Multiscale Fractal Descriptors Applied to Texture ClassificationJoao Batista Florindo1 and Odemir Martinez Bruno21)Universidade de Sao PauloInstituto de Fısica de Sao Carlos (IFSC) - http://scg.ifsc.usp.brjbflorindo@gmail.com2)Universidade de Sao PauloInstituto de Fısica de Sao Carlos (IFSC) - http://scg.ifsc.usp.brbruno@ifsc.usp.br

This work proposes the combination of multiscale transform with fractal descriptors employed in the clas-sification of gray-level texture images. We apply the space-scale transform (derivative + Gaussian filter)over the Bouligand-Minkowski fractal descriptors, followed by a threshold over the filter response, aiming atattenuating noise effects caused by the final part of this response. The method is tested in the classificationof a well-known data set (Brodatz) and compared with other classical texture descriptor techniques. Theresults demonstrate the advantage of the proposed approach, achieving a higher success rate with a reducedamount of descriptors.

Keywords: Fractal descriptors, multiscale transform, image analysis, pattern recognition

I. INTRODUCTION

Texture analysis is one of the most studied problemsin Computer Vision area1–3. The methodology employedin such problems may be divided into 4 categories2, toknow, structural, statistical, spectral and model-basedmethods. In this last category, we find the fractal tech-niques.

Fractal geometry shows to be an efficient tool to char-acterize and discriminate complex objects usually foundin texture images4–6. This precision of fractal represen-tation is explained by the flexibility of a fractal object,given its nature instrinsically complex. Moreover, frac-tals are strongly identified by their self-similarity behav-ior, that is, each part constitutes a copy only linearlytransformed of the whole structure. Such characteristicis also widely observed in many objects from the realworld, in a greater or smaller degree of intensity.

Among the most known fractal-based techniques,we can cite the multifractals7, the multiscale fractaldimension8 and the fractal descriptors. Here, we choosefractal descriptors approach, due to their high effi-ciency demonstrates in several works studying textureanalysis9,10.

Despite its efficiency, fractal descriptors still have prob-lems with the analysis of images presenting a significantamount of patterns and details, which may be observedonly at some specific scales. This is explained by thefact that, although fractal descriptors analyzes the im-age under different scales, the descriptors themselves arerepresented globally and are not able to highlight anykind of individual scale.

With the aim of overcoming such drawback, this workdevelops and studies a novel approach where a multiscaletransform11 is applied to the original descriptors. Partic-ularly, we employed a space-scale transform (derivative+ Gaussian filter) and applyed a threshold over the filterresponse in order to discard some irrelevant information.The application of a multiscale transform enabled the use

of fractal descriptors as not only a global curve whichdescribes the image, but a set of informations about de-tails, patterns and irregularities observed at a wide rangeof scales.

II. FRACTAL GEOMETRY

Fractal geometry is the area which studies fractal ob-jects. Fractals are complex geometrical structures char-acterized by an infinite degree of self-similarity, that is,each part of the object corresponds to a copy of thewhole, only affected by linear geometrical transforms(affine transforms)12,13.

As well as fractals, we may find a large number ofstructures in nature also characterized by a high levelof repetition of patterns. Moreover, we also verify thatmany of such natural object cannot find a precise enoughrepresentation through Euclidean classical geometry12,13.Such observations yield researchers to aplpy fractal geom-etry tools in a wide range of problems in several areas4–6.Among the employed fractal tools, fractal dimension isthe most used.

A. Fractal Dimension

As well as in Euclidean geometry we have metrics likeperimeter and area, fractal dimension is a fundamentalcharacteristic measure of a fractal object.

Essentially, this metric is obtained by measuring theextension of the object through a rule with a variableside-length. The dimension of an object S is given bythe following expression:

D = limε→0

log(Nε(S))

log(ε), (1)

where N is the number of units with length ε necessaryto cover the whole object.

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The above equation is only usable when we know theexact mathematical rule which gave rise to the fractalobject. In practical situations like in this work, whenwe are dealing with objects represented in a digital im-age, we must turn to estimation methods which providean approximate fractal dimension value. Most of suchmethods are based on a generalization of the Equation1, where, instead of the number of rules N , we use anycomplexity metric whether in spatial or frequency space.

Among such estimation methods, we opted to use theBouligand-Minkowski (BM) technique. This choice wasbased on its good performance in texture analysis prob-lems similar to those showed in this work9,10,14–16.

The Bouligand-Minkowski approach consists in somebasic steps. Initially, the gray-level intensity texture im-age I ∈ [1 : M ] × [1 : N ] → < is mapped onto a three-dimensional surface S in the following manner:

S = {i, j, f(i, j)|(i, j) ∈ [1 : M ]× [1 : N ]}, (2)

and

f(i, j) = {1, 2, ...,maxI}|f = I(i, j), (3)

being maxI the maximum intensity value of the image.In the following, the surface is dilated by a sphere with

a variable radius r, that is, around each point with coor-dinates (x, y, z) of the surface, we construct a sphere withradius r and center at (x, y, z). For each value of r wecan compute the dilation volume V (r) in a strightforwardmanner by:

V (r) = {⋃p∈S

Br(p)}, (4)

where Br are the spheres. An equivalent expression isthe following:

V (r) = {p1 ∈ R3|∃p2 ∈ S : |p1 − p2| ≤ r}, (5)

where p1 is a point outside the surface, while p2 is insideS.

