Computing fractal descriptors of texture images using sliding boxes: An application to the identification of Brazilian plant species

Journal Pre-proof Computing fractal descriptors of texture images using sliding boxes: An application to the identification of Brazilian plant species Giovanni Taraschi, Joao B. Florindo

PII: DOI: Reference:

S0378-4371(19)32037-0 https://doi.org/10.1016/j.physa.2019.123651 PHYSA 123651

To appear in:

Physica A

Received date : 15 August 2019 Revised date : 9 November 2019 Please cite this article as: G. Taraschi and J.B. Florindo, Computing fractal descriptors of texture images using sliding boxes: An application to the identification of Brazilian plant species, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123651. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

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Research Highlights

• Texture descriptors using a sliding box counting approach.

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• Theory based on fractal geometry and sliding detection probabilities. • Employed in the classification of gray level texture images. • Application to the identification of Brazilian plant species.

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• Better classification accuracy both on benchmark and practical tasks.

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Journal Pre-proof *Manuscript Click here to view linked References

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Giovanni Taraschia , Joao B. Florindoa,∗

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a Institute of Mathematics, Statistics and Scientific Computing - University of Campinas Rua S´ ergio Buarque de Holanda, 651, Cidade Universit´ aria ”Zeferino Vaz” - Distr. Bar˜ ao Geraldo, CEP 13083-859, Campinas, SP, Brasil

Abstract

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Computing fractal descriptors of texture images using sliding boxes: an application to the identification of Brazilian plant species

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This work proposes a new model based on fractal descriptors for the classification of grayscale texture images. The method consists of scanning the image with a sliding box and collecting statistical information about the pixel distribution. Varying the box size, an estimation of the fractality of the image can be obtained at different scales, providing a more complete description of how such parameter changes in each image. The same strategy is also applied to a especial encoding of the image based on local binary patterns. Descriptors both from

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the original image and from the local encoding are combined to provide even more precise and robust results in image classification. A statistical model based

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on the theory of sliding window detection probabilities and Markov transition processes is formulated to explain the effectiveness of the method. The descriptors were tested on the identification of Brazilian plant species using scanned images of the leaf surface. The classification accuracy was also verified on three benchmark databases (KTH-TIPS2-b, UIUC and UMD). The results obtained demonstrate the power of the proposed approach in texture classification and,

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in particular, in the practical problem of plant species identification.

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Keywords: Box-counting, Fractal descriptors, Texture classification, Automatic plant taxonomy.

∗ Corresponding

author Email addresses: [email protected] (Giovanni Taraschi), [email protected] (Joao B. Florindo) Preprint submitted to Elsevier

November 9, 2019

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1. Introduction Fractals have shown to be a powerful tool in the modeling and analysis of

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natural objects and images in general, being used in different areas such as

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Biology, Physics, Engineering, Medicine, and many others [1, 2, 3, 4, 5]. The

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high flexibility of fractal geometry and its ability to describe objects with high

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degree of complexity at different scales make it an appropriate tool for describing

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nature. Therefore, for many real world applications, fractal geometry is more

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adequate than Euclidean geometry [6].

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Most fractal-based methods use the concept of fractal dimension. This can

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be interpreted as a measure of the complexity or, equivalently, of the way that an

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object occupies the space [6]. In the analysis of materials and texture images in

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general, the fractal dimension is related to physical parameters such as roughness

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and luminance [7].

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However, in most practical situations only the fractal dimension is not suf-

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ficient to describe the object satisfactorily [8]. More elaborated approaches are

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necessary. Examples are multifractal spectrum [9], multiscale fractal dimension

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[10] and fractal descriptors [11]. In this work we focus on the last approach given

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the relevant results achieved in practical situations, especially in the analysis of

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biological images [8, 12, 13, 14, 15].

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Most methods employed to numerically estimate the fractal dimension rely

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on some sort of measure computed over a range of scales (in some sense this

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approximates the Hausdorff measure in the analytical definition). Fractal de-

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scriptors are vectors of numerical (real) values obtained from the values of such

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measures in each observed scale. Despite the success previously demonstrated,

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fractal and multifractal descriptors explore complex relations within the spatial

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structure of the object and direct approaches like the box counting dimension

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has not been sufficiently investigated for that purpose. A possible explanation

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for this is the sensitivity of classical box counting method to spatial translation,

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stemming from the use of a fixed grid of boxes.

