measuring fractal dimension: morphological estimates and...

MEASURING FRACTAL DIMENSION:MORPHOLOGICAL ESTIMATES AND ITERATIVE OPTIMIZATION

Petros Maragos

Division of Applied SciencesHarvard University

Cambridge, MA 02138

and Fang -Kuo Sun

Abstract

The Analytic Sciences Corp.55 Walkers Brook Drive

Reading, MA 01867

An important characteristic of fractal signals is their fractal dimension. For arbitrary fractals, anefficient approach to evaluate their fractal dimension is the covering method. In this paper we unifymany of the current implementations of covering methods by using morphological operations with varyingstructuring elements. Further, in the case of parametric fractals depending on a parameter that is inone -to -one correspondence with their fractal dimension, we develop an optimization method, whichstarts from an initial estimate and by iteratively minimizing a distance between the original functionand the class of all such functions, spanning the quantized parameter space, converges to the true fractaldimension.

1 IntroductionFractals are mathematical sets with a high degree of geometrical complexity that can model many naturalphenomena [7]. Examples include physical objects such as clouds, mountains, trees and coastlines, as wellas image intensity signals that emanate from them (assuming certain restrictions on the object's reflectanceand illumination [12]). Although, the fractal images are the most popularized class of fractals due to theirfantastic resemblance with natural scenes, there are also numerous natural processes described by time -series measurements (e.g., 1/ f -noises, econometric and demographic data, pitch variations in music signals)that are fractals [7,17]. The one -dimensional signals f (t) representing these measurements are fractals inthe sense that their graph G(f) = {(t, y) : y = f (t)) is a fractal set. Thus, modeling fractal signals is ofgreat interest in signal and image analysis.

An important characteristic of fractals useful for their description and classification is their fractaldimension D, which exceeds their topological dimensión. Intuitively, D measures the degree of theirboundary fragmentation or roughness. It makes meaningful the measurement of metric aspects of fractalsets such as their length or area. Specifically, given a measure unit (a "yardstick ") of length e, the lengthL(e) of a curve at scale e is equal to the number of yardsticks that can fit sequentially along the curvetimes e. For a fractal curve, L(e) increases without limit when e decreases and follows the proportionalitylaw L(e) cx el -D. Taking logarithms yields

log[L(e)] = (1 - D) log(e) -I- constant (1)

Hence, D can be measured from the slope of the (log L(e), loge) data.In this paper we deal with the problem of estimating the fractal dimension of "topologically one-

dimensional" (1 -dim) signals with discrete argument. (Extending most of the ideas in this paper to 2 -dimsignals is very straightforward and hence omitted.) We start in Section 2.1 and Section 2.2 with a briefsurvey of some existing methods, some of which are general whereas others apply only to special classes offractals. Section 2.3 focuses on the covering method, a general and efficient approach to compute the fractaldimension of arbitrary fractals. We unify and extend many of the current digital implementations of the

416 / SPIE Vol. 1199 Visual Communications and Image Processing IV (1989)

MEASURING FRACTAL DIMENSION: MORPHOLOGICAL ESTIMATES AND ITERATIVE OPTIMIZATION

Petros Maragos and Fang-Kuo Sun

Division of Applied Sciences The Analytic Sciences Corp.Harvard University 55 Walkers Brook Drive

Cambridge, MA 02138 Reading, MA 01867

Abstract

An important characteristic of fractal signals is their fractal dimension. For arbitrary fractals, an efficient approach to evaluate their fractal dimension is the covering method. In this paper we unify many of the current implementations of covering methods by using morphological operations with varying structuring elements. Further, in the case of parametric fractals depending on a parameter that is in one-to-one correspondence with their fractal dimension, we develop an optimization method, which starts from an initial estimate and by iteratively minimizing a distance between the original function and the class of all such functions, spanning the quantized parameter space, converges to the true fractal dimension.

1 Introduction

Fractals are mathematical sets with a high degree of geometrical complexity that can model many natural phenomena [7], Examples include physical objects such as clouds, mountains, trees and coastlines, as well as image intensity signals that emanate from them (assuming certain restrictions on the object's reflectance and illumination [12]). Although, the fractal images are the most popularized class of fractals due to their fantastic resemblance with natural scenes, there are also numerous natural processes described by time- series measurements (e.g., l//-noises, econometric and demographic data, pitch variations in music signals) that are fractals [7,17]. The one-dimensional signals f(t) representing these measurements are fractals in the sense that their graph G(f) — {(*,y) : y = f(t)} ls a fractal set. Thus, modeling fractal signals is of great interest in signal and image analysis.

An important characteristic of fractals useful for their description and classification is their fractal dimension D, which exceeds their topological dimension. Intuitively, D measures the degree of their boundary fragmentation or roughness. It makes meaningful the measurement of metric aspects of fractal sets such as their length or area. Specifically, given a measure unit (a "yardstick") of length e, the length L(e) of a curve at scale e is equal to the number of yardsticks that can fit sequentially along the curve times e. For a fractal curve, L(e) increases without limit when e decreases and follows the proportionality law L(e) oc el ~ D . Taking logarithms yields

log[L(e)] = (1 - £>)log(e) + constant (1)

Hence, D can be measured from the slope of the (log L(e),loge) data.In this paper we deal with the problem of estimating the fractal dimension of "topologically one-

dimensional" (1-dim) signals with discrete argument. (Extending most of the ideas in this paper to 2-dim signals is very straightforward and hence omitted.) We start in Section 2.1 and Section 2.2 with a brief survey of some existing methods, some of which are general whereas others apply only to special classes of fractals. Section 2.3 focuses on the covering method, a general and efficient approach to compute the fractal dimension of arbitrary fractals. We unify and extend many of the current digital implementations of the


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covering method by using morphological erosions and dilations with varying structuring elements. (Theerosions and dilations are the basic operators of signal and image analysis by mathematical morphology[13].) We shall refer to these unified algorithms as morphological estimates. In addition, to its conceptualusefulness as a unifying theme, the morphological approach has two practical advantages: 1) it reduces thedimensionality of the processed data from two to one, and 2) it is simple to implement. In Section 2.4 wediscuss two implementations of the variation method for estimating fractal dimension, introduced in [4].The variation method can be interpreted as stemming from a special case of the morphological approachto implement the covering method.

Both the covering and variation methods apply to arbitrary fractals. However, their actual performancecan be tested on parametric fractals, e.g., fractals depending on a single parameter that is in one -to -onecorrespondence with their fractal dimension D. Fortunately, there are numerous classes of such parametricfractal signals and related algorithms for their synthesis. Two examples used in this paper are the randomfunctions of fractional Brownian motion (FBM) [8] and the deterministic Weierstrass -Mandelbrot cosine(WMC) functions [2]. Although the performance of the covering and variation method is satisfactory forsome cases (i.e., yields reasonable estimation errors), if one is free to vary arbitrarily important parametersof the problem such as D or the signal's duration, then, as our experiments on FBM and WMC functionsindicate, their performance falls drastically in many instances. Thus in Section 3 we present the maincontribution of this paper, which is both a very effective method (i.e., it yields practically zero estimationerrors) to estimate fractal dimension and a new way of looking at this problem. It is somewhat restrictedsince it applies only to parametric fractals, but the large number of such parametric classes and theirpractical applicability motivates well our new method. Our basic idea is as follows: So far researchers startfrom an original fractal signal of true fractal dimension D, use various approaches to derive an estimate,D *, of D, and are content if the estimation error JD -D *I is small. This criterion, however, does not sayanything about how "close" is the original fractal signal to some other fractal signal of true dimension D *.Further, any degree of "closeness" should be somehow compatible with laws of visual perception since thefractal dimension is a geometrical attribute. In our approach, from an initial morphological estimate D*we synthesize the corresponding fractal function f* . Then by searching in the parameter space D we solvea nonlinear optimization problem, where a distance is iteratively minimized between the original f andeach new iteratively synthesized f*. The process terminates when we reach a local or global minimum.We call this the Iterative Optimization method. As distance metrics we have used standard 4, p = 1, 2, oo,metrics. We also introduced a Hausdorff -type distance for this iterative optimization. Our motivation forusing this distance is that it is more suitable than 4 distances to compare two signals in terms of theiroverall geometrical structure, which is a signal attribute that fractal methods attempt to capture.

Finally we conclude with an application of the above ideas to determine the optimum fractal functionfor signal interpolation among a parametric class of deterministic fractal functions stemming from thetheory of Iterated Function Systems [1].

