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 space space D D we we solve solve a nonlinear optimization problem, where aa distance is iteratively minimized betweenbetween thethe originaloriginal /f and each new iteratively synthesized /*.f*. TheThe process process terminatesterminates whenwhen wewe reachreach aa local local or or global global minimum. minimum. We callcall thisthis thethe Iterative OptimizationOptimization method. AsAs distance metrics we have usedused standardstandard 4,lp, p p = = 1, l,2,oo, 2, oo, metrics. WeWe also introduced a HausdorffHausdorff-type -type distance distance for for this this iterativeiterative optimization.optimization. Our motivationmotivation for using thisthis distance isis thatthat itit isis moremore suitablesuitable thanthan 4ip distances distances to to compare compare twotwo signalssignals inin termsterms ofof theirtheir overall geometricalgeometrical structure, structure, which which is is a a signal signal attribute attribute thatthat fractal methods attempt toto capture.capture. Finally we concludeconclude withwith anan application of the above ideas to determine the optimumoptimum fractalfractal functionfunction for signal interpolationinterpolation among a parametricparametric classclass of deterministicdeterministic fractal functionsfunctions stemming from thethe theory of Iterated FunctionFunction SystemsSystems [1].[1].
2 Covering andand Variation Methods 2.1 General Methods Descriptions ofof variousvarious approachesapproaches toto measure fractalfractal dimension cancan be found inin [7,17,4].[7,17,4]. MandelbrotMandelbrot [7] definesdefines formally formally the the fractalfractal dimension of a setset asas itsits HausdorffHausdorfF-Besicovitsch -Besicovitsch dimensiondimension DJJB*DHB. Thus, a subset of R*RE7 isis fractalfractal ifif DHB DJJB strictlystrictly exceeds exceeds itsits topologicaltopological dimensiondimension DT.DT- SomeSome veryvery closelyclosely related dimensions are the MinkowskiMinkowski-Bouligand -Bouligand dimensiondimension DMBDMB [10>3][10,3] and and thethe boxbox dimension DB [3],[3], which are obtained inin a different wayway butbut are identical; i.e.,i.e., DMB == DB.DB- InIn addition,addition, inin most most cases cases ofof practicalpractical interest DHB = DMB.DMB- InIn this this paper paper we we focus focus on on the the Minkowski Minkowski-Bouligand -Bouligand dimension,dimension, which we shall henceforth callcall fractalfractal dimension D,D, because: 1)1) it isis closely relatedrelated to DHBDJJB and hence able to quantify the
SPIE Vol. 11991199 Visual Communications and ImageImage ProcessingProcessing IV IV (1989) (1989) / / 417
Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx fractal aspects ofof a signal,signal, 2)2) itit coincidescoincides with DHBDJJB inin manymany cases;cases; 3) efficient algorithms exist to compute it; 4) it willwill be applied toto discretediscrete signalssignals wherewhere most approachesapproaches can yieldyield only approximate results.results. Let G be a planarplanar curvecurve whose dimensiondimension 1 1< < DD << 2 isis toto estimate.estimate. Although,Although, DBDB == DMBDMB inin thethe continuous case, theythey correspond toto thethe followingfollowing two different algorithms (with(with differentdifferent performances)performances) inin the discrete case.case. Box countingcounting method:method: PartitionPartition the planeplane withwith squaressquares ofof side e and count thethe numbernumber N(c)N(e) ofof squares squares that intersectintersect thethe curve. curve. Then Then D D= limE= lim ,o _>olog[N(e)]/log(l/e). log[N(E)]/ log(1 /E). If an equivalentequivalent length L(e) = EN(e)eN(e) isis defined byby dividingdividing the the totaltotal area of these squares byby e,e, thenthen L(e) behaves asas in (1). Minkowski Cover Cover method: method: ThisThis isis based conceptually on Minkowski's ideaidea of finding the lengthlength ofof irregular curves: dilate them withwith disksdisks ofof radiusradius ee (and(and thusthus createcreate aa "Minkowski "Minkowski sausagesausage"), "), find thethe area A(e)A(E) ofof thethe dilateddilated set, and setset itsits length length equal equal to to lim6_.0 lim _»o£(e), L(e), where L(E)L(c) == A(E)A(e)/2e. /2E. IfIf GG isis aa fractal, then L(e) behaves as in (1).(1). Generalized CoverCover methodmethod [4]: /4): ThisThis methodmethod unifies thethe box counting and thethe MinkowskiMinkowski cover method by viewing themthem asas special cases ofof aa generalized "cover"."cover ". This This covercover C(c) C(e) isis thethe union of sets B fromfrom a family B suchsuch that:that: GCGC C(e),G(e), andand eacheach BB EG BB intersectsintersects G,G, isis homeomorphichomeomorphic toto thethe diskdisk andand hashas diameterdiameter on the order of e.£. If A(e) is the areaarea ofof thethe cover,cover, thenthen itit waswas shownshown inin [16,4][16,4] thatthat A(e) log ^- = DD log(1)log(-) + constantconstant (2) E2 E Thus, in all the aboveabove methods, D cancan bebe estimatedestimated byby fittingfitting aa straightstraight lineline toto andand measuringmeasuring thethe slope of the plotplot ofof the the data data (log (log L(E), L(e),loge) log e) or,or, equivalently, equivalently, of of the the data data (log[A(E) (log[A(e)/e /E2],2 ],logl/e). log 1 /c).