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In this context, this work proposes new fractal descriptors for grayscale tex-

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tures based on a translation invariant version of box counting dimension. The

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gray image is mapped onto a three-dimensional cloud of points by associating

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each pixel in coordinates (x, y) with normalized gray value z to a point with

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coordinates (x, y, z). The invariance is achieved by using a sliding box with side

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length r sweeping the cloud structure and counting the number of points inside

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each cube. The descriptors are provided by the cumulated distribution of these

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points for a range of r values. In this way both global and local description are

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obtained. To make the classification performance even better we also compute

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these descriptors from an alternative local encoding of the image based on local

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binary patterns [16].

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This approach is directly related to a typical characteristic of fractals: fine

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structure. An object has fine structure if it presents the same level of details

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at any scale of observation [17]. Here we present a statistical model, based on

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Markov transition processes [18], to explain how the distribution of points can

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express such fine structure and, in particular, the influence of attributes such

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as homogeneity in the proposed distribution.

Finally, the accuracy of the method was tested in the identification of Brazil-

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ian plant species using images from the leaf surface as well as in the classification

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of three well-known benchmark databases of texture images, namely, UIUC [19],

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UMD [20], and KTH-TIPS2-b [21]. The performance in such tasks is compared

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to other state-of-the art image descriptors, to know, LBP [16], VZ-Joint [22],

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SIFT + BoVW [23], WMFS [24], PLS [25], FC-CNN VGGVD [26], and oth-

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ers. These compared approaches were outperformed by our proposal in terms of

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classification accuracy. Such promising performance can be explained to a large

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extent by the flexibility of fractal models in expressing the intrinsic richness of

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natural structures like those here represented by the real-world texture images.

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Attributes of these structures generally called “fractality” measures are known

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to be tightly related to the complexity of those materials, which for example in

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a biological context like that of the plant species correspond to a mapping of

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the evolution process of that specimen.

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2. Bibliographical Review A texture image (or visual texture) is a grayscale image that presents par-

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ticular patterns in the pixel distribution at different scales. Texture images are

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capable of expressing high amount of information about a real-world object such

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as luminosity, roughness, regularity and density. Approaches for texture analysis

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are classically categorized into 4 groups: structural, statistical, transform-based

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and model-based [27].

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Structural methods usually work on well-defined geometric primitives and

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mathematical morphology [28] is a typical example of this approach. Transform-

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based approaches derive from the image representation in other spaces (mostly

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in frequency domain). In this way this approach can faithfully describe the

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periodicity present in many images. Fourier [29] and wavelets [30] are exam-

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ples of techniques applied in this category. Statistical methods work on the

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relationship between pixels. Although they have obtained interesting results

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in practical situations, they do not have a meaningful mathematical modeling,

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which makes their results difficult to be interpreted. LBP [31], bag-of-visual-

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words [32], Scale-Invariant Feature Transform (SIFT) [33] and LPQ [34] are

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important examples of this approach. More recently, an approach that may in

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some sense be considered as a statistical solution correspond to those methods

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based on the popular deep convolutional networks adapted for texture classifi-

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cation. Examples are the methods proposed in [35, 26]. Despite the well known

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success of deep networks in general images, texture classification is still a big

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challenge due to the specificity of many problems in texture analysis as well as

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to the difficulty to acquire sufficiently large number of samples necessary for

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training these networks.

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Finally we have model-based methods, which seek to combine the precision

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and robustness of methods that explore the relationship among pixels or among

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regions of the image with an already well-established mathematical and physical

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model. In this category, fractal-based methods such as multifractal spectrum

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[9], multiscale fractal dimension [10] and fractal descriptors [11] are prominent.

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Various natural structures are related to fractal geometry, mainly with respect

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to the self-similarity idea [6]. In this way the fractal-based methods aim at

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exploring and quantifying this relation to obtain a more faithful representation

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of these objects. In the next sections we explain more about fractal theory and

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the fractal descriptors.

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In fact, several works have been recently published applying fractal geome-

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try (especially fractal dimension) in the most diverse problems in nature. For

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example, in [36] the authors employ the threedimensional fractal dimension of

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of magnetic resonance images of cortical surfaces in a system of computer aided

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diagnosis of Alzheimer’s disease. Other related applications can be found in

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Parkinson’s disease [37], epileptic seizure [38], and other medical areas. An-

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other interesting application is presented in [39] where the fractal dimension is

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used in the analysis of the structure of adsorption pore of coals. In [40] the

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fractal dimension is used to detect nonlinearity and chaos signature of a binary

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star system. Coming specifically to image analysis, in [41] the fractal dimen-

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sion of microscopy images is used to assess the soil digestibility of two types of

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pretreated biomasses. In [42] the authors establish relations between the fractal

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dimension of tissue images of pork loin and salmon with water and fat fractions.

