Numerical methods for volume preserving image registration

Transcription

1 Numerical methods for volume preserving image registration SIAM meeting on imaging science 2004 Eldad Haber with J. Modersitzki Emory University Volume preserving IR p.1/??

2 Outline What is image registration Mathematical setup Adding constraints - Volume Preserving Numerical Solution Discretization Numerical optimization Summary Volume preserving IR p.2/??

3 What is Image-Registration Given Template image Reference image T(x) = T(x 1, x 2, x 3 ) R(x) = R(x 1, x 2, x 3 ) find a transformation u(x) = [u(x), v(x), w(x)] such that T(x + u(x)) R(x) Volume preserving IR p.3/??

4 Example I R T T 0 R = =100.00% Start animation - HeadSpin Volume preserving IR p.4/??

5 Mathematical setup To find a good transformation we pose the problem as an optimization problem min u D(T(x + u(x)) R(x)) + αs(u) D( ) - distance function S( ) - regularization term. Volume preserving IR p.5/??

6 Mathematical setup - distance funct. For this talk use SSD (but other can be used e.g. MI, L1...) with derivatives D(u) = 1 T(x + u) R(x) 2 2 D u = T(u) (T(x + u) R(x)) Volume preserving IR p.6/??

7 Mathematical setup - regularization Use elastic regularizer (but other can be used e.g. fluid, curvature...) ( ) S(u) = 1 µ u = 1 2 (λ + 2µ) u Au with derivatives S u = µ u (λ + 2µ) u = Au Volume preserving IR p.7/??

8 Mathematical setup Solution by a tradeoff between fit and penalty by minimizing 1 2 T(x + u) R(x) αu Au Volume preserving IR p.8/??

9 Example - The blob But the problem is still highly ill-posed and there is no unique solution Start animation - The-Blob To decrease nonuniqueness add constraints Volume preserving IR p.9/??

10 Adding constraints Constraints can be added as Limit u(x) to a small subspace e.g. affine linear u(x) = Ax + b Landmarks [Fisher and Modersitzki 02] Other - mass or volume preserving [Rohlfing 02; Haker and Tannenbaum 01] Volume preserving IR p.10/??

11 Volume Preserving The volume of each deformed pixel in constant. Mathematically C(u) = det(i d + u) 1 = 0 The constraint is a highly nonlinear PDE for u. Note that C u(p) = (det(i d + u)(i d + u) T p ) Volume preserving IR p.11/??

12 Constraint optimization To solve the problem we pose as an equality constraint optimization problem min D(T(x + u) R) + αs(u) s.t C(u) = det(i d + u) 1 = 0 Volume preserving IR p.12/??

13 Numerical setup Two options Discretize the optimization problem and solve the discrete problem (Discretize-Optimize) Optimize in Hilbert space and only then discretize (Optimize-Discretize) Take the first approach Volume preserving IR p.13/??

14 Numerical setup - Discretization Computing S(u) Need to approximate differential operators such as, For stable discretization use staggered grids, used in CFD and CEM. Only short differences are needed Volume preserving IR p.14/??

15 Staggered grid Volume preserving IR p.15/??

16 Numerical setup - Discretization Computing T(x + u(x)) Use interpolation to evaluate T(x + u(x)) on a regular grid Note that T(x + u(x)) T(x) + T(x) u(x) Need to have a differentiable function. Use smoothing B-splines Volume preserving IR p.16/??

17 Discretization Discretizing the constraint - Finite Volume. Basic calculus of changing variables yields det(i d + u) dx = dx V V (u) The problem is reduced to finding the volume of the deformed cell. Volume preserving IR p.17/??

18 Discretization 2D Calculating volume 3D d c y x a b Obtaining C h (u) - the discrete analog of C(u). Polynomials of differences of u Volume preserving IR p.18/??

19 Discretization The discrete optimization problem min 1 2 T h(x h + u) R h 2 + αs h (u) s.t C h (u) = 0 Notes Very large number of (equality) constraints Discretization has to satisfy LBB condition Volume preserving IR p.19/??

20 Solving the optimization problem Use Sequential-Quadratic-Programming The Lagrangian of the problem is L(u, p) = 1 2 T h(u) R h 2 + α 2 u Au + C(u) p. Differentiating we obtain T (T h (u) R h ) + αau + C u (u) p = 0 C(u) = 0. Volume preserving IR p.20/??

21 Solving the optimization problem Use Newton like method, each iteration solve ( ) ( ) ( ) H C u s u L u (u, p) = C u 0 s p L p (u, p) H = T T + αa. KKT system Indefinite Need special solvers Volume preserving IR p.21/??

22 Solving the KKT system Use MINRES with the preconditioner ) (Ĥ 0 M = 0 Ŝ Ŝ approximates the Schur complement S := C u H 1 C u. [ Silvester, Elman & Wathen] Volume preserving IR p.22/??

23 Solving the KKT system To approximate the Schur complement, we use similar to [ Silvester, Elman & Wathen] Ŝ 1 = (Ĉu ) H Ĉu. where Ĉ u (Cu C u) 1 C u approximates the pseudo-inverse of C u. Volume preserving IR p.23/??

24 Solving the KKT system Basic operations for preconditioning (Approximately) solve the discrete vector equation Av = b (Approximately) solve the scalar equation (C u C u)w = z Volume preserving IR p.24/??

25 Approximating A A is a discretization of µ +(λ + 2µ) Tightly coupled system of PDE s Use a single V-cycle with box smoothers (h-ellipticity = 1/4) Volume preserving IR p.25/??

26 Approximating C u C u is a discretization of C u C u x a 1 x + x a 2 y + y a 3 x + y a 4 x The operator is elliptic... Use a simple V-cycle with GS smoothing Volume preserving IR p.26/??

27 Updating and stabilizing Updating the solution using a line search û u + γs u. But due to the high nonlinearity of C(u) we may have 0 C(û) Volume preserving IR p.27/??

28 Updating and stabilizing: projection Project to the constraint by approximately solving 0 = C(û + s u ) C(û) + C u s u or C u C uw = C(û) s u = C uw Volume preserving IR p.28/??

29 The blob revised Start animation - The-Blob NO Start animation - The-Blob WITH Volume preserving IR p.29/??

30 Example IV breast imaging To evaluate the size of a breast tumor one requires to register in and out wash images. Volume preserving IR p.30/??

31 deformed T difference difference with nodal grid no registration Volume preserving IR p.31/??

32 deformed T difference difference with nodal grid no registration Volume preserving IR p.32/??

33 deformed T difference difference with nodal grid no registration Volume preserving IR p.33/??

34 deformed T difference difference with nodal grid no registration Volume preserving IR p.34/??

35 Summary-Outline What is image registration Mathematical setup Numerical Solutions - I Adding constraints - Volume Preserving Solution techniques Summary Volume preserving IR p.35/??