Computing the Geodesic Interpolating Spline

Transcription

1 Computing the Geodesic Interpolating Spline Anna Mills 1, Tony Shardlow 1, and Stephen Marsland 2 1 The University of Manchester, UK 2 Massey University, NZ Abstract. We examine non-rigid image registration by knotpoint matching. We consider registering two images, each with a set of knotpoints marked, where one of the images is to be registered to the other by a nonlinear warp so that the knotpoints on the template image are exactly aligned with the corresponding knotpoints on the reference image. We explore two approaches for computing the registration by the Geodesic Interpolating Spline. First, we describe a method which exploits the structure of the problem in order to permit efficient optimization and second, we outline an approach using the framework of classical mechanics. 1 Introduction and Formulation of problem We develop methods for computation of the Geodesic Interpolating Spline (GIS) as described by Marsland and Twining in [1]. A comprehensive survey of registration methods is given in [2]. The GIS was developed from the thin-plate spline [3] and clamped-plate spline [4] to provide diffeomorphic mapping between images so that no folds or tears are introduced to the images and no information is lost from the image being mapped. We formulate the GIS as a minimization problem as follows. Minimize l(q i (t), v(t, x)) = 1 1 Lv(t, x) 2 IR ddx dt, (1) 2 over deformation fields, v(t, x) IR d and paths, q i (t) B for i = 1,..., n c, where B IR d is the unit-ball domain of the image and L = 2 approximating the Willmore energy. We have constraints for t 1, i = 1,..., n c, B dq i dt = v(t, q i(t)), q i () = P i, q i (1) = Q i. (2) Using techniques in [5], we can represent the minimizing vector field as v(t, x) = α i (t)g(q i (t), x), (3) i=1 where α i (t), q i (t) are respectively multipliers and knotpoint positions at time t for a set of n c knotpoints and where G(x, y) is the Green s function derived by

2 Boggio [6]. We experiment numerically with the 2-dimensional Green s function G(x, y) = x y 2 ln x y 2 for the biharmonic operator with zero Dirichlet and Neumann boundary conditions on B. In this way, we can derive min i,j=1 α i α j G(q i, q j )dt, such that dq nc i dt = j=1 α j G(q i, q j ), (4) q i () = P i, q i (1) = Q i, i = 1,..., n c. (5) We explore two methods for the minimization. In Sec. 2, we examine the structure of the discretized version of (4), and use an optimization method exploiting this structure. In Sec. 3, we reformulate the problem in a Hamiltonian framework to compute the GIS. In Sec. 4, we test the methods on brain images with knotpoints marked by clinicians. In Sec. 5, we summarize our results. 2 Numerical Optimization Exploiting Partial Separability To find the minimizer of (4), we discretize in time to achieve a finite dimensional system. This yields a constrained optimization problem with a clear structure. The numerical optimization routine, Lancelot B, from the Galahad suite [7], uses group partial separable structure to express the dependence between different variables and make the structure of the resulting Hessian matrices clear, improving the performance on large scale problems. The discretization of the GIS is partially separable, because of the dependence on time. We move to a discretized version of the problem using time step t = 1/N. We use α n i α i (n t), n =,..., N 1 and q n i q i (n t), n =,..., N to give the problem in the following form: Minimize over α n i, qn i N(q n+1 i l{α n i, qn i } = 1 2 i = 1,..., n c such that N 1 i,j=1 n= α n i α n j G(qn i, qn j ) (6) nc q n i ) = α n j G(qn i, qn j ), n =,..., N 1, i = 1,..., n c (7) j=1 with conditions q i = P i, and q N i = Q i, i = 1,..., n c. As detailed in [7], Lancelot B uses an iterative method. Outer iterations minimize augmented Lagrangian merit functions, and inner iterations minimize a quadratic model of the merit function. Lancelot B minimizes a function where the objective function and constraints can be described in terms of a set of group functions and element functions, where each element function involves only a small subset of the minimization variables. Specifically, Lancelot B solves problems with objective functions of the form f(x) = w g i g i wij e e j(x e j ), x = (x 1, x 2,..., x n ). (8) i Γ j Ei

