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Measuring Fractal Dimension MEASURING FRACTALFRACTAL DIMENSION:DIMENSION: MORPHOLOGICAL ESTIMATES AND ITERATIVE OPTIMIZATIONOPTIMIZATION Petros MaragosMaragos and FangFang-Kuo -Kuo Sun Division of Applied SciencesSciences The AnalyticAnalytic SciencesSciences Corp. Harvard UniversityUniversity 55 Walkers Brook Drive Cambridge, MAMA 0213802138 Reading, MAMA 0186701867 Abstract An important characteristiccharacteristic of fractal signals isis theirtheir fractal dimension. For arbitrary fractals, an efficient approachapproach to to evaluateevaluate theirtheir fractal dimension isis thethe covering method.method. In thisthis paperpaper wewe unifyunify many of thethe current implementations ofof covering methodsmethods by usingusing morphologicalmorphological operations operations withwith varyingvarying structuring elements. Further, in the casecase of parametric fractalsfractals dependingdepending on on a a parameterparameter thatthat is in oneone-to-one -to -one correspondence correspondence with with their their fractal fractal dimension,dimension, we we develop develop an an optimizationoptimization method,method, which starts fromfrom anan initialinitial estimateestimate andand by by iteratively iteratively minimizing minimizing aa distancedistance betweenbetween thethe originaloriginal functionfunction and the classclass of all suchsuch functions,functions, spanningspanning thethe quantizedquantized parameterparameter space,space, convergesconverges to to the the truetrue fractalfractal dimension. 1 Introduction Fractals are mathematical setssets withwith aa highhigh degreedegree ofof geometrical complexity thatthat can modelmodel many natural phenomena [7].[7], Examples include physical objectsobjects suchsuch asas clouds, mountains,mountains, trees and coastlines,coastlines, asas wellwell as image intensityintensity signalssignals thatthat emanate fromfrom them (assuming(assuming certain restrictionsrestrictions onon thethe object'sobject's reflectance reflectance and illumination [12]). Although,Although, thethe fractal images areare the most popularized class of fractalsfractals due to their fantastic resemblanceresemblance with naturalnatural scenes,scenes, therethere areare also also numerous numerous natural natural processes processes described described by by time time- - series measurementsmeasurements (e.g., 1/l//-noises, f -noises, econometriceconometric andand demographicdemographic data,data, pitch variationsvariations inin musicmusic signals)signals) that areare fractalsfractals [7,17].[7,17]. TheThe oneone-dimensional -dimensional signals f(t)f (t) representing these measurements are fractalsfractals inin the sense thatthat their graph G(f) =— {(t, {(*,y) y) : : yy == f f(t)}(t)) is ls a afractal fractal set. set. Thus, Thus, modeling modeling fractal fractal signals signals isis ofof great interest inin signalsignal andand imageimage analysis.analysis. An important characteristiccharacteristic ofof fractalsfractals usefuluseful for theirtheir descriptiondescription andand classificationclassification is their fractalfractal dimension D, whichwhich exceedsexceeds theirtheir topologicaltopological dimensión.dimension. Intuitively, D measuresmeasures the degreedegree ofof theirtheir boundary fragmentation or roughness. ItIt makesmakes meaningfulmeaningful the measurementmeasurement ofof metricmetric aspectsaspects ofof fractalfractal sets such asas their lengthlength oror area.area. Specifically,Specifically, givengiven a measure unit (a(a "yardstick"yardstick") ") ofof lengthlength e,e, thethe lengthlength L(e) of aa curve at scalescale e is equalequal toto thethe numbernumber ofof yardsticksyardsticks thatthat cancan fitfit sequentiallysequentially alongalong thethe curvecurve times e.e. For a fractal curve,curve, L(e)L(e) increasesincreases without limitlimit whenwhen ee decreasesdecreases and followsfollows thethe proportionality law L(e) occx e ell~ -D.D . TakingTaking logarithmslogarithms yields log[L(e)] = = (1(1 -- D)£>)log(e) log(e) -I-+ constant (1) Hence, DD can be measured from the slopeslope of the (log(log L(e),loge)L(e), loge) data.data. In thisthis paperpaper wewe dealdeal withwith thethe problem problem of of estimating estimating thethe fractal fractal dimension dimension ofof "topologically "topologically oneone-- dimensional" (1-dim)(1 -dim) signals signals with with discretediscrete argument.argument. (Extending most ofof the ideasideas inin thisthis paperpaper to to 2 2-dim-dim signals isis veryvery straightforwardstraightforward andand hencehence omitted.)omitted.) WeWe start inin SectionSection 2.12.1 andand SectionSection 2.22.2 withwith a briefbrief survey of some existingexisting methods, some of which are general whereas others applyapply