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3. Fractal theory

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In general, a fractal can be understood as a mathematical object that has re-

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markable properties such as self-similarity, fine structure and complexity, which

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make them unsuitable for a purely Euclidean representation [17]. In fact, there

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is no unique definition for the concept of fractal object. The most classical one

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is related to Hausdorff dimension.

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3.1. Hausdorff dimension

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Given a set U , the diameter of this set is given by |U | = sup{d(x, y) : x, y ∈

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U }, where d(x, y) is a distance defined over a metric space. We say that a family

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of sets {Ui } is a σ-cover of a set F if: 5

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i) 0 < |Ui | < σ

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ii) F ⊂ ∪∞ i=1 Ui .

Thus the Hausdorff measure of a set F is defined as: (∞ ) X s s s s H (F ) = lim Hσ (F ) where Hσ (F ) = inf |Ui | : Ui is a σ-cover of F .

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σ→0

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i=1

s

(1)

For any fractal structure, H shows a rather peculiar behavior, to know, that

Hσs = ∞ for s < D and Hσs = 0 for s > D, for some real and non-negative value

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D. Then D is defined as the Hausdorff dimension of F . Formally:

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D = inf{s : Hs (F ) = 0} = sup{s : Hs (F ) = ∞}.

(2)

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Therefore, in the classical and most accepted definition of Mandelbrot [6], a

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fractal is a mathematical object whose Hausdorff dimension strictly exceeds its

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Euclidean (topological) dimension.

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3.2. Estimates of the fractal dimension

Real world objects do not have infinite self-similarity and, in general, the

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construction rules of the object are not known, as is usual in the generation

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of geometric fractals. This makes the analytical calculation of the Hausdorff

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dimension quite complicated and often impossible [6]. There is a need to develop

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methods to estimate the fractal dimension in these situations.

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The basic definition of fractal dimension by Hausdorff measures involves an

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infinite covering by elements with diameter smaller than σ and this diameter is

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raised to an exponent s. When this idea is transfered to the discrete domain, it

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can be understood as an exponential function

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Mσ ∝ σ s ,

(3)

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where Mσ is a measure of the object at the scale σ, i.e., where any detail

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larger than σ is disregarded. This allows one to define alternative definitions to

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Hausdorff dimension and therefore to obtain estimates for the fractal dimension.

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It is important to note that alternative definitions may assume values other

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than the Hausdorff dimension, but retain the idea of measuring the complexity

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and spatial occupation of the object. Among the alternative definitions, box-

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counting dimension is one of the most popular [17].

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3.3. Fractal descriptors

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In practical situations, a scalar measure such as the fractal dimension (or its

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correspondent estimation) is not sufficient to describe all the details commonly

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found at different scales of a real object. In the analysis of textures, for example,

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there are images with visibly different aspects, but presenting the same fractal

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dimension [8]. In this context, the idea of a more complete set of fractal measures

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arises. Among the techniques that explore this gap, the most popular ones

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are the multifractal spectrum, the multiscale fractal dimension and the fractal

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descriptors. Here we are interested in the last approach.

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Fractal descriptors are based on the exponential law obeyed by fractals.

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However, unlike the estimation methods for fractal dimension where an ana-

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lytical curve is adjusted to the data points, a function u : log σ → log Mσ is

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defined and all the log × log curve is used to describe the object of interest. In

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this way the descriptors provide information on all the scales of the texture,

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giving a multiscale representation of the image. These descriptors can provide

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information that is significantly richer than that obtained from a single scalar

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measure and are advantageous when compared with other approaches [43].

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4. Proposed method

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This work proposes a new type of fractal descriptors, named Sliding Box Descriptors (SBD), based on a different strategy for box counting together with the

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respective statistical analysis of the distribution of pixels in the boxes. Whereas

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classical box-counting partitions the analyzed image into a fixed grid, here we

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“slide” a window over the image with different side lengths, each length corre-

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sponding to a particular scale of analysis. Although sliding (gliding) boxes have

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been adopted for the calculus of lacunarity [44], up to our best knowledge this

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is the first attempt to use it for fractal descriptors.

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The method begins by mapping the grayscale image I with dimensions m×n into a three-dimensional set of points B:

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Zm×n → Zm×n×Imax

(i, j) 7→ (i, j, I(i, j)),

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where 0 ≤ I(i, j) < Imax is the intensity of the pixel with coordinates (i, j). We define {ln }n∈N as a decreasing sequence such that l0 = min{m, n, Imax }   and ln+1 = l2n . For each value of the sequence, we scan the set B using a

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(4)

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sliding cubic box with side ln . As the box passes through the set it counts

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the number of points involved in the current step. To execute this scans, a

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three-dimensional discrete convolution is performed: D(j1 , j2 , j3 ) =

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k2

k3

B(k1 , k2 , k3 ) · C(j1 − k1 , j2 − k2 , j3 − k3 )

(5)

with each ki running over all valid indexes of B and C.

