Estimating fractal dimension with fractal interpolation function models

$Page 1: Estimating fractal dimension with fractal interpolation function models$
930 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 6, DECEMBER 1997

Estimating Fractal Dimension with FractalInterpolation Function Models

Alan I. Penn,*Member, IEEE, and Murray H. Loew,Senior Member, IEEE

Abstract—Fractal dimension (fd) is a feature which is widelyused to characterize medical images. Previously, researchers haveshown that fd separates important classes of images and providesdistinctive information about texture. We analyze limitationsof two principal methods of estimating fd: box-counting (BC)and power spectrum (PS). BC is ineffective when applied todata-limited, low-resolution images; PS is based on a fractionalBrownian motion (fBm) model—a model which is not universallyapplicable. We also present background information on the use offractal interpolation function (FIF) models to estimate fd of datawhich can be represented in the form of a function. We presenta new method of estimating fd in which multiple FIF models areconstructed. The mean of the fd’s of the FIF models is taken asthe estimate of the fd of the original data. The standard deviationof the fd’s of the FIF models is used as a confidence measure ofthe estimate. We demonstrate how the new method can be used tocharacterize fractal texture of medical images. In a pilot study,we generated plots of curvature values around the perimetersof images of red blood cells from normal and sickle cell subjects.The new method showed improved separation of the image classeswhen compared to BC and PS methods.

Index Terms—Feature analysis, fractal dimension, fractals,texture.

I. INTRODUCTION

A. Fractal Dimension (fd) as a Texture Feature

FRACTAL dimension (fd) has been used to characterizedata texture in a large number of physical and biological

sciences. In medical imaging, researchers have used fd tocharacterize mammographic patterns [1], colorectal polyps[2], trabecular bones [3]–[5], retinal vessels [6], renal arteries[7], Papanicolaou-stained cervical samples [8], and epitheliallesions [9]. While this paper is directed toward the applicationof fd to medical imaging, it draws upon results obtained froma variety of disciplines.

In prior studies a combination of fd and standard texturefeatures has been shown to provide better discrimination thanstandard texture features alone. As an example, in a study ofcomputed tomography (CT) lung scans Uppaluriet al. tested

Manuscript received May 20, 1996; revised May 7, 1997. The AssociateEditor responsible for coordinating the review of this paper and recommendingits publication was C. Roux.Asterisk indicates corresponding author.

*A. I. Penn is with Alan Penn & Associates, 14 Clemson Court, Rockville,MD 20850 USA. He is also with the Department of Mathematics, theDepartment of Electrical Engineering and Computer Sciencs, and the Institutefor Medical Imaging and Image analysis, The George Washington University,Washington, D.C. 20052 USA (e-mail: [email protected]).

M. H. Loew is with the Department of Electrical Engineering and ComputerScience and the Institute for Medical Imaging and Image Analysis, The GeorgeWashington University, Washington, D.C. 20052 USA.

Publisher Item Identifier S 0278-0062(97)09170-2.

17 features, including five first-order texture measures, fiverun-length statistics, six co-occurrence matrix measures andfd. They found fd to be abest featurein each experiment [10].The selection of fd as abest featureimplies that fd separatesthe images into the desired classes and that fd characterizesinformation which is not characterized by the other candidatefeatures [11].

B. Methods of Estimating fd

A variety of procedures have been proposed for estimatingthe fd of images [12]. We will refer to the procedures as fdestimators. Important categories of fd estimators include thefollowing.

• fd-estimators in which pixel size is used to represent thelower limit of the range of scale invariance. Includedare two-dimensional (2-D) and three-dimensional (3-D)box counting (BC), Minkowski sausage, variation, andmorphological estimation [1], [13]–[15].

• fd-estimators which use a fractional Brownian mo-tion (fBm) model. Included are power spectrum [16],maximum-likelihood estimation [17], and gray-scalevariation [18].

• fd-estimators which use a fractal interpolation function(FIF) model. These methods estimate the fd as the knownfd value of the model [19].

We discuss, below, a common implementation of a methodfrom each of these categories. Each fd-estimator category hasinherent limitations which restrict its utility in distinguishingimages.

1) Box Counting (BC):The basic BC paradigm is as fol-lows: The image is overlaid with a sequence of rectangulargrids, and a log-log plot is generated which plots the numberof grid segments that intersect an object of interest versus thesize of the grid. A slope value is assigned to the plot and isused as the estimate of the fd [20], [21]. This basic method hasbeen applied to both 2-D and 3-D representations of medicalimages.

