分布式源编码：理论、算法与应用实践

Source

Coding

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"分布式源编码：理论、算法与应用" 分布式源编码（DSC）是当前信息技术领域的一个重要研究方向，特别是在无线传感器技术和自组织网络的快速发展背景下，它已成为一个备受关注的领域。《Distributed Source Coding: Theory, Algorithms and Applications》这本书由该领域的领军人物编辑并撰写，为系统设计者、实现者、研究人员和研究生提供了最新理论、算法及应用的详尽介绍，堪称DSC的权威参考书籍。书中深入浅出地阐述了分布式源编码的基本原理，特别关注特定类型源的编码性能极限。它探讨了如何将标准信道编码如低密度奇偶校验码（LDPC）和Turbo码应用于DSC，并讨论了压缩感知在稀疏信号分布式压缩中的适用性。此外，书中还涵盖了一系列实际应用案例，包括分布式视频编码、麦克风阵列以及生物特征数据的安全保护。本书的关键知识点包括： 1. **分布式源编码基础**：解释了DSC的基本概念，它如何在传感器网络、自组织网络和分布式无线视频系统中发挥作用，特别是在监控等应用中的价值。 2. **性能极限分析**：针对特定源类别的DSC性能极限进行了详细研究，这有助于设计者理解在不同场景下编码的最优策略。 3. **标准信道编码的使用**：讨论了如何利用LDPC和Turbo码改进DSC，这些编码技术在提高数据传输效率和错误纠正方面具有显著优势。 4. **压缩感知与稀疏信号处理**：介绍了压缩感知在处理稀疏信号时的优势，这对于传感器网络中数据的高效压缩和恢复至关重要。 5. **实际应用**：书中包含了大量的应用实例，如分布式视频编码，它在远程监控和实时视频传输中的应用；麦克风阵列的应用，用于声音定位和噪声消除；以及在生物特征数据安全中的应用，如指纹或面部识别的数据加密和保护。 6. **算法设计与实现**：提供了实用的DSC算法设计方法，指导读者如何在现实环境中实现这些算法。 7. **索引与参考**：书后附有索引，方便读者查找相关主题，同时提供了最新的研究进展和参考文献，便于进一步学习和研究。这本书对于想要深入了解分布式源编码及其应用的专业人士来说，是一份宝贵的资源，它不仅提供了理论知识，还包含了丰富的实践经验和应用示例，可以帮助读者全面掌握这一领域的核心技术和最新动态。

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1.3 Distributed Source Coding 9

Proposition 1.2. Let S be a stationary ergodic Gaussian source with autocorrela-

tion function E[S

n⫺k

]⫽ ␾(k) and power spectral density

⌽(␻) ⫽

⬁



k⫽⫺⬁

␾

(k)

⫺jk␻

.(1.13)

Then the rate-distortion function for S under mean-squared error distortion is

given by:

R(D

␯

)⫽

4␲



␲

⫺␲

max



0, log

⌽(␻)

␯



d␻ (1.14)

␯

⫽

2␲



␲

⫺␲

min{␯, ⌽(␻)}d␻.(1.15)

PROOF. See Berger [10, p. 112]. ■

While it can be difﬁcult to evaluate this in general,upper and lower bounds can give

a better sense for its behavior. For instance,let ␴

be the variance of a stationary ergodic

Gaussian source. Then a result of Wyner and Ziv [14] shows that the rate-distortion

function can be bounded as follows:

log

␴

⫺⌬

⭐ R

(D) ⭐

log

␴

,(1.16)

where ⌬

is a constant that depends only on the power spectral density of S.

1.2.4 Some Notes on Practical Considerations

The problem formulation considered in this chapter focuses on the existence of codes

for cases in which the encoder has access to the entire source noncausally and knows

its distribution. However,in many situations of practical interest,some of these assump-

tions may not hold. For instance,several problems have considered the effects of delay

and causal access to a source [15–17]. Some work has also considered cases in which

no underlying probabilistic assumptions are made about the source [18–20]. Finally,

the work of Gersho and Gray [21] explores how one might actually go about designing

implementable vector quantizers.