2.2 Parametric FractalsFractals andand SpecialSpecial MethodsMethods The FBM and WMC fractal functionsfunctions are thethe twotwo primaryprimary classesclasses of parametric testtest signalssignals onon whichwhich we shall evaluate the various methods. The FBM Bfj(t)BH(t) with parameter 0 << HH < <1 is1 ais timea time-varying -varying random random function function withwith stationary,stationary, Gaussian-distributed,Gaussian -distributed, and and statisticallystatistically self-affine self-affine increments; increments; the the latter latter means means that that [Bjf(t[BH(t ++ T)T) —- B(t)]B(t)] isis statistically indistinguishable from r~r- HH[B(t+[B(t+ TT) rT) -— B(t)]B(t)] forfor anyany tt andand anyany rr > > O. 0. TheirTheir power power spectrumspectrum is 5(o;)S(w) oc l/o;1 /w2H2^"1"1 +1. Hence, Hence, an an efficient efficient algorithm algorithm [17] [17] to to synthesize synthesize an an FBM FBM is is to to create create aa randomrandom sampledsampled spectrum whosewhose average magnitude isis 1l/o;^" /wH 1"+o.50'5 and and its its random random phase phase is is uniformly uniformly distributed distributed over [0,2?r].[0, 24 In ourour experimentsexperiments we synthesized and then transformedtransformed thisthis spectrumspectrum viavia aa 2048 2048-point -point inverse FFTFFT to obtain the FBMFBM sequencesequence from which we retained the firstfirst (N(N ++ 1)1) samples.samples. Another example of parametric fractals isis thethe classclass ofof WMCWMC functions
00 WyyH(t)H (t) = = Ef) ry-nH[1 7- nff[l - -cos(rynt)] cos(7nt)] , 0 < H < 11 , (3) n= -cooo where 7ry > > 1.1. InIn our experiments we created WMC sequences byby samplingsampling tt GE [0,1][0, 1] at at (N(N ++ 1) equidistant points, using aa fixed 7=5,ry = 5, and by truncating thethe infiniteinfinite seriesseries so that thethe remainingremaining finitefinite numbernumber ofof terms incurs an approximation error << 101CT -6.5 . TheThe fractalfractal dimensiondimension DD ofof bothboth FBMFBM andand WMCWMC functionsfunctions is in oneone-to-one -to-one correspondence correspondence with with H H becausebecause DD = 22 - H.H. Some specialspecial methods methods (not(not coveredcovered in in this this paperpaper duedue toto theirtheir very narrow applicability) toto measure D for FBM's include: i) Fitting aa straightstraight line line to to the the data data (log (log S(w),S (a;),log log w)u) andand measuringmeasuring thethe slopeslope yieldsyields H andand hencehence DD [12].[12]. ii) The statisticalstatistical self self-affinity -affinity of FBM's yields a powerpower scalingscaling lawlaw forfor many of its moments; linearlinear regressionregression onon thesethese datadata can measuremeasure H andand hencehence DD [12].[12], iii) AA maximummaximum likelihoodlikelihood method for estimating the HH ofof fractional fractional GaussianGaussian noisenoise (derivatives(derivatives ofof FBMs)FBMs) was developeddeveloped inin [6].[6]. (Note, thatthat thethe spectralspectral methodmethod (i)(i) couldcould alsoalso bebe appliedapplied toto WMC'sWMC's becausebecause theirtheir spectrum behavesbehaves likelike 1/w2H +1 too.)too.)
418 // SPIE SPIE Vol. Vol. 11991199 Visual Visual Communications Communications and and Image Image Processing Processing IVIV (1989)(1989)
Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx 2.3 Covering Method In this paperpaper wewe deal not with arbitrary curvescurves but only with finite-lengthfinite -length signals signals /(*), f (t), 0 0 << tt < T,T, inin which casecase thethe curvecurve GG ofof thethe previous discussiondiscussion becomes becomes the the graph graph G(f) G(f) ofof f.f. In thisthis sectionsection wewe shallshall focus on a generalizedgeneralized version of the MinkowskiMinkowski covercover method.method. That is,is, oneone generalizedgeneralized covercover consistentconsistent with the definitiondefinition ofof thethe "General"General Cover"Cover" methodmethod cancan bebe obtained obtained as as follows: follows: givengiven anyany compact compact convexconvex planar set B,5, formform positivepositive homotheticshomothetics eBeB == {El) {eb : bb E£ BB}, }, and definedefine the covercover C(E)C(e) as thethe unionunion ofof sets in BB, , whichwhich contains contains all all vector vector translates translates eBeB + + zz == {et{eb + z;z\ b EE B}B} of eB centered at pointspoints zz of the graph G(f). InIn the the formalism formalism of of mathematical mathematical morphologymorphology this is equivalent to simply dilating G(f)G(f) withwith a structuring elementelement eB:eB: C(c)C(e) = = U|J eB eB + +z z= = G(f) G(f)®eB ® eB (4) zEG(f) Then (2)(2) applies.applies. The MinkowskiMinkowski covercover method method corresponds corresponds to to using using a adisk disk for for B.B. TheThe "horizontal"horizontal structuring element method" inin [4][4] corresponds toto using aa lineline segmentsegment for for BB parallelparallel toto the domain of /.f. In [4] thethe implementationsimplementations of the Minkowski covering,covering, box box counting, counting, and and horizontalhorizontal structuringstructuring methods were donedone byby viewingviewing G(f)G(f) asas aa binarybinary imageimage signalsignal andand dilatingdilating thisthis binarybinary image. image. This,This, however, however, twotwo- - dimensional processingprocessing ofof aa 1-dim1 -dim signal, signal, on on the the one one handhand isis unneseccaryunneseccary andand on the otherother handhand squaressquares the requirementsrequirements inin storagestorage spacespace andand timetime complexitycomplexity ofof implementingimplementing thethe coveringcovering method. Namely,Namely, letlet
Ypo(/)Ypo( f ) = {{(*,y)eR (t, y)E R22 ::yy <
