Multifractal Texture Analysis using a Dilation-based H

older Exponent

Joao Batista Florindo

, Odemir Martinez Bruno

and Gabriel Landini

College of Medical and Dental Sciences, University of Birmingham, St Chad’s Queensway, Birmingham, U.K.

Physics Institute of Sao Carlos, University of Sao Carlos, Sao Carlos, Brazil

Keywords:

Multifractal, Texture Classiﬁcation, Bouligand-Minkowski, Fractal Geometry.

Abstract:

We present an approach to extract descriptors for the analysis of grey-level textures in images. Similarly to

the classical multifractal analysis, the method subdivides the texture into regions according to a local H

older

exponent and computes the fractal dimension of each subset. However, instead of estimating such exponents

(by means of the mass-radius relation, wavelet leaders, etc.) we propose using a local version of Bouligand-

Minkowski dimension. At each pixel in the image, this approach provides a scaling relation which ﬁts better

to what is expected from a multifractal model than the direct use of the density function. The performance of

the classiﬁcation power of the descriptors obtained with this method was tested on the Brodatz image database

and compared to other previously published methods used for texture classiﬁcation. Our method outperforms

other approaches conﬁrming its potential for texture analysis.

1 INTRODUCTION

Since the seminal work of Julesz (Julesz, 1981),

the analysis of texture images, and particularly tex-

ture classiﬁcation, have played a fundamental role in

many applications such as remote sensing, image re-

trieval, object recognition, and others (Zhang and Tan,

2002). Methods such as textons (Varma and Zisser-

man, 2009), texels (Todorovic and Ahuja, 2009), den-

sity maps (Ardizzone et al., 2013) and several oth-

ers have been particularly successful in the solution

of highly complex problems (Farinella et al., 2014).

Among the methods of texture analysis, model-

based approaches (Materka and Strzelecki, 1998) are

especially ﬂexible to analyse a large variety of im-

ages, since they are constructed over adaptable pa-

rameters. One of the model-based methods that pro-

vided interesting results in texture classiﬁcation is

based on the use of the multifractal spectrum (Xu

et al., 2009). That approach is not only precise in

enabling discrimination between different images but

it is also robust to some changes in illumination and

viewpoint. Despite this, the method relies on local

older exponents obtained by applying a Gaussian

convolution to the pixel neighbourhood, suggesting

that more elaborated approaches might be helpful to

describe textures.

Here we propose to use a local H

older exponent

based on the measure of Bouligand-Minkowski dila-

tion volumes. That approach has shown to provide

efﬁcient texture characterisation via the “fractal de-

scriptors” (Florindo and Bruno, 2013) and may pro-

vide a more precise representation of local textures

in terms of irregularity. In this context, the local di-

mension is estimated by mapping the pixels within a

local neighbourhood into a cloud of points in a three-

dimensional space. Therefore, each point in this cloud

is progressively dilated by spheres with growing ra-

dius and the the local H

older exponent is computed

from the exponential relation between the radius of

the spheres and the volume of the dilated cloud.

We applied this approach to the classiﬁcation of

40 classes from the Brodatz images database (Bro-

datz, 1966) and the task performance was compared

to other state-of-the-art and classical descriptors pre-

viously published in the literature. The potential of

this modiﬁed version of the local dimension is demon-

strated by the ratio of images correctly classiﬁed in

the database, which was higher than with the other

reported methods.

2 FRACTAL GEOMETRY

Fractal Geometry was developed from the mid seven-

ties by Mandelbrot (Mandelbrot, 1982) to provide a

means to characterise objects which could not be de-

scribed accurately by means of Euclidean Geometry.

505

Florindo J., Bruno O. and Landini G..

Multifractal Texture Analysis using a Dilation-based Hölder Exponent.