onlyonly toto special special classes classes ofof fractals. SectionSection 2.32.3 focuses on the coveringcovering method, aa generalgeneral andand efficientefficient approach approach toto computecompute thethe fractal dimension ofof arbitraryarbitrary fractals. WeWe unify and extendextend manymany ofof the currentcurrent digitaldigital implementationsimplementations ofof thethe 416 // SPIE SPIE Vol. Vol. 11991199 Visual Visual Communications Communications and and Image Image Processing Processing IV IV (1989)(1989) Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx covering methodmethod by using morphological erosionserosions and and dilationsdilations withwith varyingvarying structuringstructuring elements. (The(The erosions andand dilations are thethe basicbasic operatorsoperators ofof signal signal andand imageimage analysisanalysis byby mathematical mathematical morphology morphology [13].) WeWe shallshall referrefer toto these unified algorithms as morphologicalmorphological estimates.estimates. InIn addition, toto itsits conceptualconceptual usefulness asas aa unifyingunifying theme,theme, the morphological approachapproach hashas twotwo practicalpractical advantages: 1)1) itit reducesreduces the dimensionality ofof thethe processedprocessed datadata from two toto one, and 2) it isis simplesimple to implement.implement. InIn SectionSection 2.42.4 wewe discuss twotwo implementationsimplementations ofof thethe variation method for estimating fractalfractal dimension,dimension, introducedintroduced inin [4].[4]. The variation method cancan bebe interpretedinterpreted asas stemmingstemming fromfrom aa specialspecial casecase ofof thethe morphologicalmorphological approach to implement the coveringcovering method. Both the covering andand variation methods apply to arbitrary fractals.fractals. However,However, theirtheir actualactual performance performance can be tested onon parametricparametric fractals,fractals, e.g., e.g., fractals fractals depending depending on on a a single single parameter parameter that that is isin inone one-to-one -to -one correspondence withwith their fractal dimensiondimension D. Fortunately,Fortunately, therethere are are numerous numerous classes classes ofof suchsuch parametricparametric fractal signals andand related algorithms for their synthesis. Two examples usedused inin thisthis paperpaper are the random functions of fractional BrownianBrownian motionmotion (FBM)(FBM) [8][8] andand the the deterministic deterministic Weierstrass Weierstrass-Mandelbrot -Mandelbrot cosinecosine (WMC) functionsfunctions [2]. Although the performanceperformance of the coveringcovering andand variation methodmethod isis satisfactorysatisfactory forfor some casescases (i.e.,(i.e., yieldsyields reasonablereasonable estimation estimation errors),errors), ifif oneone is is free free to to vary vary arbitrarily arbitrarily importantimportant parameters of the problem such asas D or the signal'ssignal's duration,duration, then,then, asas ourour experimentsexperiments onon FBMFBM andand WMCWMC functionsfunctions indicate, their performance fallsfalls drasticallydrastically inin manymany instances.instances. Thus in SectionSection 3 we present the mainmain contribution of this paper, whichwhich isis bothboth aa very very effectiveeffective method (i.e.,(i.e., itit yieldsyields practicallypractically zerozero estimationestimation errors) to estimate fractal dimensiondimension and a newnew way of lookinglooking at at thisthis problem. It isis somewhatsomewhat restricted since itit applies only to parametric fractals, but thethe largelarge numbernumber ofof suchsuch parametricparametric classesclasses and their practical applicability motivates well our new method.method. OurOur basicbasic ideaidea isis as follows: So farfar researchersresearchers startstart from anan original fractalfractal signal of true fractal dimensiondimension D, useuse variousvarious approaches to derivederive an estimate,estimate, D*,D *, of of D, D, and and areare contentcontent if the estimation error JD\D —-D D*\ *I isis small.small. ThisThis criterion,criterion, however,however, does not say anything about howhow "close""close" isis thethe originaloriginal fractalfractal signalsignal toto somesome otherother fractalfractal signal signal of of true true dimension dimension D D*. *. Further, anyany degreedegree ofof "closeness""closeness" should be somehowsomehow compatible with laws of visual perception since the fractal dimension isis aa geometrical attribute.attribute. InIn ourour approach,approach, fromfrom anan initialinitial morphological morphological estimateestimate D*D* we synthesizesynthesize thethe corresponding fractalfractal function f*/*. Then byby searching in the parameterparameter
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