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To reproduce the sliding box effect, we take C ∈ Zln ×ln ×ln , C(k1 , k2 , k3 ) = 1 (∀k1 , k2 , k3 ) and only the valid part of the convolution is taken. That is, the

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(6)

Therefore the texture SBD descriptors are obtained from D by taking the

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indices j1 , j2 , j3 belong to the ranges:     ln ln 1+ −1 ≤ j1 ≤ m − 2 2     ln ln 1+ ≤ j2 ≤ n − −1 2 2     ln ln 1+ ≤ j3 ≤ Imax − −1 . 2 2 cumulative distribution: r

C (k) =

k XXX X j1

j2

δ(D(j1 , j2 , j3 ), k 0 ),

(7)

j3 k0 =0

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where δ(x, y) is the Kronecker delta (1 if x = y, 0 otherwise). Finally the

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descriptors are provided by the cumulative distributions for r within a pre8

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specified interval. D=

[

[C r (k)]α ,

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r = 2, 4, 8, · · · , min(m, n),

(8)

where α is a constant empirically determined and specific for each database. Its

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main role is to give the appropriate weight for each possible number of points.

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Here we tested only α integer to keep the process simple, although there is no

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constraint for using real values or even to combine more than one value. Here

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we empirically found out that α = 15 yields optimal results for the analyzed

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databases.

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We also apply this methodology to maps of local binary patterns (LBP) of

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the original images. Here we adopt the LBP riu2 maps [16], which are deter-

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mined, for P interpolated points over a neighborhood with radius R by   PP −1 s(g − g ) if U (LBP ) ≤ 2 p c P,R p=0 riu2 LBPP,R =  P +1 otherwise,

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pixels and

where gc is the value of the reference (central) pixel, gp are values of neighbor

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U (LBPP,R ) = |s(gP −1 − gc ) − s(g0 − gc )| +

P −1 X p=0

|s(gp − gc ) − s(gp−1 − gc )|.

Generally speaking, each scan provides statistical data about the distribution

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of the pixels and their intensities. As the size of the cube is different for each

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scan, an image analysis is obtained in different scales: the cubes with larger

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sides provide data about the overall structure of the image while the small cubes

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capture information about local data. Here we see clearly the combination of

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the fractal approach, given by the analysis at different scales, with the statistical

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distribution. The details of the whole process can be seen in the Algorithm 1

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and Figure 4 exhibits the curves produced in each step of the method.

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Algorithm 1 Sliding-box descriptor algorithm. Input: A

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Imax = max(size(m, n))

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for i = 1 to m do

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for j = 1 to n do m l max aux ← A(i,j)·I 256

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end for

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B(i, j, aux) ← 1

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end for

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l ← min(m, n)

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while l ≤ 2 do

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C ← ones3(l)

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D ← conv3(B, C)

for k = 0 to l2 do

C ← sum(D ≤ k)α

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end while

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end for S D ← {D, C}   l ← 2l

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Output: D

Table 1: Variables used in Algorithm 1.

Matrix m × n containing the texture image in gray scales

B

Three-dimensional array resulting from mapping A

C

Three-dimensional array used as a mask for convolution

D

Valid convolution result between B and C

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Box size in each scan

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Vector of texture descriptors

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Table 2: Pre-programmed routines supposed to be available by Algorithm 1

Return the size (number of rows and columns) of the matrix A

max(m,n)

Return the maximum value between m and n

ones3(l)

Return a three-dimensional array l × l × l whose elements

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are all 1 sum(A ≤ k)

Return the sum of values in A smaller or equal to k

conv3(B,C)

Return the valid part of three-dimensional discrete

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convolution between B e C

Figure 1: Steps involved in the proposed method. From left to right, the original texture, the cloud of points and 3D sliding boxes, individual cumulated distribution and aggregated

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distribution.

4.1. Motivation

The strategy adopted here of inspecting the distribution of points inside

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each box rather than simply counting the number of boxes covering the object

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of interest leads to a more complete description about how the object occupies

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the space enclosed by the image domain. Furthermore, the use of sliding boxes

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instead of a fixed grid also makes this representation more precise and fine-

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tuned.

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To see how the distribution of the number of points impacts the image

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descriptor, first we need to understand how the classical three-dimensional box

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counting dimension works. That is obtained by counting the number of cubes 11

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covering the image cloud at each scale. In practice, we would be counting

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the number of cubes containing at least one point. This would simply be the

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cumulated distribution of each non-null possible number of points.