2 Covering and Variation Methods2.1 General Methods

Descriptions of various approaches to measure fractal dimension can be found in [7,17,4]. Mandelbrot[7] defines formally the fractal dimension of a set as its Hausdorff -Besicovitsch dimension DHB. Thus, asubset of RE is fractal if DHB strictly exceeds its topological dimension DT. Some very closely relateddimensions are the Minkowski -Bouligand dimension DMB [10,3] and the box dimension DB [3], which areobtained in a different way but are identical; i.e., DMB = DB. In addition, in most cases of practicalinterest DHB = DMB. In this paper we focus on the Minkowski -Bouligand dimension, which we shallhenceforth call fractal dimension D, because: 1) it is closely related to DHB and hence able to quantify the

SPIE Vol. 1199 Visual Communications and Image Processing IV (1989) / 417

covering method by using morphological erosions and dilations with varying structuring elements. (The erosions and dilations are the basic operators of signal and image analysis by mathematical morphology [13].) We shall refer to these unified algorithms as morphological estimates. In addition, to its conceptual usefulness as a unifying theme, the morphological approach has two practical advantages: 1) it reduces the dimensionality of the processed data from two to one, and 2) it is simple to implement. In Section 2.4 we discuss two implementations of the variation method for estimating fractal dimension, introduced in [4]. The variation method can be interpreted as stemming from a special case of the morphological approach to implement the covering method.

Both the covering and variation methods apply to arbitrary fractals. However, their actual performance can be tested on parametric fractals, e.g., fractals depending on a single parameter that is in one-to-one correspondence with their fractal dimension D. Fortunately, there are numerous classes of such parametric fractal signals and related algorithms for their synthesis. Two examples used in this paper are the random functions of fractional Brownian motion (FBM) [8] and the deterministic Weierstrass-Mandelbrot cosine (WMC) functions [2]. Although the performance of the covering and variation method is satisfactory for some cases (i.e., yields reasonable estimation errors), if one is free to vary arbitrarily important parameters of the problem such as D or the signal's duration, then, as our experiments on FBM and WMC functions indicate, their performance falls drastically in many instances. Thus in Section 3 we present the main contribution of this paper, which is both a very effective method (i.e., it yields practically zero estimation errors) to estimate fractal dimension and a new way of looking at this problem. It is somewhat restricted since it applies only to parametric fractals, but the large number of such parametric classes and their practical applicability motivates well our new method. Our basic idea is as follows: So far researchers start from an original fractal signal of true fractal dimension D, use various approaches to derive an estimate, D*, of D, and are content if the estimation error \D — D*\ is small. This criterion, however, does not say anything about how "close" is the original fractal signal to some other fractal signal of true dimension D*. Further, any degree of "closeness" should be somehow compatible with laws of visual perception since the fractal dimension is a geometrical attribute. In our approach, from an initial morphological estimate D* we synthesize the corresponding fractal function /*. Then by searching in the parameter space D we solve a nonlinear optimization problem, where a distance is iteratively minimized between the original / and each new iteratively synthesized /*. The process terminates when we reach a local or global minimum. We call this the Iterative Optimization method. As distance metrics we have used standard lp, p = l,2,oo, metrics. We also introduced a Hausdorff-type distance for this iterative optimization. Our motivation for using this distance is that it is more suitable than ip distances to compare two signals in terms of their overall geometrical structure, which is a signal attribute that fractal methods attempt to capture.

Finally we conclude with an application of the above ideas to determine the optimum fractal function for signal interpolation among a parametric class of deterministic fractal functions stemming from the theory of Iterated Function Systems [1].

2 Covering and Variation Methods

2.1 General Methods

Descriptions of various approaches to measure fractal dimension can be found in [7,17,4]. Mandelbrot [7] defines formally the fractal dimension of a set as its HausdorfF-Besicovitsch dimension DJJB* Thus, a subset of R*7 is fractal if DJJB strictly exceeds its topological dimension DT- Some very closely related dimensions are the Minkowski-Bouligand dimension DMB [10>3] and the box dimension DB [3], which are obtained in a different way but are identical; i.e., DMB = DB- In addition, in most cases of practical interest DHB = DMB- In this paper we focus on the Minkowski-Bouligand dimension, which we shall henceforth call fractal dimension D, because: 1) it is closely related to DJJB and hence able to quantify the



fractal aspects of a signal, 2) it coincides with DHB in many cases; 3) efficient algorithms exist to computeit; 4) it will be applied to discrete signals where most approaches can yield only approximate results.

Let G be a planar curve whose dimension 1 < D < 2 is to estimate. Although, DB = DMB in thecontinuous case, they correspond to the following two different algorithms (with different performances) inthe discrete case.

Box counting method: Partition the plane with squares of side e and count the number N(c) of squaresthat intersect the curve. Then D = limE ,o log[N(E)]/ log(1 /E). If an equivalent length L(e) = EN(e) isdefined by dividing the total area of these squares by e, then L(e) behaves as in (1).

Minkowski Cover method: This is based conceptually on Minkowski's idea of finding the length ofirregular curves: dilate them with disks of radius e (and thus create a "Minkowski sausage "), find the areaA(E) of the dilated set, and set its length equal to lim6_.0 L(e), where L(E) = A(E) /2E. If G is a fractal,then L(e) behaves as in (1).

Generalized Cover method /4): This method unifies the box counting and the Minkowski cover methodby viewing them as special cases of a generalized "cover ". This cover C(e) is the union of sets B from afamily B such that: GC C(e), and each B E B intersects G, is homeomorphic to the disk and has diameteron the order of e. If A(e) is the area of the cover, then it was shown in [16,4] that

logA(e)

= D log(1) + constantE2 E

(2)

Thus, in all the above methods, D can be estimated by fitting a straight line to and measuring theslope of the plot of the data (log L(E), log e) or, equivalently, of the data (log[A(E) /E2], log 1 /c).

2.2 Parametric Fractals and Special MethodsThe FBM and WMC fractal functions are the two primary classes of parametric test signals on which weshall evaluate the various methods.

The FBM BH(t) with parameter 0 < H < 1 is a time -varying random function with stationary,Gaussian -distributed, and statistically self-affine increments; the latter means that [BH(t + T) - B(t)] isstatistically indistinguishable from r- H[B(t+ TT) - B(t)] for any t and any r > O. Their power spectrumis S(w) oc 1 /w2H +1 Hence, an efficient algorithm [17] to synthesize an FBM is to create a random sampledspectrum whose average magnitude is 1 /wH +o.5 and its random phase is uniformly distributed over [0, 24In our experiments we synthesized and then transformed this spectrum via a 2048 -point inverse FFT toobtain the FBM sequence from which we retained the first (N + 1) samples.

Another example of parametric fractals is the class of WMC functions00

yyH(t) = E ry-nH[1 - cos(rynt)] , 0 < H < 1 ,

n= -co(3)

where ry > 1. In our experiments we created WMC sequences by sampling t E [0, 1] at (N + 1) equidistantpoints, using a fixed ry = 5, and by truncating the infinite series so that the remaining finite number ofterms incurs an approximation error < 10 -6. The fractal dimension D of both FBM and WMC functionsis in one -to-one correspondence with H because D = 2 - H.

Some special methods (not covered in this paper due to their very narrow applicability) to measure Dfor FBM's include: i) Fitting a straight line to the data (log S(w), log w) and measuring the slope yieldsH and hence D [12]. ii) The statistical self -affinity of FBM's yields a power scaling law for many of itsmoments; linear regression on these data can measure H and hence D [12]. iii) A maximum likelihoodmethod for estimating the H of fractional Gaussian noise (derivatives of FBMs) was developed in [6].(Note, that the spectral method (i) could also be applied to WMC's because their spectrum behaves like1/w2H +1 too.)


fractal aspects of a signal, 2) it coincides with DJJB in many cases; 3) efficient algorithms exist to compute it; 4) it will be applied to discrete signals where most approaches can yield only approximate results.

Let G be a planar curve whose dimension 1 < D < 2 is to estimate. Although, DB = DMB in the continuous case, they correspond to the following two different algorithms (with different performances) in the discrete case.

Box counting method: Partition the plane with squares of side e and count the number N(e) of squares that intersect the curve. Then D = lim _>olog[N(e)]/log(l/e). If an equivalent length L(e) = eN(e) is defined by dividing the total area of these squares by e, then L(e) behaves as in (1).

Minkowski Cover method: This is based conceptually on Minkowski's idea of finding the length of irregular curves: dilate them with disks of radius e (and thus create a "Minkowski sausage"), find the area A(e) of the dilated set, and set its length equal to lim _»o£(e), where L(c) = A(e)/2e. If G is a fractal, then L(e) behaves as in (1).

Generalized Cover method [4]: This method unifies the box counting and the Minkowski cover method by viewing them as special cases of a generalized "cover". This cover C(c) is the union of sets B from a family B such that: GC G(e), and each B G B intersects G, is homeomorphic to the disk and has diameter on the order of £. If A(e) is the area of the cover, then it was shown in [16,4] that

log ^- = D log(-) + constant (2)

Thus, in all the above methods, D can be estimated by fitting a straight line to and measuring the slope of the plot of the data (log L(e),loge) or, equivalently, of the data (log[A(e)/e2 ],logl/e).

2.2 Parametric Fractals and Special Methods

The FBM and WMC fractal functions are the two primary classes of parametric test signals on which we shall evaluate the various methods.