be the ypographypograph ofoff f (also(also knownknown asas "umbra""umbra" inin morphology). morphology). LetLet also also (f (f@g)(i) ED g)(t) == supsupz{z{/(z)+0(£-:r)} f (x) +g(t -x)} and (f(/ e© g)(t) 0)00 == infz{ infz{/(^) f (x) -~ g(x 9(x -- t)} *)} be k erespectively respectively thethe function function dilationdilation andand erosionerosion of f byby aa structuringstructuring function gg with compactcompact support.support. Then,Then, (see (see [13,15,5,9] [13,15,5,9] forfor propertiesproperties ofof thethe ypographs),ypographs), ifif we we ignoreignore the end effects aroundaround t t == 0 and t == T,T, thethe dilated dilated graphgraph C(e)C(e) cancan alternativelyalternatively bebe obtainedobtained as the set difference betweenbetween thethe ypographsypographs of thethe dilated and eroded function; i.e.,i.e., C(e)C(e) ...w Ypo(fYpo(/©6<7)\Ypo(/0e0), ®eg) \ Ypo(f e eg), where gg isis suchsuch thatthat Ypo(g) = {(t,{(t,y):y ggt[t [-l] -1] = gt[1]gt [l] == 0 ,, ggt[0]t [Q] = = h>0, h > 0, and gt[n]gt [n] == -oo `dnVn ^# -1,0,-1, 0,1. 1. (7) If B is the 33 xx 33-pixel -pixel square,square, the corresponding structuring functionfunction isis shapedshaped likelike aa rectangle:rectangle: gr9r[n] [n] = = h>0 h > 0n= n = -1,0,1,-1,0,1, andand grgr [n][n] == -oo-oo Vnbn ^0 -1,-1,0,1. 0,1. (8) If hh = 0, the structuringstructuring functionsfunctions gt and gg,.r becomebecome the the samesame flatflat (binary)(binary) structuring element.element. 2) Perform the dilations andand erosionserosions ofof f/ byby egeg atat discretediscrete scalesscales ee == 1,1,2,..., 2,... , emax. emax . For integer ee wewe define eE asas thethe e-foldE -fold dilation dilation of of g gwith with itself. itself. ThenThen f/ ED© eg eg and and f f Qe Egeg cancan bebe implemented recursively: ff®(€ ®(e ++l)g= l)g=(f®eg)®g (fe eg) eg ,, fe(E/ 0 (e ++1)g= l)g = (f(feeg)eg 0 eg) 0 g (9) SPIE Vol. 1199 Visual Communications and Image ProcessingProcessing IVIV (1989) (1989) / / 419419 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx The dashed lines in Fig. lala show show thesethese erosions erosions/ /dilations dilations byby g,.gr at scalesscales e€ = 10,10,20. 20. 3) Compute the areas A[E]A[c] = = E£=O((/>ñ o((f ® © Eg) 2.4 Variation MethodMethod As Figs. lb,ld1b,ld show,show, even with aa goodgood selectionselection of h, thethe coveringcovering method estimatesestimates for LFD(1)LFD(l) (i.e.,(i.e., the locallocal fractal dimensiondimension for the firstfirst positionposition ofof thethe scalescale windowwindow-first -first 10 scales) are not accurate.accurate. This is partly duedue toto thethe quantizationquantization ofof thethe signal's signal's domaindomain andand hencehence thethe smallsmall numbernumber ofof available available samples toto computecompute (at(at small scales) thethe local minima/minima /maxima,maxima, as as observed observed in in [4]. [4]. Thus Thus atat eE = = 1 there is a neighborhood ofof onlyonly 33 samplessamples forfor thethe min/maxmin /max operationsoperations toto createcreate a cover. The variationvariation methodmethod of Dubuc etet al. [4] attemptsattempts toto correctcorrect thisthis problemproblem byby re-re-arranging arranging thethe originaloriginal signalsignal samplessamples f[n],/[n], 0 < nn << N, N, to toretain retain a smallera smaller number number of of samples samples 00 << m m < ul[m]U![m] = maxmax{/[n]:(m-l)d At the endsends mm == 0,0, M, M, the the local local max max takes takes place place only only over over the the available available samples. samples. The The erosions erosions be b €[m][m] areare given fromfrom thethe formulaeformulae (10)(10) byby replacingreplacing maxmax withwith min and u with b.6. TheThe resultingresulting LFD(E) LFD(e) dependsdepends on d and hence on MM. . TwoTwo optimaloptimal valuesvalues ofof dd cancan bebe foundfound byby searchingsearching overover all permissiblepermissible valuesvalues and finding that dd whichwhich resultsresults inin aa betterbetter least- least-squares squares lineline fitfit toto thethe loglog-log -log plotplot either over the first 1010 scalesscales or over all eemaxmax scales. scales. Which Which of of these these two two optimal optimal valuesvalues of of d d to to useuse dependsdepends ofof coursecourse onon thethe application. Variation by SignalSignal Truncation:Truncation: Let M+lM +1 bebe thethe variablevariable number ofof samples to retain afterafter eliminatingeliminating N --M M samples samples fromfrom thethe originaloriginal signal /,f, where 2e2Emazmaz = = M Mminmt-n < < M M << N. ErosionErosion/dilation /dilation valuesvalues areare computed only only at at MM ++1 1 originaloriginal samples. For eacheach M,M, letlet d = [N[N/M /MJ J bebe aa integerinteger ratioratio factor.factor. ThenThen uuE[m] [m] and and 6 bE[m] [m] are are computed computed for for all all e eexactly exactly as as for for the the variation variation by by decimation decimation method method withwith dd interpretedinterpreted 420 / /SPIE SPIE Vol. Vol. 11991199 Visual Visual Communications Communications and and Image Image Processing Processing 1V IV (1989) Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx now asas aa ratioratio rather thanthan asas aa decimationdecimation factor.factor. (Note(Note thatthat for for several several differentdifferent MM thethe integerinteger dd maymay remain the same.) Figure 22 shows thethe results of estimating thethe LFD(E)LFD(e) forfor FBMFBM andand WMC signalssignals (with(with DD =— 1.5,1.5, N == 500,500, emaxemax —= 25)25) byby usingusing bothboth thethe variationvariation byby decimationdecimation andand byby truncationtruncation methods, methods, whichwhich hadhad similar performance.performance. Both variation methods perform better thanthan thethe covering covering methodmethod whenwhen anan optimumoptimum d or M isis selected.selected. However,However, thethe variationvariation byby truncationtruncation isis muchmuch moremore computionallycomputionally intense than the decimation method.method. ForFor example,example, with NN == 500 500 and and emax emax == 25,25, the the decimation decimation method method searches searches overover d =— 1,1,..,10 ..,10 andand hencehence overover 1010 valuesvalues of M, whereaswhereas the truncationtruncation methodmethod searchessearches over allall MM with 50 < MM << 500. 