3D Reconstruction of the Human Rib Cage from 2D Projection ... - ZIB

4.3. Statistical Shape Models
For these reasons, quaternions are used in the scope of this work. Accordingly,
the rotation component of T is denoted as q = (qw, qx, qy, qz). A disadvantage of
quaternions is that they are hard to visualize, which is why rotations are converted to
Euler angles (θx, θy, θz) to interpret results of the pose reconstructions in Chapter 6.
4.3 Statistical Shape Models
Statistical shape models (SSMs) were first introduced by Cootes et al. [CTCG95].
The idea behind SSMs is to derive deformable models that exhibit two counteractive
properties. SSMs are specific, in the way that deformations of the model are characteristic
of the specific class of shapes the model represents. At the same time, they
should be general enough to model considerable shape variations of, for instance,
the differences of an anatomical structure between different individuals.
An SSM is a linear model that captures the typical shape of a class of shapes
along with its variability. Such a model is generated or ’learned’ from a statistical
analysis of a set of shape examples (training shapes) that represent the same class of
shapes. SSMs can be used to create new and plausible instances of shapes, similar
to those of the training shapes, since only deformations according to the variability
within the set of training shapes are permitted.
4.3.1 Statistical Shape Modeling
Before an SSM can be derived, the training shapes need to be converted into a
comparable form, which allows for a statistical analysis.
Representing Training Shapes Shapes of the training set are represented by sets
of annotated points that sample the boundary of the respective shape. Each point
x possesses one corresponding point x ′ ↔ x on each of the training shapes. Point
correspondences need to be consistent throughout all the shapes in the training set.
The point annotations, however, can differ in their nature. They may describe landmarks
of distinct biological or anatomical relevancy, mark points with characteristic
features (e.g. specific curvature), or could simply be placed to ensure a dense and
regular sampling of the shapes. SSMs were also introduced under the name point
distribution models (PDMs), since the spatial distribution of the annotated points
is analyzed in order to generate such a model.
Let n be the number of training shapes and m the number of corresponding points,
then the i-th training shape is a vector
vi = (xi1, yi1, zi1, ..., xim, yim, zim) T , i = 1, ..., n (4.5)
of points xij = (xij, yij, zij) T , j = 1, ..., m, where xij ∈ R 3 and vi ∈ R 3m . Now, each
training shape vi can be seen as a single point in 3m-dimensional space.
Correspondence Problem To establish correspondences between two shapes Si
and Sl, a bijective function c : Si ↔ Sl needs to be determined, that defines a set
of corresponding points xij ↔ xlj, such that each point xij on Si is mapped to
27