Image Processing Reference

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where
X
is the unknowns organized in a 4
n
× 3 transformation matrix;
E
d
(
X
) is distance meas-

ure between all target points and transformed source points, in contrast to classical ICP.
X

is not a single rotation or translation but a collection of affine transformation for each point;

E
s
(
X
) is stiffness regularization, topology matrix is created based on points neighborhood to

preserve the shape of object during iterations; we use square matrix topology (every point has

four neighbors).
α
is stiffness vector, which influences the flexibility of cloud shape;
βE
1
(
X
) is a

factor used for guiding registration, based on known position of landmarks in source and tar-

get sets of points.
β
is a weighting factor, used to fade out the importance of landmarks toward

the end of the registration process.

The implemented nonrigid ICP algorithm consists of two iterative loops. In the outer loop,

the stiffness factor
α
is gradually decreased with uniform steps, starting from higher values,

which enables recovery of an initial rigid global alignment, to lower values, allowing for more

localized deformations. For a given value of
α
, the problem is solved iteratively in the inner

loop. The condition of changing stiffness vector is threshold norm of transformation diference

from adjoining iterations. In our implementation
β
is constant and equals 1. The above equa-

tion can be transformed into the system of linear equations, which is solved by computing

pseudo-inverse matrix (see Amberg et al. [
6
] for details). This is the iterative algorithm, where

each iteration consists of two main steps, namely finding correspondences between source and

target points and computing affine transformations for each source point. If the second step is

modiied by the solution proposed by Amberg, that causes better results, corresponding prob-

lem remains critical for final results.

2 Materials and methods

Improvement of finding correspondences was implemented. Classically finding correspond-

ences are done by searching Euclidean distance between closest points in source and target for

in normal vector of source point direction. The general idea of presented approach was to take

into account knowledge about markers' positions not only in computing transformation phase

but also in finding correspondence phase in every algorithm's iteration. Decision to test a few

approaches of finding correspondences was done:

• searching along normal vectors in source points, following the initial rigid registration

• along static marker vector displacement, where marker vectors are calculated only once at

the preliminary stage. Marker vector is defined by positions of specific marker in source

and target point cloud,

• along dynamic marker vector displacement, where marker vectors are calculated in each

iteration based on constant positions of nearest marker points in matrix topology. Trans-

formed source point in each iteration is treated as a new origin of dynamic marker vector.

Classical Euclidean distance is treated as baseline to compare the obtained results. Generally

it is challenging to verify registration approach. We used global and local criteria for evalu-

ation:

• global measurement: average distances between nearest source and target points, average

distances between correspondences,

• local measurement—quality of correspondences: average correspondence assignment error

of points nearest to the markers and the number of correspondences for every target

points.

Data set consists of abdomen surfaces acquired by six volunteers on free breathing using

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