500. NoteNote that that the the covering covering method method with with hh = = 0 becomes0 becomes identical identical withwith thethe variationvariation byby decimation methodmethod ifif dd = = 1,1, andand withwith thethe variation byby truncationtruncation methodmethod ifif MM == N.N. 3 Iterative OptimizationOptimization MethodMethod Assume a a classclass of of parametric parametric fractal fractal signals signals fp fp parameterizedparameterized by by a aparameter parameter P P thatthat isis related to theirtheir fractal dimension throughthrough anan invertibleinvertible functionfunction D D == &(P).ij>(P). For example, for FBM's or WMC'sWMC's the parameter PP isis HH andand DD = = tl)(H) ifi(H) == 2 2 - H. H. Our Our new new approach approach to to measure measure the the fractal fractal dimension dimension of of such such a signal fp consistsconsists ofof thethe followingfollowing steps: (1)(1) WeWe useuse a simplesimple and fastfast morphologicalmorphological approach (i.e., the covering method method withwith hh == 0) to comecome up with an initial estimate,estimate, D/?*, *, ofof thethe truetrue D. (2)(2) WeWe computecompute somesome distance betweenbetween thethe original fractalfractal functionfunction fpfp and anotheranother fractalfractal fp./p* which which was was synthesized synthesized toto havehave dimension exactlyexactly D*D* == ij>(P*).t,(P *). (3)(3) ByBy usingusing nonlinearnonlinear optimization, wewe search in the parameterparameter spacespace ofof P (or(or equivalentlyequivalently ofof D)D) valuesvalues ofof the the chosen chosen classclass ofof fractalsfractals byby synthesizingsynthesizing fractalsfractals whosewhose parametersparameters correspond toto a fractal dimensiondimension D* and computing their distances from the original fractal until this cycle convergesconverges to to a a locallocal oror globalglobal minimumminimum in in the the parameterparameter space.space. InIn thisthis wayway thethe resultingresulting fractalfractal dimension willwill correspondcorrespond toto aa fractal function which is also close (with(with respect to thethe specificspecific distance)distance) to the originaloriginal function. WeWe callcall thisthis thethe IterativeIterative OptimizationOptimization method.method. AsAs distancedistance metricsmetrics wewe cancan useuse standard 4,£p, p p = = 1, 1, 2, 2, oo,oo, metrics.metrics. P `'P(f1,DD f2) = Ill/i Ifl -- 12I Mlp Ip =- lfll(AW¡¡n] - f2ln.])P/2 H)¡¡ P (11) (n-L o0 Further, wewe also introduce a definition of aa HausdorffHausdorff metric metric for for this this iterative iterative optimization.optimization. Our motivation for using thisthis HausdorffHausdorff distance distance is is that that itit is better suitablesuitable than 4ip distances distances to to compare compare the the geometrical geometrical structural differencesdifferences (peak/valley(peak /valley distributions) distributions) betweenbetween two two signals signals in in a a wayway thatthat agreesagrees with humanhuman visual perception. TheThe HausdorffHausdorff metricmetric waswas soso farfar defineddefined only for sets. HereHere wewe extendextend itsits definitiondefinition toto functions and provide a morphologicalmorphological algorithm forfor its computation.computation. Thus,Thus, given given two two compact compact sets sets A1i AI, A2A2 their Hausdorff metric can be computedcomputed asas H(Ai,H(A1, AA2)2 ) == inf{inf{e : :A1C AiC A2A2 ®0 EDeD andand A2CA2 C AlAI e© ED eD}, }, (12) where eDED isis a a diskdisk ofof radius radius e.e. IfIf AIAl and A2A2 becomebecome thethe ypographsypographs of twotwo functionsfunctions /ifi andand f2/2 withwith compactcompact supports, thenthen wewe definedefine theirtheir ypograph ypograph-based -based HausdorffHausdorff distance asas Hy(/i,/HY(f1, f2)2 ) = = JT[Ypo(/i),H[Ypo(fl), Ypo(/Ypo(f2)]2)] = = inf{6inf {e : /: xfl < S /f22 © ®Eg eg andand 12/2 < fi/i ®0 eg}eg} (13) Hy compares /ifi andand 12/2 inin termsterms ofof their their protrusions protrusions (peaks).(peaks). If we want a distancedistance sensitivesensitive both to the peaks and thethe valleys,valleys, thenthen wewe cancan formform the sumsum Hs(fl,Hs(/i,/ f2)2 ) = HY(fi,Hy(/i,/ 12)2) + - He(fi,He(/i,/f2),2 ), which adds to Hy(f1Hy(/i,/ ,f2)2 ) thethe Hausdorff distancedistance He(/i,/He(fi, f2)2) = Hy(cHy(c - fl,/i,c c - - f2) /2 )of of their their epigraphs, epigraphs, i.e., i.e., of of the the negation negation- - complements ofof /ifl andand 12/2 wherewhere cc isis somesome constantconstant function.function. SPIE Vol.Vol 1199 VisualVisual Communications and Image ProcessingProcessing IV IV (1989) (1989) / / 421 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx Figure 3 reports aa seriesseries ofof experimentsexperiments whosewhose goalgoal waswas toto inestigateinestigate howhow wellwell cancan thethe Hausdorff Hausdorff or or Eplp distances find a local/globallocal /global minimumminimum inin comparingcomparing aa given parametricparametric fractal withwith anan ensemble ensemble ofof similarsimilar fractals whose parameterparameter varies over allall possiblepossible values.values. Figs.Figs. 3a,3b,3e,3f3a,3b,3e,3f show show that that bothboth the Hy and thethe t\ti distancedistance yieldyield aa veryvery clearclear globalglobal minimumminimum when comparing an original FBM or WMCWMC signalsignal of a fixed D with similarsimilar signalssignals whose D spansspans allall thethe interval interval [1, [1,2]. 