Enhancing fractal descriptors on images by combining boundary and interior of Minkowski dilation
Marcos W. S. Oliveira, Dalcimar Casanova, João B. Florindo, Odemir Martinez Bruno
EEnhancing Fractal Descriptors on Images by Combining Boundary and Interior ofMinkowski Dilation
Marcos William da S. Oliveira ∗ Institute of Mathematics and Computer Science, University of S˜ao Paulo (USP),Avenida Trabalhador s˜ao-carlense, 400 13566-590 S˜ao Carlos, S˜ao Paulo, Brazil
Dalcimar Casanova, † Jo˜ao B. Florindo, ‡ and Odemir M. Bruno § (Dated: August 6, 2018)This work proposes to obtain novel fractal descriptors from gray-level texture images by combininginformation from interior and boundary measures of the Minkowski dilation applied to the texturesurface. At first, the image is converted into a surface where the height of each point is the grayintensity of the respective pixel in that position in the image. Thus, this surface is morphologicallydilated by spheres. The radius of such spheres is ranged within an interval and the volume and theexternal area of the dilated structure are computed for each radius. The final descriptors are givenby such measures concatenated and subject to a canonical transform to reduce the dimensionality.The proposal is an enhancement to the classical Bouligand-Minkowski fractal descriptors, whereonly the volume (interior) information is considered. As different structures may have the samevolume, but not the same area, the proposal yields to more rich descriptors as confirmed by resultson the classification of benchmark databases. I. INTRODUCTION
In the last decades, Fractal Geometry has demon-strated to be a worthy tool to develop robust and precisemethods of image analysis [1–6]. Such methods have beensuccessfully applied to a number of problems comprisingthe analysis of digital images in several areas, such asPhysics [7, 8], Medicine [9, 10], Engineering [11, 12], etc.Among such fractal-based approaches, methods likemultifractals [13], multiscale fractal dimension [14], lo-cal fractal dimension [15], etc. have outperformed otherclassical and state-of-the-art image analysis methods inmany situations.More recently, a novel fractal-based imaging methodnamed fractal descriptors has been proposed in [16, 17].Roughly speaking, this approach extracts features of theimage by computing the fractal dimension at differentscales of observation and taking all these values over apredefined range. Although this approach has demon-strated to be a promissing solution for image analysisproblems, it was defined and studied only for a limitednumber of well-known techniques to estimate the fractaldimension. Among the studied possibilities, Bouligand-Minkowski has provided remarkable results both for theanalysis of natural and synthetic images [16]. These de-scriptors are obtained from the interior measures (vol-umes) of the object of interest dilated by spheres with apredefined range of radius values. ∗ [email protected] † [email protected] ‡ fl[email protected] § [email protected] Despite the great results achieved, Bouligand-Minkowski takes into account only the interior (volume)of the dilated structure. It is well-known from Geome-try that the boundary of a structure encloses informa-tion as rich as the interior and, for instance, in a three-dimensional space, two objects with the same volumemay have different areas. In this way, this work proposesto enhance the Bouligand-Minkowski descriptors by in-cluding information from the boundary, that is, the areain a three-dimensional space. The combination is accom-plished by means of a simple concatenation of measures,followed by a dimensionality reduction through the can-nonical analysis [18]. The performance of the proposalis assessed over databases of texture images and the re-sults are compared to other classical and state-of-the-artmethods in the literature.
