视觉与图像处理中的马尔可夫随机场经典指南

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《马尔可夫随机场在计算机视觉中的应用》是一本由Andrew Blake、Pushmeet Kohli和Carsten Rother合编的经典著作，由麻省理工学院出版社于2011年出版。本书专注于马尔可夫随机场(Markov Random Fields, MRF)在计算机视觉和图像处理领域的理论与实践，为读者提供了一种强大的工具和技术框架来理解和解决复杂的视觉问题。马尔可夫随机场是一种概率模型，它假设数据在局部具有统计独立性，但全局上受到邻域内其他变量的影响。这种特性使得MRF在处理图像分割、物体识别、纹理分析、边缘检测等任务时展现出强大的能力，因为它们能够捕捉到像素或区域之间的依赖关系，从而提高图像理解的精度。书中内容涵盖了以下几个核心主题： 1. **MRF基础**：介绍了马尔可夫随机场的基本概念，包括条件概率、势函数（energy function）以及最大似然估计等关键原理，帮助读者建立起对MRF理论的深入理解。 2. **视觉应用实例**：通过具体的计算机视觉任务，如图像分割、目标检测、图像配准等，展示了如何将MRF理论转化为实际算法。例如，通过定义适当的潜在变量和势函数，可以设计出有效的分割策略，同时考虑到相邻像素的连续性和一致性。 3. **计算方法**：讨论了优化技术，如Belief Propagation（BP）、Loopy Belief Propagation（LBP）和Approximate Inference，这些是求解大规模MRF问题的重要算法，对于实际应用中的性能至关重要。 4. **扩展与创新**：书中还涉及了一些前沿的研究进展，如半监督学习、深度学习与MRF的结合，以及MRF在深度网络中的潜在应用，展示了MRF在不断演进的计算机视觉领域中的发展潜力。 5. **参考文献与索引**：本书提供了详尽的参考文献，便于读者进一步探索相关研究，同时索引也方便读者快速查找特定主题的信息。《马尔可夫随机场在视觉和图像处理中的应用》不仅是一本理论教材，还是一本实践指南，它为从事计算机视觉、图像处理和机器学习的科研人员和工程师提供了宝贵的理论支持和实战案例。通过阅读这本书，读者不仅能掌握MRF的核心概念，还能了解到如何将其应用于解决现代视觉问题，推动相关领域的研究和技术创新。

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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