Transcript Brain deformation

Image Registration with
Hierarchical B-Splines
Z. Xie and G. Farin
Support
Arizona Alzheimer Disease Research Center
Motivation

Image Fusion
 Image Comparison
 Image Segmentation
 Pattern Recognition
Classification

Landmark based methods
– Point based method
– Curve based method
– Surface based method

Intensity based methods
Free Form Deformation (FFD)
FFD with Hierarchical B-Splines
r
s
t
f ( x, y, z )   bi , j ,k N i3 ( x) N 3j ( y) N k3 ( z )
i 0 j 0 k 0
1. Putting the object into the B-Spline hyperpatchs.
2. Moving the B-Spline control points to deform the object.
3. Refining the control points related to complex regions.
4. Adjusting the refined control points for detail deformation.
Point based registration
Input : n pairs of points (pi ,qi ), pi ,qi  R 2 , i  1,..., n.
Output : An C 0 function f: R 2  R 2 with f(pi )  qi ,i  1,...,n.
This problem naturally breaks down into two scattered data
approximation problems. The least squares solution of this
problem can be found by solving the linear systems.
s
t
qr   bi , j N (xr )N (yr ); r  1,..., n
i 0 j 0
3
i
3
j
How does it work?
•Local refinement by knot insertion.
•Recomputing related control points.
Why hierarchical B-Splines?

Efficiency
 Global to local influence
Example of point based registration
Source
Target
Deformed Source
Surface based registration
Input : The point set of the source surface S S and
the point set of the target surface St .
Output : A transfo rmation T such that the distance
between T(S s ) and St is minimized.
Together with the Iterative Closest Point (ICP)
approach, this problem can be converted into a
scattered data approximation problem.
Iterative Closest Point
Distance Transform
Hierarchical Deformation with
Hierarchical B-Splines

Initialize: Rigid Transformation
 Linear matching: Iterative Affine Deformation
 Nonlinear matching: Hierarchical Cubic B-Splines
– Increase level of detail iteratively
Advantage

Validity. Right matching between individual points
by matching big shape feature first, then refine the
detail gradually.
 Efficiency. Only pay attention to complex regions.
 Precision. Enough of degrees of freedom for
matching detail.
Example of 2-D registration
Example of 3D matching
Movie of 3D Deformation
Intensity-based registration
Input : The source image I s(p) and the target
image I t(p). Both I s and I t are mapping
from location to intensity.
Output : A spatial transfor mation f : R 3  R 3
such that I t(p)  I s(f(p)).
Together with optic flow, this problem can be
converted into scattered data approximation problem.
Optic flow
Optic flow is a visual displacement flow field
associated with the variation in an image
sequence. It can be used as an estimator of the
displacement of one pixel on the source image
to its matching pixel on target image.
Hierarchical Deformation vs.
multi-resolution data representation
Example of intensity based registration
Source
Target
Deformed source
Movie of intensity based registration
Future Work

Multi-resolution surface representation
 More robust displacement estimator for
intensity based registration.
 Multi-modal intensity based registration