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

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