马尔可夫随机场在计算机视觉与图像处理中的应用专著

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《马尔可夫随机场在计算机视觉与图像处理中的应用》（Markov Random Fields for Vision and Image Processing）是一本由Andrew Blake、Pushmeet Kohli和Carsten Rother合著的专业书籍，由麻省理工学院出版社于2011年出版。该书主要探讨了马尔可夫随机场在计算机视觉和图像处理领域的理论及其实际应用。马尔可夫随机场是一种概率模型，它在处理复杂图像分析任务时表现出强大的能力，如图像分割、对象识别、纹理分析和形态学处理等。本书的核心内容围绕以下几个关键知识点展开： 1. **马尔可夫随机场（Markov Random Fields, MRF）**：MRF是统计物理中随机过程的一种数学模型，其特点是局部依赖性和全局一致性。在计算机视觉中，MRF通过定义邻域像素之间的条件概率分布，使得图像中的局部结构可以影响全局决策，从而捕捉到自然图像中的空间相关性。 2. **计算机视觉中的应用**：书中详细阐述了如何将MRF理论应用于实际的计算机视觉任务，如边缘检测、图像分割（如GrabCut算法）、物体跟踪、图像重建等。MRF有助于解决这些任务中的不确定性和模糊性问题，提供了一种有效的建模和优化方法。 3. **数学基础**：书中包含深入的数学理论，涉及概率论、图论、优化方法（如最大后验估计和能量最小化）以及相关的数值算法，以便读者理解MRF的理论背景和计算实现。 4. **编辑团队**：Andrew Blake是计算机视觉领域的知名学者，他的贡献为本书提供了坚实的学术基础；Pushmeet Kohli和Carsten Rother则以其在图像处理和机器学习方面的专业知识，确保了技术内容的实用性和前沿性。 5. **版权和出版信息**：本书享有版权，未经许可不得任何形式复制或机械复制，且提供了联系特殊数量折扣的渠道。印刷和装订质量良好，适合学术研究和教学使用。《马尔可夫随机场在计算机视觉与图像处理中的应用》不仅涵盖了丰富的理论内容，还提供了实践经验，对于理解计算机视觉中的概率模型、优化方法和实际应用具有很高的价值，是该领域的重要参考资料。

1.2 The Hidden Markov Model (HMM) 7

quite straightforward and is an exact algorithm for computing the marginal posteriors. After

one complete forward pass, the ﬁnal marginal distribution is P(x

, z | ω), and so ﬁnally

P(z | ω) =



P(x

, z | ω) (1.14)

can be computed as the evidence for a known model ω that solves problem 1 above. The

forward pass (1.13) constitutes half of BP, the remaining part being a backward pass that

recurs from node N back to node 1 (details omitted here, but see [382]). Using the forward

and backward passes together, the full set of marginal posterior distributions

P(x

| z, ω), i = 1,...,N (1.15)

can be computed. This is required for problem 3 above, in order to compute the expected

values of the sufﬁcient statistics that are needed to estimate the parameters ω by expectation

maximization [121]—also known in the speech analysis literature as the Baum–Welch

method [381].

The second algorithm is the Viterbi algorithm, a dynamic programming optimization

algorithm applied to the state sequence x. It is also equivalent to a special case of max-

product belief propagation, which also is mentioned quite frequently in the book. The aim

is to solve the second problem above, computing the MAP estimate of the state vector as

x = arg max

P(x | z,ω) (1.16)

via a forward recursion:

) = P(z

| x

,ω)max

i−1

P(x

| x

i−1

,ω)P

i−1

) (1.17)

where P

is deﬁned by

) ≡ max

,...,x

i−1

P(x

,...,x

,...,z

| ω). (1.18)

Each forward step of the recursion can be viewed as a message-passing operation. In step

i −1 of the computation, node i −1 sends the message P

) to node i.

After the forward steps are complete, the ﬁnal component ˆx

of the MAP solution ˆx can

be computed as

ˆx

= arg max

). (1.19)

This is followed by a backward recursion

ˆx

i−1

= arg max

i−1

P(ˆx

| x

i−1

,ω)P

i−1

), (1.20)

after which all components of ˆx are determined.

8 1 Introduction to Markov Random Fields

The purpose of the discussion in this section has been largely to explain the nature of

hidden variables in simple Markov models, as a precursor to later discussion of hidden

variables in the more complex, two-dimensional kinds of models that are used in vision.

