小波分析之旅：计算机视觉与图像处理

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"a wavelet tour of signal processing(3 edition)" 是一本由Steven M. A. Smith撰写的关于小波分析的经典著作，主要针对计算机视觉和小波图像处理领域。本书详细介绍了小波分析这一数学工具在信号处理中的应用。小波分析是一种能够同时在时域和频域提供局部信息的分析方法，它在图像处理、数据压缩、故障检测、金融数据分析等多个领域有着广泛的应用。第三版更新了最新的研究成果和技术发展，为读者提供了更全面的理解和深入的洞察。书中的内容可能涵盖以下几个方面： 1. 小波理论基础：包括小波函数的定义、性质、生成和选择，以及小波变换的基本原理，如离散小波变换（DWT）和连续小波变换（CWT）。 2. 多分辨率分析：讲解了多分辨率表示的概念，它是小波分析的核心，允许信号在不同尺度上进行分析。 3. 图像处理应用：阐述了如何利用小波变换对图像进行去噪、压缩和特征提取，特别是在计算机视觉中的应用，如边缘检测、图像分类和目标识别。 4. 信号去噪：讨论了小波阈值去噪技术，这是去除信号噪声的一种有效方法，可以保持信号的主要特征。 5. 数据压缩：小波分析在数据压缩中的作用，通过选择重要的小波系数实现高效的数据编码。 6. 实时处理与算法实现：探讨了小波变换在实时系统中的应用及其对应的快速算法，如快速小波变换（FWT）。 7. 案例研究：可能包含实际工程问题的案例，以展示小波分析在解决具体问题中的实际应用。 8. 版权与许可信息：提醒读者尊重知识产权，书中提到的公司名称和产品可能是商标，需要遵循相关法律法规。 "a wavelet tour of signal processing(3 edition)" 是一本适合初学者和专业人士深入理解小波分析的权威教材，通过阅读此书，读者将能够掌握小波理论并学会如何将其应用于实际的信号处理和图像分析任务中。

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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