Finally, varying the values of r along a specifical in-terval obtained empirically, the dimension is estimatedthrough the following expression:

D = 3− limε→0

log(Vr(S))

log(r). (6)

The Figure 1 illustrates the dilation process.

III. FRACTAL DESCRIPTORS

Despite its importance as a characterizer of objects,fractal dimension is still limitted, mainly when we needto discriminate among a large number of objects with ahigh degree of similarity. Such difficulty can be justifiedby two main aspects. The first is that fractal dimensionis only a unique value to describe the whole complexity

(a) (b) (c)

0 0.5 1 1.5 2 2.512

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13

13.5

14

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log(r)

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FIG. 1. Dilation of the surface to obtain the Bouligand-Minkowski dimension. (a) Texture image. (b) Surface rep-resentation. (c) An example of dilation of the surface (in thiscase, with r = 10). d) Curve log(V )× log(r).

of an object. The second is that we observe a lot ofobjects with a completely distinct appearance whilst theyhave the same estimated fractal dimension15. In order toovercome such drawback, Bruno et al.9 developed a novelmethodology called fractal descriptors.

This technique consists in composing a set of features(descriptors) from the power-law fractality curve of theanalyzed object. In Bouligand-Minkowski method usedhere, such descriptors are obtained from the dilation vol-ume values. We explain the procedure to compute suchvalues in the following.

Based on the Equation 5 and on the fact that we aredealing with a discrete space, the Euclidean DistanceTransform (EDT)17 shows to be a powerful method tocompute BM dimension. In this way, we construct a dis-tance mapping for the surface S, using pre-defined valuesfor the distance:

E = 0, 1,√

2, ..., l, ..., (7)

where

l ∈ D = {d|d = (i2 + j2)1/2; i, j ∈ N}. (8)

Thus, we define the set gr(S) of points at a distance rfrom a point with coordinates (Sx, Sy, Sz) in S:

gr(S) ={

(x, y, z)|[(x− Sx)2 + (y − Sy)2 + (z − Sz)2]1/2 = E(r)}.

(9)Finally, the dilation volume is given by:

V (r) =

r∑i=1

Q(i), (10)

3

where Q(i) represents the number of points at a distancei ≤ r:

Q(i) =∑

(x,y,z)

χgr (x, y, z), (11)

where χ is the indicator function: χA(x) = 1 if x ∈ Aand χA(x) = 0 otherwise.

The descriptors are obtained from the power-law rela-tion u between V (r) and r, that is:

u : log(r)→ log(V (r)). (12)

In this way, V (r) acts as a fractality measure while rcorresponds to the representation scale.

The function u can be used directly, as in14 or aftersome kind of transform. In this work, we propose a mul-tiscale transform to enhance the results obtained withfractal descriptors.

IV. PROPOSED METHOD

In image analysis, and particularly texture analysis,we are often faced with situations where the behaviorof some analysis method varies severely according to thescale at which the image is analyzed. Multiscale trans-form was developed with the aim of attenuating suchdifficulty, providing robust information about features ofthe object which are observed only at specific scales.

The multiscale transform, applied over a curve u(t) likethat of fractal descriptors, consists in a mapping functionof u(t) onto U(b, a), where b is directly related to t anda is the scale parameter.

Basically, the literature shows three categories of mul-tiscale transform, to know, space-scale, time-frequencyand time-scale18. Here, we adopted the space-scale ap-proach, given its large use in image analysis and partic-ularly interesting results in works related to the identifi-cation of patterns in natural objects, like those studiedin this work9,10.

In this approach, the function U(b, a) is obtained byderivating the function u(t) and applying a Gaussiansmoothing filter ga, where a is the smoothing parame-ter. More formally, we have:

U(b, a) = U1(t) ∗ ga(t) = U1(t, a), (13)

where U1(t, a) is the convolution of the original signalu(t) with the first derivative of the gaussian g1a, that is:

U1(t, a) = u(t) ∗ g1a(t). (14)

Inasmuch as multiscale is a two-dimensional transformapplied over a one-dimensional signal, it adds a subtan-tial amount of redundance information. To solve thisdrawback, we employed a technique based on the ad-justment of visual system to select objects in a scene18.Such solution, named fine-tuning selection consists in se-lecting a specific value of parameter a and projecting the

transform over that point. Here, we select the value of aempirically.

Finally, we applied a threshold over the obtained signalin order to eliminate noises added by the application ofthe multiscale over the final part of Bouligand-Minkowskicurve, where the data becomes more sensible due to thehigh level of wavefront interference. The threshold pointis also determined empirically.

The Figure 2 shows a summary of the steps in theproposed method in a box diagram.