A 2-D curve is constructed by intersecting the 3-D surfacewith a horizontal plane corresponding to a gray-scale level.This dimension-reduction technique is widely used and hasbeen shown in many applications to retain the ability to distin-guish between different surfaces. Cross studied the sensitivityof fd estimate to the choice of threshold level in an analysisof renal angiograms; he concluded that the threshold level didnot significantly affect the results [22]. Dimension reduction isbased on an extension of a 1954 theorem by Marstrand [23].This theorem extension is generally believed to be valid under

0278–0062/97$10.00 1997 IEEE

PENN AND LOEW: ESTIMATING FRACTAL DIMENSION WITH FRACTAL INTERPOLATION FUNCTION MODELS 931

suitable conditions, but fails when the underlying surface isanisotropic or undulating [14].

A 3-D image is constructed as the surface of a set ofrectangular skyscrapers with heights determined by opticaldensity and bases determined by the pixel grid [1]. Since gray-scale levels are not linearly related to optical density [24],the optical density axis is effectively quantized nonuniformly.Since the fd estimate is affected by the resolution, the fdestimate will be affected by the location of the object on theoptical density axis.

The primary causes of error when using BC are resolutionand the method of assigning a representative slope value toa nonlinear log-log plot. Huanget al. varied the resolution ofimages with known fd and estimated fd using the least-squaresfit to the log-log plot using BC [25]. They found that whenthere is a very low image resolution, the fd estimates werevirtually identical, regardless of the true fd values. Specifically,they show the following results of using BC on low-resolutionimages which are generated by sampling a test image with a64 64 grid

Underlined value is true fdbracketed value is fd computed with BC

They also demonstrate that a constant fd value, fd2.47,is computed for all of the following:

Underlined value is true fdgrid size in parentheses

From these results it is clear that resolution substantiallyaffects BC estimates.

Huang et al. also studied the effect of the method ofestimating the slope of a discrete nonlinear log-log plot on thefd value. They found that using a fixed portion of points onthe log-log plot (which they called anaive estimator) did notproduce accurate estimates of the fd [25]. They concluded:“The portion of the curve that provides the best estimate isnot necessarily the most linear portion of the plot. There isno obvious way of choosing the portion of the plot that yieldsthe ‘best’ estimate.”

Dubuc et al. noted that BC estimators are very sensitiveto choice of quantization methods, and that small values ofresolution are equivalent to large values of scale, leading tosevere discontinuities in the log-log plot and yielding lessaccurate results [15].

We cite three examples in which BC has led to contradictoryor flawed estimates of fd.

a) Trabecular Bone:there have been conflicting conclu-sions about the fd of trabecular bones.

1) Cross determined that the fd of trabecular bone was1.0 and, therefore, the bone was not fractal in nature[3].

2) Majumdar, Weinstein, and Prasad determined thattrabecular bone had fd1.56 in normal subjects andfd 1.20 in osteoporotic subjects [4].

3) Ishidaet al. determined that normal trabecula had fd1.435 and abnormal trabecula had fd0.306 [26].

(Note that the Ishida value for abnormal trabecula isbelow the minimum range of fd values for curvesin Euclidean two-dimension and is indicative of aCantor set [20].)

b) Cell nuclei:MacAulay and Palcic used 3-D box countingto analyze Papanicolaou-stained cervical samples andgenerated a log-log plot with two points [8]. Theyexplained their method by noting:

“Since the resolution of a conventional imagingsystem is limited, it is usually only practical toexamine nuclei at two different scales.”

Estimating the fd on the basis of a two-point plot isan extreme case of Huang’snaive estimatordescribedabove.

c) Lung X rays: Levy Vehel studied the classification oflung diseases and used three classes of low-resolutionimages of lungs: normal, chronic disease, and pulmonaryembolism [27]. He found:

“Leaving apart extremal sub-images, we have ob-tained the remarkable result that all slices and allprojections yield fd of 2.3, regardless of the typeof lung.”

Levy Vehel’s finding is explained by Huang’s results,described above.