II. FRACTAL GEOMETRY
A number of works characterizing and analyzing im-ages using fractal geometry have been reported in theliterature [1–6]. Generally, these works model objectsand scenarios from the real world as an approximationof mathematical or statistical fractals and extract frac-tal properties of the element. The most used of theseproperties is the fractal dimension.The formal definition of the fractal dimension, alsocalled Hausdorff-Besicovitch dimension, is obtained fromthe Hausdorff measure. Let X be a geometrical set ofpoints in an N -dimensional topological space. Its Haus-dorff measure H sδ is calculated by H sδ ( X ) = inf ∞ (cid:88) i =1 | U i | s , (1) a r X i v : . [ phy s i c s . d a t a - a n ] D ec where U i is a δ -cover of X ; that is, there exists a count-able collection of sets { U i } , with | U i | ≤ δ , such that X ⊂ ∪ ∞ i =1 U i and | U i | denotes the diameter of U i , thatis, the maximum possible distance between two any ele-ments of U i : | U i | = sup {(cid:107) x − y (cid:107) : x, y ∈ U i } . (2)As the parameter δ is a superior limit for the diameterof the balls U i covering the fractal object, it can be con-sidered a scale metric and should be removed from thedimension definition once this is scale-independent. Inthis way a limit to 0 is applied over δ giving rise to themeasure H s H s ( X ) = lim δ → H sδ ( X ) . (3)As it can be demonstrated in Measure Theory, H s ( X )has a particular behavior that arises for any set of points X , that is, the value of H s is always ∞ for any s < D and0 for any s > D , where D is a non-negative real value.The point of discontinuity D is the Hausdorff-Besicovitchfractal dimension of XD ( X ) = inf { s : H s ( X ) = 0 } = sup { s : H s ( X ) = ∞} . (4)Although the above definition is the most exact andgeneralist method to calculate the dimension, by usingan infinitesimal covering, it is necessary to know the an-alytical expression of the object being measured. How-ever this is not possible when the real-world element ap-proximated by a fractal is represented in a discrete andfinite space as the digital images discussed in this work.To address these situations, several approximation meth-ods have been proposed in the literature [19, 20]. Suchmethods aim to compute a measure of self-similarity andcomplexity of the object by generalizing the definition ofthe Euclidean dimension. Thus the object is measuredby a rule unit with the same topological dimension of theobject. The length r of this unit is ranged along an in-terval to compute the number of units N ( r ) necessary tocover the object. In this context, the dimension is givenby D = − lim r → log( N ( r ))log( r ) . (5) A. Bouligand-Minkowski
One of the most commonly used techniques to estimatethe dimension based on Equation 5 is the Bouligand-Minkowski [19]. Similarly to the Hausdorff-Besicovitchdefinition, it is also derived from a measure, in this case,the Bouligand-Minkowski measure M ( X, S, τ ) of a set X ∈ R n M ( X, S, τ ) = lim r → V ( ∂X ⊕ S r ) r n − τ , (6) where −∞ < τ < + ∞ is a parameter and V ( ∂X ⊕ S r ) isthe volume of the edge of X ( ∂X ) morphologically dilatedby a structuring element S , symmetrical with respect tothe origin and with radius r .The Bouligand-Minkowski fractal dimension D BM isgiven by D BM ( X, S ) = inf { τ : M ( X, S, τ ) = 0 } . (7)In practice, the dimension is computed by a neighbor-hood strategy. Each point of the object X is replaced bya structuring element S (cid:15) , with radius (cid:15) , and the numberof points within the union of such elements is used toestimate the volume V . Thus the dimension is providedby D BM ( X ) = lim (cid:15) → (cid:18) n − logV ( X ⊕ S (cid:15) ) log(cid:15) (cid:19) . (8)For an object represented in a digital image, an ef-ficient and precise method to compute the Bouligand-Minkowski dimension is the Euclidean Distance Trans-form [21], by Saito’s Algorithm [22]. Starting from a tex-ture image I : R → R , it is mapped into a 3D structure(surface) X , such that each point with coordinate ( x, y )and pixel intensity z is converted into the point with co-ordinate ( x, y, z ) in X . To simplify the idea, the object X is supposed to be in R . The distance transform DT X is given by DT X ( i, j ) = min { d (( i, j, k ) , ( i (cid:48) , j (cid:48) , k (cid:48) )) : ( i (cid:48) , j (cid:48) , k (cid:48) ) ∈ X } , (9)for all ( i, j, k ) ∈ R , where d ( p, q ) is the Euclidean Dis-tance between p and q .The set of possible distances R (Euclidean) is given by R = { r : r = (cid:112) i + j + k ; i, j, k ∈ N } . (10)In the following, these values of r are sorted increasingly R = { r , r , r , r , r , ..., r max } = { , , √ , √ , √ , ..., r max } . (11)Thus the dilation volume V ( r ) is computed by V ( r ) = { ( i, j, k ) : DT X ( i, j, k ) = r } (12)and the dimension is given by D BM = 3 − lim r → log( V ( r ))log( r ) . (13)Numerically, this limit uses to be given by the slope of astraight line fit to the curve log( r ) × log( V ( r )). Figure 1illustrates the dilation process. l og ( V (r)) FIG. 1. Bouligand-Minkowski fractal dimension for three-dimensional objects. A texture image is mapped onto a sur-face and handled as a three-dimensional structure [16]. Thelog( r ) × log( V ( r )) curve can be used to estimate the fractaldimention of the object. B. Fractal Descriptors
