Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

Chapter 3
Optimization and computer
graphics
In this chapter we will show the importance of optimization methods in computer
graphics. We will pose several problems in the area, whose solutions use
optimization techniques. The problems we will pose here will be solved in the
following chapters with the appropriate optimization technique: continuous, discrete,
combinatorial or variational.
3.1 A unified view of computer graphics problems
Computer graphics problems are, in general, posed by considering two spaces O 1
and O 2 of graphical objects, and relations between them. 1 In several practical
situations, the spaces O 1 and O 2 are topological spaces and the relation is an
operator (i.e. a continuous function) from O 1 to O 2 . An even more well behaved
condition occurs when O 1 and O 2 are vector spaces and the operator is linear.
We will consider the intermediate case when we have two topological spaces
of graphical objects O 1 and O 2 , and an operator R: O 1 → O 2 from O 1 to O 2 .
From this we have the following generic problems:
• For a given graphical object x ∈ O 1 , compute T (x) (direct problem);
1 A relation is a subset of the cartesian product O 1 × O 2 .
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