III WVC 2007 - Iris.sel.eesc.sc.usp.br - USP

More documents

Recommendations

Info

WVC'2007 - III Workshop de Visão Computacional, 22 a 24 de Outubro de 2007, São José do Rio Preto, SP.An Efficient Method for Parameter Estimation on the Multi-Level LogisticMRF Image Model using Maximum Pseudo-Likelihood ApproachAlexandre L. M. Levada 1 , Nelson D. A. Mascarenhas 21 Instituto de Física de São Carlos - Universidade de São Paulo, São Carlos, SP, Brasil.2 Departmento de Computação - Universidade Federal de São Carlos, São Carlos, SP, Brasil.alexandreluis@ursa.ifsc.usp.br, nelson@dc.ufscar.brAbstractThis paper addresses the problem of maximumpseudo-likelihood (MPL) estimation of the nonisotropicMarkov Random Field image model knownas Multi-Level Logistic (MLL) in a computationallyefficient way. The proposed method consists in writingthe pseudo-likelihood function on a feasible way, interms of variables a kand r k(where k denotes theclique type), obtained by a unique scanning of theimage. The objective function is then maximized usingan implementation of the simplex search method,which does not require the computation of numericalor analytical gradient. Experiments using tomographicimages show that the proposed method is consistent inthe presence of random noise.1. IntroductionMarkov Random Fields define a robust approachfor contextual modeling. However, there are manyopen problems in MRF parameter estimation, and inmost applications, the model parameters are stillchosen by trial and error [1-2]. The main difficult inMRF parameter estimation is that the traditionalmethods cannot be directly applied to the problems.For example, the most general estimation method,maximum likelihood, is computationally intractable,due to the partition function on the Gibbs jointdistribution (global model). To overcome thisdifficulty, Besag [3] proposed the maximum pseudolikelihoodapproach (MPL), which uses the localconditional density functions to define a pseudolikelihoodfunction. Our contribution is the proposal ofan efficient methodology for estimation of Multi-LevelLogistic MRF model parameters based on rewriting thepseudo-likelihood function on a feasible way, and byavoiding intensive global optimization algorithms.This paper is organized as follows. Section 2 isconcerned about the MLL MRF image model and thepseudo-likelihood approach. Section 3 presents theproposed estimation method. The experiments andresults are described on Section 4. Finally, Section 5presents the conclusions and final remarks.2. MRF Image ModelsThe fundamental notion associated with Markovproperty is the conditional independence, since theknowledge of a local region (neighborhood system)isolates a single element from the entire field. A MRFdefined on a lattice S is a collection of randomvariables for which the probability of a single elementgiven the entire lattice is equal to the probability of thiselement given a finite support region of the lattice,called neighborhood [4]. Non-causal neighborhoodsystems are referred as zero-order, first-order, secondorderand so on. Figure 1 shows the finite supportregions for first to fifth order systems:Figure 1. Finite support regions for severalneighborhood systemsIn this work, we assume a second-order non-causalnon-isotropic and homogeneous model, called Multi-Level Logistic (the spatial dependency parameters arenot the same for all directions, but they are constantalong the entire image).2.1 The Multi-Level Logistic Image ModelIn stochastic image modeling, we need to define aprobability distribution over possible images whichreflect our prior knowledge about the desired solution.One of such models is the MLL model, which assumesdifferent spatial dependency parameters, each one for aspecific direction: horizontal ( β ), vertical (1β ) and2diagonals ( β3, β ). The standard model assumes that4the spatial parameters are globally constant throughoutthe field. The inhomogeneous model allows the29
WVC'2007 - III Workshop de Visão Computacional, 22 a 24 de Outubro de 2007, São José do Rio Preto, SP.parameters to vary spatially. Thus, it can be defined bya set of local conditional density functions (LCDF’s).For a general neighborhood system N , the LCDF of aMulti-Level Logistic pairwise interaction model isdefined as [5]:( i,j|xN)i,jp x=exp V x , xk kexp V x , xk∈ xijG k( , + ε,+ η)C i j i j( , + ε,+ η)C i j i j(1)where N denotes the neighborhood around pixeli,jx ,i,jxi ε, j ηNi,j∈ , + + G { 1, 2,..., K}= is the set ofpossible values of x (K is the number of classes) andi,j( ,,+ ε,+ η)V x xCki j i j β , x = x= − β , x ≠ xk i, j i+ ε,j+ηk i, j i+ ε,j+ηwhere ckdefines the clique type, as shows Figure 2.(2)Figure 2. Possible clique types for a MLL pairwiseinteraction process2.2 Maximum pseudo-likelihood estimationThe main difficult in MRF parameter estimation isthat the traditional methods cannot be directly appliedto the problems. The most general estimation method,maximum likelihood approach, is computationallyintractable. The main advantage of the MPL estimatoris its computational simplicity. Fortunately, as themaximum likelihood (ML) estimator, the MPLestimator has also a series of desirable properties, asconsistency, that is, it converges to the true parameterwith probability one as the size of image growsinfinitely, and asymptotic normality [6]. The MPLestimator satisfies:ˆ βarg max | ,( ,x β),MPL= ∏ p xi j N i j (3)β ( i,j)∈SOften, maximizing the log-pseudo-likelihoodfunction is more convenient (log is a monotonicallyincreasing function), which results in:log PLi,j∈S( β )k= ( ,,+ ε,+ η)V x xCki j i j log exp V x , x k i,j∈S xi,j∈G k −( , + ε,+ η)C i j i j(4)Thus, in order to obtain ˆMPLβ , we need to maximizeequation (4). In the following sections we propose anefficient and computationally feasible method to solvethis problem.3. An approach for MLL parameterestimation using MPL estimationThe log-pseudo-likelihood function can beinterpreted as the difference between 2 main terms.Expanding the summation on first term, gives:( ,+ ε + η) ( ,+ ε + η)T = V x x + V x x +1 C1 i, j i , j C2i, j i , j( i,j)∈S( ,,+ ε, + η) + ( ,,+ ε,+ η)V x x V x xC3 i j i j C4i j i j(5)Denoting by g the number of cliques of type kkwith equal labels and e the number cliques of type kkwith different labels, expression (5) becomes:( β β ) ( β β )( g3β3− e3β3) + ( g4β4 −e4β4) ( gk ek)βkk akβkT = g − e + g − e +1 1 1 1 1 2 2 2 2= −=k(6)where a represents, for the entire image, the totalknumber of cliques of type k with equal labels minus thetotal number of cliques of type k with different labels.Considering a second-order neighborhood system, theneach pixel xi,jbelongs to 2 cliques of type k, asillustrates Figure 3.Figure 3. Cliques for a central pixel on a secondorderMLL pairwise interaction process30
Page 1 and 2: III Workshop de VisãoComputacional
Page 3 and 4: Instituto de Biociências, Letras e
Page 5 and 6: WVC 2007 - III Workshop de Visão C
Page 7 and 8: ApresentaçãoA área de Visão Com
Page 10 and 11: Automatic Pattern Recognition of Bi
Page 12 and 13: WVC'2007 - III Workshop de Visão C
Page 16: WVC'2007 - III Workshop de Visão C
Page 91 and 92:
WVC'2007 - III Workshop de Visão C
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
Page 243 and 244:
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Page 253 and 254:
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363:
show all

III WVC 2007 - Iris.sel.eesc.sc.usp.br - USP

Create successful ePaper yourself

Delete template?

Save as template?