2.2.4
Surface reconstruction also recovers surface values but from sparse data. It consists of two main components: restoration and interpolation. The situation of sparse data is typical with stereopsis [Marr and Poggio 1979 ; Grimson 1981 ; Mayhew and Frisby 1981]. There, a pair of images taken from two slightly different viewpoints are compared and matched to give the corresponding points. The depth values at the matched points are computed using triangulation. Since the matched points are sparsely distributed, so are the computed depth values. The sparseness brings about more uncertainties which have to be resolved by interpolation.
Figure 2.3: Surface reconstruction from sparse data.
Let 
be a set of available depth values where
is the depth value at location 
and
is the set of subscripts for the available data. Assume
the observation model
where 
is identical independent Gaussian noise and 
is the true value of the
underlying surface. Fig.2.3 illustrates 
,
and 
. Our aim is to recover the underlying surface
from the sparse data d. When continuous solutions are
considered, it may be reasonable to look at the problem from the
analytic viewpoint.