Image Registration Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 199

involved in the registration is large. The purpose of the registration is to realign the
slices in order to reconstruct the correct shape of the organ under examination. A
typical example is in cardiovascular imaging, in which the shape of the heart or of
valves should be reconstructed from parallel sections, correcting artifacts due to
patient breath. In the present problem, the registration method must be robust to
missing data or outliers. Registering the slices sequentially (the second with respect
to the first, the third with respect to the second, etc.) sometimes leads to misregis-
tration. In fact, if an error occurs in the registration of a slice with respect to the
preceding slice, this error will propagate through the entire volume so that a global
offset of the volume may be observed due to error accumulation. If all the slices are
registered in respect to the one taken as reference, differences between slices can be
too large to allow correct registration.
Krinidis et al. proposed a solution introducing the use of a global energy
function having as variables the rigid transformation parameters of each slice
[30]. The global energy function is minimized with the ICP algorithm, which is
able to register multiple views of a 3-D structure. The implemented global energy
function is associated with a pixel similarity metric based on the Euclidean
distance transform.
Consider a set of N slices ( I 1 , , I N ). A pixel in a slice is represented by

p = ( x , y ). The alignment of all images in the sequence can be achieved by
maximizing an energy function E (

…

⋅ ), which expresses the similarity between the

2-D images:

N N

∑ 1

E () Θ= f T I (( , ( ) T I (7.12)

i i j j

i = 1 j =

where f ( )is a similarity metric, denotes a rigid transformation matrix. The

chosen metric energy function accumulates the similarity between each trans-
formed image and all of the other already-transformed images. Assuming that
the similarity function is symmetric leads to the following global maximization
problem, where W ij 's are appropriate weights for the similarity between each
slice pair:

⋅

N N

∑ ∑

ˆ

Θ= arg max

T

WfT I T I j ( ( ), ( )) (7.13)

ij i i j j

i = 1 ji =

Without any additional constraints, the optimization problem has an infinite
number of solutions. If the transformation applied to an arbitrary chosen image
is constrained to be the identity transformation, we have 3(N 1) parameters to

estimate in the case of 2-D rigid transformation. If the transformation is nonrigid,
the number of parameters to estimate becomes virtually infinite. In this case,

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