However, even in vision the discrete HMM structure has some direct applications. It has

proved useful for representing temporal problems that are somewhat analogous to speech

analysis, but in which the audio input is replaced by a time sequence of visual features.

Well-known examples include the recognition of American Sign Language [449] and the

recognition of hand gestures for command and control [522]. This book deals mainly with

discrete Markov models—that is, ones in which the states of each X

belong to a ﬁnite set L.

However, in vision by far the greater application of timelike HMMs employs continuous

state-space to represent position, attitude, and shape in visual tracking [333, 477, 50, 253,

209, 124]. In such continuous settings the HMM becomes a classical or nonclassical form

of Kalman ﬁlter. Both exact solutions to the estimation problems that arise, and efﬁcient

approximate solutions, are much studied, but are outside the scope of this book.

1.3 Markov Models on Trees

In the following section 1.4, Markov Random Fields (MRFs) are deﬁned as probabilistic

models over undirected graphs. On the way there, we now consider undirected models on

trees as intermediate in complexity between the linear graphs—chains and HMMs—of sec-

tion 1.2, and graphs of unrestricted connectivity. Clearly the HMM graph (ﬁgure 1.3b) is a

special case of an undirected model on a tree. Trees appear to be of intermediate complexity

but, perhaps surprisingly, turn out to be closer to HMMs, in that inference can be performed

exactly. The Viterbi and forward-backward algorithms for HMMs generalize to two differ-

ent kinds of message passing on trees. However, once the nodes on two leaves of a tree are

coalesced into a single leaf—for example, leaves b and d in ﬁgure 1.4—a circuit may be

formed in the resulting graph, and message-passing algorithms are no longer an exact solu-

tion to the problems of inference.

As with Markov chains and HMMs, in undirected trees the topological structure conveys

two aspects of the underlying probabilistic model. First are the conditional independence

properties, that:

P(x

|{x

,j= i}) = P(x

|{x

,(i,j)∈ E}). (1.21)

The set B

={j : (i, j ) ∈ E} is known as the Markov blanket of i, its neighbors in the tree

(or generally graph) G. The second aspect is the decomposition of the joint distribution, the

generalization of (1.5) and (1.10). How can a distribution with this independence property

be constructed? The answer is, as a distribution that is factorized over the edges of the tree:

P(x ) =



(i,j)∈E

i,j

). (1.22)

1.4 MRFs 11

1.4 MRFs: Markov Models on Graphs

At the start of the chapter, the choice of graphs for image processing was motivated by the

need to establish dependencies between pixels that are nearby in an image. The graphs that

were proposed for that purpose are not trees, but contain cycles of edges, typically many

cycles, as in ﬁgure 1.1. The representation of independence in undirected graphs follows

the methodology given above for trees (1.21). The random variable at a node is dependent

directly only on random variables in its Markov blanket B

. The Hammersley–Clifford

theorem [98] gives the general form for a distribution with these Markov properties:

P(x ) =



c∈C

(x) (1.38)

where C is the set of maximal cliques of G—deﬁned to be the set of maximal subgraphs

of G that are fully connected in E, and maximal in the sense that no further node can be

added to a clique without removing the full connectedness property. Note that for a tree the

cliques are simply the edges of the graph, and so the decomposition (1.38) simpliﬁes to the

decomposition (1.22) for trees, above. Usually the decomposition is expressed in terms of

an energy or cost function E(x):

P(x ) ∝ exp

(

−E(x)

)

where E(x) =



c∈C



(x). (1.39)

More generally, there may be a dependence on parameters, so that

P(x ) =

Z(ω)

exp

(

−E(x,ω)

)

where E(x,ω) =



c∈C



,ω), (1.40)

and now the partition function

Z(ω) =



exp

(

−E(x,ω)

)

(1.41)

is included to maintain the normalization condition for the distribution,



P(x ) = 1.

An alternative representation of the undirected graph in ﬁgure 1.1 is the factor graph

[276]. The undirected graph (or Markov network) makes conditional dependencies explicit,

but factorization properties are somewhat implicit—they need to be computed in terms

of maximal cliques. Factor graphs are a little more complex in that they introduce a sec-

ond type of node, the function node, in addition to the nodes for variables but have the

advantage of making factorization explicit. In many of the cases used in this book—for

example, ﬁgure 1.1a, with 4-way connectivity between pixels—the factorization structure

is straightforward. Each edge in that example is a maximal clique, and therefore the factors

are functions of the two variables on the nodes belonging to each edge. On the other hand,

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