边缘检测技术综述：噪声平滑与图像融合应用

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本文是一篇深入探讨边缘检测技术的综述文章，发表于2000年6月，来源于CiteSeer数据库。边缘检测作为图像处理中的关键步骤，对于计算机视觉、机器学习以及信号处理等领域具有重要意义。文章由两位作者——Djemel Ziou和Salvatore Antoine Tabbone共同完成，其中Djemel Ziou来自Lorraine大学的数学与信息技术研究所，是计算机科学研究中心/法国国家科研中心的一部分，而Salvatore Tabbone则在Lorrain de Recherche en Informatique et Ses Applications项目上有所贡献。边缘检测技术的目标是识别图像中的边界或轮廓，这对于物体识别、目标跟踪、图像分割等任务至关重要。文章可能会涵盖各种经典边缘检测算法，如Sobel算子、Canny算子、Roberts算子，以及它们各自的特点、适用场景和优缺点。此外，可能还会讨论边缘检测中的噪声抑制问题，这是提高边缘检测准确性的关键环节，如文中提到的"noisesmoothing"项目。在边缘检测的过程中，通常会涉及到图像预处理，包括平滑滤波（如高斯滤波）、阈值处理和边缘连接。Canny算子以其多级处理流程（高通滤波、计算梯度幅值和方向、非极大值抑制以及双阈值检测）被认为是一种更高级别的边缘检测方法，能够有效减少伪边缘并保留真实边缘。图像融合也是边缘检测技术的一个相关应用领域，这可能意味着将多个视角或模态的图像结合在一起，通过边缘信息来增强整体图像理解和解析。例如，"ImageFusion"项目可能就是在这个背景下提及的。作者们在文章中分享了他们的研究成果和实践经验，强调了边缘检测技术在实际应用中的挑战和可能的解决方案。由于文章上传于2015年5月，可以推测后续的研究也围绕着这些基础理论和技术进行了发展和优化。这篇综述提供了边缘检测技术的全面概述，涵盖了算法原理、实践应用和研究进展，对于想要深入了解该领域的读者来说，是一份宝贵的参考资料。对于那些对边缘检测感兴趣，或者正在进行相关研究的工程师、学生和研究人员，这篇文章提供了丰富的学习材料和研究灵感。

w to cho ose the scale (Multi-scale) and how to select an edge detector (multi-detector) for

a target application. In what follows, we will present the three op erations, the multi-detector

and the multi-scale approaches in more details. The reader can nd a study of smo othing lters

and dierentiation op erators used in edge detection in the pap er of Torre and Poggio [114].

3.1 Smo othing of the Image

It should b e recalled that smo othing has a p ositive eect to reduce noise and to ensure robust

edge detection; and a negative eect, information loss. Clearly, we have a fundamental trade-

o here b etween loss of information and noise reduction. The ultimate goal is to nd optimal

detectors that ensure a favorable compromise b etween noise reduction and edge conservation.

Regularization theory is the formalization, as an optimization problem, of the search for an

optimal lter. In fact, image dierentiation, like many other problems in early computational

vision, is ill-p osed in the sense that the existence, uniqueness, and stability of a solution cannot

b e guaranteed in the absence of additional constraints. Poggio [86] suggests that the class of

admissible solutions to an ill-p osed problem is restricted by imp osing additional constraints.

Usually, these constraints concern the appropriate compromise b etween noise reduction and

edge conservation. For example, in standard Tikhonov's regularization, a problem is rendered

well-p osed by restricting the acceptable solutions to the space of smo oth functions. In the

edge detection context, Poggio and Torre [85, 86] show that regularizing dierentiation can

b e accomplished by convolution of the image with the cubic spline (or its derivatives), with

area controlled by a regularization parameter. The regularization parameter determines the

compromise b etween noise elimination and the preservation of image structure. In ltering

terminology, the regularization parameter is called the

scale

. In addition to the cubic spline,

two other regularization lters have b een prop osed in [86], the Green function and the Gaussian,

whose resp ective impulse resp onses are:

(







(

cos

(



sin

(



)) (1)

(







(2)

where

>

0 and

>

0. Although these lters ensure a compromise b etween noise elimina-

tion and the preservation of image structure, we are faced with the problem of choosing the

regularization parameter. This is imp ortant since judicious selection of this parameter reduces

information loss. This problem will b e presented b elow (cf. 3.4).

The attributes of a smo othing lter that inuence the p erformance of the edge detector are

its linearity, the duration of its impulse resp onse, and its invariance to rotation. Non-linear

ltering has b een proven to b e more successful than linear ltering b ecause it removes certain

kinds

of noise b etter (e.g., impulse noise) while preserving edge information [84]. However,

in this section we will focus on linear ltering b ecause it is more common in edge detection.

The duration of the impulse resp onse characterizes the supp ort of the lter in the spatial or

frequential domain. For instance, in edge detection, three kinds of linear low-pass lters have

b een used: band-limited lters, supp ort-limited lters and lters with minimal uncertainty.

The invariance to rotation prop erty ensures that the eect of smo othing is the same regardless

of edge orientation.

A sucient condition for a function or op erator (functional) to b e rotationally invariant is

that its p olar form dep ends only on the radial distance



and not on its direction

tan

(



y=x

. Formally, a function

(

x; y

) is said to b e invariant to rotation (we may also

call it a radial or rotationally symmetric function) i for all



(

x; y

(

X; Y

), where

xcos

(



y sin

(



) and



xsin

(



y cos

(



). For example, the Gaussian is invariant

to rotation, whereas the 2D version of the lter dened in (eq. 1) is not. This prop erty will

b e examined in more detail in section 4.2. Other information on this topic can b e found

in [6, 15, 125, 129].

3.2 Image Dierentiation

It should b e recalled that the purp ose of edge detection is to localize variations of the image

grey level and to identify the physical phenomena which pro duced them. Dierentiation is the

computation of the necessary derivatives to lo calize these edges. the dierentiation operator is

characterized by its order, its invariance to rotation and its linearity.

The order of the dierentiation op erator is dened by the order of its partial derivatives.

An op erator

x;y

is invariant to rotation i

X;Y

x;y

, where

is the rotation matrix.

x;y

is linear i for all p ositive scalars



and



and for all functions

(

x; y

) and

(

x; y

), we

h ave

x;y

(

f

(

x; y

g

(

x; y

)) =



x;y

(

x; y

)) +



x;y

(

x; y

)). The most commonly used

op erators are the gradient, the Laplacian and the second-order directional derivative. As we

will show, the prop erties of the op erator to b e used are determined by the characteristics of

the image and the subsequent use of edges.

The gradient is a rst-order op erator dened as the vector (

;

). The mo dulus and the

direction of the gradient are dened by:

grad

(

)

and

ar ctg

(3)

The gradient direction is p erp endicular to the edge orientation. In many prop osed schemes,

the gradient direction is used to lo calize edges. The gradient mo dulus op erator is non-linear

and invariant to rotation. It is only computed by using derivatives in

and

. In a noisy

image, the use of several directional derivatives may b e useful for increasing the signal-to-noise

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