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# 基于 EM 的 MRF 彩色图像分割 - PowerPoint PPT Presentation

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### 基于EM的MRF彩色图像分割

• Markov Random Fields
• Markov-Gibbs 等价性
• 有用的MRF模型
• 多级GRF模型和MML 模型
• MAP-MRF标记
• 观察模型

• MRF 参数估计
• 基于EM和MRF的彩色图像分割
• 图像特征的提取
• 聚类的个数的分析

• Sites 和 Labels
• A labeling of the sites in S in terms of the labels in L: f = { }
• Sites S= {1,…m}
Cliques
• A cliquec for (S, N) is defined as a subset of sites in S .在c中所有的sites都是相邻的。
• 对于（S，N）所有势团的集合是：
Markov-Gibbs 等价性(证明省略)
• An MRF is characterized by its local property (the Markovianity)
• GRF is characterized by its global property (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 N
Gibbs Random Field----definition
• F is said to be a Gibbs Random Field on S with respect to N if and only if its configurations obey a Gibbs distribution:

• 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 模型和多级GRF模型
• There are M (>2) discrete labels in the label set ,L={1,2,…,M}.

• The higher level Gibbs distribution uses an isotropic random field （MLL）
• A lower level Gibbs distribution describes the filling-in in each region
• 在纹理分割中：

blob-like regions are modeled by a high level MRF which is an isotropic MLL

these regions are filled in by patterns generated according to MRFs at the lower level

MAP-MRF标记
• 1.贝叶斯估计：

2. d:观察的数据

C( , f)是费用函数

p(f | d)is the posterior distribution

• 根据（1），贝叶斯风险为：

• where k is the volume of the space containing all points f for which

MAP-MRF approach 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.

• 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 参数估计
• EM算法：一种迭代的标记-估计算法

• 对图像中的每个像素，计算一个d维的特征向量X, X可以包含各种不同的颜色表示，以及一序列滤波器的输出。
• 我们将图像模型表示如下：图像中的每个像素均是由g个图像分割中的某一个的密度函数计算得到的。因此为产生一个像素，首先选择一个图像分割区域，然后通过该区域的密度函数生成所需的像素

• 1.每一个分割（块）的参数
• 2.混合权重
• 3.各个像素来源于模型中的哪个分量（从而实现图像分割）

• 1 . 如果我们已经知道了各个像素分别来源于哪个分量，那么确定参数将会变得容易

2. 如果知道了参数， 那么对于每个像素，就能够确定最可能产生那个像素的分量（这样就确定了图像分割）

3.但问题是两者都不知道。

The expectation-maximization (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.通过用期望值来替代丢失的（隐藏的）数据，为丢失的数据获取工作变量的集合2.接着将计算出的不完备数据的期望值代入到完备数据的似然函数中，用这个函数计算相对要简单一些3.然后最大化这个函数获得参数的值。EM算法的主要思想是1.通过用期望值来替代丢失的（隐藏的）数据，为丢失的数据获取工作变量的集合2.接着将计算出的不完备数据的期望值代入到完备数据的似然函数中，用这个函数计算相对要简单一些3.然后最大化这个函数获得参数的值。
• 这时不完备数据的期望值可能已经改变了。
• 通过交替执行期望阶段和最大化阶段，迭代直致收敛
EM算法的形式化描述
• 1.使用不完备的数据以及参数的当前值来计算完备数据的期望值（E步）
• 2.使用E步计算出的完备数据的期望值，最大化完备数据关于参数的对数似然函数（M步）。
• 1，2步交替直到收敛。

• However, we cannot work directly with this complete-data log likelihood because it is a random function of the missing 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 process
• The label process w is modeled as a MRF

with respect to a second order neighborhood system

Image process
• 多元高斯密度分布是一种典型的适合大多数分类问题的模型。其中，对于某个给定的类m，特征向量d是连续取值的。
EM算法
• 假设存在r个像素，丢失（隐藏）的数据形成一个r×L的数组表示的指示变量Z.
• 在每一行，除了一个像素，其他的值均为0，这个值表示每个像素的特征向量来源于哪个块（分割）

• The brightness and texture features are extracted 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 intensity; and two position features（x,y） coordinates of the pixels

• 基于直方图的聚类个数分析