3 In the above, we have Γ, a set of indices of group functions g i and we have E i, a set of nonlinear element functions e j and group and element weight parameters, respectively w g i and wij e. The constraints must be of the form c i (x) = w g i g i wije e j (x e j) + a N i x =, (9) j Ei for i in the set of indices of constraints Γ c. Examining our objective function, we see that there is a natural division into N groups, each group being given by, for the n th group w g i g i wnje e j (x e j) = 1 (α n i G(q n i, q n j ))α n j. (1) 2 j E n i,j=1 In this notation, x e j is the vector containing the optimization variables (qn i, αn i ), for i = 1,..., n c. Similarly, the 2n c N velocity constraints in (7) can be characterized by 2n c N groups. We require that the start and end points of each control point path coincide with the landmarks on, respectively, the floating and reference images. This gives 4n c constraints of the form = (q 1 i P i )w, = (q N+1 i Q i )w, i = 1,..., n c. (11) Hence we have 4n c groups characterizing the landmark constraints, each group being weighted by some w 1. In total, we have N + 2n c N + 4n c groups characterizing the problem. We see the block diagonal sparsity structure of the Hessian for the augmented Lagrangian function for our problem in Fig. 2 where the Hessian for a problem involving 5 time steps and 8 knotpoints is shown, calculated using a numerical scheme on the augmented Lagrangian function as used in Lancelot B, L A (x, λ; µ) = f(x) λ i c i (x) + 1 c 2 i (x), (12) 2µ i Γ c i Γ c where µ is the penalty parameter and λ i for i Γ c are Lagrangian multipliers. There are 5 time blocks on the main diagonal, each with coupling to the adjacent time blocks. There is a banded off-diagonal structure, the two bands of A being due to the constraints being divided into x-dimension and y-dimension constraints. Lancelot B uses Newton methods, which require Hessian approximations. The routine exploits group partial separability of the problem to make calculation and storage of the Hessian approximations more efficient. 3 Classical Mechanics Approach We present a novel formulation of the GIS for image registration as a problem in Hamiltonian dynamics. The GIS problem is given by the minimization

4 Fig. 1. The sparsity structure of the Hessian 15 2 problem (4). We can treat this as a Lagrangian by setting L(q, q) = 1 2 i,j=1 α i α j G(q i, q j ), (13) where q = (q 1,..., q nc ), and q = ( dq1 dt,..., dqnc dt ) represent position and velocity, respectively. Following Arnold [8], we see that the Hamiltonian of the system, H(p, q) = p q L(q, q), is the Legendre transform of the Lagrangian function as a function of the velocity, q, where p is the generalized momentum, L q, and so q = H α, H p = q, α = H q, H q = L q. (14) Construct a matrix A IR nc nc so that A ij = G(q i, q j ), i, j = 1,..., n c, and a matrix G IR dnc dnc as G = A I, (where I is the d-dimensional identity matrix). We can use (4) and define a vector α = [α 1, α 2,..., α n c ] to derive dq dt = Gα. and hence we see that the generalized momentum is given by L q = G 1 Gα = α. Substituting this expression into the standard Euler- Lagrange equations [8] gives L/ q = dα/dt. Hence, with (14), we have the coupled system of Hamiltonian equations [ ] [ ] q() P = = Y, (15) α() A where P is the vector of initial knotpoint positions in (4) and A is the initial vector of generalized momentum. Substituting (13) and the derivatives from (15) into the expression for the Hamiltonian of the system gives H(α, q) = α q L(q, q) = 1 2 i,j=1 α i α jg(q i, q j ), (16) so that H is now a function of α and q. We have shown that the solutions, q i (t) and α i (t) of (4) are solutions of (15). We solve the nonlinear system of equations

5 Φ(A; P ) = Q for A as a shooting problem, where Φ(A; P ) := q(1), the position component of the solution of the Hamiltonian system d dt q i = j=1 α j G(q i, q j ), with initial conditions given in (15). d dt α i = α i α j G(q i, q j ), i = 1,..., n c, q j j=1 (17) Numerical Implementation To solve (17), we discretize in time. We choose to discretize using the Forward Euler method. Experiments with symplectic methods have shown no advantage for this problem, principally because it is a boundary value problem where long time simulations are not of interest, and no suitable explicit symplectic integrators are available. Using the notation q n i q i (n t), α i(n t), n =,..., N, t = 1/N, we have α n i [ ] q n+1 = α n+1 [ q n α n ] + t H(q n, α n ) α H(qn, α n ) q [ ] P, = A [ q α ] = Y. (18) We wish to examine the variation with respect to the initial momentum, A in order to provide Jacobians for the nonlinear solver. The initial positions, P remain fixed. Using the Forward Euler scheme for some function f, we have X n+1 = X n + tf(x n ) with initial condition X = Y. Differentiating with respect to A gives dx n+1 da = dxn da + ) tdf(xn dx n dx n da, dx da = [, I], (19) where I is the 2n c 2n c identity matrix. Let J n be the Jacobian and solve numerically a coupled system of equations J n+1 = J n + t df(xn ) dx n J n, X n+1 = X n + tf(x n ) (2) with initial conditions J = [, I], X = Y. f(x) = H q H α, X = [ q α ], df(x n ) dx n = In our problem, we have 2 H 2 H q α α 2 2 H q 2 2 H α q. (21) The entries of the Jacobian in (21) can be calculated explicitly. The analytic calculation of the Jacobian permits efficient solution of the nonlinear equation Φ(A; P ) = Q using the NAG nonlinear solver nag_nlin_sys_sol [9]. 4 Comparison of Techniques Numerical Optimization Approach The inner iterations of the optimization method of Lancelot B use a minimization of a quadratic model function for which an

