Docteur de l'université Automatic Segmentation and Shape Analysis ...

34 Chapter 2 Literature Review
Procrustes analysis (for review, see Gower and Dijksterhuis, 2004; Gower, 2010)
is used to solve the alignment problem of corresponding points (Goodall, 1991).
The ordinary Procrustes analysis refers to the least square estimation of alignment
between two configurations. While the translation can be estimated by difference
between the two gravity centres, the rotation matrix can be solved by a singu-
lar value decomposition (SVD), and the scale can be calculated by the centroid
size of two configurations. In the Generalized Procrustes Analysis (GPA, Gower,
1975), a group of configurations are aligned collectively to minimize the sum of
squared distances between pairwise configurations. The GPA starts with an ar-
bitrary configuration in the collection as the initial mean, and iteratively aligns
all the configurations to the current estimate of the mean, which is updated at
each iteration until convergence. The configurations can be aligned via similarity
transformations, or rigid transformations depending on the application. Alter-
native criteria of L1- and L∞-norms have been proposed for an alignment more
resistant to noises (Larsen et al., 2001).
In contrast to Procrustes analysis, the ICP algorithm aligns two point sets without
the assumption of given correspondence between them. The basic version of ICP
was originally proposed by Chen and Medioni (1992), Besl and McKay (1992),
and Zhang (1994), and became widely used and adapted by the imaging and com-
puter vision community due to its simplicity. It establishes the correspondence
between two point sets by searching for the closest point of each point from an-
other set, and computes the transformation between the two sets minimizing the
distances between the corresponding points. This process is iterative. The search
for correspondence and the computation of the transformation are repeated until
convergence. The correspondence by closest point may lead to the algorithm in
the suboptimal local minimum. Instead of matching the whole point sets, subset
matching has been developed by Zhang (1994) in order to make the algorithm
more robust against outliers, disappearance, and occlusion. Multitude of variants
have been proposed for ICP (Rusinkiewicz and Levoy, 2001), along with improve-
ments in robustness and efficiency (e.g. Simon et al., 1995; Jost and Hugli, 2003;
Sharp et al., 2002; Granger and Pennec, 2002; Fitzgibbon, 2003).