The FBM Bfj(t) with parameter 0 < H < 1 is a time-varying random function with stationary, Gaussian-distributed, and statistically self-affine increments; the latter means that [Bjf(t + T) — B(t)] is statistically indistinguishable from r~H [B(t+ rT) — B(t)] for any t and any r > 0. Their power spectrum is 5(o;) oc l/o;2^"1" 1 . Hence, an efficient algorithm [17] to synthesize an FBM is to create a random sampled spectrum whose average magnitude is l/o;^"1" 0' 5 and its random phase is uniformly distributed over [0,2?r]. In our experiments we synthesized and then transformed this spectrum via a 2048-point inverse FFT to obtain the FBM sequence from which we retained the first (N + 1) samples.

Another example of parametric fractals is the class of WMC functions

WH (t) = f) 7-nff [l - cos(7nt)] , 0 < H < 1 , (3)n= oo

where 7 > 1. In our experiments we created WMC sequences by sampling t G [0,1] at (N + 1) equidistant points, using a fixed 7=5, and by truncating the infinite series so that the remaining finite number of terms incurs an approximation error < 1CT 5 . The fractal dimension D of both FBM and WMC functions is in one-to-one correspondence with H because D = 2 - H.

Some special methods (not covered in this paper due to their very narrow applicability) to measure D for FBM's include: i) Fitting a straight line to the data (log S (a;),log u) and measuring the slope yields H and hence D [12]. ii) The statistical self-affinity of FBM's yields a power scaling law for many of its moments; linear regression on these data can measure H and hence D [12], iii) A maximum likelihood method for estimating the H of fractional Gaussian noise (derivatives of FBMs) was developed in [6]. (Note, that the spectral method (i) could also be applied to WMC's because their spectrum behaves like

too.)



2.3 Covering MethodIn this paper we deal not with arbitrary curves but only with finite -length signals f (t), 0 < t < T, inwhich case the curve G of the previous discussion becomes the graph G(f) of f . In this section we shallfocus on a generalized version of the Minkowski cover method. That is, one generalized cover consistentwith the definition of the "General Cover" method can be obtained as follows: given any compact convexplanar set B, form positive homothetics eB = {El) : b E B }, and define the cover C(E) as the union ofsets in B , which contains all vector translates eB + z = {et + z; b E B} of eB centered at points z of thegraph G(f). In the formalism of mathematical morphology this is equivalent to simply dilating G(f) witha structuring element eB:

C(e) = U eB + z = G(f) ® eB (4)zEG(f)

Then (2) applies. The Minkowski cover method corresponds to using a disk for B. The "horizontalstructuring element method" in [4] corresponds to using a line segment for B parallel to the domain of f.

In [4] the implementations of the Minkowski covering, box counting, and horizontal structuring methodswere done by viewing G(f) as a binary image signal and dilating this binary image. This, however, two -dimensional processing of a 1 -dim signal, on the one hand is unneseccary and on the other hand squaresthe requirements in storage space and time complexity of implementing the covering method. Namely, let

Ypo( f ) = { (t, y) E R2 : y < f (t) } (5)

be the ypograph off (also known as "umbra" in morphology). Let also (f ED g)(t) = supz{ f (x) +g(t -x)} and(f e g)(t) = infz{ f (x) - g(x - t)} be respectively the function dilation and erosion of f by a structuringfunction g with compact support. Then, (see [13,15,5,9] for properties of the ypographs), if we ignorethe end effects around t = 0 and t = T, the dilated graph C(e) can alternatively be obtained as the setdifference between the ypographs of the dilated and eroded function; i.e., C(e) ... Ypo(f ®eg) \ Ypo(f e eg),where g is such that

Ypo(g) = {(t, y) : y < b, (t, b) E B} . (6)

Further, Ypo(eg) = EYpo(g). Then, the cover area will be A(e) fo (f ® eg -f e eg) (t)dt. Thus, instead ofcreating the cover of a one -dimensional signal by expanding its graph in the plane (which means processinga two -dimensional signal), the original one -dimensional signal can be filtered with an erosion and a dilationsystem.

Putting all the above ideas together leads to what we shall henceforth call the covering method forestimating the fractal dimension of a discrete -argument finite -length signal fin], n = 0, 1, ... , N. Thisconsists of the following steps:1) Select a structuring function g[n] with a 3- sample support (unit size) such that (6) is true, i.e., thegraph of g is the upper boundary of B. B should be a unit -size discrete disk. For example, if B is a 5 -pixelrhombus, then the structuring function is shaped like a triangle, defined by

gt[ -1] = gt[1] = 0 , gt[0] = h > 0, and gt[n] = -oo `dn # -1, 0,1. (7)

If B is the 3 x 3 -pixel square, the corresponding structuring function is shaped like a rectangle:

gr [n] = h > 0 n = -1,0,1, and gr [n] = -oo bn 0 -1, 0,1. (8)

If h = 0, the structuring functions gt and g,. become the same flat (binary) structuring element.2) Perform the dilations and erosions of f by eg at discrete scales e = 1, 2, ... , emax. For integer e we defineE as the E -fold dilation of g with itself. Then f ED eg and f e Eg can be implemented recursively:

f ®(e +l)g= (fe eg) eg , fe(E +1)g= (feeg)eg (9)


2.3 Covering Method

In this paper we deal not with arbitrary curves but only with finite-length signals /(*), 0 < t < T, in which case the curve G of the previous discussion becomes the graph G(f) of f. In this section we shall focus on a generalized version of the Minkowski cover method. That is, one generalized cover consistent with the definition of the "General Cover" method can be obtained as follows: given any compact convex planar set 5, form positive homothetics eB = {eb : b £ B}, and define the cover C(e) as the union of sets in B, which contains all vector translates eB + z = {eb + z\ b E B} of eB centered at points z of the graph G(f). In the formalism of mathematical morphology this is equivalent to simply dilating G(f) with a structuring element eB:

C(c) = |J eB + z = G(f)®eB (4)

Then (2) applies. The Minkowski cover method corresponds to using a disk for B. The "horizontal structuring element method" in [4] corresponds to using a line segment for B parallel to the domain of /.

In [4] the implementations of the Minkowski covering, box counting, and horizontal structuring methods were done by viewing G(f) as a binary image signal and dilating this binary image. This, however, two- dimensional processing of a 1-dim signal, on the one hand is unneseccary and on the other hand squares the requirements in storage space and time complexity of implementing the covering method. Namely, let

Ypo(/) = {(*,y)eR2 :y </(*)} (5)

be the ypograph of f (also known as "umbra" in morphology). Let also (f@g)(i) = supz {/(z)+0(£-:r)} and (/ © 0)00 = infz{/(^) ~ 9(x - *)} ke respectively the function dilation and erosion of f by a structuring function g with compact support. Then, (see [13,15,5,9] for properties of the ypographs), if we ignore the end effects around t = 0 and t = T, the dilated graph C(e) can alternatively be obtained as the set difference between the ypographs of the dilated and eroded function; i.e., C(e) w Ypo(/©6<7)\Ypo(/0e0), where g is such that

Ypo(g) = {(t,y):y<b,(t,b)eB}. (6)

Further, Ypo(e0) = eYpo(0). Then, the cover area will be A(e) » /0 (f®eg - fQeg)(i)dt. Thus, instead of creating the cover of a one-dimensional signal by expanding its graph in the plane (which means processing a two-dimensional signal), the original one-dimensional signal can be filtered with an erosion and a dilation system.

Putting all the above ideas together leads to what we shall henceforth call the covering method for estimating the fractal dimension of a discrete-argument finite-length signal /[n], n — 0,1,..., N. This consists of the following steps:1) Select a structuring function g[n] with a 3-sample support (unit size) such that (6) is true, i.e., the graph of g is the upper boundary of B. B should be a unit-size discrete disk. For example, if B is a 5-pixel rhombus, then the structuring function is shaped like a triangle, defined by

gt [-l] = gt [l] = 0 , gt [Q] = h>0, and gt [n] = -oo Vn ^ -1,0, 1. (7)

If B is the 3 x 3-pixel square, the corresponding structuring function is shaped like a rectangle:

9r[n] = h>0 n= -1,0,1, and gr [n] = -oo Vn ^ -1,0,1. (8)

If h = 0, the structuring functions gt and gr become the same flat (binary) structuring element.2) Perform the dilations and erosions of / by eg at discrete scales e = 1,2,..., emax . For integer e we define e as the e-fold dilation of g with itself. Then / © eg and f Q eg can be implemented recursively:

f®(€ + l)g=(f®eg)®g , / 0 (e + l)g = (f 0 eg) 0 g (9)



The dashed lines in Fig. la show these erosions /dilations by g,. at scales e = 10, 20.3) Compute the areas A[E] = >ñ o((f ® Eg) - (f e Eg))[n].4) Fit a straight line using least- squares linear regression to the plot of (log A[E] /2, log 1 /e). The slopeof this line gives us the fractal dimension of f . For "real world" signals with some fractal structure, theassumption of exact self -similarity at all scales is not true. Hence, as in [11], instead of a global dimension,we estimate the local fractal dimension LFD(E), which for each e is equal to the slope of a line segmentfitted to the log -log plot of (2) over a moving window [e, c + 91 of 10 scales, where e = 1, 2, ..., Emaz - 9.