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To simplify the idea we illustrate the case of n points randomly placed within

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a line segment with length L and partitioned into subintervals (boxes), each one

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with length r. The probability of k boxes being non-empty is equivalent to

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the probability of n points being distributed over sr = L/r boxes with sr − k

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boxes left empty. This is a classical problem in combinatorics and the solution

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is provided by

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S(n, k)k! snr

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P (k) =

sr sr −k

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(9)

where S(n, k) are Stirling numbers of second kind, defined as S(n, k) =

  k k n 1 X (−1)k−j j . k! j=0 j

(10)

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Figure 2 shows a couple of simulated examples illustrating how this distribution

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is accurately confirmed in practical tests.

On the other hand, when we have sliding windows and wish to count the

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distribution of the number of points in each window position, we should resort to

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the theory of sliding window detection probabilities [18]. The underlying theory

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was originally developed to analyze radar/sonar signals in naval surveillance and

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models the sliding windows as an automaton, associating the current state with

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the number of points within that window.

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In particular, our line segment can be represented by a binary vector with

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L components (points correspond to 1’s and empty spaces to 0’s). Hence the

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probability of k points falling within the sliding window is obtained with the

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help of a transition matrix for the underlying finite automaton.

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The states are binary numbers whose decimal representation ranges between

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0 and 2r −1. As the points are randomly placed following a uniform distribution,

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the probability of a ’1’ arising in the binary vector is a constant p = 1/L.

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The transition table also has an accepting state “binary number containing k
1’s”. There are na = kr states that already are accepting states and these are 12

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0.2

1

0.4

0.8

0.3

0.6

0.2 Simulated Predicted

0 0

0.4

0.1

5

10

Simulated Predicted

0.2

0

15

0

5

10

k

n = 16 0.5

0.6

Simulated Predicted

0.2

0 15

0.2

Simulated Predicted

0

5

10

15

0

2

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k

r=6

Simulated Predicted

0

0

20

k

0.6 0.4

0.4

0.2

15

n = 64

P(k)

P(k)

P(k)

0.3

10

k

0.8

0.8

10

5

1

0.4

5

0

n = 36 1

0

15

k

0.6

0.1

Simulated Predicted

0

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0.1

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P(k)

P(k)

0.3

0.5

P(k)

0.4

r=8

4

6

8

10

12

k

r = 10

Figure 2: Simulations of the distribution of the number of boxes intersecting the object (classical box counting) using different values for the number of points n and box size r. The simulation considers L = 120. In the first row we fixed r = 8 and in the second one we used n = 64.

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removed from the table.

We illustrate with r = 3 and k = 2. The transition probabilities correspond-

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ing to going from state abc to bcd are given in Table 3. This table also has an

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accepting state k = 2. States 011, 101 and 110 already are accepting states.

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The automaton probability also depends on the initial probability of each state

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S0 in the transition matrix (including the accepting state). In general:

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S0 =

h

1/L 1/L

1/L · · ·

(L − sa )/L

i

(11)

The final probabilities (for all states) are determined by the repeated multipli-

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cation by T :

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Pf = S0 T L−r = S0 T L−3 .

(12)

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Finally, the probability of entering into an accepting state, which implies the ex-

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istence of k points within the sliding window, is provided by the last component

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of Pf , i.e.,

P (k) = Pf (L − na + 1). 13

(13)

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Table 3: Transition matrix T for sliding probabilities in a simple case: r = 3 and k = 2.

001

1−p

p

010

111

1−p

001

p

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1−p

k=2

p

1−p

010 100

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000

000

p

111

p

k=2

1−p 1

Figure 3 compares the hypothesized distribution with some simulated situations.

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In comparison with Figure 2 one can notice that the sliding distribution captures

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more variance in the data than the distribution of fixed boxes. Moreover, the

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influence of the box size r is also more evident in the sliding distribution. While

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a smaller value of r yields a nearly flat uniform distribution, larger ones result

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in more normal-like curves. Similar behavior is observed with respect to the

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number of points n. This is an immediate consequence of the central limit

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theory in statistics and the natural trend to normal distributions.

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We notice that the curves in Figure 3 encompass larger part of the variance

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in the Gaussian-like distribution than exhibited by the fixed box distribution in

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Figure 2. The distribution variance is directly related to physical properties like

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homogeneity that are well known to be fundamental in texture discrimination.

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The ability of expressing larger part of the variance basically attests that the

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descriptors provided by the proposed method are more complete than those

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possibly provided by classical box counting dimension. The use of sliding boxes

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allowed a statistical description that could not be carried out in the original

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context at the same time that we preserve the straightforwardness of the analysis

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of boxes covering objects of interest at different scales, as usual in any fractal

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analysis.

