小波理论与信号处理：Stephane Mallat著英文第三版概览

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"Signal Processing with Wavelets - A Beginner's Guide (3rd Edition) by Stéphane Mallat" 这本书，"信号处理的小波导引（英文第三版）"由Stéphane Mallat撰写，深入浅出地介绍了信号处理领域中的小波理论与应用。小波分析是一种数学工具，它结合了频域和时域分析的优点，能够对信号进行多尺度、多分辨率的分析。这一版可能更新了最新的研究成果和技术发展，特别强调了稀疏表示在信号处理中的重要作用。稀疏表示是现代信号处理中的一个核心概念，它指的是用尽可能少的基函数来精确表示复杂信号。在小波框架下，许多信号可以被表示为少数几个小波系数的线性组合，这种表示方式对于数据压缩、去噪、图像恢复以及模式识别等应用具有极大价值。书中可能涵盖了以下几个关键知识点： 1. **小波基础**：解释了小波函数的基本性质，如正交性、紧凑支撑和时间-频率局部化特性。这有助于理解小波如何在不同尺度上捕获信号的特征。 2. **小波变换**：讨论了连续和离散小波变换，包括Morlet小波、Haar小波等常见类型，以及它们的计算方法和特性。 3. **多分辨率分析**：阐述了小波如何通过多分辨率分析提供对信号细节的逐级细化视图，这对于理解和重建信号至关重要。 4. **稀疏表示与压缩感知**：深入探讨了如何利用小波进行信号的稀疏表示，以及与压缩感知理论的关联，解释如何在较少的采样点下重构信号。 5. **应用案例**：书中可能包含各种实际应用示例，如图像处理、音频信号分析、地震数据处理和医学成像等，展示了小波分析在这些领域的实用性。 6. **算法实现**：可能会介绍用于执行小波变换和稀疏表示的算法，包括快速算法，如快速傅里叶变换（FFT）在小波计算中的应用。 7. **版权与许可信息**：提醒读者，书中的内容受到版权保护，未经许可不得复制或传播。如果需要使用其中的内容，应按照出版商提供的联系方式申请许可。通过这本书，读者将能系统地学习小波分析的基础知识，并掌握如何运用这些理论到实际的信号处理问题中，从而提升解决问题的能力。对于初学者，这是一个很好的起点，对于有经验的研究者，也提供了深入研究的参考。

6 CHAPTER 1 Sparse Representations

(a) (b)

FIGURE 1.1

(a) Discrete image f [n] of N ⫽ 256

pixels. (b) Array of N orthogonal wavelet coefﬁcients

f , ␺

j,n

 for k ⫽ 1, 2, 3, and 4 scales 2

; black points correspond to |f , ␺

j,n

|⬎ T . (c) Linear

approximation from the N /16 wavelet coefﬁcients at the three largest scales. (d) Nonlinear

approximation from the M ⫽ N /16 wavelet coefﬁcients of largest amplitude shown in (b).

Stochastic versus Deterministic Signal Models

A representation is optimized relative to a signal class, corresponding to all poten-

tial signals encountered in an application.This requires building signal models that

carry available prior information.

A signal f can be modeled as a realization of a random process F ,the probability

distribution of which is known a priori. A Bayesian approach then tries to minimize

1.2 Approximation and Processing in Bases 7

the expected approximation error. Linear approximations are simpler because they

only depend on the covariance. Chapter 9 shows that optimal linear approxima-

tions are obtained on the basis of principal components that are the eigenvectors

of the covariance matrix. However,the expected error of nonlinear approximations

depends on the full probability distribution of F . This distribution is most often

not known for complex signals, such as images or sounds, because their transient

structures are not adequately modeled as realizations of known processes such as

Gaussian ones.

To optimize nonlinear representations, weaker but sufﬁciently powerful deter-

ministic models can be elaborated. A deterministic model speciﬁes a set ⌰, where

the signal belongs.This set is deﬁned by any prior information—for example,on the

time-frequency localization of transients in musical recordings or on the geometric

regularity of edges in images. Simple models can also deﬁne ⌰ as a ball in a functional

space, with a speciﬁc regularity norm such as a total variation norm. A stochastic

model is richer because it provides the probability distribution in ⌰. When this dis-

tribution is not available, the average error cannot be calculated and is replaced by

the maximum error over ⌰. Optimizing the representation then amounts to mini-

mizing this maximum error, which is called a minimax optimization.

