《Markov随机场在视觉与图像处理中的应用》- 深入理解人体姿态估计

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"《Markov Random Fields for Vision and Image Processing》是由微软剑桥研究院的专家们共同编撰的一本书，详细探讨了马尔科夫随机场在视觉和图像处理领域的应用。这本书的主编包括Andrew Blake，他也是Kinect项目的关键人物。书中主要关注如何利用马尔科夫随机场技术解决人体姿态估计等关键问题，特别是在实时鲁棒性方面。" 在计算机视觉和图像处理领域，马尔科夫随机场（Markov Random Fields，MRFs）是一种强大的统计建模工具，它能够捕捉图像或场景中的局部和全局依赖关系。MRFs在处理图像中的噪声、边缘检测、图像分类、图像恢复以及对象识别等方面有着广泛的应用。在本书中，作者深入浅出地介绍了这一理论，并将其与实际的视觉任务相结合。 Andrew Blake作为Kinect项目的主要负责人，他的团队利用先进的摄像设备和MRFs技术将人体姿态估计问题转化为像素级别的预测问题，从而实现了实时的人体运动跟踪。这种方法显著降低了计算复杂性，使得实时体感交互成为可能。在书中，读者可以了解到如何构建和优化MRF模型，以及如何利用这些模型进行高效的人体姿态估计和其他视觉任务。书中的内容涵盖了MRFs的基本理论，包括能量函数、图割（graph cuts）、信念传播（belief propagation）等优化算法，同时结合实例展示了它们在实际图像处理问题中的应用。此外，书中还讨论了如何将马尔科夫随机场与其他机器学习方法，如条件随机场（CRFs）和深度学习结合，以提升模型的性能和泛化能力。此外，书中包含的参考文献和索引为读者提供了进一步研究的路径，帮助他们深入理解该领域的最新进展和技术挑战。通过阅读这本书，读者不仅可以掌握马尔科夫随机场的理论基础，还能了解到如何将这些理论应用于解决实际的视觉和图像处理问题，对于研究人员和工程师来说，是一本不可多得的参考资料。

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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