Among previous approaches, the work in [11] and [14] (transposed to 1 -dim signals) corresponds tousing gt with h = 1. The "horizontal structuring element method" in [4] corresponds to using h = 0.

Figure 1 shows results from experiments on evaluating the fractal dimension of FBM and WMC signalsvia the covering method using g,. with three different heights h. The shape of the structuring function g isnot very crucial; gt yields a slightly finer multiscale area distribution than gr. The height h, however, playsan important role neglected by previous researchers. Although h does not affect the continuous versionof the log -log plot (2), in the discrete case large h will sample this plot very coarsely and produce poorresults. Thus small h are preferred for finer multiscale covering area distributions. However, the smaller his, the more computations are needed to span a given signal's range. A good practical rule is to set h lessthan or equal to the signal's dynamic range divided by the number of its samples. This rule attempts toconsider the quantization grid in the domain and range of the function as square as possible. Note that ifh = 0, the erosions /dilations by g can be performed faster.

2.4 Variation MethodAs Figs. 1b,ld show, even with a good selection of h, the covering method estimates for LFD(1) (i.e.,the local fractal dimension for the first position of the scale window -first 10 scales) are not accurate.This is partly due to the quantization of the signal's domain and hence the small number of availablesamples to compute (at small scales) the local minima /maxima, as observed in [4]. Thus at E = 1 thereis a neighborhood of only 3 samples for the min /max operations to create a cover. The variation methodof Dubuc et al. [4] attempts to correct this problem by re- arranging the original signal samples f[n],0 < n < N, to retain a smaller number of samples 0 < m < M _< N and form the dilations uE [m]and erosions be [m] at all scales e, and by using essentially the covering method with a binary structuringfunction (i.e., h = 0) that has 3- sample domain for e > 1 and 2(N /M) + 1- sample domain for e = 1. Wehave implemented the concept of the variation method in the following two different ways:

Variation by Signal Decimation: Let d be an integer variable decimation factor where 1 < d < dmax =LN /2Emaxj and [xi denotes the greatest integer < x. Erosion /dilation values are computed only everyother d -th original sample. Thus, for 0 < m < M = LN /d],

ul[m] = max {f[n]:(m- 1)d<n<(m +1)d }, a =1u[m] = max(ue_i[m - 1], uE_i[m + 1]) , E = 2, 3, ..., Emaz

At the ends m = 0, M, the local max takes place only over the available samples. The erosions be [m] aregiven from the formulae (10) by replacing max with min and u with b. The resulting LFD(E) depends on dand hence on M. Two optimal values of d can be found by searching over all permissible values and findingthat d which results in a better least- squares line fit to the log -log plot either over the first 10 scales orover all emax scales. Which of these two optimal values of d to use depends of course on the application.

Variation by Signal Truncation: Let M +1 be the variable number of samples to retain after eliminatingN -M samples from the original signal f , where 2Emaz = Mmin < M < N. Erosion /dilation values arecomputed only at M +1 original samples. For each M, let d = [N /MJ be a integer ratio factor. ThenuE[m] and bE[m] are computed for all e exactly as for the variation by decimation method with d interpreted

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420 / SPIE Vol. 1199 Visual Communications and Image Processing 1V (1989)

The dashed lines in Fig. la show these erosions/ dilations by gr at scales € = 10,20.3) Compute the areas A[c] = E£=O((/ © <tf) ~ (f Q <#)) [*].4) Fit a straight line using least-squares linear regression to the plot of (log A[e]/e2 ,log 1/e). The slope of this line gives us the fractal dimension of /. For "real world" signals with some fractal structure, the assumption of exact self-similarity at all scales is not true. Hence, as in [11], instead of a global dimension, we estimate the local fractal dimension LFD(e), which for each e is equal to the slope of a line segment fitted to the log-log plot of (2) over a moving window [e,e + 9] of 10 scales, where e = 1,2, ...,emaz - 9.

Among previous approaches, the work in [11] and [14] (transposed to 1-dim signals) corresponds to using gt with h = 1. The "horizontal structuring element method" in [4] corresponds to using h = 0.

Figure 1 shows results from experiments on evaluating the fractal dimension of FBM and WMC signals via the covering method using gr with three different heights h. The shape of the structuring function g is not very crucial; gt yields a slightly finer multiscale area distribution than gr . The height /i, however, plays an important role neglected by previous researchers. Although h does not affect the continuous version of the log-log plot (2), in the discrete case large h will sample this plot very coarsely and produce poor results. Thus small h are preferred for finer multiscale covering area distributions. However, the smaller h is, the more computations are needed to span a given signal's range. A good practical rule is to set h less than or equal to the signal's dynamic range divided by the number of its samples. This rule attempts to consider the quantization grid in the domain and range of the function as square as possible. Note that if h = 0, the erosions/dilations by g can be performed faster.

2.4 Variation Method

As Figs. lb,ld show, even with a good selection of h, the covering method estimates for LFD(l) (i.e., the local fractal dimension for the first position of the scale window-first 10 scales) are not accurate. This is partly due to the quantization of the signal's domain and hence the small number of available samples to compute (at small scales) the local minima/ maxima, as observed in [4]. Thus at e = 1 there is a neighborhood of only 3 samples for the min/max operations to create a cover. The variation method of Dubuc et al. [4] attempts to correct this problem by re-arranging the original signal samples /[n], 0 < n < N, to retain a smaller number of samples 0 < m < M < N and form the dilations u€ [m] and erosions b€ [m] at all scales e, and by using essentially the covering method with a binary structuring function (i.e., h = 0) that has 3-sample domain for e > 1 and 2(N/M) + 1-sample domain for e = 1. We have implemented the concept of the variation method in the following two different ways:

Variation by Signal Decimation: Let d be an integer variable decimation factor where 1 < d < dmax = [N/2cmax \ and [x\ denotes the greatest integer < x. Erosion/dilation values are computed only every other cf-th original sample. Thus, for 0 < m < M = [N/d\,

U![m] = max{/[n]:(m-l)d<n<(m + l)d}, e=lue [m] = max(u _i[m- l],u _i[m + l]) , e = 2,3, ...,emaz . l '

At the ends m = 0, M, the local max takes place only over the available samples. The erosions b€ [m] are given from the formulae (10) by replacing max with min and u with 6. The resulting LFD(e) depends on d and hence on M . Two optimal values of d can be found by searching over all permissible values and finding that d which results in a better least-squares line fit to the log-log plot either over the first 10 scales or over all emax scales. Which of these two optimal values of d to use depends of course on the application.

Variation by Signal Truncation: Let M+l be the variable number of samples to retain after eliminating N - M samples from the original signal /, where 2emaz = Mmt-n < M < N. Erosion/dilation values are computed only at M + 1 original samples. For each M, let d = [N/M J be a integer ratio factor. Then u [m] and 6 [m] are computed for all e exactly as for the variation by decimation method with d interpreted



now as a ratio rather than as a decimation factor. (Note that for several different M the integer d mayremain the same.)

Figure 2 shows the results of estimating the LFD(E) for FBM and WMC signals (with D = 1.5,N = 500, emax = 25) by using both the variation by decimation and by truncation methods, which hadsimilar performance. Both variation methods perform better than the covering method when an optimumd or M is selected. However, the variation by truncation is much more computionally intense than thedecimation method. For example, with N = 500 and emax = 25, the decimation method searches overd = 1, ..,10 and hence over 10 values of M, whereas the truncation method searches over all M with50 < M < 500. Note that the covering method with h = 0 becomes identical with the variation bydecimation method if d = 1, and with the variation by truncation method if M = N.

3 Iterative Optimization MethodAssume a class of parametric fractal signals fp parameterized by a parameter P that is related to theirfractal dimension through an invertible function D = &(P). For example, for FBM's or WMC's theparameter P is H and D = tl)(H) = 2 - H. Our new approach to measure the fractal dimension of sucha signal fp consists of the following steps: (1) We use a simple and fast morphological approach (i.e., thecovering method with h = 0) to come up with an initial estimate, D *, of the true D. (2) We compute somedistance between the original fractal function fp and another fractal fp. which was synthesized to havedimension exactly D* = t,(P *). (3) By using nonlinear optimization, we search in the parameter space ofP (or equivalently of D) values of the chosen class of fractals by synthesizing fractals whose parameterscorrespond to a fractal dimension D* and computing their distances from the original fractal until thiscycle converges to a local or global minimum in the parameter space. In this way the resulting fractaldimension will correspond to a fractal function which is also close (with respect to the specific distance)to the original function. We call this the Iterative Optimization method. As distance metrics we can usestandard 4, p = 1, 2, oo, metrics.

