Graphics Reference
In-Depth Information
Chapter 18
Processing
This chapter introduces the mathematics needed to understand what happens when
we perform various operations on images like scaling, rotating, blurring, sharpen-
ing, etc., and how to avoid certain unpleasant artifacts when we do these opera-
tions. It's a long chapter with lots of mathematics; we've done our best to keep it
to a minimum without telling any lies. We begin with a very concise summary of
the chapter, and gradually expand on the themes presented there.
The entire chapter can be regarded as an application of the Coordinate-
System/Basis principle: Always use a basis that is well suited to your work. In
this case, the objects we're working with are not geometric shapes, as in Chap-
ter 2, but images, or more accurately, real-valued functions on a rectangle or a line
segment.
Even with the goal of minimal mathematics with no lies, it can be difficult to
see the forest for the trees, so in this section we present an informal description
of the keys ideas of this chapter.
Much of what we say in this section is deliber-
ately wrong.
Usually there's a corresponding true statement, which unfortunately
has so many preconditions that it's difficult to see the essential ideas. You should
therefore consider this as a high-level guide to the remainder of the chapter.
We'll be looking at the light arriving at one row of an image sensor, because
almost all the interesting questions arise when we look at a single row. We'll say
that the amount of light arriving at position
x
is
S
(
x
)
. If we're ray tracing, we might
determine the value
S
(
x
)
by tracing a ray starting at location
x
. If we're using a
real-world camera, the value
S
(
x
)
is provided by nature. In either case,
S
is a real-
valued function on an interval, and we'll assume it's continuous. So we'll begin
by looking at continuous functions on an interval.
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