Graphics Reference
In-Depth Information
CHAPTER 21
Digital Image Processing Topics
Prerequisites:
Chapters 4 (integration) and Chapter 5 (metric spaces) in [AgoM05]
21.1
Introduction
This brief chapter looks at the topic of signal processing and its application to digital
image processing. Complex numbers play an important role in this because of various
transforms that get involved. On the other hand, except for the topic of antialiasing,
digital image processing as a whole falls outside the scope of the geometric modeling
topics central to this topic. Our discussion of the Fourier transform and signal pro-
cessing will be very limited. The only reason that we take up these topics at all is so
that the reader can make some sense out of the various approaches to dealing with
the antialiasing problem. The reader interested in learning more about the large field
of digital image processing should consult [Glas95], [GonW87], or [RosK76], where
additional references can be found. For the mathematics behind Fourier series and
the Fourier transform see [Brac86], [Seel66], or [Apos58].
To motivate some of the mathematics we start, in Section 21.2, with the Laplace
equation that historically has been the driving force behind a great many develop-
ments. Section 21.3 shows its connection to Fourier series. After defining the impor-
tant L
p
spaces in Section 21.4, we define the Fourier series for periodic functions in
Section 21.5 and state a few of the important results about them. Sections 21.6 and
21.7 define the Fourier transform and convolution, respectively. For us, the main
application of all these mathematics comes in Section 21.8. The reader who is not
interested in the mathematics of this chapter should at least look at that section, espe-
cially Figures 21.9 and 21.10. These show the main points about the sampling problem
behind aliasing.
Before getting started, however, we need to address an integration issue. Through-
out this topic all our integrals were implicitly assumed to be Riemann integrals, which
is the integral that one learns about in calculus classes. There is another integral, the
Lebesgue integral, which is more general and better for advanced mathematical topics
for technical reasons. In particular it would be better for the Fourier transform and
related topics because the Riemann integral would sometimes give incomplete and