2]. Resolutions of anywhere between 0.01 and 0.1 sufficesuffice to to sample sample the the parameterparameter spacespace ofof DD andand still observe aa clear minimum.minimum. The HyHy distancedistance has a higher computationalcomputational complexitycomplexity than than £1, 4, andand bothboth yieldyield similarsimilar results. ForFor thethe HyHy distancedistance wewe usedused the structuring function gg,.r withwith hh = 0.01.0.01. UsingUsing aa smallersmaller hh oror gtgt insteadinstead ofof g,. gr refinesrefines Hy butbut requiresrequires more iterationsiterations of erosions/dilations.erosions /dilations. InIn Figs.Figs. 3a,3b3a,3ó thethe FBM's are viewed as deterministic functions (the random number generator waswas re-re-initialized initialized forfor each each D D withwith thethe same seed). From a statisticalstatistical viewpoint,viewpoint, however, they they areare randomrandom functions,functions, andand hencehence theirtheir HausdorffHausdorff and and £p4 distancesdistances areare actuallyactually randomrandom variables. Unfortunately, as Figs. 3c,3d show, thethe average HyHy andand t\4 distancesdistances (the(the averageaverage waswas takentaken overover 50 independent FBMFBM realizationsrealizations forfor eacheach D) D) dodo notnot have aa global minimum,minimum, exceptexcept whenwhen DD >> 0.5 and the Hy distance is used (the Hs distance performs slightlyslightly better).better). Therefore,Therefore, inin ourour iterativeiterative optimizationoptimization method, the FBMFBM signalssignals areare viewedviewed henceforth only deterministically. In all thethe aboveabove experimentsexperiments wewe used both types of Hausdorff distancesdistances (Hy (Hy and and Hs) Hs) and and threethree typestypes ofof ip4 distancesdistances (p(p = 1,l,2,oo). 2, oo). They all had similar performanceperformance for (deterministic) FBMFBM and WMCWMC signals.signals. HenceHence wewe focusfocus henceforthhenceforth only on the Hy andand thethe t\4 distances.distances. Figure 4 compares the estimated locallocal fractalfractal dimensiondimension LFD(1)LFD(l) forfor FBMFBM andand WMCWMC signalssignals ofof varyingvarying dimension D D andand fixed durationduration NN == 500500 byby usingusing thethe covering,covering, the variationvariation byby decimation,decimation, and ourour iterative optimization method.method. TheThe iterativeiterative optimizationoptimization was done by using the initialinitial estimateestimate fromfrom the covering method,method, andand thenthen improving itit by first proceeding inin thethe D space at stepssteps ofof 0.010.01 and thenthen (when inin the neighborhoodneighborhood ofof thethe globalglobal minimum)minimum) byby refiningrefining itit withwith optimization optimization steps steps of of OS= OS=0.001. 0.001. As the Fig. 4 shows thethe iterative optimization method gives superiorsuperior resultsresults than anyany otherother methodmethod sincesince it achieves estimationestimation errors errors that that are practically zerozero for for all all D. D. (The errors are guaranteedguaranteed to bebe inin thethe order of OS; inin our experiments, they were almostalmost alwaysalways in in thethe orderorder ofof 10~10 -44 or or 10~10 5-5.).) The The t\4 distancedistance was used, but thethe HyHy distancedistance performedperformed veryvery similarly.similarly. The same conclusions can be reached from Fig. 5 which shows thethe estimated dimensiondimension LFD(1)LFD(l) forfor FBMFBM and WMC signals withwith varying duration N andand fixedfixed dimensiondimension D,D y byby usingusing allall thethe previousprevious approaches.approaches. Again, for allall NN the iterative optimizationoptimization methodmethod outperforms outperforms all all others others and and yields yields practically practically zero zero errors. errors. Comparing the covering and variation methods, wewe see from Figs.Figs. 44 andand 5 that: 1)1) thethe covering covering methodmethod performs worseworse than than thethe variation,variation, except forfor veryvery smallsmall D D << 1.2 andand smallsmall N.N. 2)2) InIn thethe variationvariation method,method, optimizing the decimation factor dd overover the firstfirst 1010 scales performs better thanthan optimizing optimizing itit over over allall scalesscales for D < 1.5,1.5, whereas for D > 1.51.5 thethe oppositeopposite itir true.true. 3)3) Changing Changing thethe signal signal lengthlength NN affects affects thethe above above conclusions forfor smallsmall NN < 300.300. 4 AnAn Application Application to to Fractal Fractal Interpolation Interpolation FunctionsFunctions Given datadata points (xi,(xt-,y y;),t-), ii = 0,0,1,..,/, 1,..,I, wewe areare concernedconcerned withwith continuouscontinuous functions functions / f :: S -->—> R onon aa compact intervalinterval 5,S, whichwhich interpolateinterpolate thethe datadata as /(zf (xi)t-) = ytyt* and whosewhose graphs are attractors ofof Iterated Function Systems (IFS) [1].[1]. TheseThese systemssystems consistconsist ofof aa finitefinite numbernumber ofof affineaffine contractivecontractive mapsmaps whichwhich cancan model well self-similarself -similar sets sets as as the the union union (collage)(collage) of of small small patches patches (each (each patch patch isis thethe transformationtransformation of the original setset by an affine map).map). AA generalgeneral affineaffine map for an IFS isis Wn , n = 1,l,2,...,/. 2, (14) y cndn y +fn 422 // 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 We havehave aa finite number I/ ofof suchsuch maps,maps, eacheach associatedassociated withwith somesome probabilityprobability pn.pn . If thethe mapsmaps areare contractive, thenthen there exists a randomrandom algorithmalgorithm [1][1] in whichwhich aa pointpoint (z,y)(x, y) isis iterativelyiteratively mappedmapped toto otherother points by these affine maps (randomly chosen withwith probabilityprobability ppn)n) andand thus fillsfills the pointspoints ofof aa uniqueunique attractor. ForFor suchsuch anan IFSIFS toto have have asas attractorattractor the the graph graph of of a afunction function its its parameters parameters must must have have somesome restrictions. Namely,Namely, forfor thethe nn-th -th affineaffine map, an = (xn - xn- 1)/(x7 - x0),bnbn = 00 ,, en€n == xnXn _i-1 -- anxo.dnXQ. - . (15) dodn ee ((-1, -1, 1)1) , cCnn = = [y[ynn - - yynn-i -1 - - ddn(y7n(yi -- yo)1/(x7 -- x0),fn = yn -1 - dny0 -- Cnxo. Here wewe assume assume that that ddon == V is constant forfor all n, and we call itit the "vertical"vertical scale parameter" (contraction(contraction factor) V. ThusThus givengiven the initialinitial data pointspoints (xi,(xt-, yi),yt-), the fractal interpolated functionfunction (FIF)(FIF) fv isis aa uniqueunique fractal signal parameterized byby V.V. FigureFigure 6a6a showsshows anan originaloriginal functionfunction