DOI: 10.5220/0005302305050511

In Proceedings of the 10th International Conference on Computer Vision Theory and Applications (VISAPP-2015), pages 505-511

ISBN: 978-989-758-089-5

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Many natural structures are irregular and despite that

natural objects are not strictly fractals in a mathemat-

ical sense (e.g. they are not inﬁnitely detailed), they

have, nevertheless, characteristics that suggest that

they are better modelled by Fractal Geometry con-

cepts (such as their complexity and self-similarity) at

least within a limited interval of scales.

The most widely used measure of an object in

Fractal Geometry is the fractal dimension and one

of its mathematical deﬁnitions is called Hausdorff-

Besicovitch dimension, formally deﬁned by:

D(E) = lim

δ→0

N(δ, E)

logδ

, (1)

where N(δ, E) is the minimum number of sets with

diameter δ necessary to cover E. For real-world ob-

jects the deﬁnition of an inﬁnite covering is not feasi-

ble and several approximations for N(δ, E) have been

proposed in the literature (Falconer, 2003), such as

box-counting, Bouligand-Minkowski dilation, mass-

radius relation, and so on.

3 MULTIFRACTALS

Multifractal theory assumes that certain class of ob-

jects (multifractals) have different degrees of self-

similarity in different sub-parts and therefore they

should be approximated by several fractal-like mod-

ellings (with different fractal dimensions) instead of

using only one global dimensional value.

The analysis is performed in a few steps. First,

the image is divided into local subsets, then the scal-

ing of a measure in this subsets (i.e. the local H

older

exponent α) is computed. Finally, the fractal dimen-

sions f (α) of the sets of locations with local H

older

exponent α is computed. That relation of values is the

multifractal spectrum of the image and can be used as

descriptors of the image texture.

Whereas there are different approaches to obtain

the multifractal spectrum, here we focus on the so-

lution proposed in (Xu et al., 2009) because it is

straightforward in implementation, it achieved good

results in texture classiﬁcation and showed to be

relatively invariant to certain scale and illumination

changes.

The local measure µ employed in (Xu et al., 2009)

is named “density function” and it is deﬁned at each

point p in an intensity image I by

µ(p) =

Z Z

B(p,r)

∗I)d p, (2)

where B(p, r) is a disk centred at p, with radius r, and

is a circular Gaussian kernel with radius r:

rσ

√

2π

−kpk

2σ

, (3)

for some smoothing parameter σ. The H

older expo-

nent is computed by

α(p) = lim

r→0

logµ(B(p, r))

logr

. (4)

The limit is estimated from the slope of a straight line

ﬁt to the curve of logr ×log µ(B(p, r)).

Secondly, the computation of the multifractal

spectrum (MFS), is achieved by dividing the image

into subsets E

according to

= {p ∈ℜ

: α(p) = k} (5)

and computing the respective fractal dimension D,

such that the MFS is obtained by:

MFS = {D(E

) : k ∈ ℜ}, (6)

where D can be computed by Equation 1.

4 PROPOSED METHOD

The dividing process makes possible to think of mul-

tifractal analysis as an elaborated histogram of the

image, where the homogeneity of the texture is as-

sessed under two perspectives: one local, given by the

older exponent and one global, provided by the frac-

tal dimensions of each subset resulting from the divi-

sion of the image. As consequence, the local H

older

exponent plays a fundamental role in the analysis and

therefore should be carefully deﬁned.

The neighbourhood value α(p), which in (Xu

et al., 2009) is computed via the density function

µ(B(p, r)), is an example of a H

older exponent (Bed-

ford, 1989). Such parameter is more generically de-

ﬁned in the context of H

older condition. A function f

satisﬁes this condition if, for any x and y in its domain,

the following inequality holds:

|f (x) − f (y)|≤Ckx −yk

, (7)

where C is a real constant and α is the H

older expo-

nent. An especially interesting case occurs when α =

1, and the function is said to be Lipschitz-invariant.