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3

0.8

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k

P(k)

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r=8

2

4

6

8

10

k

r = 10

Figure 3: Simulations of the distribution of points within a sliding window (proposed descriptors) using different values for the number of points n and box size r. The simulation considers L = 120. In the first row we fixed r = 8 and in the second one we used n = 64.

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5. Experiments

The classification accuracy of the proposed method was assessed on three

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texture databases frequently used in the literature for benchmark purposes.

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The proposed descriptors were also employed for the identification of species of

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Brazilian plants, based on scanned images of their leaf surfaces.

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The first database in our comparison is KTHTIPS-2b [21], a collection of

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4752 images evenly divided into 11 categories (materials). The classification

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problem in this data set should focus on the material represented in the image

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rather than on the instance of the photographed object. Each material is divided

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into 4 samples, each one possessing particular settings of illumination, scale and

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pose. The experimental protocol adopted here is the most frequently found

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in the literature [21, 50, 23, 26, 48], i.e., 1 sample used for training and the

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remaining 3 samples used for testing. The rationale behind this protocol is

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the focus on categories rather than on exemplars, in this way the algorithm

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should be capable of recognizing an image without seeing any exemplar of that 15

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particular sample. In general, this poses more challenge to the classification

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process as different categories share similarities, in their material composition,

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for instance, but those similarities are not necessarily expressed in the visual

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aspect. The accuracy (percentage of images assigned to the correct class) and

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standard deviation are obtained by averaging out the results for the 4 possible

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combinations of training/testing.

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The second data set is UIUC [19]. This contains 1000 images equally di-

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vided into 25 texture categories (classes). The images were photographed under

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non-controlled conditions, which makes them susceptible to variation in scale,

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perspective, illumination, and albedo. The training/testing division protocol

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follows the usual procedure in the literature, i.e., 20 images of each texture

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randomly selected for training and the remaining 20 images employed for test-

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ing. Such procedure is repeated 10 times to provide the average accuracy and

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respective deviation.

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The third texture database is UMD [20]. This is composed by 1000 high-

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resolution images, also collected under uncontrolled conditions. The images are

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categorized into 25 classes, each one with 40 images and each image has a resolu-

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tion of 1280 × 960. The main particularity of this database is the high variation

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in viewpoint and scale, turning the classification process into a challenging task.

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With the aim of reducing the number of features and attenuating the effects

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of the “dimensionality curse”, the proposed descriptors are processed by prin-

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cipal component analysis [45]. Following that, the descriptors are finally used

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as the input of the classifier. Here we verified the use of two classifiers: sup-

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port vector machines (SVM) with settings as those employed in [26], i.e., linear

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kernel, C = 1 and L2 normalization, and linear discriminant analysis (LDA)

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[46].

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6. Results and Discussion

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The first test was carried out to assess the classification accuracy (percentage

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of images correctly classified) of the proposed method when two different classi-

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Table 4: Classification accuracy using SVM and LDA classifier.

SVM

LDA

KTHTIPS-2b

40.1±4.6

61.9±3.1

UIUC

59.7±2.8

88.9±0.9

UMD

78.9±1.9

99.2±0.4

1200Tex

61.2±1.9

86.6±1.1

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fiers (LDA and SVM) are employed. Table 4 shows the accuracy in each texture

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database together with the corresponding standard deviation for the most com-

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plete version of the descriptors, i.e., for feature vectors combining descriptors

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D in Equation (8) both for the gray level image and for the LBP mapping.

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LDA yielded the highest accuracies in all the data sets. Based on this, all the

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remaining tests, presented in Tables 5, 6 (Proposed), and 7 (Proposed), and

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Figures 4 and 5, were also accomplished by using this classifier.

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In Table 5 we exhibit the accuracy of the proposed fractal descriptors com-

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puted over the original gray-valued image, the LBP encoding and combining

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both feature vectors. Notice that we used LDA classifier for the same complete

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feature vector in Table 4 and that is why the second column of Table 4 and the

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third column of Table 5 are identical. We can observe that each approach can

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be more or less interesting depending on the specific database being analyzed.

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In most cases, however, the sliding box method applied to the LBP encoding

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provides higher accuracy than the direct application to the original image. We

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also notice that combining fractal features over the gray values and the LBP

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codes can provide even better classification results.

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Table 6 lists the accuracy of the proposed descriptors in the benchmark tex-

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ture databases, compared with results published in the literature following sim-

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ilar protocols. The proposed method outperforms state-of-the-art approaches

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like SIFT + KCB or SIFT + BoVW in KTHTIPS-2b. Our proposal also pre-

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sented better results than classical texture descriptors like LBP/VAR and VZ-

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Joint in UIUC and even methods based on automatic learning like FC-CNN are

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Table 5: Classification accuracy of the sliding fractal descriptors computed over the original

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gray level image, the LBP encoding, and combining both approaches.