DD ¡¡

`'P(f1, f2) = I Ifl - 12I Ip - ( L lfll¡¡

n] - f2ln.])PP

n- o0

Further, we also introduce a definition of a Hausdorff metric for this iterative optimization. Our motivationfor using this Hausdorff distance is that it is better suitable than 4 distances to compare the geometricalstructural differences (peak /valley distributions) between two signals in a way that agrees with humanvisual perception. The Hausdorff metric was so far defined only for sets. Here we extend its definition tofunctions and provide a morphological algorithm for its computation. Thus, given two compact sets A1i A2their Hausdorff metric can be computed as

H(A1, A2) = inf{ : A1C A2 ® ED and A2C Al e ED }, (12)

where ED is a disk of radius e. If Al and A2 become the ypographs of two functions fi and f2 with compactsupports, then we define their ypograph -based Hausdorff distance as

HY(f1, f2) = H[Ypo(fl), Ypo(f2)] = inf {e : fl S f2 ®Eg and 12 < fi ® eg} (13)

Hy compares fi and 12 in terms of their protrusions (peaks). If we want a distance sensitive both tothe peaks and the valleys, then we can form the sum Hs(fl, f2) = HY(fi, 12) - He(fi, f2), which adds toHy(f1 ,f2) the Hausdorff distance He(fi, f2) = Hy(c - fl, c - f2) of their epigraphs, i.e., of the negation -complements of fl and 12 where c is some constant function.

SPIE Vol 1199 Visual Communications and Image Processing IV (1989) / 421

now as a ratio rather than as a decimation factor. (Note that for several different M the integer d may remain the same.)

Figure 2 shows the results of estimating the LFD(e) for FBM and WMC signals (with D — 1.5, N = 500, emax — 25) by using both the variation by decimation and by truncation methods, which had similar performance. Both variation methods perform better than the covering method when an optimum d or M is selected. However, the variation by truncation is much more computionally intense than the decimation method. For example, with N = 500 and emax = 25, the decimation method searches over d — 1,..,10 and hence over 10 values of M, whereas the truncation method searches over all M with 50 < M < 500. Note that the covering method with h = 0 becomes identical with the variation by decimation method if d = 1, and with the variation by truncation method if M = N.

3 Iterative Optimization Method

Assume a class of parametric fractal signals fp parameterized by a parameter P that is related to their fractal dimension through an invertible function D = ij>(P). For example, for FBM's or WMC's the parameter P is H and D = ifi(H) = 2 H. Our new approach to measure the fractal dimension of such a signal fp consists of the following steps: (1) We use a simple and fast morphological approach (i.e., the covering method with h = 0) to come up with an initial estimate, /?*, of the true D. (2) We compute some distance between the original fractal function fp and another fractal /p* which was synthesized to have dimension exactly D* = ij>(P*). (3) By using nonlinear optimization, we search in the parameter space of P (or equivalently of D) values of the chosen class of fractals by synthesizing fractals whose parameters correspond to a fractal dimension D* and computing their distances from the original fractal until this cycle converges to a local or global minimum in the parameter space. In this way the resulting fractal dimension will correspond to a fractal function which is also close (with respect to the specific distance) to the original function. We call this the Iterative Optimization method. As distance metrics we can use standard £p, p = 1, 2, oo, metrics.

= ll/i - Mlp = (AW - /2 H)P (11)

Further, we also introduce a definition of a Hausdorff metric for this iterative optimization. Our motivation for using this Hausdorff distance is that it is better suitable than ip distances to compare the geometrical structural differences (peak/valley distributions) between two signals in a way that agrees with human visual perception. The Hausdorff metric was so far defined only for sets. Here we extend its definition to functions and provide a morphological algorithm for its computation. Thus, given two compact sets AI, A2 their Hausdorff metric can be computed as

H(Ai, A2 ) = inf{e : AiC A2 0 eD and A2 C AI © eD}, (12)

where eD is a disk of radius e. If AI and A2 become the ypographs of two functions /i and /2 with compact supports, then we define their ypograph-based Hausdorff distance as

Hy(/i,/2 ) = JT[Ypo(/i), Ypo(/2 )] = inf{6 : /x < /2 © eg and /2 < /i 0 eg} (13)

Hy compares /i and /2 in terms of their protrusions (peaks). If we want a distance sensitive both to the peaks and the valleys, then we can form the sum Hs(/i,/2 ) = Hy(/i,/2 ) + He(/i,/2 ), which adds to Hy(/i,/2 ) the Hausdorff distance He(/i,/2 ) = Hy(c - /i,c - /2 ) of their epigraphs, i.e., of the negation- complements of /i and /2 where c is some constant function.



Figure 3 reports a series of experiments whose goal was to inestigate how well can the Hausdorff or Epdistances find a local /global minimum in comparing a given parametric fractal with an ensemble of similarfractals whose parameter varies over all possible values. Figs. 3a,3b,3e,3f show that both the Hy and theti distance yield a very clear global minimum when comparing an original FBM or WMC signal of a fixedD with similar signals whose D spans all the interval [1, 2]. Resolutions of anywhere between 0.01 and0.1 suffice to sample the parameter space of D and still observe a clear minimum. The Hy distance hasa higher computational complexity than 4, and both yield similar results. For the Hy distance we usedthe structuring function g,. with h = 0.01. Using a smaller h or gt instead of g,. refines Hy but requiresmore iterations of erosions /dilations. In Figs. 3a,3ó the FBM's are viewed as deterministic functions (therandom number generator was re- initialized for each D with the same seed). From a statistical viewpoint,however, they are random functions, and hence their Hausdorff and 4 distances are actually randomvariables. Unfortunately, as Figs. 3c,3d show, the average Hy and 4 distances (the average was taken over50 independent FBM realizations for each D) do not have a global minimum, except when D > 0.5 andthe Hy distance is used (the Hs distance performs slightly better). Therefore, in our iterative optimizationmethod, the FBM signals are viewed henceforth only deterministically. In all the above experiments weused both types of Hausdorff distances (Hy and Hs) and three types of 4 distances (p = 1, 2, oo). Theyall had similar performance for (deterministic) FBM and WMC signals. Hence we focus henceforth onlyon the Hy and the 4 distances.

Figure 4 compares the estimated local fractal dimension LFD(1) for FBM and WMC signals of varyingdimension D and fixed duration N = 500 by using the covering, the variation by decimation, and ouriterative optimization method. The iterative optimization was done by using the initial estimate fromthe covering method, and then improving it by first proceeding in the D space at steps of 0.01 and then(when in the neighborhood of the global minimum) by refining it with optimization steps of OS= 0.001.As the Fig. 4 shows the iterative optimization method gives superior results than any other method sinceit achieves estimation errors that are practically zero for all D. (The errors are guaranteed to be in theorder of OS; in our experiments, they were almost always in the order of 10 -4 or 10 -5.) The 4 distancewas used, but the Hy distance performed very similarly.

The same conclusions can be reached from Fig. 5 which shows the estimated dimension LFD(1) for FBMand WMC signals with varying duration N and fixed dimension D, by using all the previous approaches.Again, for all N the iterative optimization method outperforms all others and yields practically zero errors.

Comparing the covering and variation methods, we see from Figs. 4 and 5 that: 1) the covering methodperforms worse than the variation, except for very small D < 1.2 and small N. 2) In the variation method,optimizing the decimation factor d over the first 10 scales performs better than optimizing it over all scalesfor D < 1.5, whereas for D > 1.5 the opposite it true. 3) Changing the signal length N affects the aboveconclusions for small N < 300.

4 An Application to Fractal Interpolation FunctionsGiven data points (xi, y;), i = 0, 1, .., I, we are concerned with continuous functions f : S --> R on acompact interval S, which interpolate the data as f (xi) = yt and whose graphs are attractors of IteratedFunction Systems (IFS) [1]. These systems consist of a finite number of affine contractive maps which canmodel well self -similar sets as the union (collage) of small patches (each patch is the transformation of theoriginal set by an affine map). A general affine map for an IFS is

Wny cn dn y + fn

n 1, 2,


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Figure 3 reports a series of experiments whose goal was to inestigate how well can the Hausdorff or lp distances find a local/global minimum in comparing a given parametric fractal with an ensemble of similar fractals whose parameter varies over all possible values. Figs. 3a,3b,3e,3f show that both the Hy and the t\ distance yield a very clear global minimum when comparing an original FBM or WMC signal of a fixed D with similar signals whose D spans all the interval [1,2]. Resolutions of anywhere between 0.01 and 0.1 suffice to sample the parameter space of D and still observe a clear minimum. The Hy distance has a higher computational complexity than £1, and both yield similar results. For the Hy distance we used the structuring function gr with h = 0.01. Using a smaller h or gt instead of gr refines Hy but requires more iterations of erosions/dilations. In Figs. 3a,3b the FBM's are viewed as deterministic functions (the random number generator was re-initialized for each D with the same seed). From a statistical viewpoint, however, they are random functions, and hence their Hausdorff and £p distances are actually random variables. Unfortunately, as Figs. 3c,3d show, the average Hy and t\ distances (the average was taken over 50 independent FBM realizations for each D) do not have a global minimum, except when D > 0.5 and the Hy distance is used (the Hs distance performs slightly better). Therefore, in our iterative optimization method, the FBM signals are viewed henceforth only deterministically. In all the above experiments we used both types of Hausdorff distances (Hy and Hs) and three types of ip distances (p = l,2,oo). They all had similar performance for (deterministic) FBM and WMC signals. Hence we focus henceforth only on the Hy and the t\ distances.