This is a key property in the theory of multifractals

because it assures the local invariance often desired

in image analysis. The density function is an exam-

ple of this category but some other functions invariant

under self-afﬁne transforms could be employed.

Based on this assumption, here, a local H

older ex-

ponent based on the Bouligand-Minkowski dilations

VISAPP2015-InternationalConferenceonComputerVisionTheoryandApplications

506

is proposed. Similarly to α(p) the new exponent here-

after called β(p) is parametrized by the size r of the

neighbourhood. At each point p ∈I, with coordinate

(i, j), a window w : I([i −r, i + r], [ j −r, j + r]) is ex-

tracted and mapped onto a cloud of points C(w), sat-

isfying:

(i, j, k) ∈ C(w) ↔ w(i, j) = k. (8)

In the following, each point p ∈ C(w) is dilated by

a sphere B(p, d), centred at p with radius d and the

dilated volume is computed through

V (d) =

[

p∈C(w)

B(p, d). (9)

The local H

older exponent β(p) is computed by

β(p) = lim

d→0

logV (d)

logd

, (10)

and this limit is estimated by ﬁtting a straight line to

the curve log d ×logV (d) and computing the corre-

sponding slope for each point in the image. Finally,

the exponent is locally assigned to the pixel p. Figure

1 illustrates the steps involved in the dilation process

of the pixels within w.

In order to verify how well β(p) estimates the

older exponent and consequently the multifractal

spectrum of a grey-level image, Figure 2 compares

the proposed f (β) with the method in (Xu et al.,

2009) over a synthetic multifractal texture. Figure

2(a) shows the image generated by the Meakin model

(Meakin, 1987). Figure 2(b) exhibits f (α) curve ob-

tained through (Xu et al., 2009) and the proposed one

compared to the theoretical spectrum, obtained by a

procedure described in (Chhabra et al., 1989). Figure

2(c) shows a histogram of the average error when ﬁt-

ting a straight line to the log−log curve at each point

The f (α) and f (β) curves have a similar shape to

that of the expected curve, although the experimen-

tal values obtained depart from the theoretical values,

due to the data being a discrete image. In the partic-

ular case of f (β) this is also caused by the sparsity

of the spheres in the space. An isolated point that is

dilated and suddenly becomes connected to other di-

lated points has its volume raised very quickly leading

to overestimated values of the exponent. However, the

logd ×logV (d) curve in the proposed method showed

the best ﬁt to a straight line than logµ(B(p, r)) ×logr

in (Xu et al., 2009), as illustrated by the smaller er-

ror in the histogram. Such behaviour is due largely

to the smoother growing of the dilation volume when

the radius is gradually increased.

Finally, Figure 3 illustrates the obtained H

older

exponents for a texture of example from the real-

world and the respective multifractal spectrum. Here,

even the shape of the spectrum curve is different from

those found in well-deﬁned multifractals like in the

Meakin model, which again can be explained by the

limited number of scales allowed by the image do-

main. Another important remark here is that the val-

ues of α (or β) are expected to stay between 2 and

3, since they are basically the fractal dimension of an

object immersed in the three-dimensional topological

space . However, here the sparsity of spheres in more

discontinuous regions of the image causes the expo-

nent to extrapolate such interval, as previously ex-

plained. Such extrapolation is also common in other

applied works on multifractals, like (Xu et al., 2009;

Meakin, 1987; Bedford, 1989)and several others.

Following (Xu et al., 2009), this exponent is also

computed not only on the intensity image, but on the

image gradient, and on its Laplacian. Therefore, three

variants of β(p) are deﬁned:

(p) = lim

d→0

log

p∈C(w)

B(p, d)

logd

, (11)

(p) = lim

d→0

log

p∈C(|∇w|)

B(p, d)

logd

, (12)

and

(p) = lim

d→0

log

p∈C(|∇

w|)

B(p, d)

logd

, (13)

From equations 5 and 6, three MFS vectors are

obtained (M

, M

and M

) and they are concatenated

to be used as the texture descriptors. The descriptors

based on β(p) were also interchanged and merged

with those from α(p) to obtain even more precise

classiﬁcation.

