Nonrigid Registration

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.

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.

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

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

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.

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

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

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

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

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

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

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 …

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

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

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)

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.

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.

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.

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.

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)

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.)

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))

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.

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

End of nonrigid registration and Beginning of Course Review

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

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

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

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

All Done!