于灰度级间断的检测是最为普遍的检测方法。本节中，我们讨论实现一阶和二阶数字导数检测一幅图像中边缘的方法。在节介绍图像增强的内容中介绍过这些导数。本节的重点将放在边缘检测的特性上。某些前面介绍的概念在这里为了叙述的连续性将进行简要的重述。 基本说明
在节中我们非正式地介绍过边缘。本节中我们更进一步地了解数字化边缘的概念。直观上，一条边缘是一组相连的像素集合。这些像素位于两个区域的边界上。然而，我们已经在节中用一定的篇幅解释了一条边缘和一条边界的区别。从根本上讲，如我们将要看到的，一条边缘是一个“局部”概念，而由于其定义的方式，一个区域的边界是一个更具有整体性的概念。给边缘下一个更合理的定义需要具有以某种有意义的方式测量灰度级跃变的能力。
我们先从直观上对边缘建模开始。这样做可以将我们引领至一个能测量灰度级有意义的跃变的形式体系中。从感觉上说，一条理想的边缘具有如图10-5(a)所示模型的特性。依据这个模型生成的完美边缘是一组相连的像素的集合(此处为在垂直方向上)，每个像素都处在灰度级跃变的一个垂直的台阶上(如图形中所示的水平剖面图)。
实际上，光学系统、取样和其他图像采集的不完善性使得到的边缘是模糊的，模糊的程度取决于诸如图像采集系统的性能、取样率和获得图像的照明条件等因素。结果，边缘被更精确地模拟成具有“类斜面”的剖面，如图10-5(b)所示。斜坡部分与边缘的模糊程度成比例。在这个模型中，不再有细线(一个像素宽的线条)。相反，现在边缘的点是包含于斜坡中的任意点，并且边缘成为一组彼此相连接的点集。边缘的“宽度”取决于从初始灰度级跃变到最终灰度级的斜坡的长度。这个长度又取决于斜度，斜度又取决于模糊程度。这使我们明白:模糊的边缘使其变粗而清晰的边缘使其变得较细。
图10-6(a)显示的图像是从图10-5(b)的放大特写中提取出来的。图10-6(b)显示了两个区域之间边缘的一条水平的灰度级剖面线。这个图形也显示出灰度级剖面线的一阶和二阶导数。当我们沿着剖面线从左到右经过时，在进人和离开斜面的变化点，一阶导数为正。在灰度级不变的区域一阶导数为零。在边缘与黑色一边相关的跃变点二阶导数为正，在边缘与亮色一边相关的跃变点二阶导数为负，沿着斜坡和灰度为常数的区域为零。在图10-6(b)中导数的符号在从亮到暗的跃
变边缘处取反。
（a） （b）
图10-5 （a）理想的数字边缘模型，（b）斜坡数字边缘模型。
斜坡部分与边缘的模糊程度成正比
图10-6 (a)由一条垂直边缘分开的两个不同区域,(b)边界附近的细
节显示了一个灰度级剖面图和一阶与二阶导数的剖面图
由这些现象我们可以得到的结论是:一阶导数可以用于检测图像中的一个点是否是边缘的点(也就是判断一个点是否在斜坡上)。同样，二阶导数的符号可以用于判断一个边缘像素是在边缘亮的一边还是暗的一边。我们注意到围绕一条边缘，二阶导数的两条附加性质(1)对图像中的每条边缘二阶导数生成两个值(一个
不希望得到的特点);(2)一条连接二阶导数正极值和负极值的虚构直线将在边缘中点附近穿过零点。将在本节后面说明，二阶导数的这个过零点的性质对于确定粗边线的中心非常有用。
最后，注意到某些边缘模型利用了在进人和离开斜坡地方的平滑过渡(习题。然而，我们在接下来的讨论中将得出同样的结论。而且，这一点从我们使用局部检测进行处理就可以很明显地看出(因此，节中对于边缘的局部性质进行了说明)。 尽管到此为止我们的注意力被限制在一维水平剖面线范围内，但同样的结论可以应用于图像中的任何方向上。我们仅仅定义了一条与任何需要考察的点所在的边缘方向相垂直的剖面线，并如前面讨论的那样，对结果进行了解释。
注：出自
Digital Image Processing 2nd Edition . Prentice Hall
Image Segmentation
The material in the previous chapter began a transition from image processing methods whose input and output are images, to methods in which the inputs are images, but the outputs are attributes extracted from those images (in the sense defined is Section . Segmentation is another major step in that direction.
Segmentation subdivides an image into its constituent regions or objects. The level to which the subdivision is carried depends on the problem being solved. That is, segmentation should stop when the objects of interest in an application have been isolated. For example, in the automated inspection of electronic assemblies, interest lies in analyzing images of the products with the objective of determining the presence or absence of specific anomalies, such as missing components or broken connection paths. There is no point in carrying segmentation past the level of detail required to identify those elements.
Segmentation of nontrivial images is one of the most difficult tasks in image processing. Segmentation accuracy determines the eventual success or failure of computerized analysis procedures. For this reason, considerable care should be taken to improve the probability of rugged segmentation. In some situations , such as industrial inspection applications, at least some measure of control over the environment is possible at times. The experienced image processing system designer invariably pays considerable attention to such opportunities. In other applications, such as autonomous target acquisition, the system designer has no control of the environment. Then the usual approach is to focus on selecting the types of sensors most likely to enhance the objects of interest while diminishing the contribution of irrelevant image detail. A good example is the use of infrared imaging by the military to detect objects with strong heat signatures , such as equipment and troops in motion.
Image segmentation algorithms generally are based on one of two basic properties of intensity values: discontinuity and similarity. In the first category, the approach is to partition an image based on abrupt changes in intensity, such as edges
in an image. The principal approaches in the second category are based on partitioning an image into regions that are similar according to a set of predefined criteria. Thresholding, region growing, and region splitting and merging are examples of methods in this category.
In this chapter we discuss a number of approaches in the two categories just mentioned. We begin the development with methods suitable for detecting gray level discontinuities such as points, lines, and edges. Edge detection in particular has been a staple of segmentation algorithms for many years. In addition to edge detection per se, we also discuss methods for connecting edge segments and for \edges into region boundaries. The discussion on edge detection is followed by the introduction of various thresholding techniques . Thresholding also is a fundamental approach to segmentation that enjoys a significant degree of popularity, especially in applications where speed is an important factor. The discussion on thresholding is followed by the development of several region-oriented segmentation approaches. We then discuss a morphological approach to segmentation called watershed segmentation. This approach is particularly attractive because it combines several of the positive attributes of segmentation based on the techniques presented in the first part of the chapter. We conclude the chapter with a discussion on the use of motion cues for image segmentation. of Discontinuities
In this section we present several techniques for detecting the three basic types of gray-level discontinuities in a digital image: points, lines, and edges. The most common way to look for discontinuities is to run a mask through the image in the manner described in Section . For the 3 x 3 mask shown in Fig. , this procedure involves computing the sum of products of the coefficients with the gray levels contained in the region encompassed by the mask. That is. with reference to Eq. . the response of the mask at anv point in the image is given by
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