Chung et al. provided an insightful analysis of problemswhich can arise from BC [28]. They noted that the log-logplot consists of many points at large box sizes, and few points(on the tail of the curve) at small box sizes, with the physicallymeaningful points being those at small box sizes. With arelatively small number of physically meaningful points, linearregression yields a linear fit which reflects the slope of thelarge-box data points. Chunget al. also demonstrated howerroneous linearization could result in a nonfractal structureappearing to be fractal.

2) Power Spectrum (PS):The basic PS paradigm is as fol-lows: a power spectrum is generated and a slope value isassigned to a log-log plot of power versus frequency. If thepower spectrum follows an inverse power law

(1)

then the log-log plot will be a straight line with slope equalto .

This model implicitly assumes that all data sets which obeyan exponential power law are fractals and that the power spec-trum is the unique determinant of the fd. These assumptionsare, in general, not valid. As Osborne and Provenzale noted:“... in general, the sole observation of power-law spectra doesnot allow for immediately inferring the fractal nature of thecorresponding signals and the value of the fractal dimension.”[29]

One reason PS is popular is the ease with which the powerlaw exponent is converted into an estimate of the fd. It isgenerally held that if the power spectrum of an image inEuclidean two-dimension obeys the form of (1), then the fd


can be computed using the following:

fd (2)

For example, see [30]. Equation (2) cannot, however, beindiscriminately applied to any time series or function thatobeys an inverse power law. A justifiable application of (2)requires that there is a random process and random phases.The interested reader is referred to [31] and [32] for analysesof limitations of this formula. Hough suggests that the fd of1.5 which is so common in observations of natural scenes may,in fact, be a limiting value which is obtained by summing setswith different fd’s [32].

The fd estimators which are based on the fBm modelrequire astatistical self-affinity with global scaling[33]. Thisglobal self-affinity with global scaling is violated by iteratedfunction systems (IFS) since the various affine transformations,in general, have different scaling factors. The various affinetransformations produce a nonuniform scaling. IFS attractorsform an important class of fractals which are frequently usedto generate realistic models of biological systems [34].

An example of how PS can generate poor results is providedin [13]. This example is particularly noteworthy since the testimages were generated using midpoint displacement, a well-known method of generating fractals which is, itself, based onthe fBm model.

3) Fractal Interpolation Function (FIF): A third, lesscommon, method of estimating fd uses the fractal interpolationfunction (FIF). The FIF is a form of IFS in which the fractalis the attractor of a Hutchinson operator which is defined bya set of affine transformations [35]. In an IFS, each affinetransformation is defined by six parameters, usually presentedas the entries of a 2 2 matrix and a 2 1 translation vector

Parameters are constrained to the following:have values described in [36]; is the vertical stretchingfactor.

The fd of an FIF is implicitly computed as a function ofusing an equation given in [35]. When the interpolation

points are evenly spaced, the fd is explicitly computed [35]

fd (3)

Strahle proposed the following method for estimating the fdbased on an FIF model: Interpolate between every other point,setting free parameters to ensure that the FIF attractor has thecorrect values at the midpoints [19]. The fd of the originaldata is estimated to be the fd of the FIF attractor.

In Strahle’s FIF model, the intrapixel structure is similarto the interpixel structure. The transformation parameters arechosen to ensure that the curve passes through all midpointsand interpolation points. The vertical scaling parameters areset by the following:

(4)

where are the function (-coordinate) values at thefirst, middle, and last points of the curve in theth interpolationinterval, and are function values at the first, middle,and last points of the entire curve.

Chau and Leu [36] extended Strahle’s method to allow forestimating the fd of a signal with unequally spaced interpo-lation points. Marvasti and Strahle [37] extended Strahle’smethod of estimating fd by allowing multiple points betweeninterpolation points, but still using a single vertical scalingfactor between each pair of adjacent interpolation points.

Strahle’s method, as well as the extensions in [36] and[37], generate well-defined fd estimates since the interpolationalgorithms use a unique vertical stretching factor betweeneach pair of adjacent interpolation points. Limitations of FIFmethods are discussed in [37].

FIF estimators implicitly assume an infinite range of scale-invariance—the same assumption used in fBm-based estima-tors. While this assumption is false in real-world applications,it provides an idealized model for comparative analysis.

In general, the pixel size and the true range of scale-invariance are unrelated. In BC, on the other hand, the range ofscale-invariance is implicitly assumed to be set by the physicalsize of the pixel.