Equation 5 can be generalized by replacing N ( r ) byany self-similarity measure M ( (cid:15) ), where (cid:15) is a scale pa-rameter.This generalization results in the proposal and studyof several methods to approximate the dimension [19,20], each one providing results more or less close to thetheoretical value, depending on the specific application.However, the outcome still is only a single real value todescribe all the complexity of an object. Moreover, whenthis object is not a mathematical fractal, its dimensionis highly scale-dependent and a global dimension may beof little or no usefulness.To address these points and make possible a more com-plete fractal-based analysis of real-world structures, thefractal descriptors were proposed in [16, 17]. Basically,instead of computing only the fractal dimension, the frac-tal descriptors are composed by all the values of dimen-sion at each scale along a range of observation. Consid-ering that (cid:15) is a scale parameter, such set of descriptors u can be obtained from the self-similarity curve: u : (cid:15) → M ( (cid:15) ) . (14)Using the Bouligand-Minkowski method described previ-ously, in thres dimensions, these descriptors are given bythe logarithm of the dilation volumes u = [log( V ( r )) , log( V ( r )) , ..., log( V ( r max )))] . (15) In an image analysis task, the descriptors u can beused directly [16] or after some type of transform [23] aswell as they can be extracted from the entire image [17]or from disjoint regions [24]. III. BOUNDARY MEASURE
VERSUS
INTERIOR MEASURE
On the above discussion, the Equation 15 and the pre-vious works [16, 17] talk about the use of interior in-formation (volume in three dimensional space) to obtainthe fractal descriptors. This work proposes to analise thedifference between the information comprised within theboundary and the interior of a region. In two dimensions,this analysis consists of studying measures of perimeterand area of a flat shape. In the three-dimensional case,the discussion concerns area measures of surfaces andmeasures of volumes.The main idea depicted here is that two different ob-jects, with a dilatated radius r , can have the same interiormeasure, but different boundary measures.To support this theory, the dilation of circles is anal-ysed in two dimensions. Figure 2 illustrates three differ-ent circle arrangements, with three different center pointsfor each case and the same radius. It is an exemple thatthe interior areas of three circles, including the intersec-tions, are the same but the the boundary measures of theregions, i.e. the perimeters, are different. FIG. 2. Different arrangements of three circles with the sameradius. The center points, areas and perimeter measures areexplained on Table I.
In Image Processing, the perimeter of a flat region Θcan be estimated by P = n e + n o √ , (16)where n e and n o are the number of even and odd codes,respectively, in chain-code representation [25, 26]. Equa-tion 16 simply performs a count of the number of pixelson the boundary of a region and estimates the arc lengthof its contour.In the same way, the area measure is estimated by thecount of pixels that represent the object Θ. Therefore,the area is obtained by A = (cid:88) p ∈ Θ , (17)where p denotes any pixel on the image.After applying Equations 17 and 16 for the three ar-rangements on Figure 2, similar area measures and differ-ent perimeters are obtained. Table I shows these resultsto the arrangements of the Figure 2.These results support our theory, we can verify thatthe two different images (or pixels arrangement) can havethe same dilated areas (to a given dilatated radius r ), butdifferent perimeters. In this situation only the perimeteris useful to discriminate the images. In this way we canuse the bondary measure as an complementary featureto compose our fractal descriptor.For three-dimensional objects, the surface area mea-sure and volume measure are estimated in a similar wayto that of Equation 16 and 17. Furthermore, the behaviorof sphere dilation is also similar to that of circle dilation. IV. APPLICATIONS ON IMAGES
