1. Nonrigid Registration

2. Transformations are more complex
Rigid has only 6 DOF—three shifts and three angles
Important non-rigid transformations
Similarity: 7 DOF
Affine: 12 DOF
Curved: Typically DOF = 100 to 1000.

3. Popular Curved Transformations
Thin-plate splines
Belong to the set of radial-basis functions
Cubic B-splines
Belong to the set of B-splines
Both are better than the polynomial transformations
Polynomial requires too many terms to produce a “well behaved” transformation.

4. Form of the Polynomial Transformation
Two dimensional example:
which can be written this way:

5. Point Registration with Polynomials…
Localize a set of old and new points.

6. Point Registration with Polynomials…
For N pairs of points (x,y)  (x’,y’)
Find the coefficients
and
that satisfy
for all N pairs.
3. Use them to compute (x’,y’) for every point (x,y) in the image.

7. Method for Finding Coefficients
Requires solution of Ax = b
A depends on the initial points
b depends on final points
x contains the coefficients
See handout
Polynomial Transformation.doc

8. Polynomials behave badly!
Matlab demonstration:
get_test_images
test_warp(im,M,type)

9. Thin-plate splines behave well
Suggested for image registration by Ardi Goshtasby in 1988 [IEEE Trans. Geosci. and Remote Sensing, vol 26, no. 1, 1988].
Based on an analogy to the approximate shape of thin metal plates deflected by normal forces at discrete points
Uses logs

10. Form of the Thin-plate Spline
Consider a point x,y, other than the N localized ones:

11. Form of the Thin-plate Spline
Find its distance to each of the N localized points:

12. Form of the Thin-plate Spline
x’ and y’ have this form (two dimensional example):
radial basis
where

13. Point Registration with TPS
For N pairs of points (x,y)  (x’,y’)
Find the 6 + 2N coefficients
that satisfy
for all N pairs (2N equations) and also satisfy …

14. Point Registration with TPS
… these 6 equations:
3. Use them to compute (x’,y’) for every point (x,y) in the image.

15. Why does TPS behave well?
As x moves away from the N fiducial points,
the terms in the sum
begin to cancel out. The sum  0. The same thing happens for y, so

16. Method for Finding Coefficients
Requires solution of Ax = b
A depends on the initial points
b depends on final points
x contains the coefficients
See handout
Thin-Plate Spline Transformation.doc
Example of use in medical image registration: Meyer, Med. Im. Analy, vol 1, no. 3, pp (1996/7)
(same as for polynomials)

17. Cubic B-Splines
Also determines the motion of all points on the basis of a few “control” points.
Not suitable for point registration because the control points must lie on a regular grid.

18. Cubic B-Splines Behave Well
Good behavior (i.e., small effect from motion of remote control points) is due to the fact that it uses only “local support”:
All voxels inside a small square are affected only by motion in the large square
of the same color.

19. Cubic B-Splines Behave Well
Each large square is divided into 25 regions (125 in 3D), and within each of them a polynomial transformation is used.

20. Cubic B-Splines Behave Well
Each large square is divided into 25 regions (125 in 3D), and within each of them a polynomial transformation is used.
The polynomial coefficients are chosen so that all derivatives up to 2nd order are continuous.

21. Method for Finding Coefficients
Requires solution of Ax = b
A depends on the initial points
b depends on final points
x contains the coefficients
See handout
B-Spline Transformation.doc
Example of use in medical image registration: Rueckert, IEEE TRANS. MED. IMAG., VOL. 18, NO. 8, AUGUST 1999
(same as for polynomials and TPS)

22. Continuous Derivatives
Continuous derivatives reduce “kinks”.
Both polynomials and thin-place-splines are continuous in all derivatives.
Cubic B-splines are continuous in all derivatives except 3rd order.
(All derivatives higher than 3rd order are zero.)

23. Nonrigid Intensity Registration
Let B’(x,y) = B(x’,y’) be a transformed version of image B(x,y).
Let D(A,B’) be a measure of the dissimilarity of A and B’
e.g., SAD, or –I(A,B’)
Search for the transformation, (x’,y’) = T(x,y), that makes D(A,B’) small.
Loop: Adjust control points, find coefficients, calculate D(A,B(T(x,y))

24. Regularization
It may be desired to limit the variation of the transformation T in some way.
keep some derivatives small
keep the Jacobian close to 1
Define a variation function, V(T) that is large when the variation in T large
Search for T that makes C = D(A,B’) + V(T) small.

25. Search Techniques
Grid, steepest descent, Powell’s method, simplex method [Numerical Recipes]
Stochastic methods (use randomness)
Simulated annealing
Genetic algorithms
Course-to-fine search
change discretization of coefficients of T
change discretization of images

26. End of nonrigid registration and Beginning of Course Review

27. Geometrical Transformations
Rigid transformations
All distances remain constant
x’ = Rx + t
Nonrigid transformations
Distances change but lines remain straight
Curved transformations
Polynomial
Thin-plate spline
B-spline

28. Registration Dichotomy
Prospective
Something is done to objects before imaging.
Fiducials may be added to objects and point registration done.
Retrospective
Nothing is done to objects before imaging.
Anatomical points (rarely reliable)
Surfaces (rarely reliable)
Intensity

29. Rigid Point Registration
Minimize Square of Fiducial Registration Error:
Closed-form solution with SVD
Error triad: FLE, FRE, TRE

30. Intensity Registration
For intramodality
Sum absolute differences
If B = aA + c: Correlation coefficient
For all modalities
Entropy
Mutual Information
Normalized Mutual Information

31. All Done!