Database

Gray level

LBP

Gray level + LBP

KTHTIPS-2b

45.7±3.1

61.2±3.0

61.9±3.1

UIUC

81.1±1.5

69.1±2.2

UMD

92.4±1.1

99.0±0.4

99.2±0.4

1200Tex

77.1±1.4

84.4±1.2

86.6±1.1

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88.9±0.9

also outperformed in UMD. UMD and UIUC are classical examples of “textures

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in their strict sense” and such results confirm the suitability of fractal-based

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methods to analyze such types of images.

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Figure 4 depicts the confusion matrices for the benchmark textures. Gen-

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erally speaking, these diagrams confirm the results in Table 6, but they also

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provide a rather important information, which is the accuracy per class, al-

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lowing in this way a more complete analysis of the classification outcomes. In

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Figure 4 (a), KTHTIPS-2b presents lower accuracy in classes 3 (“corduroy”), 5

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(“cotton”), and 10 (“wool”). These are actually materials highly susceptible to

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confusion as they all correspond to types of clothing fabrics and are composed by

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similar texture patterns. In UIUC, the method achieves much higher accuracy,

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with some relevant misclassification only in class 19 - carpet ( confused with 8

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- granite). These are materials characterized by a similar granular appearance,

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which poses some difficulties even for visual discrimination.

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Another important aspect to be pointed out here is the computational com-

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plexity of the compared approaches. As can be inferred from Algorithm 1,

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the complexity bottleneck of the proposed method is the convolution. However

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efficient algorithms for that can perform in order O(n log n) or even O(n) con-

364

sidering overlapping [56]. In our case, this basically corresponds to the number

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of pixels in the image. Other “handcrafted” compared features like LBP and

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VZ-Joint also have similar performance as they essentially rely on local-based

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comparisons. On the other hand, estimating complexity of methods involving

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some sort of learning-based process, such as FC-CNN or PCANet is actually a

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Table 6: Accuracy of the proposed method compared with other texture descriptors in the literature.

UIUC

Accuracy (%)

VZ-MR8 [47]

46.3

LBP [16]

50.5

VZ-Joint [22]

53.3

LBP-FH [48]

54.6

CLBP [49] ELBP [50] SIFT + KCB [23] SIFT + BoVW [23] Proposed

Method

Accuracy (%)

RandNet (NNC) [51]

56.6

PCANet (NNC) [51]

57.7

BSIF [52]

73.4

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Method

p ro

KTHTIPS-2b

57.3

VZ-Joint [22]

78.4

58.1

LBPriu2 /VAR [16]

84.4

58.3

ScatNet (NNC) [53]

88.6

58.4

Proposed

88.9

61.9

UMD

Accuracy (%)

FC-CNN AlexNet [26]

95.9

DeCAF [23]

96.4

al

Method

1

96.6

(H+L)(S+R) [19]

97.0

FC-CNN VGGM [26]

97.2

FC-CNN VGGVD [26]

97.7

SIFT+BoVW [23]

98.1

SIFT+LLC [26]

98.4

WMFS [24]

98.7

OTF [55]

98.8

PLS [25]

99.0

Proposed

99.2

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Expected class

(a) 5

(b) 20

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Expected class

(c)

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Figure 4: Confusion matrices. (a) KTHTIPS-2b. (b) UIUC. (c) UMD.

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hard task, as it depends on the number of layers and operators. What can be

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empirically determined is that, in general, those approaches usually consumes

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much more computational resources than the traditional descriptors.

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In general, the proposed method presents important advantages like the good

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performance without requiring large amounts of data for training, as usual in

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modern learning-based approaches, an algorithm that performs in reasonable

375

computational time even with not so advanced hardware and the interpretabil-

376

ity associated with a fractal-based model, as fractals are known for long time to

377

be a natural mathematical tool to describe nature. In terms of disadvantages,

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the most significant one is the unsuitability of this approach for the analysis of

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general purpose images, for example for object recognition. In this particular

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task, neural networks are more adequate and recommended as a more general-

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izable model.

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6.1. Identification of Plant Species

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Table 7 shows the accuracy of the proposed descriptors in 1200Tex database

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[57], compared with other results recently published in the literature on this

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same data set. 1200Tex is a collection of images from the leaf surface of 20

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Brazilian species photographed in vivo. For each species 20 samples were col-

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lected, cleaned, registered (alignment with respect to the vertical axis) and pho-

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tographed by a commercial scanner. The original image of each photographed

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sample was split into 3 non-overlapping windows, each one with resolution

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128 × 128. Such windows were extracted from regions of the leaf less affected by

391

texture variance caused by spurious elements. Before classification, all the im-

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ages were converted into gray values, resulting in a database with 1200 images.