As showed by the above discussion, the area andperimeter of a dilated object express complementary in-formation. In this way the most natural approach is touse both features together. The synergy achieved by thisapproach can bring a better discrimination power to thefractal descriptors. To do that, a simple concatenationof both feature vectors is made in order to obtain a morerich fractal descriptors.In order to verify our theory an application of textureanalysis is carried out using the novel proposed fractaldescriptor. Such analisys is performed over a supervisedclassification task, though the proposed method can bealso used to perform a CBIR (Content-based image re-trieval), segmentation or other kinds of image analysis.For this application, the both signatures (area andvolume) are computed for each image, concatened andthe supervised classification is carried out by applying aCanonical Analysis [18] followed by a Linear Discrimi-nant Analysis (LDA) (also called Fisher linear discrim-inant) [27]. The 10-fold cross-validation scheme is usedin all experiments and over benchmark databases. Thecanonical analysis is, basically, a geometric transforma-tion of the feature space in order to generate new uncor-rellated features based on linear combinations. The ideaof this method is to find a new projection of the datawhere the class separation is maximized. From p originalfeatures, p -cannonical variables can be obtained. How-ever, a reduction in the number of variables to be eval-uated is usually desired. Therefore, a LDA supervisedclassification is accomplished by using the most signifi-cant p cannonical variables. A. Texture
Three texture sets are used in the experiments: Bro-datz, Vistex and Outex. 1. Brodatz texture database is derived from the Bro-datz Album [28] and has become the standard forevaluating texture algorithms, with hundreds ofstudies having been applied to this set of images.This database is composed by 1776 texture samplesgrouped into 111 classes. Each image is 128 × × × × × p -cannonical variables used in LDA classifier. Figure 3 il-lustrates the behavior of the success rates when the num-ber of p -cannonical variables is ranged. We observe that,for Vistex data set, the rate increases at a first moment,achieves an optimal rate and thus stabilizes with a verysmall decrease when we consider more descriptors. Thisbehavior was expected since the high dimensionality ofthe feature vectors damage the efficiency of the classifier.For all considered fractal descriptors this same behaviorcan be observed.We also can verify that the volumetric information isa little bit more effective than area counterpart. How-ever, we see that the synergy between such two kinds ofinfomation yields a improvement on the perfomace of thetexture analysis.Based on the behavior observed in Figure 3, we setupan experiment with a total of 40 p -cannonical variables.Table II shows the achieved results. For all data sets, thebest sucess rate is provided by the combination of vol-umetric and surface features, the average performanceimprovement is close to 2%. This result corroborates ourtheory that the area can be useful as a complementaryinformation in the studied fractal descriptor. It is im-portant to emphasize that combining different featuresdoes not increase the number of descriptors used in theclassifier.Arrangements Center Points Area PerimeterLeft (6,6); (8,6); (15,10) 152 49.799Center (6,6); (6,12); (10,19) 152 44.627Right (6,6); (12,9); (14,7) 152 46.627 TABLE I. Center points, interior areas and perimeters of three arrangements of circles on Figure 2.
Success rate (%) and stardart deviation (%)Methods Brodatz VisTex OutexVolumetric 88.15(0.26) 91.28(0.67) 80.66(0.43)Area 87.93(0.29) 89.98(0.52) 80.91(0.37)Volumetric + Area 89.29(0.21) 93.09(0.31) 82.57(0.33)
TABLE II. Results of fractal descriptors on texture databases using 40 p -cannonical variables. Note that the synergy betweenthe volumetric and area features can improve the final results. S u c e ss r a t e ( % ) AreaVolumetricVolumetric + Area
FIG. 3. Accuracy versus number of p -cannonical variables.The optimal p -cannonical variables of Vistex datasets isnearly 30-40 features. V. CONCLUSION
This work proposed a new way of computing frac-tal descriptors from gray-level fractal descriptors. Themethod combines information from the interior (volume)and boundary (area) of a surface representation of theimage dilated by spheres with variable radii (Minkowski dilation).The results of applying the proposal to the classi-fication of benchmark data sets showed that by us-ing a reasonable number of descriptors, the volumeachieved higher correctness rates than the area used inthe Bouligand-Minkowski approach. However the per-formance is enhanced when combining area and volumeinformation.In fact, the area enriched the descriptors in that itprovides a different and complementary viewpoint of thetexture. Thus different images can have the same vol-ume for a specific dilation, but its area may be different,contributing for a more robust discrimination.
ACKNOWLEDGMENTS
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