Page - 377 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 (a) (b) (c) Figure1.ExampleofatexturefromtheVisTexdatabase(a),oneofitspatches(b)andthecorresponding outlier (c). Once thedatabase is built, it is 15-timesequally andrandomlydivided inorder toobtain the trainingand the testing sets that are furtherused in the supervisedclassificationalgorithm. Then, foreachpatch inbothdatabases,a featurevectorhas tobecomputed. The luminancechannel isfirst extractedand thennormalized in intensity. Thegrayscalepatchesarefilteredusing the stationary wavelet transformDaubechiesdb4filter (see [39]),with twoscalesandthreeorientations. Tomodel thewavelet sub-bands,variousstochasticmodelshavebeenproposedin the literature.Amongthem, theunivariategeneralizedGaussiandistributionhasbeenfoundtoaccuratelymodel theempirical histogramofwavelet sub-bands [40]. Recently, ithasbeenproposedtomodel thespatialdependency ofwavelet coefficients. Tothisaim, thewavelet coefficients located ina p×q spatialneighborhood of the current spatial position are clustered in a randomvector. The realizations of these vectors canbe furthermodeledbyelliptical distributions [41,42], copula-basedmodels [43,44], etc. In this paper, thewavelet coefficientsareconsideredasbeingrealizationsofzero-meanmultivariateGaussian distributions. Inaddition, for thisexperiment thespatial information iscapturedbyusingavertical (2×1)andahorizontal (1×2)neighborhood.Next, the2×2samplecovariancematricesareestimated foreachwaveletsub-bandandeachneighborhood. Finally,eachpatchisrepresentedbyasetofF=12 covariancematrices (2scales×3orientations×2neighborhoods)denotedY=[Y1, · · · ,YF]. TheestimatedcovariancematricesareelementsofPm,withm= 2, andtherefore, theycanbe modeledbyRiemannianLaplacedistributions.Moreprecisely, inorder to take intoconsiderationthe within-classdiversity,eachclass inthetrainingset isviewedasarealizationofamixtureofRiemannian Laplacedistributions (Equation(27))withMmixturecomponents, characterizedby ( μ,Y¯μ,f ,σμ,f), having Y¯μ,f ∈ P2, with μ = 1, · · · ,M and f = 1, · · · ,F. Since the sub-bands are assumed to be independent, theprobabilitydensity function isgivenby: p(Y|( μ,Y¯μ,f ,σμ,f)1≤μ≤M,1≤f≤F)= M ∑ μ=1 μ F ∏ f=1 p(Yf |Y¯μ,f ,σμ,f) (37) The learningstepof theclassification isperformedusingtheEMalgorithmpresented inSection4, andthenumberofmixturecomponents isdeterminedusingtheBICcriterionrecalled inEquation(31). Note that for theconsideredmodelgiven inEquation(37), thedegreeof freedomisexpressedas: DF=M−1+M×F× ( m(m+1) 2 +1 ) (38) sinceonecentroidandonedispersionparametershouldbeestimatedper featureandpercomponent of themixturemodel. Inpractice, thenumberofmixturecomponentsMvariesbetweentwoandfive, andtheMyieldingto thehighestBICcriterion is retained.Asmentionedearlier, theEMalgorithm issensitive to the initial conditions. Inorder tominimize this influence, for thisexperiment, theEM 377