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In terms of validation protocol, we randomly selected 30 images per species for

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training and the remaining images were employed for testing. This procedure

395

was repeated 10 times, allowing in this way the computation of the average

396

accuracy and standard deviation.

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Figure 5 provides more detailed scenario than Table 7 by showing the confu-

398

sion matrix of the proposed method in 1200Tex. The accuracies in most classes 21

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the identification of plant species.

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Table 7: Accuracy of the sliding fractal descriptors compared with other literature results in

Accuracy (%)

LBPV [49]

70.8

Network diffusion [58]

75.8

FC-CNN VGGM [26]

78.0

Gabor [57]

p ro

Method

84.0

FC-CNN VGGVD [26]

84.2

Schroedinger [59]

85.3

86.0

Pr e-

SIFT + BoVW [23] Proposed

Predicted class

5

10

15

86.6

25 20 15 10 5

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20 5

10

15

20

Expected class

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Figure 5: Confusion matrix for 1200Tex.

are good and the most critical case is class 8, which is mostly confused with class

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6. In fact, these correspond to species whose leaf textures look rather similar,

401

especially in terms of the arrangement of nervures and microtextures, which are

402

known to be prominent elements for the discrimination among samples from

403

different species.

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In overall terms, the results corroborate that fractal descriptors can still be

405

considered as competitive in texture classification. This is in fact expected given

406

the way that many materials are usually formed in nature and the well known

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adequacy of fractal geometry in modeling such processes. As also expected, the

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effectiveness of fractal modeling is even more evident in “pure” textures (like 22

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UIUC and UMD), or in practical problems, like those involving the classification

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of biological images, here illustrated with the identification of plant species.

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The obtained results not only suggest more in-depth research on this topic,

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but they also presents fractal descriptors as a useful alternative that should

413

be verified in practice. Such practical interest is justified by a competitive

414

performance associated with the fact that “hand-crafted” approaches like the

415

one proposed here do not require neither large amounts of training data nor high

416

computational power. Fractal descriptors also provide more straightforward

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interpretation of the model, given that fractal geometry has been associated for

418

a long time with a suitable model of nature.

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7. Conclusions

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This work proposed and studied the applicability of an image descriptor,

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with focus on grayscale texture images, based on the fractal geometry theory.

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The method relies on classical techniques to numerically estimate the box count-

423

ing fractal dimension. As usual, the grayscale image is mapped onto a cloud

424

of points in the three-dimensional space. However, instead of using boxes with

425

fixed position, we adopted a scheme where the boxes with different sizes slide

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over the image. The proposed descriptors combine in this way the multiscale

427

analysis by using different box sizes with local features by quantifying the dis-

428

tribution of pixels within each box in each position.

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The performance of the proposed descriptors was assessed in the classifica-

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tion of three benchmark data sets of grayscale images (KTHTIPS-2b, UIUC

431

and UMD) and in a real-world problem: the identification of species of Brazil-

432

ian plants. In both situations, the proposal achieved highest ratios of images

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correctly classified in comparison with other classical and state-of-the-art tex-

434

ture descriptors. We also developed a theoretical statistical model to explain

435

how the distribution of covering boxes are different from the distribution of the

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number of points within the sliding window. Such model confirmed that the

437

proposed strategy yields a more complete description of the texture image.

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Summarizing, the obtained results confirmed that straightforward approaches to generate fractal descriptors can be competitive with more complex alterna-

440

tives where the statistics of the measure expressing the concept of “fractality” in

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fractal geometry is not clearly stated. Moreover, it also validates that different

442

strategies potentially derived from the same idea in fractal geometry can lead

443

to different results when such strategies are adapted for algorithms in digital

444

images. This is the case of box-counting, a technique that theoretically is known

445

to be equivalent to Bouligand-Minkowski dimension, whereas the respective de-

446

scriptors present results significantly different.

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Acknowledgements

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G. T. gratefully acknowledges the financial support of The University of

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Campinas Fund for Research and Extension Studies (FAEPEX) Proc. 2300/17.

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J. B. F. gratefully acknowledges the financial support of The State of S˜ ao

451

Paulo Research Foundation (FAPESP) (Proc. 2016/16060-0) and from Na-

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tional Council for Scientific and Technological Development, Brazil (CNPq)

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(Grant #301480/2016-8).

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Declarations of interest: none