CS 294-2, Visual Grouping and Recognition èprof. Jitendra Malikè Sept. 8, 1999

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1 CS 294-2, Visual Grouping and Recognition èprof. Jitendra Malikè Sept. 8, 999 Lecture è6 èbayesian estimation and MRFè DRAFT Note by Xunlei Wu æ Bayesian Philosophy æ Markov Random Fields Introduction Hammersley-Cliæord Theorem Dual Lattice Number In last lecture, we discussed Maximum Likelihood Estimation and its application in Mixture Model. The core of Maximum Likelihood Estimation is to ænd æxed but unknown parameter ç that maximizes the likelihood function P èyjçè, Where y is our observation. Today, we begin to discuss another statistic concept that has broad applications in Computer Vision, Bayesian Estimation. Bayesian Philosophy Formal deænition: in the Bayesian approach it is assumed that wehave some probability density function for parameter ç. Then, the given the obsevations y the conditional probability density for that parameter is given by Eq. P èçjyè = P èyjçèp èçè P èyè where P èçè is prior probability, which is get before receiving any data. P èçjyè is called posterior probability, which is obtained after those data coming. The computer vision analogy is given as èè P èworld j Imageè = P èimage j WorldèP èworldè P èimageè where P èworldè is our instinct guess with our eyes shut. For instance, we æip a coin n trials and observe y times of heads. P èçè is the èpriorè probability of seeing heads. If P èçè is speciæed to be uniformly distributed over interval ë; ë, the postperior probability P èçjyè is obtained as P èçjyè = "è n y! P èçè y è, P èçèè n,y è =z è2è where z is normalizing constant tomake P èç j yè within ë; ë. With diæerent sets of n and y, wehave Figure. We see the conditional distribution function becomes more steeper

2 P( θ y) n = 5, y = 3 P( θ y).6 n = 2, y = 2 P( θ y).6 n =, y = 6.6 Figure : Coin æipping examples for Bayesian estimation. as the number of trials rises up. This example demonstrate that how conditional probability èbayesian estimationè can help us to determine objects in vision. Suppose that we have 2 trials, then what is the probability ofç 5 heads show up? Z 2 Z P èç 5headsè = x=5 ç P èxjçèdxp èçjyèdç In vision, we generally assume that we have piecewise constant è slowly varying cues in our images. The cues can be pixel brightness, texture conæguration and even motion. As in Figure 2, we see 5 diæerent texture regions. Within each region, the texture structure maintains the same. In the real world, what we will observe is the real clean image I and additional noise ç. Our next target is trying to eliminate the eæect of ç. 2 Markov Random Fields 2. Introduction Markove Random Field èmrfè is introduced by Dobrushin in 968 and reæned by Stuart Geman and Donald Geman in 984. This concept is an extension of Markov Chain. In an -D example, we put a drunk person on a line. At each time step he randomly walks to either the left or the right. As we show in Figure 3, if he reaches position n at time t, then 2

3 Figure 2: One image contains æve diæerent texture regions. The texture structure remains the same inside each region. Xt- = n P(Xt = n+) =.5 P(Xt = n+) =.5 Figure 3: Drunk person walks on a line. An illustration of Markov Chain x probabilities of reaching position n + or n, are the same, :5. The probability that the person stay on position x at time t is given by P èx t jx t,;x t,2;:::è=p èx t jx t,è We state that ëthe future is independent of the past given the present". The deænition of MRF consists of 2 properties: è3è. P èx =!è é ; 8! 2 æ where! is a conæguration as in the right half of Figure P èx s = x s jx r = x r ;r 6= sè =P èx s = x s jx r = x r ;r 2 G s è where G s is a neighborhood of s. For example, P èx4 j x; x2; x3; x5; :::; x9è =P èx4 j x; x5; x7è. MRF is also called undirected graphical model. 2.2 Hammersley-Cliæord Theorem Deiæntion of Hammersley-Cliæord Theorem is given by P èx =!è = expè, P c2c V c è!èè è4è z where z is a normalizing constant to conæne P èx =!è inside of ë; ë, C is the set of all cliques and V c is a function whose value depends only on the nodes in clique c. There are 3

4 X X2 X3 X4 X5 X6 X7 X8 X9 Figure 4: a 3-by-3 area with elements x to x9. x i can only be or in binary setup. Thus this area can have2 9 diæerent conæguration. The right part of the ægure is one conæguration. MRF technique is employed to show the conditional independency given the value from its neighbors. X X2 X3 X4 X5 X6 X7 X8 X9 X X X2 X3 X4 X5 X6 Figure 5: Illustration of clique. èx2;x7è, èx6;x3;x7è and èx2;x3;x7;x6è are cliques. Notice that there is no clique including 5 nodes. broad variations of V c è!è playing as weighting functions for diæerent situations. Clique is a set of nodes in a graph such that any pair of them are connected by an arc. We illustrate the deænition by Figure Dual Lattice Number Before get into details about Hammersley-Cliæord Theorem, we introduce the concept of Dual Lattice Number. It is deæned as virtual nodes on the edge between two adjacent pixels and only takes binary values, and. Dual Lattice Number of indicates the boundaries of groups in an image. states that the two surrounding pixels belong to one group as shown in Figure 6. This procedure is called Line Process. There are more examples on page 735 of Sturat Geman and Donald Geman's paper. 4

5 X X2 X5 X6 Figure 6: Line Process. Black and white disks are Dual Lattice Numbers. Black disks indicate the value of and white disks represent the value of. After line process, above 4-by-4 area is divided into two regions. 5

Markov Random Fields

Markov Random Fields Umamahesh Srinivas ipal Group Meeting February 25, 2011 Outline 1 Basic graph-theoretic concepts 2 Markov chain 3 Markov random field (MRF) 4 Gauss-Markov random field (GMRF), and