There are two primary differences between fBM-basedestimators and FIF: 1) fBm-based estimators limit the spectrumto an exponential shape, whereas FIF does not have this lim-itation, and 2) fBm-based estimators require global statisticalscaling factors, whereas FIF permits nonuniform scaling.

C. Self Affinity and Sampled GeneralizedKiesswetter (SGK) Curves

Falconer defined a self-affine set as follows [20]: Ifare affine contractions on , the unique compact

invariant set generated by is termed a self-affine set.The corresponding definition of a self-affine plot is: A plotis self-affine if it can be modeled as the compact invariantset (attractor) of a set of affine contractions. Unfortunately,the termself affine setis also used to define a construct inwhich the concept of self-similarity is extended to permit adifferent scaling factor for each coordinate axis [33], [38]. Thedifference between these two definitions is significant; theFalconer definition allows for nonuniform scaling, whereasthe second definition requires a global scaling factor for eachcoordinate axis. We will use the termself-affinein the senseof Falconer.

Dubuc and Elqortobi introduced a generalization of Kiess-wetter curves on as follows [39]: Each function in theclass is bounded and satisfies the following:

where on each interval, , the following holds:

Kiesswetter curves represent an important class of fractals andwere used as test fractals in [15] which defined the variationmethod of computing fd.


A curve which matches a Generalized Kiesswetter curveat can be thought of as being “almost invariant”or “almost self-affine” in the sense of Falconer. We refer to adiscrete function which is defined on the points and whichmatches a Generalized Kiesswetter curve at these points, as aSampled Generalized Kiesswetter (SGK) curve.

D. Representation of Fractal Images by Functions

A variety of methods have been studied for transformingimage data to permit fractal analysis. Examples include an-alyzing data along lines [5] and data along parabolic spirals[40].

There are serious questions about the validity of using suchmethods for arbitrary data sets [14]. Since there may not bea clear relationship between the fd value of a plot derivedfrom an image and the fd value of the image itself, it isnot clear exactly what is being measured. As an example,Samarabanduet al. analyzed fd along the axis of bones(using a morphological estimator) and on the image using thepower spectrum; the former method showed that osteoporosisgenerally lowered the fd, while the latter method showed thatosteoporosis generally raised the fd [30].

In spite of these problems, dimension reduction by analyzingthe fd of a derived plot is commonly believed to preservethe fractal nature, and is frequently used as an experimentalmethod [14].

II. M ETHOD

A. Overview

Suppose Strahle’s method is used on an SGK curve definedon the points . We use the points to be interpola-tion points, allowing multiple data points between adjacentinterpolation points. We refer to the noninterpolation points as“interior points” within interpolation intervals. For each affinetransformation, the stretching factor,, is set by mapping themidpoint of to the midpoint of using (4). (This issimilar to the method of [37], but we follow [19] in using thevalue of the central interior point to determine the stretchingfactor.) Once the stretching factors are set for the midpoints ofthe interpolation intervals, the form of the SGK curve assuresthat the interpolation curve also runs through all interior points.

Continuing with our assumption of an SGK curve, wegeneralize Strahle’s method, using nonmidpoints to determinethe stretching factors. When (4) is generalized to an arbi-trary interior point, the -coordinate of which is a fraction

, between -coordinates of adjacent interpolationpoints, and , we obtain

(5)

If we select an arbitrary interior point from each interpolationinterval, indicated by superscript, we generate an estimateof the fd. By using combinations of interior points (one fromeach interval) we generate a space of fd estimates.

Fig. 1 illustrates how multiple FIF models are generated. InFig. 1, corresponds to interior point and ;

Fig. 1.

corresponds to interior point andare -coordinate values of , respectively;are -coordinate values of , respectively. We use twoaffine transformations from the entire graph to the subgraphs

: The first is fixed by mappings ; the second isfixed by mappings .

Each fd estimate generated in this way is a true fd valueof a fractal interpolation curve which can be discretized to thegiven SGK curve. Thus, each FIF model represents a candidateunderlying model from which the discretized data could havebeen generated.

While arbitrary data plots will not, in general, have the formof an SGK curve, we define below a procedure that will findthose subplots which are approximated well by an SGK curve.Each SGK-like subplot is used to generate a sample space offd estimates, and estimates from the collection of subplots arecombined to form a larger subspace of estimates.

