刘易斯·杜博斯博士的多维信号与色彩图像处理新论：基于格子理论

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"《多维信号与彩色图像处理中的格子理论》是一本由埃里克·杜博斯编著的创新性著作，专为图像和视频处理领域的研究生、博士生以及高级本科生和专业人士设计。本书深入探讨了多维信号处理的新理论，特别关注于输入和输出为向量值信号的情况。作者在书中首先介绍了连续和离散域的信号与系统，以及离散域周期信号的概念，这些都是信号处理的基础。传统的信号处理方法往往采用归一化表示非矩形采样，这会隐藏信号的空间、时间和频率的几何及尺度信息。杜博斯教授则选择使用实际的时空和频率单位，尽可能地采用基无关的表示，并在需要计算或实现时引入特定的基。因此，格子理论自始至终贯穿全书，而矩形采样被视为其特殊情形。这种处理方式对于颜色和彩色图像处理至关重要，因为它们依赖于基于对称群的离散变换，这包括高效的计算算法。本书的一大亮点是专门章节——《格子在信号处理中的应用》，帮助读者深入理解格子在信号处理中的作用。此外，还详细讨论了格子在描述离散域信号和周期性、采样与重构、随机场模型、对称不变信号与系统以及多维傅立叶变换性质等方面的应用。为了增强理解和实践，书中穿插了大量的MATLAB示例和可下载的源代码。《多维信号与彩色图像处理中的格子理论》不仅为读者提供了一个全面的多维信号系统理论入门，也为未来的研究提供了新的视角和灵感。无论是在信号处理的专业教育还是在实际应用中，这本书都具有重要的参考价值。"

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6 2 Continuous-Domain Signals and Systems

Figure 2.1 Illustration of image window  with aspect ratio ar ∶ 1pw= ar ph.

distance, as has long been common in the broadcast video industry. However, we are free

to choose any convenient unit of length in a given application, for example, the size of

a sensor or display element, or an absolute measure of distance such as the meter or

micron. e ratio ar = pw∕ph is called the aspect ratio, the most common values being

4/3 for standard  and 16/9 for . With this notation, 1 pw = ar ph (see Figure 2.1).

Time is measured in seconds, denoted s. Examples of continuous-domain space-time

images include the illumination on the sensor of a video camera, or the luminance of

the light reﬂected by a cinema screen or emitted by a television display.

Since the image is undeﬁned outside the spatial window , we are free to extend it

outside the window as we see ﬁt to include all of ℝ

as the domain. Some possibilities

are to set the image to zero outside , to periodically repeat the image, or to extrapolate

it in some way. Which of these is chosen depends on the application.

ere are two common ways to attach an xy coordinate system to the image window,

involving the location of the origin and the orientation of the x and y axes, as shown in

Figure 2.2. e standard orientation used in mathematics to graph functions would place

the origin at the lower left corner of the image with the y-axis pointing upward. However,

(b)

(a)

Figure 2.2 Orientation of xy-axes. (a) Common bottom-to-top orientation in mathematics.

(b) Scanning-based top-to-bottom orientation.

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2.2 Multidimensional Signals 7

because traditionally images have been scanned from top to bottom, most image ﬁle

formats store the image line-by-line, with the top line ﬁrst, and line numbers increasing

from top to bottom of the image. is makes the orientation shown in Figure 2.2(b) more

convenient, with the origin in the upper left corner of the image and the y-axis pointing

downward. For this reason, we will generally use the orientation of Figure 2.2(b).

2.2 Multidimensional Signals

A multiD signal can be considered to be a function from the domain, here ℝ

,tothe

range. In this and the next few chapters, we consider only real and complex valued

signals, which we call scalar signals. In later chapters, where we consider color signals,

we will take the range to be a suitable vector space. In addition to naturally occurring

continuous space-time images, many analytically deﬁned D-dimensional functions are

useful in image processing theory. A few of these are introduced here.

2.2.1 Zero–One Functions

Let be a region in the D-dimensional space, ⊂ ℝ

. We deﬁne the zero–one function



(x) as illustrated in Figure 2.3(a) by



(x)=

{

1, if x ∈ ;

0, otherwise.

