Image Processing Reference

In-Depth Information

1.4

0.7

QO

QP

SOCP

QO

QP

SOCP

1.2

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1

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0.1

0

0

0

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frame #

frame #

Figure 4.5:
Comparison of the accuracies obtained with the SOCP formulation based on
Salzmann
et al.

[
2007a
] with the QP and unconstrained (QO) formulations of
Zhu
et al.
[
2008
]. Reconstructions were

obtained with image noise variance 1 (left) and 2 (right). Note that the QP and QO approaches yield

better results than the SOCP one. Courtesy of J. Zhu.

4.2

IMPOSINGGEOMETRICCONSTRAINTS

The methods discussed in the previous section are very generic in that they make very few assumptions

on the smoothness or physical properties of the surface. However, they are all limited by the fact

that they involve frame-to-frame tracking and are therefore subject to drift and irrecoverable failure

if there are too few valid correspondences in any given frame. Furthermore, they require a full video

sequence, as well as an initial shape estimate for the first frame, either of which may not be available.

A useful alternative is therefore to replace temporal consistency constraints by geometric

ones that allow reconstruction using a single input image or a very short sequence of consecutive

ones. The difficulty then is to design the constraints so as to make as few unwarranted assumptions

on the allowable surface deformations as possible. In the remainder of this section, we classify

approaches according to how stringent the constraints are. We start with developable surfaces, whose

deformations are very strongly constrained. We then move on to surfaces that deform smoothly,

including those that remain globally smooth and those that need only be locally smooth and can

therefore develop creases. We conclude by discussing inextensible surfaces.

4.2.1 DEVELOPABLE SURFACES

Developable surfaces are surfaces with zero Gaussian curvature, meaning that, for all points and

all possible deformations, one of the principal curvatures must be zero. Such 3D surfaces can be

flattened onto a plane without distortion and are ruled surfaces. For example, initially flat pieces of

paper are developable and are often used to demonstrate techniques that rely on this property.

As shown in
Gumerov
et al.
[
2004
], given only surface boundaries in both the reference and

input image acquired by a calibrated camera, it is possible to recover the 3D structure by solving

Ordinary Differential Equations. Another approach is to explicitly parameterize the reference surface

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