The new method is briefly described as follows: 1) Trans-form source data to an- plot; 2) Decompose the- plotinto SGK-like subplots; 3) Model each SGK-like subplot witha set of FIF curves; 4) Compute the fd for each FIF modelusing (3); 5) Estimate fd from the values generated in step 4).

The subplots are not affine images of the entire plot; ratherthey are self-affine components of the entire plot. Thus thismethod is fundamentally different from algorithms whichattempt to solve the IFS inverse problem [41].

B. Implementation Steps

Step 1) Transform image data into a function: As notedabove, a variety of methods have been proposedfor transforming image data into a function. Inthe pilot study presented in Section III, we selecta representative threshold cutoff image and plotcurvature values versus perimeter index when thecell perimeter is traced in a clockwise direction.

Step 2) Identify a set of subplots which are approximatelyself-affine (SGK-like):We perform a global searchof all contiguous subplots which exceed minimumsize and interpolation criteria. In the pilot study, werequired a minimum of seven interpolation pointsand a minimum of five interior points within eachinterpolation interval.Let be the minimum bounding rectangleof a contiguous subplot. forms a just-touching set of parallelograms. The parallelogram


is illustrated in Fig. 1. Since we are search-ing for SGK-like subplots, we consider only FIF’sfor which the have equal widths. We do,however, permit the number of interior points perinterval to differ from the number of interpolationintervals. For each contiguous subplot we analyzestretching factors computed from combinations ofinterior points, using (5). We then identify the FIFmodel which is minimal distance from the subplotusing mean-square error (MSE) as the measure.We restrict the analysis to subplots for which thisminimum distance is small and is achieved bymultiple FIF models. These qualifying subplots arethe self-affine (SGK-like) curves used in Step 3.The minimum distance and the minimum numberof models that meet the distance requirement areparameters of the implementation.

In the pilot study reported in Section III, the-plots contained fewer than 200 points. For plots ofthis order of magnitude, exhaustive searches of allqualifying subplots can be performed in real timeon a workstation.

Step 3) Generate FIF models which approximate qualifyingsubplots: The following description refers to asingle subplot. The affine transformations of Step 2partition the -axis under the subplot into segmentsof equal width. Let be the range of values forthe subplot; be the range of values for theth segment of the subplot; be the set of points

constituting the subplot; be restricted tovalues in ; be the range of -coordinate valuesin . For each segment index,, we generate a setof affine transformationsand form FIF models by selecting a for each .

consists of the points . Thecorresponding set of points in the subplot consistsof . For eachinterior point has the form specifiedin Section I-B, with the vertical stretching factorsatisfying (5).

The stretching factors generated by thevarious interior points do not have a commonvalue. Since is approximately affinely similarto using the construct of Step 2, there is aconsistency of the vertical stretching factors. Con-sequently, many of the map to a set ofpoints which are near to (using MSE). Ifhas substantial contraction relative to the size ofthe pixel, vertical position is determined primar-ily by the transformation which carries to .The primary effect of the secondary transformationwhich maps to is to control the verticalposition within the pixel quantization range. Foreach , we evaluate MSE between and

, and only use which generate a modelwhich is within a prescribed distance from.

The stretching factor is the ratio of the vari-ation within a segment [numerator of (5)] to the

variation over the entire plot [denominator of (5)].For each interior point in a segment, thedenominator is computed using a point which isin the same position relative to the-limits of thewhole plot as is the point to the -limits ofthe segment plot. Since all segments of a subplotcontain the same number of interior points, allsegments generate the same set of denominators.To generate a larger set of representative stretchingcoefficients, we process multiple subplots of thedata.

Step 4) Determine fd for each FIF model:fd values areevaluated using (3).

Step 5) Generate overall estimate of fd from distribution offd estimates of subplots:The fd of each FIF modelis an approximation of the fd of the subplot. Be-cause of quantization and sampling error, the fd ofany single FIF model is an unreliable estimator ofthe fd of the subplot which is being modeled. Sincethe new method generates multiple FIF models, itgenerates a distribution of fd estimates for eachsubplot. We use the mean of the fd estimates ofthe FIF models as the fd estimate of the subplot.The mean of the subplot fd estimates over the setof qualifying subplots is used as the overall fdestimate.