ff (an(an FBMFBM ofof 256256 pointspoints withwith H == 0.3).0.3). AsAs data data (xi, (z tyi)-,yt-) we we select select 17 17 points points from from this this function function f,f, whosewhose xixt - areare equally equally spaced;spaced; hencehence all aann = 1/16.1/16. Figs. 6b,6c show twotwo signalssignals thatthat resulted from the IFSIFS interpolationinterpolation algorithmalgorithm with V = -0.6, 0.6, +0.6; +0.6; the the dotted dotted line line shows shows the the piecewise piecewise linear linear interpolation interpolation between between thethe original original 1717 points.points. The larger the V, thethe rougherrougher lookslooks thethe interpolatedinterpolated functionfunction fy./y. AllAll these these interpolated interpolated functions functions werewere computed at lintIint == 256256 samplesample pointspoints equallyequally spacedspaced inin theirtheir domain.domain. Fig.Fig. 6d 6d shows shows thethe fractal fractal dimension dimension LFD(l)LFD(1) (computed usingusing thethe covering methodmethod with with gg,.r and and h h == 0.01) of fv forfor VV spanningspanning thethe intervalinterval [-0.9,0.7][- 0.9,0.7] at at steps steps of of 0.032. 0.032. We We see see that that thethe relationrelation between VV and thethe fractalfractal dimensiondimension DD isis one one-to-one -to-one over eacheach halfhalf ofof the V parameterparameter spacespace ((-1,1). -1,1). (We(We areare currentlycurrently workingworking toto provideprovide anan approximateapproximate analytic formula for this relation.) FigsFigs 6e6e andand 6f 6f show, show, respectively,respectively, the HausdorffHausdorfFand and eptp distances (mean absolute error,error, rmsrms error,error, andand max absoluteabsolute error)error) betweenbetween the the original original / f andand thethe interpolatedinterpolated /V,fv, as a function of the parameterparameter V.V. SelectingSelecting one one local local minimum minimum for for the the symmetric symmetric Hausdorff Hausdorff distance distance givesgives us V = -0.8,-0.8, whereas whereas thethe minimumminimum forfor thethe 40£<» distancedistance givesgives usus VV = -0.1. 0.1. PlottingPlotting the the corresponding corresponding interpolated functions in Figs. 6g andand 6h6h showsshows that that thethe optimal parameterparameter V extracted viavia thethe HausdorffHausdorff-- distance minimizatiqnminimizatign approximatesapproximates /f with an interpolated function that isis closer to /f inin aa wayway that agreesagrees more withwith the human perceptionperception ofof the roughnessroughness of the function's graph. Finally instead of searching through the whole parameterparameter space V,V, we cancan insteadinstead find first the fractal dimension whichwhich willwill givegive usus anan initial estimate ofof V,V, duedue toto the the one one-to-one -to -one relation between D andand V,V, and then improveimprove this initialinitial estimateestimate byby searchingsearching locallylocally aroundaround it.it. Acknowledgements. P. P. Maragos Maragos was was supported supported by by the the National National Science Science FoundationFoundation underunder GrantGrant MIPS-86-58150MIPS -86 -58150 with with matching matching funds funds from from Bellcore, Bellcore, Xerox, Xerox, and and an an IBM IBM Departmental Departmental Grant, Grant, and in part by ARO underunder Grant Grant DAALO3 DAALO3-86-K-0171. -86-K -0171. References [1] M. F. Barnsley,Barnsley, "Fractal"Fractal InterpolationInterpolation Functions Functions", ", Constr.Constr. Approx., Approx., 2, 2, pp. pp. 303 303-329, -329, 1986.1986. [2] M. V. Berry and Z.Z. V.V. Lewis,Lewis, "On"On thethe Weierstrass- Weierstrass-Mandelbrot Mandelbrot fractal functionfunction", ", Proc. R.R. Soc.Soc. Lond.Lond. A, 370,370, pp.459pp.459-484, -484, 1980.1980. [3] G.G. Bouligand, "Sur lala notionnotion d'ordred'ordre de de mesure mesure d'un d'un ensemble ensemble plan plan", ", Bull. Bull. Sci. Sci. Math., Math., II 11-53, -53, pp.185-pp.185- 192, 1929.1929. [4] B.B. Dubuc,Dubuc, J. F.F. Quiniou,Quiniou, C.C. Roques-Roques-Carmes, Carmes, C.C. TricotTricot andand S. W. Zucker,Zucker, "Evaluating"Evaluating the fractalfractal dimension ofof profilesprofiles", ", Phys.Phys. Rev.Rev. A,A, vol.39, vol.39, pp.1500 pp.1500-1512, -1512, Feb.Feb. 1989.1989. [5] R. M. Haralick, S. R. Sternberg, andand X.X. Zhuang,Zhuang, "Image"Image AnalysisAnalysis UsingUsing Mathematical MorphologyMorphology",", IEEE Trans.Trans. PatternPattern Anal. Anal. Mach. Mach. Intell., Intell., PAMI PAMI-9, -9, pp.523 pp.523-550, -550, July 1987.1987. SPIE Vol.Vol. 11991199 Visual CommunicationsCommunications andand ImageImage ProcessingProcessing IV IV (1989) (1989) / / 423 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx [6] T.T. Lundahl, W.J. Ohley,Ohley, S.M.S.M. Kay,Kay, and R.R. Siffert,Siffert, "Fractional"Fractional BrownianBrownian Motion:Motion: AA Maximum Maximum Likeli- Likeli hood Estimator andand Its Its Application Application to to Image Image Texture Texture", ", IEEE IEEE Trans. Trans. Med. Med. Imag., Imag., MI MI-5, -5, pp.152 pp.152-160, -160, Sep. 1986.1986. [7] B. B. Mandelbrot, TheThe FractalFractal Geometry Geometry of of Nature, Nature, NY: NY: W.H.W.H. Freeman, Freeman, 1982,1983. 