Page 126 - Artificial Intelligence for Computational Modeling of the Heart

98 Chapter 3 Learning cardiac anatomy

3.2 Parsing of cardiac and vascular
structures

3.2.1 From shallow to deep marginal space learning
3.2.1.1 Problem formulation
Let us reformulate object localization as a classification prob-
lem. Boxes of image intensities, i.e., constrained axis-aligned sub-
regions of the image, parameterized as h ∈ U are sampled from
the image. The variable U denotes the parameter space. We dis-
tinguish between positive samples (centered around the ground-
truth object position), and negative samples which are extracted
from the rest of the image. A classifier is trained to distinguish be-
tween these two categories. At test time, the classifier is applied
exhaustively to scan the complete parameter space and yield the
most probable positions of an object of interest. In practice, how-
ever, this strategy is subject to severe computational limitations,
as the scanning effort grows exponentially with the dimensional-
ity of the parameter space. Given an arbitrary anatomical object
of interest, the goal is to estimate its restricted affine transforma-
tion which is defined by nine parameters. They define the location
T = (t x ,t y ,t z ), the orientation R = (φ x ,φ y ,φ z ), and the anisotropic
scale S = (s x ,s y ,s z ) of the considered object. One can observe that
a simple coarse discretization of d = 10 possible values per param-
9
eter results in highly prohibitive number of d = 1,000,000,000
hypotheses that need to be evaluated by the classifier. Marginal
Space Learning and its deep learning based extension aim at dras-
tically reducing this computational complexity by restricting the
portion of the parameter space in which candidates are searched.
Given a bounding box computed with marginal space deep
learning (MSDL), one can use this information to estimate the
nonrigid deformation of the object. An initial estimation of the
shape is obtained by rigidly transforming the mean object shape
according to the estimated pose. This initial estimate is then itera-
tively refined using an active shape model based on deep learned
image features. In more detail, a boundary classifier is trained to
decide whether there is a boundary point at a given position and
under a given orientation. To solve this problem, Zheng et al. [31]
proposed steerable features combined with the PBT [203].
Once the pose of the object has been estimated, its shape needs
to be determined. Statistical shape modeling is used to obtain a
parametric description of the nonrigid deformation of the object,
where c 1 ,c 2 ,...,c K ∈ R are the coefficients of the major deforma-
tion modes, with K selected so that the object shape can be closely
approximated. Using a boundary classifier that is trained to dis-