基于EM的MRF彩色圖像分割的課件

1、基于EM的MRF彩色圖像分割 領(lǐng)域系統(tǒng)和勢(shì)團(tuán)Markov Random FieldsMarkov-Gibbs 等價(jià)性有用的MRF模型多級(jí)GRF模型和MML 模型MAP-MRF標(biāo)記觀察模型一個(gè)簡(jiǎn)單的例子：圖像紋理分割MRF 參數(shù)估計(jì)基于EM和MRF的彩色圖像分割圖像特征的提取聚類的個(gè)數(shù)的分析領(lǐng)域系統(tǒng)和勢(shì)團(tuán)Sites 和 LabelsA labeling of the sites in S in terms of the labels in L: f = Sites S= 1,m The labeling problem is to assign a label from the label se

2、t L to each of the sites in S.領(lǐng)域系統(tǒng)被定義為：CliquesA clique c for (S, N) is defined as a subset of sites in S .在c中所有的sites都是相鄰的。對(duì)于（S，N）所有勢(shì)團(tuán)的集合是：Markov Random FieldsDefinitionMarkov-Gibbs 等價(jià)性(證明省略)An MRF is characterized by its local property (the Markovianity) GRF is characterized by its global property

3、(the Gibbs distribution). The Hammersley-Clifford theorem establishes the equivalence of these two types of properties The theorem states that F is an MRF on S with respect to N if and only if F is a GRF on S with respect to NGibbs Random Field-definitionF is said to be a Gibbs Random Field on S wit

4、h respect to N if and only if its configurations obey a Gibbs distribution:有用的MRF模型Auto-Models auto-logistic model （Ising model）auto-binomial model auto-normal model （Gaussian MRF ）multi-level logistic (MLL) model (potts model)Hierarchical GRF Model MLL 模型和多級(jí)GRF模型There are M (2) discrete labels in t

5、he label set ,L=1,2,M.在多級(jí)兩層Gibbs模型中：The higher level Gibbs distribution uses an isotropic random field （MLL）A lower level Gibbs distribution describes the filling-in in each region 在紋理分割中：blob-like regions are modeled by a high level MRF which is an isotropic MLL these regions are filled in by patte

6、rns generated according to MRFs at the lower level MAP-MRF標(biāo)記1.貝葉斯估計(jì)： 估計(jì) 的貝葉斯風(fēng)險(xiǎn)被定義為：2. d:觀察的數(shù)據(jù) C( , f)是費(fèi)用函數(shù) p(f | d)is the posterior distribution 費(fèi)用函數(shù)：根據(jù)（1），貝葉斯風(fēng)險(xiǎn)為：根據(jù)（2）貝葉斯風(fēng)險(xiǎn)為：where k is the volume of the space containing all points f for which 因此：最小化風(fēng)險(xiǎn)就相當(dāng)于最大化后驗(yàn)概率p(f|d).這就是我們所知的最大后驗(yàn)概率估計(jì)。MAP-MRF appro

7、ach for solving computer vision problems :Pose a vision problem as one of labeling in categories LP1-LP4 and choose an appropriate MRF representation f. Derive the posterior energy to define the MAP solution to a problem. Find the MAP solution. The process of deriving the posterior energy觀察模型一個(gè)簡(jiǎn)單的例子

8、：圖像紋理分割Texture segmentation is to segment an image into regions according to the textures of the regions Texture segmentation, as other labeling problems, is usually performed in an optimization sense, such as MAP MRF 參數(shù)估計(jì)EM算法：一種迭代的標(biāo)記-估計(jì)算法基于EM和MRF的彩色圖像分割對(duì)圖像中的每個(gè)像素，計(jì)算一個(gè)d維的特征向量X, X可以包含各種不同的顏色表示，以及一序列濾波

9、器的輸出。我們將圖像模型表示如下：圖像中的每個(gè)像素均是由g個(gè)圖像分割中的某一個(gè)的密度函數(shù)計(jì)算得到的。因此為產(chǎn)生一個(gè)像素，首先選擇一個(gè)圖像分割區(qū)域，然后通過該區(qū)域的密度函數(shù)生成所需的像素我們希望確定以下參數(shù)：1.每一個(gè)分割（塊）的參數(shù)2.混合權(quán)重3.各個(gè)像素來源于模型中的哪個(gè)分量（從而實(shí)現(xiàn)圖像分割）一個(gè)兩難問題的提出： 1 . 如果我們已經(jīng)知道了各個(gè)像素分別來源于哪個(gè)分量，那么確定參數(shù)將會(huì)變得容易 2. 如果知道了參數(shù)， 那么對(duì)于每個(gè)像素，就能夠確定最可能產(chǎn)生那個(gè)像素的分量（這樣就確定了圖像分割） 3.但問題是兩者都不知道。 圖像的似然函數(shù)為：The expectation-maximizati

10、on (EM) algorithm is a general technique for finding maximum likelihood (ML) estimates with incomplete data In EM, the complete data is considered to consist of the two parts：EM算法的主要思想是1.通過用期望值來替代丟失的（隱藏的）數(shù)據(jù)，為丟失的數(shù)據(jù)獲取工作變量的集合2.接著將計(jì)算出的不完備數(shù)據(jù)的期望值代入到完備數(shù)據(jù)的似然函數(shù)中，用這個(gè)函數(shù)計(jì)算相對(duì)要簡(jiǎn)單一些3.然后最大化這個(gè)函數(shù)獲得參數(shù)的值。這時(shí)不完備數(shù)據(jù)的期望值可能已

11、經(jīng)改變了。通過交替執(zhí)行期望階段和最大化階段，迭代直致收斂EM算法的形式化描述1.使用不完備的數(shù)據(jù)以及參數(shù)的當(dāng)前值來計(jì)算完備數(shù)據(jù)的期望值（E步）2.使用E步計(jì)算出的完備數(shù)據(jù)的期望值，最大化完備數(shù)據(jù)關(guān)于參數(shù)的對(duì)數(shù)似然函數(shù)（M步）。1，2步交替直到收斂?？梢宰C明，不完備數(shù)據(jù)的對(duì)數(shù)似然函數(shù)在每個(gè)階段都是增長的，也就說參數(shù)序列收斂到不完備數(shù)據(jù)對(duì)數(shù)似然函數(shù)的某個(gè)局部最大值。However, we cannot work directly with this complete-data log likelihood because it is a random function of the missing

12、 variables f. The idea of the EM algorithm is to use the expectation of the complete-data log likelihood which will formalize EM The M-step performs maximum likelihood estimation as if there were no missing data as it had been filled in by the expectations Label processThe label process w is modeled

13、 as a MRF with respect to a second order neighborhood system Image process多元高斯密度分布是一種典型的適合大多數(shù)分類問題的模型。其中，對(duì)于某個(gè)給定的類m，特征向量d是連續(xù)取值的。Posterior energySo the energy functionEM算法假設(shè)存在r個(gè)像素，丟失（隱藏）的數(shù)據(jù)形成一個(gè)rL的數(shù)組表示的指示變量Z.在每一行，除了一個(gè)像素，其他的值均為0，這個(gè)值表示每個(gè)像素的特征向量來源于哪個(gè)塊（分割）圖像特征的提取The brightness and texture features are extra

14、cted from the L* component and the color features are extracted from the a* and b* components.two brightness features: brightness gradient and local energy content of the L* component; three color features: color gradient, local energy content of the a* and b* components;three texture features: phase divergence, homogeneity and homogeneous i