Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- A word on notation
- List of symbols
- Part I The plane
- Part II Matrix structures
- Part III Here's to probability
- Part IV Information, error and belief
- Part V Transforming the image
- 14 The Fourier Transform
- 15 Transforming images
- 16 Scaling
- Part VI See, edit, reconstruct
- References
- Index
16 - Scaling
from Part V - Transforming the image
Published online by Cambridge University Press: 05 November 2012
- Frontmatter
- Contents
- Preface
- Introduction
- A word on notation
- List of symbols
- Part I The plane
- Part II Matrix structures
- Part III Here's to probability
- Part IV Information, error and belief
- Part V Transforming the image
- 14 The Fourier Transform
- 15 Transforming images
- 16 Scaling
- Part VI See, edit, reconstruct
- References
- Index
Summary
Both wavelets and fractals (even just fractal dimension) have seen an explosion of applications in recent years. This text will point mainly to the vision side, but at the end of this chapter we give references that indicate something of the range. The story begins with Section 16.1 which is about fractals; this points to the value of the scaling theme, thereafter explored through wavelets and multiresolution. The present chapter concentrates on wavelets with the most structure: the Haar and Daubechies types. In the next this is relaxed for application to B-splines, then to surface wavelets.
Nature, fractals and compression
The potential to compress an image has to do with its degree of redundancy, and so comes broadly under image analysis. Here, more specifically, we are interested in the redundancy that may come from self-similarity of different parts of the image, rather than from more general correlation. This idea arose essentially from Mandelbrot's observations about the nature of natural images. For example, one part may be approximately congruent to another at a different scale, as in Figure 16.1. The basic way to turn this into a method of compression goes back to Barnsley and Sloan (1988) and is the subject of Section 16.1.3. For subsequent exploitation see Section 16.1.4 plus Barnsley (1992), Barnsley and Hurd (1994), Peitgen et al. (1992), Fisher (1995) and Lu (1997).
- Type
- Chapter
- Information
- Mathematics of Digital ImagesCreation, Compression, Restoration, Recognition, pp. 637 - 684Publisher: Cambridge University PressPrint publication year: 2006