III. RESULTS

In a pilot study, we attempted to distinguish between roundred blood cells (discocytes) taken from normal subjects (AA)and round red blood cells taken from sickle cell subjects(SS). This pilot study was suggested by a study by Robinsonthat noted: “...changes in discoid cells [may] antedate themorphological changes commonly perceived and may haveutility in predicting clinical course and in the testing ofantisickling agents.” [42]

SS cells contain a polymerization of the hemoglobin whichforces a structured, geometric shape. Since geometric struc-tures have fd 1, we conjecture that SS subjects have alower fd than the AA subjects. Thus, we hypothesize that the fdestimate is a feature which distinguishes SS from AA subjects.We also conjecture that the structure of the polymerizedhemoglobin in SS subjects results in greater consistency in theset of fd estimates which we generate. Thus we hypothesizethat the standard deviation of the fd subplot estimates is asecond feature that distinguishes SS from AA subjects.

While this limited pilot study is designed to demonstratethe new fd estimator and does not formally test the hypothesesthat the features distinguish SS from AA cells, the preliminaryresults are consistent with the above conjectures.

Cells were imaged at a resolution of 0.22m/pixel and 8bits of depth. Diameters of selected cells were 34–44 pixels. Toassure a high degree of roundness and uniformity, we restrictedthe study to cell images which satisfied the following criteria:

1) bounding box is square (i.e., width length);2) FormFactor .92,

whereFormFactor area/perimeter[42];


TABLE I

3) thresholded image at gray-scale level 110 contains aconnected island, spanning at least 1/2 the cell.

We selected ten discocytes from each of two AA sub-jects and two SS subjects and thresholded the images ata suitable intermediate gray-scale level (110). We used thefollowing method to generate the curvature plot for the thresh-olded images: The perimeter pixels were indexed using aboundary-following algorithm, starting with the top-left pixeland proceeding with the interior of the cell to the right. Acurvature value was assigned to each perimeter pixel usinga diffusion-based algorithm proposed by Loew (Iterations

100; Diffusion Parameter .01) [43]. We then plottedcurvature value versus perimeter index to obtain the- plotfor step 1 of the implementation.

For each subplot of the curvature-index plot we computedthe minimum MSE between the subplot and the transformedoriginal plot. Qualifying subplots were those for which aminimum of ten candidate FIF models generated the minimumMSE. Thus, each fd estimate of each subplot was evaluated asthe mean of a minimum of ten samples. The overall fd estimatefor each image was the mean over the qualifying subplots. TheMSE defined a measure of self-affinity for the subplot. In thispilot study we did not weight the subplots according to theMSE; rather, all subplots were weighted equally.

Results of the pilot study are given in Table I. The fourcolumns correspond to the four test subjects. For each image,Table I shows the image size , the computed fd calculatedas the mean of fd values of the subplots, the number ofqualifying subplots, and the standard deviation of the fd valuesof the subplots.

We generate two features for each image—the mean and thestandard deviation of the subplot fd estimates. In this study,however, we are interested in evaluating statistics for a subjector class of subjects, rather than for a single image. Table Ialso shows the mean and standard deviation values for these

Fig. 2.

features over the set of images for each subject (AA1, AA2,SS1, SS2) and over the set of images for each subject-class(AA, SS).

For individual images, the high values indicate a signifi-cant spread in fd estimates derived for the qualifying subplots.When a subject (entries in a column) or subject-class (entriesin first two, or last two columns) is used as the basic unit,there is a much tighter distribution. Table I shows that there isnearly a 1- separation between the mean fd estimates of theAA (1.535) and SS (1.452) subject-classes. Similar results arederived using the standard deviation of subplot fd estimatesas the feature. There is nearly a 1-separation between themean standard deviation values of the AA (.175) and SS (.150)subject classes.

Cells corresponding to the first entry in each column ofTable I (bold) were used as representative cells for the subjectand were analyzed further as described below. Fig. 2 showsthe multi gray-scale image and the thresholded image for therepresentative cells.

Using the thresholded images shown in Fig. 2, we evaluatedthe fd using BC (fd ) and PS (fd ). The value of fdwas computed using the procedure described in [21]. In orderto reduce the effect of partial blocks, we restricted scales toapproximate divisors of the length and height of the plot. Fig. 3shows the log-log plots for each of the representative threshold


Fig. 3.

Fig. 4.

images. The following constant offsets were added to ordinatevalues so the slopes of the curves could be readily compared:AA1 AA2 SS1 SS2 . Note thatthe fd plots can be interpreted to produce arbitrary slopesin the full range of possible values, . Accordingly, fdis ineffective for distinguishing these images.