(2.1)

Sometimes, p



(x) is called the indicator function of the region . Diﬀerent functions are

obtained with diﬀerent choices of the region . Such functions arise frequently in mod-

eling sensor elements or display elements (sub-pixels). e most commonly used ones

in image processing are the rect and the circ functions in two dimensions. Speciﬁcally,

for a unit-square region we obtain (Figure 2.3(b))

rect(x, y)=

{

1, if |x|≤0.5 and |y|≤0.5;

0, otherwise.

(2.2)

For a circular region of unit radius we have (Figure 2.3(c))

circ(x, y)=

{

1, if x

+ y

≤1;

0, otherwise.

(2.3)

ese deﬁnitions can be extended to the three-dimensional case (where the region 

is a cube or a sphere) or to higher dimensions in a straightforward fashion, and the

single notation rect(x) or circ(x) canbeusedtocoverallcases.Wewillseelaterhow

these basic signals can be shifted, scaled, rotated or otherwise transformed to generate a

much richer set of zero–one functions. Other zero–one functions that we will encounter

correspond to various polygonal regions such as triangles, hexagons, octagons, etc.

2.2.2 Sinusoidal Signals

As in one-dimensional signals and systems, real and complex-exponential sinusoidal sig-

nals play an important role in the analysis of image processing systems. ere are several

reasons for this but a principal one is that if a complex-exponential sinusoidal signal is

applied as input to a linear shift-invariant system (to be introduced shortly), the output

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10 2 Continuous-Domain Signals and Systems

If f

(x) isgivenasinEquation(2.6),thenforanyﬁxedx

∈ ℝ

,wehave

(x − x

)=B exp(j2𝜋u ⋅ (x − x

))

= B exp(j2𝜋u ⋅ x) exp(−j2𝜋u ⋅ x

)

= exp(−j2𝜋u ⋅ x

(x). (2.8)

In other words, the shifted complex exponential is equal to the original complex expo-

nential multiplied by the complex constant exp(−j2𝜋u ⋅ x

). is in turn leads to the key

property of linear shift-invariant systems mentioned earlier in this section. is will be

analyzed in Section 2.5.5 but is mentioned here to motivate the importance of sinusoidal

signals in multidimensional signal processing.

2.2.3 Real Exponential Functions

Real exponential signals also have wide applicability in multidimensional signal process-

ing. First-order exponential signals are given by

f (x, y)=A exp(−(a|x|+ b|y|)) (2.9)

and second-order, or Gaussian, signals by

f (x, y)=A exp(−(x

+ y

)∕2r

). (2.10)

Gaussian signals, similar in form to the Gaussian probability density, are widely used

in image processing. Some illustrations can be seen later, in Figure 2.6. We deﬁne the

standard versions of these signals as follows:

(x, y)=exp(−2𝜋

√

+ y

), standard ﬁrst-order exponential signal; (2.11)

(x, y)=exp(−(x

+ y

)∕2), standard Gaussian signal. (2.12)

2.2.4 Zone Plate

A very useful two-dimensional function that is often used as a test pattern in imaging

systems is the zone plate, or Fresnel zone plate. e sinusoidal zone plate is based on the

function

f (x, y)=cos(𝜋(x

+ y

)∕r

) (2.13)

where r

is a parameter that sets the scale. Figure 2.5 illustrates the zone plate 0.5 +

0.5 cos(𝜋(x

+ y

)∕r

) with r

= 1∕250 ph. It is often convenient to consider the zone

plate as the real part of the complex exponential function exp(j𝜋(x

+ y

)∕r

Examining Figure 2.5, it can be seen that locally (say, within a small square window) the

function is a sinusoidal signal with a horizontal frequency that increases with horizontal

distance from the origin, and similarly a vertical frequency that increases with vertical

distance from the origin. To make this concept of local frequency more precise, consider

a conventional two-dimensional sinusoidal signal

cos(2𝜋(ux + 𝑣y)) = cos(𝜙(x, y)) (2.14)

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刘易斯·杜博斯博士的多维信号与色彩图像处理新论：基于格子理论

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