Image Processing Reference

In-Depth Information

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

models.

4.2.2.1 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

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