Multiscale Representation

Scale

Introduction

The neighborhood operations discussed in Chapter 4 can only be the starting point for image analysis. This class of operators can only extract local features at scales of at most a few pixels distance. It is obvious that images contain information also at larger scales. To extract object fea- tures at these larger scales, we need correspondingly larger fi lter masks. The use of large masks, however, results in a signifi cant increase in com- putational costs. If we use a mask of size RW in a W -dimensional image the number of operations is proportional to RW . Thus a doubling of the scale leads to a four- and eight-fold increase in the number of operations in 2- and 3-dimensional images, respectively. For a ten times larger scale, the number of computations increases by a factor of 100 and 1000 for 2- and 3-dimensional images, respectively.

The explosion in computational cost is only the superfi cial expression of a problem with deeper roots. We illustrate it with a simple task, the detection of edges and lines at diff erent resolutions. To this end, we use the same image row but blur it by diff erent degrees (Fig. 5.1). We defi ne the corresponding scale as the distance over which the image has been blurred and analyze the gray value diff erences over this distance.

We fi rst investigate gray value diff erences at high resolution, a scale of just one pixel distance (Fig. 5.1a, b). At this fi ne scale, the change in gray values is dominated by the noisy background of the image. Any detection of gray value changes caused by the contrast between objects and background is inaccurate and erroneous. The problem is caused by a scale mismatch: the gray values only vary on larger scales than the operators used to detect them.

If we take instead a low resolution (Fig. 5.1e, f), the lines are blurred so much that the contrast has signifi cantly decreased. Moreover, two closely spaced lines in the left part of the signal have merged into one object at this coarse resolution. Therefore the detection of edges and lines is suboptimal again. At a resolution comparable to the line width, however, the line detection seems to be optimal (Fig. 5.1c, d). Noise is signifi cantly reduced compared to the fi nest scale (Fig. 5.1a) but the

 

125

B. Jä hne, Digital Image Processing                                                                                                       Copyright © 2002 by Springer-Verlag

ISBN 3–540–67754–2                                                                                                    All rights of reproduction in any form reserved.

126                                                                              5 Multiscale Representation

 

a

220

200

180

160

140

120

100 0     50   100  150  200  250

c

220

200

180

160

140

120

b

60

40

20

0

-20

-40

-60

 

d

40

30

20

10

0

-10

-20

-30

 

0     50   100  150  200  250

100 0     50   100  150  200  250

e

220

200

180

160

140

120

100 0     50   100  150  200  250

-40 0     50   100  150  200  250

f

20

15

10

5

0

-5

-10

-15

-20 0     50   100  150  200  250

 

Figure 5.1: Lines and edges at a high, c medium, and e low resolution. b, d, and

f Subtraction of neighboring pixels for edge detection for a, c, and e, respectively.

 

 

contrast between the line and the background is not yet diminished as in Fig. 5.1e.

From the discussion of this example we can conclude that the detec- tion of certain features in an image is optimal at a certain scale. This scale depends, of course, on the characteristic scales contained in the object to be detected. Optimal processing of an image thus requires the representation of an image at diff erent scales. In order to meet this demand, we need a multiscale representation of images. In this chap- ter, we will fi rst illuminate the relation between the spatial and wave number representation of images under this perspective (Section 5.1.2). Then we will introduce the scale space (Section 5.2), discuss how it can be generated by a diff usion process, and describe its basic properties. Finally, in Section 5.3 we will turn to effi cient multigrid representations such as the Gaussian pyramid (Section 5.3.2) and the Laplacian pyramid (Section 5.3.3).

5.1 Scale                                                                                     127

 

⇐ Предыдущая21222324252627282930Следующая ⇒

Поделиться: