压缩感知：低于奈奎斯特定理的高效采样技术

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"这篇文章是关于压缩感知的一篇深入浅出的论述，由Richard G. Baraniuk撰写，发表在2007年7月的IEEE SIGNAL PROCESSING MAGAZINE上。" 压缩感知（Compressive Sensing）是信号处理领域的一个重要概念，它打破了传统的香农/奈奎斯特定理（Shannon/Nyquist sampling theorem）的限制。根据奈奎斯特定理，为了不失真地捕获信号，采样率必须至少是信号带宽的两倍。然而，在许多应用中，如数字图像和视频摄像机，这种高采样率导致了大量的样本数据，使得在存储或传输前必须进行压缩。在其他应用，如医疗扫描仪、雷达系统和高速模数转换器中，提高采样率成本高昂。压缩感知提出了一种新的方法，能够在远低于奈奎斯特采样率的速率下捕获和表示可压缩信号。这种方法依赖于非适应性的线性投影，这些投影保留了信号的结构。然后，通过优化过程从这些投影中重构信号。这种方法不仅节省了采样硬件的成本，还减少了数据存储和传输的需求。其相关性在于，压缩感知的概念可以应用于各种场景。例如，在医学成像中，可以减少扫描时间和辐射剂量；在无线通信中，降低带宽需求，提高频谱效率；在遥感和地球观测中，可以处理大量数据而无需高分辨率传感器。此外，压缩感知还被用于图像和视频的快速编码与解码，以及在机器学习和数据科学中，帮助高效地处理高维数据。压缩感知的核心理论包括稀疏表示（Sparse Representation）和恢复算法。稀疏表示是指信号可以被表示为一个稀疏向量，即大部分元素为零，只有少数元素非零。恢复算法如正则化最小二乘（LASSO）、迭代硬阈值（Iterative Hard Thresholding）等，它们旨在找到满足稀疏约束的最优信号解。压缩感知提供了一种革命性的信号处理方式，通过创新的采样和重构策略，大大降低了数据处理的复杂性和成本，对于现代信息技术的发展具有重大意义。

IEEE SIGNAL PROCESSING MAGAZINE [118] JULY 2007

[

lecture NOTES

]

Richard G. Baraniuk

Compressive Sensing

IEEE SIGNAL PROCESSING MAGAZINE [118] JULY 2007

he Shannon/Nyquist sam-

pling theorem specifies that

to avoid losing information

when capturing a signal, one

must sample at least two

times faster than the signal bandwidth. In

many applications, including digital

image and video cameras, the Nyquist rate

is so high that too many samples result,

making compression a necessity prior to

storage or transmission. In other applica-

tions, including imaging systems (medical

scanners and radars) and high-speed ana-

log-to-digital converters, increasing the

sampling rate is very expensive.

This lecture note presents a new

method to capture and represent com-

pressible signals at a rate significantly

below the Nyquist rate. This method,

called compressive sensing, employs

nonadaptive linear projections that pre-

serve the structure of the signal; the sig-

nal is then reconstructed from these

projections using an optimization

process [1], [2].

RELEVANCE

The ideas presented here can be used to

illustrate the links between data acquisi-

tion, compression, dimensionality reduc-

tion, and optimization in undergraduate

and graduate digital signal processing,

statistics, and applied mathematics

courses.

PREREQUISITES

The prerequisites for understanding this

lecture note material are linear algebra,

basic optimization, and basic probability.

PROBLEM STATEMENT

COMPRESSIBLE SIGNALS

Consider a real-valued, finite-length,

one-dimensional, discrete-time signal

which can be viewed as an

N × 1

column

vector in

with elements

x[n]

n = 1, 2,...,N

. (We treat an image or

higher-dimensional data by vectorizing it

into a long one-dimensional vector.) Any

signal in

can be represented in terms

of a basis of

N × 1

vectors

{ψ

}

i=1

. For

simplicity, assume that the basis is

orthonormal. Using the

N × N

basis

matrix

 = [ψ

|ψ

| ...|ψ

]

with the

vectors

{ψ

}

as columns, a signal

can

be expressed as

x =



i =1

or x = ψs (1)

where

is the

N × 1

column vector of

weighting coefficients

=x,ψ

=ψ

and

denotes transposition. Clearly,

and

are equivalent representations of

the signal, with

in the time or space

domain and

in the



domain.

The signal

-sparse if it is a linear

combination of only

basis vectors; that

is, only

of the

coefficients in (1) are

nonzero and

(N − K)

are zero. The case

of interest is when

K  N

. The signal

is compressible if the representation (1)

has just a few large coefficients and many

small coefficients.

TRANSFORM CODING

AND ITS INEFFICIENCIES

The fact that compressible signals are

well approximated by

-sparse represen-

tations forms the foundation of trans-

form coding [3]. In data acquisition

systems (for example, digital cameras)

transform coding plays a central role: the

full

-sample signal

is acquired; the

complete set of transform coefficients

}

is computed via

s = 

; the

largest coefficients are located and the

(N − K)

smallest coefficients are dis-

carded; and the

values and locations of

the largest coefficients are encoded.

Unfortunately, this sample-then-com-

press framework suffers from three

inherent inefficiencies. First, the initial

number of samples

may be large even

if the desired

is small. Second, the set

of all

transform coefficients

}

must

be computed even though all but

them will be discarded. Third, the loca-

tions of the large coefficients must be

encoded, thus introducing an overhead.

THE COMPRESSIVE

SENSING PROBLEM

Compressive sensing address these ineffi-

ciencies by directly acquiring a com-

pressed signal representation without

going through the intermediate stage of

acquiring

samples [1], [2]. Consider a

general linear measurement process that

computes

M < N

inner products

between

and a collection of vectors

{φ

}

j=1

as in

=x,φ



. Arrange the

measurements

in an

M × 1

vector

and the measurement vectors

as rows

in an

M × N

matrix



. Then, by substi-

tuting



from (1),

can be written as

y = x = s = s (2)

where

 = 

is an

M × N

matrix. The

measurement process is not adaptive,

meaning that



is fixed and does not

depend on the signal

. The problem

consists of designing a) a stable meas-

urement matrix



such that the salient

information in any

-sparse or com-

pressible signal is not damaged by the

dimensionality reduction from

x ∈

y ∈

and b) a reconstruction algo-

rithm to recover

from only

M ≈ K

measurements

(or about as many

measurements as the number of coeffi-

cients recorded by a traditional trans-

form coder).

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