5 EXPERIMENTS

Some parameters in the proposed method were empir-

ically chosen. Thus to estimate the H

older exponent

by the Bouligand-Minkowski method a maximum di-

lation radius of 5 and a 5 ×5 neighbourhood were em-

ployed at each pixel. Moreover, the values of β were

limited within the interval [1, 4] and sampled into 33

bins, so that each M

, M

and M

is a vector of 33

real values.

The proposed method was used for the classiﬁca-

tion of Brodatz textures (Brodatz, 1966). The ﬁrst

40 images (D1-D40) at a resolution of 512x512 pix-

els were used. All images were divided into 16 non-

overlapping windows giving rise to a ﬁnal set of 40

classes with 16 images in each class. The Brodatz

collection is made of a set of texture images contain-

ing a wide variety of levels of regularity, granularity,

MultifractalTextureAnalysisusingaDilation-basedHölderExponent

507

Figure 1: A simpliﬁed 3 ×3 texture image, the cloud being dilated by variable radius and the slope of the log −log curve

used to estimate the H

older exponent. Such exponent is locally assigned to the central pixel. The dilation is showed as being

continuous only for illustration purposes.

periodicity and scale, making it ideal for image classi-

ﬁcation benchmarking purposes, specially with mul-

tiscale approaches such the method presented here.

The performance of the proposed descriptors as

classiﬁers was compared to other previous work us-

ing Local Binary Patterns (LBP) (Ojala et al., 1996),

Grey-Level Co-occurrence Matrix (GLCM) (Haral-

ick, 1979), the multifractal approach in (Xu et al.,

2009), Fourier (Gonzalez and Woods, 2002) and frac-

tal descriptors (Backes et al., 2009). To allow for fair

comparisons and to reduce the internal correlations all

the descriptors were submitted to a Principal Compo-

nent Analysis (PCA) (Duda and Hart, 1973) for di-

mensionality reduction.

Finally, the classiﬁcation was carried out by Lin-

ear Discriminant Analysis (LDA) (Duda and Hart,

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508

0.5

1.5

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

f(a )

Xu et al. 2009

Proposed

Theoretical

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

number of points

average error

Xu et al. 2009

Proposed

(a) (b) (c)

Figure 2: Multifractal analysis of a synthesized multifractal image. (a) Grey-level multifractal image generated by the proce-

dure described in (Meakin, 1987). (b) f (α) curves comparing the proposed method to that in (Xu et al., 2009) and the expected

theoretical curve provided by the method in (Chhabra et al., 1989). (c) Histogram of the average error in the log−log ﬁtting

at each point of the image.

0.5

1.5

1 1.5 2 2.5 3 3.5 4

f(a)

(a) (b) (c)

Figure 3: Multifractal analysis of a real-world texture. (a) Original image (sample c002 002 from Brodatz database (Brodatz,

1966)). (b) Local dilation-based H

older exponent. (c) Multifractal spectrum.

1973), following a 10-fold scheme, that is, the

database was split into 10 groups of equal sizes and 10

rounds of classiﬁcation were executed. In each round,

9 groups were used to train and the remaining one to

test. The ﬁnal correctness rate is given by the average

correctness over all the rounds.

6 RESULTS AND DISCUSSION

Table 1 shows the correctness rates achieved during

the classiﬁcation process when using different combi-

nations of MFS vectors, either using the original α(p)

older exponent or the proposed β(p) value. From

the table it can be observed that M

performs better on

the intensity and on the gradient of the image rather

than on the Laplacian. Besides, a higher percentage

of the images (95.78%) was correctly discriminated

when M

and M

descriptors were combined through

the PCA technique.