1982,1983. [8] B. B. Mandelbrot and J.J. vanvan Ness,Ness, "Fractional"Fractional Brownian Brownian motion,motion, fractionalfractional noise noise andand applications applications", ", SIAM Review,Review, 10(4),10(4), pp.pp. 422422-437, -437, 1968.1968. [9] P.P. MaragosMaragos andand R.R. W. Schafer,Schafer, "Morphological"Morphological FiltersFilters -- Part I:I: TheirTheir SetSet-Theoretic -Theoretic Analysis andand Relations to LinearLinear ShiftShift-Invariant -Invariant Filters," IEEEIEEE Trans. Trans. Acoust. Acoust. Speech, Speech, SignalSignal Processing, Processing, pp.1153-pp.1153- 1169, Aug. 1987.1987. [10] H.H. Minkowski,Minkowski, "Uber"Uber diedie Begriffe Lange,Lange, OberflacheOberflache undund Volumen",Volumen", Jahresber.Jahresber. Deutch. Mathematik- erverein., 9,9, pp.pp. 115115-121, -121, 1901.1901. [11] S.S. Peleg,Peleg, J.J. Naor, R. Hartley and D.D. Avnir,Avnir, "Multiple"Multiple ResolutionResolution TextureTexture Analysis Analysis andand Classification Classification", ", IEEE Trans.Trans. Pattern.Pattern. Anal. Anal. Mach. Mach. Intell., Intell, PAMI PAMI-6, -6, pp.518 pp.518-523, -523, JulyJuly 1984.1984. [12] A.A. P.P. Pentland, "Fractal"Fractal-Based -Based DescriptionDescription ofof NaturalNatural Scenes",Scenes ", IEEE IEEE Trans.Trans. PatternPattern Anal.Anal. Mach.Mach. Intell,Intell., PAMIPAMI-6, -6, pp. 661661-674, -674, Nov. 1984.1984. [13] J.J. Serra, ImageImage AnalysisAnalysis andand MathematicalMathematical Morphology,Morphology, NY: Acad. Press, 1982.1982. [14] M.M. C.C. Stein,Stein, "Fractal image models andand objectobject detection",detection ", inin Proc.Proc. SPIESPIE 845: VisualVisual Communications and ImageImage ProcessingProcessing II,II, 1987.1987. [15] S. R.R. Sternberg, "Grayscale"Grayscale Morphology,"Morphology," Comput.Comput. Vision,Vision, Graph.,Graph., Image Image Proc. Proc. 35, 35, pp.333 pp.333-355, -355, 1986.1986. [16] C. Tricot, J. Quiniou,Quiniou, D.D. Wehbi,Wehbi, C.C. Roques-Roques-Carmes, Carmes, et B. Dubuc, "Evaluation"Evaluation dede lala dimensiondimension fractalefractale d'un graphegraphe", ", RevueRevue Phys.Phys. Appl., Appl., 23, 23, pp.111 pp.111-124, -124, 1988.1988. [17] R.R. F.F. Voss, "Fractals in nature:nature: FromFrom characterization characterization toto simulation simulation", ", inin TheThe ScienceScience ofof FractalFractal Images, H.H.-O. -O. PeitgenPeitgen and D.D. Saupe,Saupe, Eds,Eds, Springer Springer-Verlag, -Verlag, 1988.1988. 424 / /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 (a) FBM (D=1.5)(D=1.5) (c) WMC (D=1.5(D=1.5 g=5)g=5) 1 T 1.61.6" - 0.2 Á1.2 - H 0.8 -• - 0.4 -• 4\IA\v,h -2.2 0.4 - I I 0 I I I I 100 200 300 400 500 0 100 200 300 400 500 SAMPLE SAMPLE in [0,1][0,1] (b) Covering Method on FBM f^\(d) Covering Method on WMC 2.0- 2.0 - 2.02.0- ——— sfh-0sfh=0 ——— sfh=0 sfh=0.01 I-.*-1.8-- ———- - sfll=°-01 g 1.8- . —-- —- sfh=0.001 sfh=0.001 Z sfh=.1 ...... Ssfh=0.01fh=0.01 QA 1.61.6-- 0 1.6 • §1.4-1.4 ^^^=^r ^ d u L21.2 " - U 1.2 - """""••••--.. o o h—1 1.0- —————1 ————— 1 ————— 1 ————— 1 ————— 1 i.o- 1.0 I I I I I 1.0 ]1l 4 7 10 13 16 ]11 4 7 10 13 16 SCALE SCALE FIGURE 1. 1. (a) (a) An An FBM FBM function function (solid (solid line) line) with with H H = = 0.5, 0.5, N N = =500 500 and and its its erosions erosions/dilations /dilations (dashed(dashed lines) by cgegrr (c(e == 10,10,20, 20, hh —= 0.01). (b)(b) EstimationEstimation ofof LFD LFD ofof FBMFBM viavia thethe covering covering methodmethod usingusing grgr with 3 different heights h = 0,0.01,0.10,0.01,0.1 (emaz (cmax = 25). (The(The thinthin solidsolid straightstraight lineline shows thethe true D = 1.5.)1.5.) (c)(c) and (d) samesame asas (a)(a) andand (b)(b) butbut for for a aWMC WMC function function and and with with h h= = 0, 0,0.001,0.01. 0.001, 0.01. SPIE Vol.Vol 1199 VisualVisual Communications and Image ProcessingProcessing IV IV (1989) (1989) / / 425 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx (a) VAR-DVAR -D onon FBM (c) VAR-DVAR -D onon WMCWMC 2.0 - 2.0-2.0 - ——— OA:d=10 OA: d=10 ——— OA:OA:d=9 d=9 Z --— -— OF: OF:d=3 d=3 z —- - —OF: OF:d=4 d=4 ...... NO:d-l O 2 1.8-- NO: d=1 1.8bON -• ...... NO:d=lNO: d=1 £ \LDIMENSz - 5 1.6 - 1.6»—'H - < ——— **~""——————————————— -- --- .. b , 4 . r -...„..-.....-..-.--.------%------"^ ^- i-...... ^^^^^^_____-^— ---- 1"Q1.4- •-- &1< M1.4- ¡ ------ <« 1 9 -• (J 1.Z1.2 - UÚ 1.2 - - O O J 1.0- ————1 ———— 1 ———— 1 ———— 1 ————1 1.0 i.o1.0 -1 11 4 7 10 1313 16 1]I 4 7 10 13 16 SCALE SCALESPAT P (b;(b)) VAR-T on FBM (d) VARVAR-T -T onon WMCWMC 2.0- 2.02.0- 2.0 - ——— OA:M=51 ——— OA:M=51OA: M=51 OA: M=51 § -— - —OF: OF:M=90 M=90 fc —- - —OF: OF:M=153 M=153 ——— . -MO- M *>00 5o 1.8 • [• ...... NO:M-500 01.81.8 • - 1>V-/.NO: 1V1M=500 +J\J\J ^*'FA' NO: M=500 zg -.-g 5 l.b-1.6 -r...------...------A 1.6 Q /^ -^""^ ^ L6 __ __ — — _ ^ J SCALE SCALE FIGURE 2. 2. (a) (a) Estimation Estimation of of LFD LFD of of an an FBM FBM function function (H (H = = 0.5, 0.5, N N = = 500) 500) via via the the variation variation by by decima- decima tion method.method. The thinthin solidsolid lineline shows thethe truetrue D = 1.5.1.5. TheThe OAOA andand OFOF lineslines showshow thethe estimatesestimates resultedresulted from optimizing the decimationdecimation factorfactor dd overover allall emaz emaz = 2525 scalesscales oror overover thethe firstfirst 10 10 scales, scales, respectively. respectively. The NO lineline correspondscorresponds to to not not optimizingoptimizing and and not not decimating decimating (d (d = = 1).1). (b)(b) same asas in (a) but usingusing the variation byby truncationtruncation method. method. (The(The NONO lineline correspondscorresponds to to not not truncating truncating since since M M = = NN = 500.)500.) (c) and (d) same as (a) and (b)(b) butbut forfor aa WMCWMC function.function. 426 // SPIE SPIE Vol.Vol. 11991199 VisualVisual CommunicationsCommunications andand Image Processing IV (1989) Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx FBM (N=500R=1)R=1) (a) FBM (N=500(N=500R=1) R=1) (b) FBM (N=500 00 `Wn le+03 D=1.2 - - D=1.5 I 1e +03 - d< 800 D=1.8 b 800 - 600 600 ...... 400 CO O H-« 400 - QA Q C/3(9) 3 200 200 --- ffi 0 o 1.0i.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 FRACTAL DIMENSIONDIMENSION FRACTAL DIMENSIONDIMENSION CO W 900900 T 630 au FBM (N=200(N=200R=50) R=50) 560 .. FBM (N=200(N=200R=50) R=50) 700 (I. 490 ci) I. (c) / 420 500 / 350 // 280 300 : .. (d) -- ...-...... I I I I —— 1—— —— i—— —— i—— —— i—— ——— ! I i I I I I 1.0i.o 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 (e) WMC (N=500)(N=500) (f) WMC (N=500)(N=500) 800 - D=1.2 2.5e+03 Z - - D=1.5 D=1.8 2e+03 r"600 - Á 1.5e+03 " 400 - ,E- le+03 O 200 o_ I I--... I 't I 1.0 1.2 1.4 1.6 1.8 2.0 FBM functions/< f; withwith D,Di = FIGURE 3. (a) HausdorffHausdorff distancesdistances Hy(f;,Hy(/,-,/z>) fD) betweenbetween threethree fixedfixed FBM functions FIGURE 3. the interval [1.01,1.99][1.01,1.99] at stepssteps ofof 0.01.0.01. (b) samesame 1.2,1.5,1.81 2 1 5 1 8 and variablevariable FBM functions IDfD whose D spans the distances, (c) and (d) are same as (a) and (b) but thethe distancesdistances areare averaged as (a)fa) but using t^fijo)4(h) ID) distances.