1.3 DISTRIBUTED SOURCE CODING

The problem of source coding becomes signiﬁcantly more interesting and challenging

in a network context. Several new scenarios arise:

■ Different parts of the source information may be available to separate encoding

terminals that cannot cooperate.

■ Decoders may have access to additional side information about the source infor-

mation; or they may only obtain a part of the description provided by the

encoders.

10 CHAPTER 1 Foundations of Distributed Source Coding

We start our discussion by an example illustrating the classical problem of source

coding with side information at the decoder.

Example 1.1

Let {S(n)}

n⬎0

be a source where source samples S (n) are uniform over an alphabet of size 8,

which we choose to think of as binary vectors of length 3. The decoder has access to a

corrupted version of the source {

S(n)}

n⬎0

where each sample

S(n) takes values in a set of

ternary sequences {0, ∗,1}

of length 3 with the property that



S(n) ⫽ (c

, c

)



S(n) ⫽ (b

, b

)



⫽

⎧

⎪

⎨

⎪

⎩

, if (c

, c

)⫽(b

, b

)

, if (c

, c

)⫽(∗, b

, b

)

, if (c

, c

)⫽(b

, ∗, b

)

, if (c

, c

)⫽(b

, b

, ∗)

(1.17)

Thus, the decoder has access to at least two of every three bits per source symbol, but the

encoder is unaware of which ones. Consider the partition of the alphabet S into

⫽



(0, 0, 0), S

⫽



(1, 1, 1),

(0, 1, 1),

(1, 0, 0 ),

(1, 1, 0),

(0, 0, 1),

(1, 0, 1)



(0, 1, 0 )



.(1.18)

If the decoder knows which of these partitions a particular sample S(n)isin,

S(n) is sufﬁcient

to determine the exact value of S (n). Thus, for each source output, the encoder can use one

bit to indicate to which of the two partitions the source sample belongs. Thus, at a rate of

1 bit per source sample, the decoder can perfectly reconstruct the output. However, in the

absence of {

S(n)}

n⬎0

at the decoder, Theorem 1.1 implies the best possible rate is H(S

)⫽3

bits per source sample.

Example 1.1 illustrates a simple version of a strategy known as binning. It turns

out that binning can be applied more generally and is used to prove many of the results

in distributed source coding.

1.3.1 Lossless Source Coding

More generally,we now consider the scenario illustrated in Figure 1.3,where two sepa-

rate encoding terminals each observe part of the data.That is,with respect to Figure 1.3,

the source streams S

and S

are dependent on each other. The coding question now

involves two separate source codes that appear at rates R

and R

, respectively,and a

receiver where the source codes are jointly decoded. Formal deﬁnitions are provided

in Appendix A (Deﬁnitions A.12 and A.13). Since the sources are dependent, the rates

and R

constrain one another. That is, if more bits are used to describe one of the

sources, typically, the number of bits for the other can be reduced. Speciﬁcally, if we

1.3 Distributed Source Coding 11

ˆˆ

FIGURE 1.3

Distributed source coding problem.

assume R

⬎ log |S

|, we can assume that the decoder knows S

without error, and

thus this problem also includes the special case of side information at the decoder.

Theorem 1.4. Given discrete memoryless sources S

and S

, deﬁne R as

R ⫽



, R

):R

⫹ R

⭓ H(S

, S

⭓ H(S

), R

⭓ H(S

)



.(1.19)

Furthermore, let

be the interior of R. Then (R

)∈R

are achievable for the two-

terminal lossless source coding problem, and (R

)/∈R are not.