Table 2 compares the best combination in Table 1

with other texture descriptors widely used in the lit-

erature. The proposed approach correctly classiﬁed

a larger number of images even outperforming meth-

ods whose efﬁciency in this task is well established.

Such result is consequence of using a richer local de-

scription of the texture, yielding to more meaningful

multifractal spectra

Figure 4 exhibits the confusion matrices for the

method as originally proposed in (Xu et al., 2009) to-

gether with the approach here proposed. In this type

of representation a minimum of light blue points out-

side the diagonal is expected and, although the differ-

ence is not so clear, the proposed descriptors presents

fewer misclassiﬁcation over the classes.

The method in its current version also presents a

drawback which is the high computational time in-

volved since the dimension is computed at each pixel

(O(N

) for an N ×N image), although this is atten-

uated by using optimized algorithms as those used

in (Backes et al., 2009). Anyway this computational

time should be improved in the future by means of

MultifractalTextureAnalysisusingaDilation-basedHölderExponent

509

Table 1: Correctness rate obtained by combining α and β in different ways.

Name Combination Correctness rate (%)

MFS1 M

70.00

MFS2 M

80.94

MFS3 M

57.66

MFS4 M

+ M

89.06

MFS5 M

+ M

95.00

MFS6 M

+ M

95.78

Table 2: Correctness rate and respective cross-validation errors obtained by the compared descriptors.

Method Correctness rate (%) Error

Fourier 88.75 0.05

GLCM 91.67 0.04

Fractal Descriptors 94.37 0.03

LBP 95.00 0.04

(Xu et al., 2009) 93.75 0.05

Proposed MFS6 95.78 0.03

Expected class

Predicted class

5 10 15 20 25 30 35 40

Expected class

Predicted class

20 40

(a) (b)

Figure 4: Confusion matrices. (a) (Xu et al., 2009). (b) Proposed.

a more parallelized procedure. Another point is that

PCA is a simple approach and more advanced fea-

ture selection algorithms also can be applied provid-

ing even better results.

Generally speaking, M

descriptors enhanced the

classiﬁcation performance of the multifractal analy-

sis and added meaningful information to M

descrip-

tors, such that when they are combined more images

are correctly classiﬁed. Nevertheless, the proposed

older exponent did not succeed when applied over

the Laplacian of the image. Such behaviour can be

explained by the dilation process, which propagates

the discontinuities on this type of image and makes

the local exponent more unstable than the smoothing

effect of the Gaussian convolution in (Xu et al., 2009).

7 CONCLUSIONS

We presented a method to estimate the local H

older

exponent in grey-level images and applied it in the

division step of the multifractal analysis of these im-

ages. Our approach is an adaptation of the Bouligand-

Minkowski fractal dimension analysis, here computed

over the neighbourhood of pixels. It was introduced

into the pipeline of the multifractal process and com-

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510

puted over the intensity grey-level image as well as

over the gradient and the Laplacian of the same im-

age.

The efﬁciency of the proposed approach was

tested in the classiﬁcation of 40 classes of Brodatz

database, and compared to other classical texture de-

scriptors and the multifractal approach described in

(Xu et al., 2009). The results showed that the new de-

scriptors can be combined to the previous multifractal

analaysis and in this way they are capable of outper-

forming the classiﬁcation results of other state-of-the-

art methods in the literature.

Such promising results suggest that the dilation

process successfully employed in a multiscale de-

scription of textures can also be a reliable method

to locally characterize a neighbourhood providing

meaningful descriptors in the context of a multifractal

analysis.

ACKNOWLEDGEMENTS

Joao Batista Florindo acknowledges support from

FAPESP (The State of S

ao Paulo Research Foun-

dation) (Grant No. 2013/22205-3). Odemir Mar-

tinez Bruno acknowledges support from CNPq (Na-

tional Council for Scientiﬁc and Technological De-

velopment, Brazil) (Grant Nos. 484312/2013 and

308449/2010).

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