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8.5 PATIENT-SPECIFIC CARDIAC PODI COMPUTATION 161

found in each geometry and are interpolated for. Accordingly, there are missing column entries in the dataset matrix,
U i , which get zero values for lack of a better choice. Although increasing the grid mesh density decreases the number of
missing entries relative to the overall number, the cube grid standardization method clearly lacks the flexibility to deal
with patient-specific hearts.

8.5.1.2 Heart Template Standardization
A better alternative is to make use of a heart template for the standardization procedure and morph every heart
dataset onto the same heart-shaped template. After the morphing process, the dataset nodes are clustered around
the template nodes, ensuring that all of them belong to one of the elements of every dataset heart. Accordingly,
the dataset matrix, U i , has no missing entries. As a suitable template heart geometry, the statistical average of all
selected heart geometries is suggested to minimize the movement of dataset nodes during the morphing process. This
optimizes the accuracy of the morphing process and the subsequent interpolation at the template nodes.
Within the field of computer vision, the morphing process is commonly known as registration. It is an important
tool for mapping two clouds of points onto each other. Three different techniques are commonly used. The first one is
rigid registration, which consists of pure translation; the second is an affine registration that in addition to translation,
allows for scaling and rotation. Finally, the third type of registration is called nonrigid registration. Rigid and affine
registration methods allow for the preservation of the overall geometric details and affect the whole geometry. In the
nonrigid scenario, the geometry can be completely morphed into another one, and the registration can be localized
across the geometry. Various examples of heart registration can be found in the literature. In Sermesant et al. [65], a
mass center alignment followed by a principal axes-based registration together with an affine registration was
employed to deform an idealized BV heart model according to CMR scans. Toussaint et al. [87] made use of a
log-diffeomorphic registration to project CMR images of a ventricle onto a perfectly truncated ellipsoid. Lamata
et al. [88] introduced a mesh warping technique that required the conversion of the idealized geometry to a binary
file, similar to CMR image data before the registration took place. A common aspect of these registration methods is
their CMR or binary image requirements. In this research, however, the source geometry and the template geometry
are both given in terms of meshes consisting of interconnected nodes. For morphing of the source geometry onto the
template one, the CPD method [47] is utilized, which is a nonrigid-based registration method. CPD is a probabilistic
method based on the maximum likelihood estimation technique and involves a motion coherence constraint over a
velocity field in order to allow for the smooth movement of points from one spatial location to another.
Consider two sets of points, the template points set, Y, and data points set, X, both stored as matrices. The number of rows
ofthese matricescorresponds tothe numberof points while thenumberof columnsrelates tothe dimension,D,ofthe points:
T
Y ¼ðy ,…,y Þ ,
1 M
T
X ¼ðx 1 ,…,x N Þ ,
where M is the number of template points and N is the number of data points in their respective set. Then assume the
validity of the Gaussian mixture model (GMM) and associate each point in Y with a Gaussian probability density func-
tion with the template point as its centroid:
M +1
X
PðmÞpðxjmÞ, (8.35)
pðxÞ¼
m¼1
where
1 kx y m k 2
e 2σ 2 (8.36)
2 D=2
pðxjmÞ¼
ð2πσ Þ
for m ¼ 1, M and
1
, (8.37)
pðxjM +1Þ¼
N
the latter accounting for noise and outliers. Imposing independent and identically distributed data assumptions, σ 2
1 CPD
denotes equal isotropic covariance, PðmÞ¼ M equals membership probabilities for every GMM centroid, and w
is a weight constant.
M
1 CPD X 1
pðxÞ¼ w CPD + ð1 w Þ pðxjmÞ: (8.38)
N M
m¼1

I. BIOMECHANICS