Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

1.5. DESCRIPTION, REPRESENTATION AND RECONSTRUCTION 15
reconstruction problem is an inverse problem. 2 A representation that allows more
than one possibility of reconstruction is called an ambiguous representation. 3 . In
this case, the reconstruction problem is ill-posed (it has more than one solution).
Although the representation problem constitutes a direct problem, the computation
of the representation operator could lead to an inverse problem. An example
of such a situation can be seen in the point sampling representation of an implicit
shape, F −1 (0), where F : R 3 → R. In this case, the representation consists in
obtaining solutions of the equation F (x, y, z) = 0, which constitutes an inverse
problem.
Figure 1.9: Representation and reconstruction.
The representation takes a continuous graphical object O and associates to it
a discrete representation O d . The reconstruction process does the opposite, that
is, from the discrete representation O d we obtain a reconstructed graphical object
O r . When O d = O r we say that we have exact reconstruction, and the representation
is also called exact. We should remark however that in general we do
not have exact reconstruction, that is, in general we obtain upon reconstruction
2 Direct and inverse problems are discussed in Chapter 1.
3 It would be more correct to adopt the term “ambiguous reconstruction”, but the term “ambiguous
representation” is already widely adopted in computer graphics.