatial datato gain a better understand-ing of the impact of scale onthe analysis of environmentalphenomena.ACKNOWLEDGMENTSThis research was supportedby a grant from NASA(Awardnumber: NAGW-4221) andby NASA Summer FacultyFellowships awards to NinaLam and Charles Emerson.Additional support was alsoprovided through a NASAIntelligent Systems researchgrant (NAS-2-37143).Wethank~abushananum CherukuriandWei Zhao, then doctoral gradu-ate assistants, for their techni-cal assistance.

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34

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UJI. 29, No.1 35

XXII FIGInternational

CongressWashington 2002

ACSM/ASPRS Conferenceand Technology Exhibition2002Marriott Wardman Park Hotel.Washington, D.C., USA.April 19~26, 2002

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