I u(t)

U(b,a)

U(b)D(k)

Bouligand-Minkowski

Multiscale

Smoothing selection

Threshold

FIG. 2. A diagram illustrating the essential steps involved inthe proposed method.

The Figure 3 depicts the discrimination power of theproposed descriptors, distinguishing visually among sam-ples from two classes.

0 5 10 15 20 25 300

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0.12

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k

D(k

)class 1class 2

Class 1 Class 2

FIG. 3. Discrimination ability of proposed descriptors D(k).

V. EXPERIMENTS

The efficiency of the proposed method is verified bya comparison with other classical and state-of-the-artmethods in the classification of a texture data set.

Here, we used the well-known Brodatz19 texture data.This corresponds to 1110 images, each one divided into10 windows. Each image is a class and each window is asample.

The classification itself is carried out by the LinearDiscriminant Analysis (LDA) classifier20 using a Hold-

4

Out scheme, with one half of the data to train and theother half to validate.

We compare the success rate to the use of Bouligand-Minkowski descriptors (without multiscale)14, Gaborwavelets descriptors21, Co-occurrence matrix1 and a mul-tiscale approach described in22.

VI. RESULTS

The Table I shows the success rate of the compareddescriptors in the classification task of Brodatz data. Wealso show the number of descriptors used to provide thebest result and the κ index20, a measure of the reliabilityof the result. The value of 1−κ may also be used as an er-ror measure of the experiment. For the propoed method,we employed a parameter a = 0.7 and a threshold afterthe 51th element.

We observe that the proposed technique provided thebest results with a reasonable number of descriptors.Such aspect is interesting once beyond the computationalefficiency, the obtainment of a high success rate using lessdescriptors turns possible the attenuation of dimensional-ity curse effects and produce a more reliable result. Alsoit is important to emphasize the high κ value obtainedby the multiscale approach, implying in a robust classi-fication process.

Complementing the results in the previous table, weshow the confusion matrices for each method in the Fig-ure 4. In this figure, we have a surface where the heightat each point corresponds to the number of samples clas-sified as being from a predicted class (obtained) whilepertains to an actual class (waited). In this graphical,we must observe the diagonal (corresponding to the sam-ples classified correctly) and peaks outside the diagonal,corresponding to the misclassifications. We see that suchmatrices reinforce the Table I. The best results corre-spond to matrices with less “holes” inside the diagonaland less peaks outside. For instance, the Mikowski andthe proposed method have a similar matrix, except forthe fact that Minkowski shows 3 little peaks outside thediagonal, while the proposed technique presents only one.

In summary, we observe that the combination of fractaldescriptors and space-scale transform constituted a pow-erful and robust method to provide texture descriptors.Although fractal descriptors provide a good result, as at-tested by the success rate of Minkowski method, it is stilllimitted to an analysis more global of the image, whichlimits its efficiency in more complex situations as, for in-stance, when we have a high degree of similarity fromdifferent classes and/or a high dissimilarity intra-class.The global fractal descriptors approach also has some di-ficulties in dealing with noised images. The multiscaleapproach allows the highlighting of scale-dependent nu-ances, characterizing the image in a richer way, unveilingimportant structures which only appear at some specificscale as well as isolating artifacts and noises inherent to

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FIG. 4. Confusion matrices. (a) Multifractal. (b) Co-occurrence. (c) Gabor. d) Minkowski. e) Proposed method.

the original image.

VII. CONCLUSION

This work proposed the application of a multiscaletransform to the fractal descriptors method for the clas-sification of texture images. We applied the space-scaletransform (derivative followed by Gaussian filtering) tothe Bouligand-Minkowski fractal descriptors. After that,we also applied a threshold to the filter response to elim-inate the less significant information.

We compared the method to other well-known texturedescriptors and the results showed that the multiscalecombination porvided the best success rate even with areduced number of descriptors. Such performance con-firmed the efficiency of multiscale transform associatedto fractal geometry, once it allows the study of detailswhich only can be detected at specific scales, turning, inthis way, the analysis more complete and robust.

The results also suggest the application of the proposedapproach to other problems from the real world, involving

5

Method Correctness Rate (%) κ Number of descriptors

Co-occurrence 92.0721 0.9201 84

Gabor 90.0901 0.9001 20

Multifractal 37.4775 0.3732 101

Minkowski 98.9189 0.9891 85

Proposed 99.4595 0.9945 51

TABLE I. Correctness rate for the Brodatz dataset.

the discrimination of objects in complex scenarios.

ACKNOWLEDGMENTS

Joao B. Florindo gratefully acknowledges the finan-cial support of CNPq (National Council for Scien-tific and Technological Development, Brazil) (Grant#140624/2009-0). Odemir M. Bruno gratefully ac-knowledges the financial support of CNPq (NationalCouncil for Scientific and Technological Development,Brazil) (Grant #308449/2010-0 and #473893/2010-0)and FAPESP (The State of Sao Paulo Research Foun-dation) (Grant # 2011/01523-1).

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