03-10-2012, 01:41 PM
Efficient Registration of Nonrigid 3-D Bodies
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INTRODUCTION
WITH THE increasing availability of 3-D imaging systems
such as computed tomography (CT) and magnetic
resonance imaging (MRI), multidimensional image analysis has
become a key topic for research. In most image processing tasks
that involve combining data from multiple sources, accurate estimation
of motion between data sets is of great importance. The
objective of image registration is to align geometrically multiple
images of a similar scene, which are acquired at different times
and positions and with different imaging devices. Registration is
widely used in many applications such as medical imaging (e.g.,
aligning data sets for disease diagnosis and treatment planning),
computer vision (e.g., object tracking and structure-from-motion),
and remote sensing (e.g., change detection, mosaicing,
image fusion, and super resolution). In this paper, we mainly
consider applying the registration algorithm to medical images.
OVERVIEW OF MOTION ESTIMATION
Description of Motion: The Affine Model
Each motion vector is in the direction of the displacement,
with amplitude equal to the amount of shift. For nonrigid registration
purposes, the motion vectors should be described locally.
The full set of motion vectors, which contains the motion of the
data set at every location, is called the motion field.
In this paper, we describe the 3-D motion field by the affine
transform [25], which can model typical motions such as
translation, rotation, scaling, and shear. A major advantage of
using the affine model lies in the fact that, if the motions at two
locations are from the same affine model (e.g., shift, rotation,
scaling, or shear, typically belonging to the same object), their
affine parameters should also be the same. This property is
important when overcoming the problems of ill-conditioning
due to limited aperture (known as the aperture problem [26]),
for estimating motions of rigid bodies (see Section V-A) and
smoothing the motion estimates across a region of the data set
(see Section IV-E).
Shift Within Subbands: Interpolation of the DT- WT
Coefficients
The feasibility of interpolating DT- WT coefficients within
each subband separately relies on the transform’s shift-invariant
property, as introduced in Section II-A. The DT- WT coefficients
of data set A need to be shifted and interpolated using
estimated motion from previous iterations. After interpolation,
the complex coefficients of data set A should look more similar
to those of data set B because the amount of motion between
A and B has been reduced. Finer level coefficients may then be
used to perform motion estimation in the subsequent iterations.
One may argue that data set A could be shifted with the estimated
motion in the spatial domain and then transformed with
the DT- WT to get the wavelet coefficients for the shifted data
set. This method is feasible but tends to be slow since shifting
a 3-D data set is computationally demanding. The advantage
of shifting in the complex wavelet domain is that it provides a
fast and smooth way of aligning the data sets as the number of
DT- WT coefficients at any level above 1 is much smaller than
the sample size of the original data set, and the coefficients are
well bandlimited. Moreover, the computations of performing
the DT- WT are avoided within the iterative loop.
CONCLUSION
We have shown how to perform accurate 3-D registration
using the phase information of the DT- WT. Our algorithm
adopts an efficient iterative coarse-to-fine approach, which estimates
large motion first and then refines the motion field. It
relies on shift-invariance and good directional filtering properties,
which are key features of the DT- WT. Nonrigid motion
is well modeled by a locally affine parametric model, whose parameters
are obtained by minimizing the squared errors of the
model. The weighting factors of the motion constraints are designed
to reduce the perturbations of the motion estimates due
to inconsistent features and noise. From the final estimated motion
field, the sensed data set can be accurately registered to the
reference data set by spatial-domain interpolation.