Nonrigid Registration

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. radial basis Form of the Thin-plate Spline • x’ and y’ have this form (two dimensional example): 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. 195-206 (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. of the same color. 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

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 • Sum absolute differences • If B = aA + c: Correlation coefficient • For all modalities • Entropy • Mutual Information • Normalized Mutual Information

31. All Done!