Figure 4 compares the estimated local fractal dimension LFD(l) for FBM and WMC signals of varying dimension D and fixed duration N = 500 by using the covering, the variation by decimation, and our iterative optimization method. The iterative optimization was done by using the initial estimate from the covering method, and then improving it by first proceeding in the D space at steps of 0.01 and then (when in the neighborhood of the global minimum) by refining it with optimization steps of OS=0.001. As the Fig. 4 shows the iterative optimization method gives superior results than any other method since it achieves estimation errors that are practically zero for all D. (The errors are guaranteed to be in the order of OS; in our experiments, they were almost always in the order of 10~ 4 or 10~ 5 .) The t\ distance was used, but the Hy distance performed very similarly.

The same conclusions can be reached from Fig. 5 which shows the estimated dimension LFD(l) for FBM and WMC signals with varying duration N and fixed dimension D y by using all the previous approaches. Again, for all N the iterative optimization method outperforms all others and yields practically zero errors.

Comparing the covering and variation methods, we see from Figs. 4 and 5 that: 1) the covering method performs worse than the variation, except for very small D < 1.2 and small N. 2) In the variation method, optimizing the decimation factor d over the first 10 scales performs better than optimizing it over all scales for D < 1.5, whereas for D > 1.5 the opposite ir true. 3) Changing the signal length N affects the above conclusions for small N < 300.

4 An Application to Fractal Interpolation Functions

Given data points (xt-,yt-), i = 0,1,..,/, we are concerned with continuous functions / : S —> R on a compact interval 5, which interpolate the data as /(zt-) = yt* and whose graphs are attractors of Iterated Function Systems (IFS) [1]. These systems consist of a finite number of affine contractive maps which can model well self-similar sets as the union (collage) of small patches (each patch is the transformation of the original set by an affine map). A general affine map for an IFS is

, n = l,2,...,/. (14)



We have a finite number I of such maps, each associated with some probability pn. If the maps arecontractive, then there exists a random algorithm [1] in which a point (x, y) is iteratively mapped to otherpoints by these affine maps (randomly chosen with probability pn) and thus fills the points of a uniqueattractor. For such an IFS to have as attractor the graph of a function its parameters must have somerestrictions. Namely, for the n -th affine map,

an = (xn - xn- 1)/(x7 - x0) , bn = 0 , en = xn -1 - anxo.do e ( -1, 1) , Cn = [yn - yn -1 - dn(y7 - yo)1/(x7 - x0) , fn = yn -1 - dny0 - Cnxo.

Here we assume that do = V is constant for all n, and we call it the "vertical scale parameter" (contractionfactor) V. Thus given the initial data points (xi, yi), the fractal interpolated function (FIF) fv is a uniquefractal signal parameterized by V. Figure 6a shows an original function f (an FBM of 256 points withH = 0.3). As data (xi, yi) we select 17 points from this function f, whose xi are equally spaced; henceall an = 1/16. Figs. 6b,6c show two signals that resulted from the IFS interpolation algorithm withV = -0.6, +0.6; the dotted line shows the piecewise linear interpolation between the original 17 points.The larger the V, the rougher looks the interpolated function fy. All these interpolated functions werecomputed at lint = 256 sample points equally spaced in their domain. Fig. 6d shows the fractal dimensionLFD(1) (computed using the covering method with g,. and h = 0.01) of fv for V spanning the interval[- 0.9,0.7] at steps of 0.032. We see that the relation between V and the fractal dimension D is one -to-oneover each half of the V parameter space ( -1,1). (We are currently working to provide an approximateanalytic formula for this relation.) Figs 6e and 6f show, respectively, the Hausdorff and ep distances (meanabsolute error, rms error, and max absolute error) between the original f and the interpolated fv, as afunction of the parameter V. Selecting one local minimum for the symmetric Hausdorff distance givesus V = -0.8, whereas the minimum for the 40 distance gives us V = -0.1. Plotting the correspondinginterpolated functions in Figs. 6g and 6h shows that the optimal parameter V extracted via the Hausdorff-distance minimizatign approximates f with an interpolated function that is closer to f in a way that agreesmore with the human perception of the roughness of the function's graph.

Finally instead of searching through the whole parameter space V, we can instead find first the fractaldimension which will give us an initial estimate of V, due to the one -to -one relation between D and V,and then improve this initial estimate by searching locally around it.

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Acknowledgements. P. Maragos was supported by the National Science Foundation under GrantMIPS -86 -58150 with matching funds from Bellcore, Xerox, and an IBM Departmental Grant, and in partby ARO under Grant DAALO3 -86-K -0171.

References

[1] M. F. Barnsley, "Fractal Interpolation Functions ", Constr. Approx., 2, pp. 303 -329, 1986.

[2] M. V. Berry and Z. V. Lewis, "On the Weierstrass- Mandelbrot fractal function ", Proc. R. Soc. Lond.A, 370, pp.459 -484, 1980.

[3] G. Bouligand, "Sur la notion d'ordre de mesure d'un ensemble plan ", Bull. Sci. Math., II -53, pp.185-192, 1929.

[4] B. Dubuc, J. F. Quiniou, C. Roques- Carmes, C. Tricot and S. W. Zucker, "Evaluating the fractaldimension of profiles ", Phys. Rev. A, vol.39, pp.1500 -1512, Feb. 1989.

[5] R. M. Haralick, S. R. Sternberg, and X. Zhuang, "Image Analysis Using Mathematical Morphology",IEEE Trans. Pattern Anal. Mach. Intell., PAMI -9, pp.523 -550, July 1987.


We have a finite number / of such maps, each associated with some probability pn . If the maps are contractive, then there exists a random algorithm [1] in which a point (z,y) is iteratively mapped to other points by these affine maps (randomly chosen with probability pn) and thus fills the points of a unique attractor. For such an IFS to have as attractor the graph of a function its parameters must have some restrictions. Namely, for the n-th affine map,

bn = 0 , €n = Xn_i - dnXQ. - .

dn e (-1, 1) , cn = [yn - yn-i - dn (yi - - - -Here we assume that dn = V is constant for all n, and we call it the "vertical scale parameter" (contraction factor) V. Thus given the initial data points (xt-, yt-), the fractal interpolated function (FIF) fv is a unique fractal signal parameterized by V. Figure 6a shows an original function f (an FBM of 256 points with H = 0.3). As data (zt-,yt-) we select 17 points from this function f, whose xt- are equally spaced; hence all an = 1/16. Figs. 6b,6c show two signals that resulted from the IFS interpolation algorithm with V = 0.6, +0.6; the dotted line shows the piecewise linear interpolation between the original 17 points. The larger the V, the rougher looks the interpolated function /y. All these interpolated functions were computed at Iint = 256 sample points equally spaced in their domain. Fig. 6d shows the fractal dimension LFD(l) (computed using the covering method with gr and h = 0.01) of fv for V spanning the interval [-0.9,0.7] at steps of 0.032. We see that the relation between V and the fractal dimension D is one-to-one over each half of the V parameter space (-1,1). (We are currently working to provide an approximate analytic formula for this relation.) Figs 6e and 6f show, respectively, the HausdorfFand tp distances (mean absolute error, rms error, and max absolute error) between the original / and the interpolated /V, as a function of the parameter V. Selecting one local minimum for the symmetric Hausdorff distance gives us V = -0.8, whereas the minimum for the £<» distance gives us V = 0.1. Plotting the corresponding interpolated functions in Figs. 6g and 6h shows that the optimal parameter V extracted via the Hausdorff- distance minimizatiqn approximates / with an interpolated function that is closer to / in a way that agrees more with the human perception of the roughness of the function's graph.

Finally instead of searching through the whole parameter space V, we can instead find first the fractal dimension which will give us an initial estimate of V, due to the one-to-one relation between D and V, and then improve this initial estimate by searching locally around it.

Acknowledgements. P. Maragos was supported by the National Science Foundation under Grant MIPS-86-58150 with matching funds from Bellcore, Xerox, and an IBM Departmental Grant, and in part by ARO under Grant DAALO3-86-K-0171.

References

[1] M. F. Barnsley, "Fractal Interpolation Functions", Constr. Approx., 2, pp. 303-329, 1986.

[2] M. V. Berry and Z. V. Lewis, "On the Weierstrass-Mandelbrot fractal function", Proc. R. Soc. Lond. A, 370, pp.459-484, 1980.

[3] G. Bouligand, "Sur la notion d'ordre de mesure d'un ensemble plan", Bull. Sci. Math., 11-53, pp.185- 192, 1929.

[4] B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot and S. W. Zucker, "Evaluating the fractal dimension of profiles", Phys. Rev. A, vol.39, pp.1500-1512, Feb. 1989.

[5] R. M. Haralick, S. R. Sternberg, and X. Zhuang, "Image Analysis Using Mathematical Morphology", IEEE Trans. Pattern Anal. Mach. Intell., PAMI-9, pp.523-550, July 1987.