(c) and (d) are same as (a) oJ theVBM fD for each D. (e) and (f) are same as (a) and (b) but forfor WMCWMC functions.funct.ons. "elover 50i^realizatLs50 realizations of the FBM ID for each D. SPIE Vol.Vol 1199 Visual Communications and Image ProcessingProcessing IV IV (1989) (1989) / / 427 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx (a) FBM (N=500)(N=500) WMC (N=500)(N=500) (b) 2.0 -(a) 2.0 - —— IO:L1IO: LI ...... VAR-D-F g0 ——• VAR-D-A So'ci) 1,81.8 +- VAR-D-A — - COV-FED 1.6 •• 1.4 •• Id 1.4 / 1.2 - cn W 1.0 1.0 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 FRACTAL DIMENSION D FRACTAL DIMENSION D FIGURE 4. 4. Estimating Estimating the the LFD(1) LFD(l) of of [FBM [FBM in in (a) (a) and and WMC WMC in in (b)] (b)] signalssignals whosewhose DD spansspans thethe intervalinterval [1.05,1.95] at at steps steps of of 0.01. 0.01. The IO lineline correspondscorresponds toto the iterativeiterative optimizationoptimization methodmethod usingusing thethe Li£1 distance. TheThe VAR VAR-D-A -D -A andand VARVAR-D-F -D -F lines lines correspond correspond to to the the variationvariation by decimation methodmethod wherewhere the decimation factor is optimized over all cmaz(max == 25 scales oror overover thethe first 10 scales, respectively. The COV-FEDCOV -FED line line correponds correponds to to thethe covering methodmethod with g,.gr andand h = 0.0. (a) FBM (D=1.5)(D=1.5) (b) WMC (D=1.5)(D=1.5) 2.0 - 2.0z.u - ——— IO:L1IO: Li -— — VAR-D-F\7VAR A T> -DT\ -FT7 g/^iz 0 O l—'l—*1.8 bec\ i -...... VAR-D-AVAR -D -A 55 1.8 -- DIMENS1\Lz -— - COV- COV-FED -FED |z 1.6- Q 1.6 - - . <^ ::` ,,.. - -_--- ~ - - ^ — s_^" V — •- —.. •- •—.-...-..... <; 14 . 1 ———I1 ———1I ———I1 — ——— I 1.01 n - ... - J—————— l —————— l —————— 1 ——— 1.0 - 100 300 500 700 900 100 300 500 700 900 NUMBER OF SAMPLESSAMPLES ((N) N ) NUMBER OF SAMPLESSAMPLES ((N) N ) FIGURE 5.5. Estimates Estimates of of the the fractal fractal dimension dimension LFD(1)LFD(l) ofof [FBM[FBM in (a)(a) andand WMC in (b)](b)] signalssignals withwith D = 1.51.5 and NN spanningspanning thethe intervalinterval [100,1000][100,1000] atat steps of 20.20. The line identities are as in Fig.Fig. 4.4. 428 // SPIE SPIE Vol. Vol. 11991199 Visual Visual Communications Communications and and Image Image Processing Processing IVIV (1.989)(1'989) Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx ^ ii n h-^ h— n • ^ u> 3 s ^ 8 B F^i™ ^^^ ^fy •^^^f ^ FBM (D=1.3) WMC (D=1.3) to b to t (d) H C j-i « 5^ 2.0 Tcj ^ ^d > t ———— ——— IO: L1 1 ^> IO: Ll ^ >^ > ?d « H- bo H-* bo VAR -D -F -H 3 — VAR -D -F 5 "5 a?y c 3^^" <; o VAR -D -A n zO 1.8 VAR -D -A * ——— » i - - COV -FED DIMENSION - - COV -FED DIMENSION H" a\ H- ON z 1.6 \ \ ————— / » i / 4^ I—* 4^ I— 4^ / '» { / 1 \ / \ — <'''/ / FRACTAL *r* FRACTAL J ' ^* ! \ • • ^ 1.4 • • * i • i \ » t % / • i ( / i j ) \\ i: jj ! I I \ ^ v- ) ! d I » * j h— *° - to t— ------.. . .----- ...... ESTIM. ESTIM. 1.2 - ...... --, -- _ _ ,- - - - _ * ' J " ' • H- . ^ - • H-I b " ^ 8 r^ ^*^ -j 8 o U> o 1 ^ 8 ss H- § S 1.0 N 100 300 500 700 „. 900 100 300 500 700 900 — ,. 5 S 3 ^8" ^ 00 X ^ > ^ C^ § O ^ S | 2 /— v^^ NUMBER OF SAMPLES (N ) k NUMBER OF SAMPLES ( N ) II ^ . s a t" 1 oo ^ O >!L rv^ "^ '*. /-N _ ro ^J> . ^* ^T FBM (D=1.8) i i i WMC (D=1.8) tO b ... (e) 2.0 (f) b - ————— • H-» bo bo t O j ^* 1.8 ' J ! } \ * /* ' z (j t ^ 1 1 "U * , • \ x * t { • -*' *' * 1 : t >' 1 t \ 1 /, | \ * » i/ ft ii **v % * , . 1 DIMENSION <* 1 DIMENSION H^ - as - \ ( « ON / , 1 .. x I / . - - ( i ( ,± .-- . ; ^ / ^ / f \'V> i :,--;:::-;%' 1.6 t ' .. á i ...: .;; -- ' \"*"`; --.--. { -.; . : t_i '^ ( 42,. "RACTAL s P ^ F f-i U FRACTAL 3 1.4 < >P ^3 C? < ^ » IO: L1 — •< » IO: L1 i ?3 C3 h— to > >„» VAR -D -F to VAR -D -F f^^? 399 n o o> •< 53 F, d ESTIM. VAR -D -A ESTIM. 1.2 VAR -D -A ' - COV -FED —— - - COV -FED " H— " b ' - H )__* ' b - I I I I 1 I I - S <-* Q c_J o ° ^ i O £? O 8 L^l 0 ° ^j 8 8 1.0 100 300 500 700 900 1.0 100 C^ S | •Tl O £ « r 300 03 S 2 500 700 S g HH W 900 'Tl O /**"N ^ .^ ^^ fS w C/3 ^ NUMBER OF SAMPLES ( N ) s^/ Jl NUMBER OF SAMPLES ( N ) I-H W W hj Cl ^ 1 ii QJ 5 3 AJ M CD ^' P 0° 3«. &- CT ^ e* ^ ^ CO S p 0 0 3 0> P CQ S >£!, / V ^2 ^^^ ^ ^^"^ s> **** '^ ^^ . 113 E3 p p sT b • andFIGURE (b) but 5 with(cont'd). D = 1.8.(c) and (d) are same as (a) and 'c? (b) but with D = 1.3. (e) and (f) are same as (a) •& s- \ ^ CO S Q) * O 8 S S §' * 1 1 o 5? ^ SJ. -»* CO Co o* " i SPIE Vol. 1199 Visual Communications and Image Processing IV (1989) / 429 CQ Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx Use: of Terms 06/21/2016 on http://proceedings.spiedigitallibrary.org/ From: Downloaded D (d) (a) (b) fBm (Npts=256,(Npts=256 , H=.3) (c) I 5 4-.- .46 3 - 4 V=0.6 Nk.. -0.6 137 2 - sto' 3 I 128 S> 2 1.19 -0.9-0.9 -0.6 -0.3 0 01 ' V 0.6 I VerticalVenice! ScaleScale Parameter ofcif FIFFIT IT 790 50 100 150 200 250 50 11X1 150' 200 250 50 100 150 250 I Sample Index Sampk index Sample Index 1 I HAUSDORFF DISTANCESDISTANCES tLPp DISTANCESDISTANCES I'I - SYMMETRIC - MAE -- HYPOGRAPH s - --- RMSE 390 oo EPIGRAPH Lag 2.9 3 300 V= -D.8 3 .. 2 ./ 2 ----- ...... 120 .4. / 1,1 // ...... -.4 4-1' I 30 I 1 I 0 -010.9 -0.6-0.6 -0 3 0 0.3 0.60.0 .0.9-0.9 .0.6.0.6 -0.3-03 03 0.6 -2 Verdes/Vertical ScaleScale ParameterParamelcr ofof FIF V V _0. 3 50 im IL 200 so Sample index (e) (f) o 50 .00 150 200 Sample index (g) (h) FIGURE 6. 6. (a) (a) Original Original 256-sample 256-sample FBMFBM function /f withwith H == 0.3.0.3. (b) Synthesized FIF FIF withwith V V = = -0.6.-0.6. (c) FIF with V = 0.6.0.6. (d)(d) FractalFractal dimensiondimension ofof FIFs fvfv withwith varyingvarying V.V. (e) Hausdorff distances betweenbetween /f andand all fv.fv . (f)(f) Ltp distances distances betweenbetween /f and all ffv.v . (g)(g) FIFFIF with VV == -0.8.-0.8. (h)(h) FIF with VV == -0.1.-0.1.