1.2.1 Sampling with Linear Approximations

Analog-to-digital signal conversion is most often implemented with a linear approxi-

mation operator that ﬁlters and samples the input analog signal. From these samples,

a linear digital-to-analog converter recovers a projection of the original analog signal

over an approximation space whose dimension depends on the sampling density.

Linear approximations project signals in spaces of lowest possible dimensions to

reduce computations and storage cost, while controlling the resulting error.

Sampling Theorems

Let us consider ﬁnite energy signals 

f 

⫽



f (x)|

dx of ﬁnite support, which is

normalized to [0, 1] or [0, 1]

for images. A sampling process implements a ﬁltering

f (x) with a low-pass impulse response

␾

(x) and a uniform sampling to output

a discrete signal:

f [n]⫽

f 

␾

(ns) for 0 ⭐ n ⬍ N .

In two dimensions, n ⫽ (n

, n

) and x ⫽ (x

, x

). These ﬁltered samples can also be

written as inner products:

f 

␾

(ns)⫽



f (u)

␾

(ns ⫺ u) du⫽ f (x), ␾

(x ⫺ ns)

with ␾

(x) ⫽

␾

(⫺x). Chapter 3 explains that ␾

is chosen, like in the clas-

sic Shannon–Whittaker sampling theorem, so that a family of functions {␾

(x ⫺ ns)}

1⭐n⭐N

is a basis of an appropriate approximation space U

. The best lin-

ear approximation of

f in U

recovered from these samples is the orthogonal

8 CHAPTER 1 Sparse Representations

projection

of f in U

, and if the basis is orthonormal, then

(x) ⫽

N⫺1



n⫽0

f [n]␾

(x ⫺ ns). (1.4)

A sampling theorem states that if

f ∈U

then

f ⫽

so (1.4) recovers

f (x)

from the measured samples. Most often,

f does not belong to this approximation

space. It is called aliasing in the context of Shannon–Whittaker sampling, where

is the space of functions having a frequency support restricted to the N lower

frequencies. The approximation error 

f ⫺



must then be controlled.

Linear Approximation Error

The approximation error is computed by ﬁnding an orthogonal basis B ⫽

{¯g

(x)}

0⭐m⬍⫹⬁

of the whole analog signal space L

[0, 1]

, with the ﬁrst N vec-

tor {¯g

(x)}

0⭐m⬍N

that deﬁnes an orthogonal basis of U

. Thus, the orthogonal

projection on U

can be rewritten as

(x) ⫽

N⫺1



m⫽0



f , ¯g

¯g

(x).

Since

f ⫽



⫹⬁

m⫽0



f , ¯g

¯g

, the approximation error is the energy of the removed

inner products:

␧

(N, f ) ⫽ 

f ⫺



⫽

⫹⬁



m⫽N

|

f , ¯g

|

This error decreases quickly when N increases if the coefﬁcient amplitudes |

f , ¯g

|

have a fast decay when the index m increases. The dimension N is adjusted to the

desired approximation error.

Figure 1.1(a) shows a discrete image f [n] approximated with N ⫽ 256

pixels.

Figure 1.1(c) displays a lower-resolution image f

N/16

projected on a space U

N/16

dimension N/16, generated by N/16 large-scale wavelets. It is calculated by setting

all the wavelet coefﬁcients to zero at the ﬁrst two smaller scales.The approximation

error is f ⫺ f

N/16



/f 

⫽ 14 ⫻ 10

⫺3

. Reducing the resolution introduces more

blur and errors. A linear approximation space U

corresponds to a uniform grid

that approximates precisely uniform regular signals. Since images

f are often not

uniformly regular, it is necessary to measure it at a high-resolution N . This is why

digital cameras have a resolution that increases as technology improves.

1.2.2 Sparse Nonlinear Approximations

Linear approximations reduce the space dimensionality but can introduce important

errors when reducing the resolution if the signal is not uniformly regular, as shown

by Figure 1.1(c). To improve such approximations, more coefﬁcients should be

kept where needed—not in regular regions but near sharp transitions and edges.

1.2 Approximation and Processing in Bases 9

This requires deﬁning an irregular sampling adapted to the local signal regularity.

This optimized irregular sampling has a simple equivalent solution through nonlinear

approximations in wavelet bases.

Nonlinear approximations operate in two stages. First, a linear operator approx-

imates the analog signal

f with N samples written f [n]⫽

f 

␾

(ns). Then, a

nonlinear approximation of f [n] is computed to reduce the N coefﬁcients f [n]

to M N coefﬁcients in a sparse representation.