The value of fd was computed following the pro-cedure described in [16], using fast Fourier transform(FFT) (Quattro Pro). The total number of points aroundthe perimeter of each representative threshold image is:AA1 AA2 SS1 SS2 . Wecomputed the PS based on the first 64 points of each perimeter,suppressed the dc term, and linearly fit points 2–32 in thefrequency domain. Fig. 4 shows the log-log plots of thepower series for both the raw data and the linear fit. Thefd values were computed using (2). Note that all computedfd values were near or above the maximum feasible valueof 2.0. Accordingly, fd is also ineffective for distinguishingthese images.

IV. DISCUSSION

Texture is used to characterize images for both segmentationand discrimination. fd has been found to be useful as a texturemeasurement. BC has been shown to be nonrobust whenapplied to data-limited applications such as low-resolutionimages. The power spectrum is related directly to the fd onlyunder a restrictive set of conditions which requires random

processes, random phases, and statistical self-affinity. Thus,there is a need for a robust feature which characterizes thefractal quality of low-resolution medical images.

Our proposed method generates fd estimates of small datasets with limited self-affinity. We treat the measurement of fdas an estimation problem, using samples from an underlyingdistribution. Each sample is generated from a FIF which,upon discretization, models a subplot of the data. We restrictthe analysis to the most self-affine (SGK-like) subplots andgenerate multiple samples for each selected subplot. We usemultiple subplots to generate a large set of samples. Thenumber of samples and the MSE between the fractal modelsand the subplots are measures of the reliability of the fdestimate for each subplot.

The standard deviation of the fd values of the modelsprovides a measure of the precision of the results. Thisapproach gives an additional tool to the investigator to modeldata-limited medical images. Previously, when investigatorsused small data sets, they had no way to evaluate the precisionof their results. Our new method characterizes the uncertaintyand provides researchers with a measure of confidence of thederived estimate of fd. The standard deviation also has utilityas a distinguishing feature.

In our pilot study, we demonstrate how the proposed methodof estimating fd is applied to the analysis of round red bloodcells. We show that our new fd estimator, defined as themean value over a sample space, is successful at separatingrepresentative low-resolution cell images for which BC andPS methods produce unreliable results. The large standarddeviations for single images, shown in Table I, demonstratethat fd estimates which are generated using a single FIF model(i.e., a point in our sample space) are also unreliable.

This study is intended to demonstrate the new method; wemake no claim as to the clinical or diagnostic value of the classseparation which is achieved. Additional research is needed inthis and other applications in which improved estimation of fdmay lead to clinical and diagnostic advances.

A major question still remains over the justification of mea-suring variation along a threshold boundary as a representationof variation on a surface. It is quite possible that while there isoverlap of information content between the two measurements,there may also be some independence between these features.Thus it is possible that both a surface estimate of the fd anda threshold perimeter estimate of the fd could find utility inimage discrimination.

One question which always arises is, “Which thresholdshould be used”? It has been argued theoretically and, undercertain circumstances demonstrated empirically, that simi-lar results are obtained for a range of threshold levels. Inreal-world applications, however, we are dealing with noise,anisotropy, and a variety of other factors, which underminethose arguments. While the study presented here used a singlethreshold level, the proposed method is amenable to the useof multiple threshold levels—the sample space of fd estimatessimply is enlarged to include models for a range of thresholdlevels.

The proposed method is applicable to a variety of fieldsin which there is a need to characterize the variability of


small sets of data. We have separated the concept of fdfrom the concept of extent of self-affinity. The estimateof the fd is derived from the computed values of the fdof the underlying models; the measure of self-affinity isderived from the closeness-of-fit of the fractal models to theoriginal data values. Using this method,tissue classification,which can be based on fd, can be evaluated in images withlimited self-affinity. Image compression, which can be basedon self-affinity, can be implemented irrespective of fd.Imagesegmentationcan be implemented using fd and degree ofself-affinity as distinguishing features.

In conclusion, the new method shows promise of beingable to distinguish among low-resolution images that are noteffectively distinguished by BC or fBm estimators. Further,since low-resolution images have limited data, it is practical toimplement the exhaustive-search techniques used in the newmethod on a workstation.

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Estimating fractal dimension with fractal interpolation function models