[6] T. Lundahl, W.J. Ohley, S.M. Kay, and R. Siffert, "Fractional Brownian Motion: A Maximum Likeli-hood Estimator and Its Application to Image Texture ", IEEE Trans. Med. Imag., MI -5, pp.152 -160,Sep. 1986.

[7] B. B. Mandelbrot, The Fractal Geometry of Nature, NY: W.H. Freeman, 1982,1983.

[8] B. B. Mandelbrot and J. van Ness, "Fractional Brownian motion, fractional noise and applications ",SIAM Review, 10(4), pp. 422 -437, 1968.

[9] P. Maragos and R. W. Schafer, "Morphological Filters - Part I: Their Set -Theoretic Analysis andRelations to Linear Shift -Invariant Filters," IEEE Trans. Acoust. Speech, Signal Processing, pp.1153-1169, Aug. 1987.

[10] H. Minkowski, "Uber die Begriffe Lange, Oberflache und Volumen", Jahresber. Deutch. Mathematik-erverein., 9, pp. 115 -121, 1901.

[11] S. Peleg, J. Naor, R. Hartley and D. Avnir, "Multiple Resolution Texture Analysis and Classification ",IEEE Trans. Pattern. Anal. Mach. Intell., PAMI -6, pp.518 -523, July 1984.

[12] A. P. Pentland, "Fractal -Based Description of Natural Scenes ", IEEE Trans. Pattern Anal. Mach.Intell., PAMI -6, pp. 661 -674, Nov. 1984.

[13] J. Serra, Image Analysis and Mathematical Morphology, NY: Acad. Press, 1982.

[14] M. C. Stein, "Fractal image models and object detection ", in Proc. SPIE 845: Visual Communicationsand Image Processing II, 1987.

[15] S. R. Sternberg, "Grayscale Morphology," Comput. Vision, Graph., Image Proc. 35, pp.333 -355, 1986.

[16] C. Tricot, J. Quiniou, D. Wehbi, C. Roques- Carmes, et B. Dubuc, "Evaluation de la dimension fractaled'un graphe ", Revue Phys. Appl., 23, pp.111 -124, 1988.

[17] R. F. Voss, "Fractals in nature: From characterization to simulation ", in The Science of FractalImages, H. -O. Peitgen and D. Saupe, Eds, Springer -Verlag, 1988.


[6] T. Lundahl, W.J. Ohley, S.M. Kay, and R. Siffert, "Fractional Brownian Motion: A Maximum Likeli hood Estimator and Its Application to Image Texture", IEEE Trans. Med. Imag., MI-5, pp.152-160, Sep. 1986.

[7] B. B. Mandelbrot, The Fractal Geometry of Nature, NY: W.H. Freeman, 1982,1983.

[8] B. B. Mandelbrot and J. van Ness, "Fractional Brownian motion, fractional noise and applications", SIAM Review, 10(4), pp. 422-437, 1968.

[9] P. Maragos and R. W. Schafer, "Morphological Filters - Part I: Their Set-Theoretic Analysis and Relations to Linear Shift-Invariant Filters," IEEE Trans. Acoust. Speech, Signal Processing, pp.1153- 1169, Aug. 1987.

[10] H. Minkowski, "Uber die Begriffe Lange, Oberflache und Volumen", Jahresber. Deutch. Mathematik- erverein., 9, pp. 115-121, 1901.

[11] S. Peleg, J. Naor, R. Hartley and D. Avnir, "Multiple Resolution Texture Analysis and Classification", IEEE Trans. Pattern. Anal. Mach. Intell, PAMI-6, pp.518-523, July 1984.

[12] A. P. Pentland, "Fractal-Based Description of Natural Scenes", IEEE Trans. Pattern Anal. Mach. Intell, PAMI-6, pp. 661-674, Nov. 1984.

[13] J. Serra, Image Analysis and Mathematical Morphology, NY: Acad. Press, 1982.

[14] M. C. Stein, "Fractal image models and object detection", in Proc. SPIE 845: Visual Communications and Image Processing II, 1987.

[15] S. R. Sternberg, "Grayscale Morphology," Comput. Vision, Graph., Image Proc. 35, pp.333-355, 1986.

[16] C. Tricot, J. Quiniou, D. Wehbi, C. Roques-Carmes, et B. Dubuc, "Evaluation de la dimension fractale d'un graphe", Revue Phys. Appl., 23, pp.111-124, 1988.

[17] R. F. Voss, "Fractals in nature: From characterization to simulation", in The Science of Fractal Images, H.-O. Peitgen and D. Saupe, Eds, Springer-Verlag, 1988.



(a) FBM (D=1.5)

1 T

0.2

-2.2

100

(c) WMC (D=1.5 g=5)

1.6 -

Á 1.2 -

H

0.8 -

0.4 - 4\IA\v,h

I 0 I I I I I

400 500 0 100 200 300 400 500

(b) Covering Method on FBM

2.0 -sfh=0

1.8 - - - sfh=0.01

Z sfh=.1

A 1.6 -

1.4

1.2 -o

(d)2.0

SAMPLE in [0,1]

Covering Method on WMC

sfh=0

-- - sfh=0.001sfh=0.01

1.0 I I I I I 1.01 4 7 10 13 16

SCALE

1 4 7 10

SCALE

13 16

FIGURE 1. (a) An FBM function (solid line) with H = 0.5, N = 500 and its erosions /dilations (dashedlines) by egr (e = 10, 20, h = 0.01). (b) Estimation of LFD of FBM via the covering method using gr with3 different heights h = 0,0.01,0.1 (emaz = 25). (The thin solid straight line shows the true D = 1.5.) (c)and (d) same as (a) and (b) but for a WMC function and with h = 0, 0.001, 0.01.


FBM (D=1.5) WMC (D=1.5 g=5)

1.6"

0.8 -•

0.4 -•

0100 200 300 400 500 0

SAMPLE

100 200 300 400

SAMPLE in [0,1]

500

(b)2.0-

I-.*-Q 1.6-

§1.4-

u L2 "oh— 1

1.0-]

Covering Method on FBM f^\2.0-

——— sfh-0

- ——— sfll=°-01 g 1.8-

0 1.6 •

^ dU 1.2 -o

————— 1 ————— 1 ————— 1 ————— 1 ————— 1 i.o- l 4 7 10 13 16 ]

SCALE

Covering Method on WMC

——— sfh=0

. — — sfh=0.001........ Sfh=0.01

^^^=^r

"""""••••--..

1 4 7 10 13 16

SCALE

FIGURE 1. (a) An FBM function (solid line) with H = 0.5, N = 500 and its erosions/dilations (dashed lines) by cgr (c = 10,20, h — 0.01). (b) Estimation of LFD of FBM via the covering method using gr with 3 different heights h = 0,0.01,0.1 (cmax = 25). (The thin solid straight line shows the true D = 1.5.) (c) and (d) same as (a) and (b) but for a WMC function and with h = 0,0.001,0.01.



(a) VAR -D on FBM2.0 -

OA: d=10

Z -- - OF: d=3NO: d=1

5 1.6 --

----- %------------------------------

..

Q 1.4-

1.2 -O

1.0

(c) VAR -D on WMC

2.0 -

zO

z 1.8 -

i-...... --< 1.4 ¡

--- --------

1.6 -

OA: d=9- - OF: d=4

NO: d=1

Ú 1.2 -O

1.01 4 7 10 13 16 1 4 7 10 13 16

SCALE

(b) VAR-T on FBM (d)2.0 - 2.0

OA: M=51 OA: M=51

- - OF: M=153 - - OF: M=901.8 - NO: M=500 'FA' 1.8 NO: M=500

zg-.-

Q 1.6 -r...------ ...-- ---- A 1.6

Q i '' I ... t i -----t 1.4 -- ...................................... ,< 1.4< M .

SCALE

VAR -T on WMC

1.2 - o1.2

Uo aa1.0 I I I I I 1.0

1 4 7 10 13 16

SCALE

1 4 7

SCALE

10 13 16

FIGURE 2. (a) Estimation of LFD of an FBM function (H = 0.5, N = 500) via the variation by decima-tion method. The thin solid line shows the true D = 1.5. The OA and OF lines show the estimates resultedfrom optimizing the decimation factor d over all emaz = 25 scales or over the first 10 scales, respectively.The NO line corresponds to not optimizing and not decimating (d = 1). (b) same as in (a) but using thevariation by truncation method. (The NO line corresponds to not truncating since M = N = 500.) (c)

and (d) same as (a) and (b) but for a WMC function.


