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WVC'2007 - III Workshop de Visão Computacional, 22 a 24 de Outubro de 2007, São José do Rio Preto, SP.For the second term, we should not adopt a globalapproach, but instead use the local informationrestricted only to each pixel’s neighborhood system.Similarly to the first term, we can write:T2= p −q( i,j)∈S x i,j∈G k ln exp ( )k k k = ln exp rkβk( i,j)∈S x i,j∈G k β (7)where p andkq represent the number of neighboringkpixels equal and different on direction k, respectivelyand rk=− 2,0, 2 . Thus, the 4-D function to beβ = β , β , β , β is:maximized on ( )1 2 3 4 L( X, β)= akβk − ln exprkβk(8)k ( i,j) ∈S xi , j∈G k where akis an element of a 1× 4 vector A and r iskan element of a 3-D K× 4× NM matrix R , with Krepresenting the number of possible labels and N , Mare the image spatial dimensions. This 3-D matrixstores one 2-D K × 4 matriz for each image pixel.Note that both A and R can be obtained through asingle image scanning.Algorithm:1. Define the number of regions/classes (K).2. Scan the image once (all NM pixels)a. Calculate A[k], the total number ofcliques of type k with equal labelsminus the total number of cliques oftype k with different labels (k = 1..4)b. Define the 3-D matrix R, containingthe difference on the number ofequal and different neighboringpixels on direction k, for all possiblecentral pixel values3. Define log-pseudo-likelihood function L( β )terms of vector A and matrix R.4. Maximize L( β )using a numerical method(preferably one without analytical or numericalgradients).5. Output parameter vector ˆ β ( ˆ ˆ ˆ ˆMPL= β1, β2, β3,β4)inFigure 4. Example of the R matrix for a 3class problemInitially, to solve the optimization problem, anynumerical algorithm could be used. Our choice for aMATLAB implementation of the simplex searchmethod [7] is motivated by the fact that it is a directsearch method and does not make use of numerical oranalytical gradients. Such class of numerical methodsdoes not require the computation (or even theexistence) of derivatives. Therefore, the proposedmethod provides a computationally feasible way toestimate the parameter vector β by MPL for anygiven number of classes M. Obviously, the price for anincrease on the number of classes is a highercomputational cost. If the number of classes is smallthe problem is simplified since the size of matrix Rsignificantly reduced.4. Contextual ClassificationBasically, the main application of this work is thedevelopment of a new methodology for contextualclassification that uses combination of sub-optimal fastiterative algorithms for multispectral imageclassification (Figure 5). The use of multispectralimages causes a significant increase on the number ofunknown MRF spatial dependency parameters,requiring a robust and accurate parameter estimationstage. The combination approach is motivated by thefact that local iterative optimization algorithms likeICM, MPM and GSA [5] are very dependent on theinitial conditions, converging to local maximasolutions. On the other side, although global methodsas Simulated Annealing (SA) provide optimalsolutions, they are very traditional approaches,computationally expensive and have low convergencespeed. Our methodology for contextual classificationproposes to investigate the use of local methods withmultiple initializations to avoid sub-optimal solutionsby adopting different classifier combination strategies.The estimation methods described here will be part ofthe MRF model parameter estimation module from thefollowing contextual classification scheme.31
WVC'2007 - III Workshop de Visão Computacional, 22 a 24 de Outubro de 2007, São José do Rio Preto, SP.variance and on Poisson distribution the variance isequal to the expectedvalue [8].Original imagePoisson noiseGaussian noiseFigure 1. Multispectral imagecontextualclassification using combination ofiterative MRFalgorithms5. Experiments and resultsTo test andd evaluate the proposed method, we presentthe resultss in 4 experiments with real image datacomparingg the values of ˆkβ for first, second and third-oforder neighborhood systems:a.) nuclearr magnetic resonance (NMR) images.b.) computerized tomography images.c.) labeled classified images.d.) Lena image.On the first experiment we considered a samplea multispectral NMR mango image, formed by T1, T2and P D bands, provided by Embrapa AgriculturalInstrumentation. These images have been used on thedevelopment of a non-invasive fruit quality assessmentsystem. In order to run the proposed method, we chosea mango transversal section P D band image withdimensionss of 256 x 256pixels, 255 gray levels (255classes). To test theestimation method againstdifferent types and levels of noise, we degraded theimage withh a signal dependent Poisson noise and aindependent additive gaussian noise, with zero mean2and σ = 0.02 . The images and obtained results areshown in Figure 5 andTable 1. Note that carefulexamination of the Poisson noise degraded imagereveals that clearer areas have more noise than darkareas. Thisis because brighter areas have higher noiseFigure 5. NMR P D band mango imagesTable 1. MPL MLL parameter values forsecond order neighborhood systems on NMR P Dband mango imagesOriginalPoissonGaussianHorizontal ˆβ 1= 1.021 ˆ β1= 0.8900 ˆ β1= 0.6565Vertical ˆ β2=0.7470 ˆ β2= 0.7448 ˆ β2= 0.6372Diagonal: / ˆ β3=0.7849 ˆ β3= 0.7753 ˆ β3= 0.6256Diagonal: \ ˆ β4=0.7882 ˆ β4= 0.7784 ˆ β4= 0.6276The second experiment usedone band of amultispectral CT image containing several materialsfound in soil (plexiglass, aluminum, water,phosphorus, calcium, besides the image background)obtained by an X and γ-ray CT scanner developed byEmbrapa Agricultural Instrumentation to exploreapplications on soilscience images. To acquire theimages, two X-ray sources and two γ-ray sources(Cesium and Americium) were used [9]. The X-rayenergies were 40keV and 85keV. The γ-ray were60keVV (Americium) and 662keV (Cesium).Onthe followinganalysis we considered one lowSNR ( Signal-to-Noise Ratio) image, contaminated withhigh levels of noise, due to the low exposuree timeduringdata acquisition (3 seconds of exposure) andanother smoother high SNR image, obtained with ahigherexposure time (20 secondss of exposure time),both images with size of 65 x65 pixels. The objectiveis to analyze the performance of the estimation methodon CT-images with different levels of noise.32
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