The proof of this result was shown by Slepian and Wolf [22];to show achievability

involves a random binning argument reminiscent of Example 1.1. However,by contrast

to that example, the encoders now bin over the entire vector of source symbols, and

they only get probabilistic guarantees that successful decoding is possible.

Variations and extensions of the lossless coding problem have been considered by

Ahlswede and Körner [23],who examine a similar setup case in which the decoder is

only interested in S

;Wyner [24], who considers a setup in which one again wants to

reconstruct sources S

and S

, but there is now an additional encoding terminal with

access to another correlated information source S

;and Gel’fand and Pinsker [25],who

consider perfect reconstruction of a source to which each encoding terminal has only

a corrupted version. In all of these cases, a random binning strategy can be used to

establish optimality. However, one notable exception to this is a paper by Körner and

Marton [26], which shows that when one wants to reconstruct the modulo-2 sum of

correlated sources,there exists a strategy that performs better than binning.

1.3.2 Lossy Source Coding

By analogy to the centralized compression problem, it is again natural to study the

problem where instead of perfect recovery, the decoder is only required to provide

estimates of the original source sequences to within some distortions. Reconsidering

Figure 1.3,we now ask that the source S

be recovered at distortion D

, when assessed

with distortion measure d

(·, ·), and S

at distortion D

, when assessed with distortion

12 CHAPTER 1 Foundations of Distributed Source Coding

measure d

(·, ·). The question is again to determine the necessary rates, R

and R

respectively, as well as the coding schemes that per mit satisfying the distortion con-

straints. Formal deﬁnitions are provided in Appendix A (Deﬁnitions A.14 and A.15).

For this problem,a natural achievable strategy arises. One can ﬁrst quantize the sources

at each encoder as in the centralized lossy coding case and then bin the quantized val-

ues as in the distributed lossless case. The work of Berger and Tung [27–29] provides

an elegant way to combine these two techniques that leads to the following result.

The result was independently discovered by Housewright [30].

Theorem 1.5: “Quantize-and-bin.” Given sources S

and S

with distortion functions d

⫻ U

→

⫹

, k ∈{1, 2}, the achievable rate-distortion region includes the following set:

R ⫽ {(



D):∃U

, U

s.t. U

⫺ S

⫺ U

E[d

, U

)]⭐ D

, E[d

, U

)]⭐ D

,(1.20)

⬎ I(S

), R

⬎ I(S

⫹ R

⬎ I(S

, S

)}.

A proof for the setting of more than two users is given by Han and Kobayashi

[31,Theorem 1, pp. 280–284]. A major open question stems from the optimality of

the “quantize-and-bin” achievable strategy. While work by Servetto [32] suggests it

may be tight for the two-user setting, the only case for which it is known is the

quadratic Gaussian setting, which is based on an outer bound developed by Wagner

and Anantharam [33, 34].

Theorem 1.6: Wagner, Tavildar, and Viswanath [35]. Given sources S

and S

that are

jointly Gaussian and i.i.d. in time, that is,



(k),S

(k)



∼

N (0,⌺), with covariance matrix

⌺⫽



␴

␳␴

␴

␳␴

␴



(1.21)

for D

, D

⬎ 0 deﬁne R(D

)as

R(D

, D

)⫽



, R

):R

⭓

log

⫹

(1⫺ ␳

⫹ ␳

⫺2R

)␴

⭓

log

⫹

(1⫺ ␳

⫹ ␳

⫺2R

)␴

⫹ R

⭓

log

⫹

(1⫺ ␳

)␴

␴

␤(D

, D

)



,(1.22)

where

␤(D

, D

)⫽ 1⫹



1⫹

4␳

(1⫺ ␳

)

␴

.(1.23)

Furthermore, let R

be the interior of R. Then for distortions D

, D

⬎ 0, (R

, R

)∈

, D

) are achievable for the two-terminal quadratic Gaussian source coding

problem and (R

, R

)/∈R(D

, D

) are not.

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分布式源编码：理论、算法与应用实践

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