(a) VAR-D on FBM

——— OA:d=10 Z — — OF:d=3 2 1.8-- ........ NO:d-l£

< ——— **~"" ———————————————b , 4 . r -...„..-.....-..-.--.------1" •--<« 1 9 -• (J 1.ZOJ

1.0- ———— 1 ———— 1 ———— 1 ———— 1 ———— 1

1 4 7 10 13 16

SCALE

(c)2.0-

\L DIMENS»— ' H

ON b

&1 M -U 1.2 -O

i.o -1]

VAR-D on WMC

——— OA:d=9 — — OF:d=4

• ........ NO:d=l

"^ ^- ̂ ^^^^^_____-^—

I 4 7 10 13 16

SPAT P

(b;2.0-

fc01.8 •g

5 l.b-J<

U 1.4 -<!e3 1.2-U O

i.o -1]

) VAR-T on FBM (d)2.0-

——— OA:M=51 — — OF:M=153 § ——— . -MO- M *>00 5o 1.8 •

1>V-/. 1V1 +J\J\J ^*

/^ -^""^ ^ L6^-^^ ......................... b-,..-——""" < 1.4-

U 1.2 -s————— 1 ————— I ————— 1 ————— 1 ————— 1 1.0-

1 4 7 10 13 16 1

SCALE

VAR-T on WMC

——— OA:M=51 — — OF:M=90

[• ........ NO:M-500

__ __ — — _ ^

-^r^^,"^^^-*^^. ^-^

I 4 7 10 13 16

SCALE

FIGURE 2. (a) Estimation of LFD of an FBM function (H = 0.5, N = 500) via the variation by decima tion method. The thin solid line shows the true D = 1.5. The OA and OF lines show the estimates resulted from optimizing the decimation factor d over all emaz = 25 scales or over the first 10 scales, respectively. The NO line corresponds to not optimizing and not decimating (d = 1). (b) same as in (a) but using the variation by truncation method. (The NO line corresponds to not truncating since M = N = 500.) (c) and (d) same as (a) and (b) but for a WMC function.



(a)`Wn le+03

d 800

600

O 400A(9)

200

W 900

700ci)

500

300

--d 100

1.0

(e)

800 -Zr" 600 -Á

O200

0

FBM (N=500 R=1)

1.0 1.2 1.4 1.6 1.8

FRACTAL DIMENSION

FBM (N=200 R=50)

I.////

.:

...-.. .....

(c)

I I I I

1.2 1.4 1.6 1.8

WMC (N=500)

D=1.2- - D=1.5

D=1.8

400 -

(b) FBM (N=500 R=1)

1e +03 -

800 -

600 ......

D=1.2- - D=1.5

D=1.8

400 -

200 ---

o

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2.0

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1.0 1.2 1.4 1.6 1.8 2.0

E-,

630

560

490

420

350

280

(f)

FRACTAL DIMENSION

FBM (N=200 R=50)

(d)

I i I I I I

1.0 1.2 1.4 1.6 1.8 2.0

2.5e+03

2e+03

1.5e+03

le+03

WMC (N=500)

FIGURE 3. (a) Hausdorff distances Hy(f;, fD) between three fixed FBM functions f; with Di =

1.2,1.5,1.8 and variable FBM functions ID whose D spans the interval [1.01,1.99] at steps of 0.01. (b) same

as (a) but using 4(h) ID) distances. (c) and (d) are same as (a) and (b) but the distances are averaged

over 50 realizations of the FBM ID for each D. (e) and (f) are same as (a) and (b) but for WMC functions.


FBM (N=500R=1)00I<b

OQC/3

3ffi

au

i

900 T

.0 1.2 1.4 1.6 1.8

FRACTAL DIMENSION

FBM (N=200R=50)(I.

< 100-1-

i.o 1.2 1.4 1.6 1.8 2.0

WMC (N=500)

FBM (N=500R=1)

COH-«

Q

CO

630560490420350280

1.2 1.4 1.6 1.8

FRACTAL DIMENSION

.. FBM (N=200R=50)

(d)

(f)2.5e+03

2e+03

1.5e+03 "

WMC (N=500)

2.0

1.0—— 1 ——

1.2

—— i ——

1.4

—— i ——

1.6—— i ——

1.8——— !

2.0

FIGURE 3. (a) Hausdorff distances Hy(/,-,/z>) between three fixed FBM functions /< with D, = 1 2 1 5 1 8 and variable FBM functions fD whose D spans the interval [1.01,1.99] at steps of 0.01. (b) same as fa) but using t^fijo) distances, (c) and (d) are same as (a) and (b) but the distances are averaged "el 50i^realizatLs oJ theVBM fD for each D. (e) and (f) are same as (a) and (b) but for WMC funct.ons.



2.0 - (a) FBM (N=500)

IO: LI

0 VAR-D-F'ci) 1,8 - VAR-D-A

COV-FED

1.6

d 1.4

1.2 -cnW

1.0

2.0 -WMC (N=500) (b)

/

1.0 1.2 1.4 1.6 1.8

1.0

2.0 1.0 1.2 1.4 1.6 1.8 2.0

FRACTAL DIMENSION D FRACTAL DIMENSION D

FIGURE 4. Estimating the LFD(1) of [FBM in (a) and WMC in (b)] signals whose D spans the interval[1.05,1.95] at steps of 0.01. The IO line corresponds to the iterative optimization method using the Lidistance. The VAR -D -A and VAR -D -F lines correspond to the variation by decimation method wherethe decimation factor is optimized over all cmaz = 25 scales or over the first 10 scales, respectively. TheCOV -FED line correponds to the covering method with g,. and h = 0.

z

d1.4

E.:

W

2.0

1.8 -

1.6-

1.2

(a) FBM (D=1.5)-

.-

zO

z

2.0 -

1.8 -

1.6 -

1.4

1.2

(b) WMC (D=1.5)

IO: LiVAR -D -FVAR -D -A

- - COV -FED

..::` ,," -..- '

_-

-_---

- .. -

V-- -.-

. ._

I

\`-.

I

_ .---'--'--

I I1.0 1.0

100 300 500 700 900

NUMBER OF SAMPLES ( N )

100 300 500 700 900


FIGURE 5. Estimates of the fractal dimension LFD(1) of [FBM in (a) and WMC in (b)] signals withD = 1.5 and N spanning the interval [100,1000] at steps of 20. The line identities are as in Fig. 4.

428 / SPIE Vol. 1199 Visual Communications and Image Processing IV (1.989)

2.0 (a) FBM (N=500) WMC (N=500)

gSo 1.8 +

—— IO:L1 ....... VAR-D-F——• VAR-D-A— - COV-FED

1.6 ••

I 1.4 ••

FRACTAL DIMENSION D FRACTAL DIMENSION D

FIGURE 4. Estimating the LFD(l) of [FBM in (a) and WMC in (b)] signals whose D spans the interval [1.05,1.95] at steps of 0.01. The IO line corresponds to the iterative optimization method using the £1 distance. The VAR-D-A and VAR-D-F lines correspond to the variation by decimation method where the decimation factor is optimized over all (max = 25 scales or over the first 10 scales, respectively. The COV-FED line correponds to the covering method with gr and h = 0.

(a) FBM (D=1.5) (b) WMC (D=1.5)2.0 -

0

\L DIMENS1l— * l— ' c\ be i i

<^

< 1.4 -

H 1.2-

1 n -

z.u——— IO:L1

\7 A T> T\ T7 /^i-— — VAR-D-F g. ....... VAR-D-A 55 1.8 -

— - COV-FED |

Q 1.6 -

~ - - ^— s_^" V — •- — • •—.-...-..... <; 14 .

sH 1-2 'W

... - J —————— l —————— l —————— 1 ——— 1.0 -

•«*»«• ••

——— 1 ——— 1 ——— 1 ——— 1 —100 300 500 700 900 100 300 500 700 900

NUMBER OF SAMPLES (N) NUMBER OF SAMPLES (N)

FIGURE 5. Estimates of the fractal dimension LFD(l) of [FBM in (a) and WMC in (b)] signals with D = 1.5 and N spanning the interval [100,1000] at steps of 20. The line identities are as in Fig. 4.

428 / SPIE Vol. 1199 Visual Communications and Image Processing IV (1'989)


FBM (D=1.3)

IO: L1VAR -D -FVAR -D -A

- - COV -FED

100 300 500 700 900

(e)

1.0

NUMBER OF SAMPLES (N )

FBM (D=1.8)

« ...: .;;,± . ;.. á i

IO: L1VAR -D -FVAR -D -A

- COV -FED

I I i

100 300 500 700

(d)

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1.6

d 1.4

1.2

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IO: LlVAR -D -FVAR -D -A

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- .......................--- ---.. . .----- .........." - --, -- _ _ ,- - - -

100 300 500 700 900


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1.8

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F, 1.2 VAR -D -A- - COV -FED

I 1.0 I 1 I I

900 100 300 500 700 900

NUMBER OF SAMPLES ( N ) NUMBER OF SAMPLES ( N )

FIGURE 5 (cont'd). (c) and (d) are same as (a) and (b) but with D = 1.3. (e) and (f) are same as (a)and (b) but with D = 1.8.


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n /

and

all f

v.

(f) t

p di

stan

ces

betw

een

/ an

d al

l fv

. (g

) FI

F w

ith V

= -

0.8.

(h)

FIF

with

V =

-0.

1.


measuring fractal dimension: morphological estimates and...

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