The discrete signal f can be considered as a vector of C

. Inner products and

norms in C

are written

f , g⫽

N⫺1



n⫽0

f [n]g

∗

[n] and f 

⫽

N⫺1



n⫽0

|f [n]|

To obtain a sparse representation with a nonlinear approximation,we choose a new

orthonormal basis B ⫽ {g

[n]}

m∈⌫

of C

, which concentrates the signal energy as

much as possible over few coefﬁcients. Signal coefﬁcients {f , g

}

m∈⌫

are com-

puted from the N input sample values f [n] with an orthogonal change of basis

that takes N

operations in nonstructured bases. In a wavelet or Fourier bases, fast

algorithms require, respectively, O(N ) and O(N log

N) operations.

Approximation by Thresholding

For M ⬍ N ,an approximation f

is computed by selecting the“best”M ⬍ N vectors

within B. The orthogonal projection of f on the space V

⌳

generated by M vectors

}

m∈⌳

in B is

⌳

⫽



m∈⌳

f , g

g

. (1.5)

Since f ⫽



m∈⌫

f , g

g

, the resulting error is

f ⫺ f

⌳



⫽



m/∈⌳

|f , g

|

. (1.6)

We write |⌳| the size of the set ⌳. The best M ⫽ |⌳| term approximation, which

minimizes f ⫺ f

⌳



, is thus obtained by selecting the M coefﬁcients of largest

amplitude. These coefﬁcients are above a threshold T that depends on M:

⫽ f

⌳

⫽



m∈⌳

f , g

g

with ⌳

⫽ {m ∈⌫ :|f , g

|⭓ T }. (1.7)

This approximation is nonlinear because the approximation set ⌳

changes with

f . The resulting approximation error is:

␧

(M, f ) ⫽ f ⫺ f



⫽



m/∈⌳

|f , g

|

. (1.8)

Figure 1.1(b) shows that the approximation support ⌳

of an image in a wavelet

orthonormal basis depends on the geometry of edges and textures. Keeping large

10 CHAPTER 1 Sparse Representations

wavelet coefﬁcients is equivalent to constructing an adaptive approximation grid

speciﬁed by the scale–space support ⌳

. It increases the approximation resolution

where the signal is irregular. The geometry of ⌳

gives the spatial distribution of

sharp image transitions and edges, and their propagation across scales. Chapter 6

proves that wavelet coefﬁcients give important information about singularities

and local Lipschitz regularity. This example illustrates how approximation support

provides“geometric”information on f ,relative to a dictionary,that is a wavelet basis

in this example.

Figure 1.1(d) gives the nonlinear wavelet approximation f

recovered from the

M ⫽ N /16 large-amplitude wavelet coefﬁcients, with an error f ⫺ f



/f 

⫽

5 ⫻ 10

⫺3

. This error is nearly three times smaller than the linear approximation

error obtained with the same number of wavelet coefﬁcients,and the image quality

is much better.

An analog signal can be recovered from the discrete nonlinear approxima-

tion f

(x) ⫽

N⫺1



n⫽0

[n]␾

(x ⫺ ns).

Since all projections are orthogonal, the overall approximation error on the orig-

inal analog signal

f (x) is the sum of the analog sampling error and the discrete

nonlinear error:



f ⫺



⫽ 

f ⫺



⫹ f ⫺ f



⫽ ␧

(N, f ) ⫹ ␧

(M, f ).

In practice, N is imposed by the resolution of the signal-acquisition hardware, and

M is typically adjusted so that ␧

(M, f ) ⭓ ␧

(N, f ).

Sparsity with Regularity

Sparse representations are obtained in a basis that takes advantage of some form

of regularity of the input signals, creating many small-amplitude coefﬁcients. Since

wavelets have localized support, functions with isolated singularities produce few

large-amplitude wavelet coefﬁcients in the neighborhood of these singularities. Non-

linear wavelet approximation produces a small error over spaces of functions that

do not have “too many” sharp transitions and singularities. Chapter 9 shows that

functions having a bounded total variation norm are useful models for images with

nonfractal (ﬁnite length) edges.

Edges often deﬁne regular geometric curves. Wavelets detect the location of

edges but their square support cannot take advantage of their potential geometric

regularity. More sparse representations are deﬁned in dictionaries of curvelets or

bandlets, which have elongated support in multiple directions, that can be adapted

to this geometrical regularity. In such dictionaries,the approximation support ⌳

smaller but provides explicit information about edges’local geometrical properties

such as their orientation. In this context, geometry does not just apply to multidi-

mensional signals. Audio signals, such as musical recordings, also have a complex

geometric regularity in time-frequency dictionaries.

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