Image Processing Reference
Flat mesh Bent mesh
Figure 4.6: In Perriollat and Bartoli [ 2007 ], a developable surface is parameterized with a set of guiding
rules, drawn in pink, and their corresponding bending angle. Courtesy of A. Bartoli. © 2007 IEEE.
in terms of guiding rules and their bending angles Perriollat and Bartoli [ 2007 ], as depicted in Fig. 4.6 .
The resulting model can then be fit to the image by minimizing the reprojection error of matching
points in the reference and input images.
The fact that sheets of paper are developable surfaces has been extensively used in the document
processing community, for example to synthetically flatten the images of curved documents and
remove shadows. The resulting approaches do not necessarily rely on correspondences. Because of
the very specific layout of printed pages, they can take advantage of shading Zhang et al. [ 2004 ]or
of textural information Liang et al. [ 2005 ] to infer 3D shape.
4.2.2 SMOOTH SURFACES
While the methods that assume the surfaces to be developable may be effective in the specific
context they have been designed for, they do not generalize naturally to broader classes of surface
deformations such as those of cloth. One way to achieve such generalization is to replace the zero
Gaussian curvature constraint by weaker ones that only force the curvature to remain small and the
deformations to be smooth.
Such regularization constraints can be introduced by enforcing a uniform level of smoothness
across the whole surface, which is simple to do but tends to preclude the modeling of sharp folds
and creases. A powerful alternative is to only force the surface to be locally or piecewise smooth,
which increases the algorithms' descriptive power at the cost of introducing slightly more complex
188.8.131.52 Global Smoothness
As discussed in Chapter 2 , a well-known approach to enforcing smoothness is to regularize shape
deformations with a linear subspace model. While, in essence, assuming that the shape is generated
with a small number of deformation modes does not necessarily enforce smoothness, the usual ways of
obtaining these modes, such as Modal Analysis or PCA of a reprentative set of deformed versions of