6 approximate minimum is found. This leads to a model reduction by solving one or more quadratic minimization problems, requiring a solution of a sequence of linear systems. The Lancelot optimization package allows a choice of linear solver from 12 available options as described in [7]. The tests use 1 time steps 1 Spread Points 1 Close Points 2 Spread Points 2 Close Points Fig. 2. Test cases for testing the linear solvers showing P i as circles and Q i as points on four selected test cases, as illustrated in Fig. 2, two comprising knotpoints taken from adjacent points in a data-set of 123 knotpoints hand-annotated on an MRI brain image, and two comprising points taken equally spaced throughout the data-set. We see the results of the experiment in Table 1. In general, the problems with the points closer together in the domain are hardest to solve. However, the method using the Munksgaard preconditioner, for instance, performs better with 1 close points than with 1 points spread through the domain. The expanding band preconditioner performs the best over the four test cases, but still shows unexpected behaviour in the 2 point tests. It is clear from these experiments that it is difficult to predict how an individual linear solver will perform on a particular problem. None of the linear solvers resulted in convergence for the case involving all of the 123 knotpoints. Solver 1 Spread 1 Close 2 Spread 2 Close Total Conjugate Gradient Diagonal Band Expanding Band Munksgaard Lin-More Multifrontal Table 1. Time taken to converge (in seconds) In order to understand this behaviour better, we explore the change in condition number of the interpolation matrix with respect to the minimum separation between knotpoints. First, we examine the interpolation matrix for the clampedplate spline. Computing the clamped-plate spline requires solving a linear system involving an interpolation matrix, G, constructed of biharmonic Green s func-

7 tions in the same manner as that in which we constructed matrix A in Sec. 3, namely G ij = G(q i, q j ), where G(, ) denotes the d-dimensional biharmonic Green s function and q i, q j are d-dimensional knotpoint position vectors. This interpolation matrix is also a key feature of the Hessian of the augmented Lagrangian, as discussed in Sec. 2. In Fig. 3, we see the change in condition number for the interpolation matrix for two parallel knotpoint paths. From the literature, [1], we expect the condition number in the 2-norm, κ 2 (G) of the interpolation matrix to vary with the minimum separation of the knotpoints in a manner bounded above by min i,j q i q j β, β IR +. Accordingly, for comparison, we calculate and plot α min i,j q i q j β. We suspect the poor performance of the linear solvers in the test cases is due to very large condition numbers for small separations. 3.5 x 14 3 condn numbers.12322q Cond Number 2 Fig. 3. Condition numbers for knotpoints with decreasing separation Distance Between Paths x 1 3 Classical Mechanics Approach Experiments show that the Hamiltonian implementation can solve all of the test cases illustrated in Fig. 2 in less than one second, whereas the Lancelot B implementation takes over 18 seconds to solve the test cases. We see the effect of an increase in the number of time steps in Fig. 4, comparing results using a numerical Jacobian with those using a true Jacobian. We see that the performance of the method using the true Jacobian is much superior to that using a numerical Jacobian, both in terms of function evaluations and of time. These tests used the first 6 knot points of the 123 point set. In Fig. 4, we see the effect on performance of the Hamiltonian implementation under an increase in the number of knotpoints, both with a numerical Jacobian and with a user-supplied analytic Jacobian, as described above. The knotpoints are taken as consecutive excerpts from the same set of 123 points as for the test sets above and 2 time steps are used. The speed of convergence of the method improves by a factor of approximately 4 when there is a user-supplied Jacobian. Notice that solving for the full 123 point set takes less than 4 seconds using this method. 5 Conclusions The Galahad implementation was developed by the authors as an improvement to Marsland and Twining s original MATLAB method [1] for calculating the GIS and showed significant improvement over the MATLAB implementation.

8 Time (secs) Increasing the Number of Time Steps time fals time true Number of Time Steps Time (secs) Increasing the Number of Knotpoints Number of Knotpoints Fig. 4. Varying the number of time steps (l) and knotpoints (r) for the Hamiltonian The Hamiltonian method shows an impressive improvement over both of these methods. We believe that the Galahad method shows disappointing performance due to the lack of a preconditioner suitable for the ill-conditioning of the interpolation matrix. The Hamiltonian method for computing the GIS dramatically outperforms the Lancelot B implementation over the test set of real data. It is clear that exact Jacobians should be supplied to the Hamiltonian implementation to give efficient performance. We see from the experiments carried out that, with exact Jacobians provided, the performance of the Hamiltonian method is superior to the performance of previous methods. References 1. Marsland, S., Twining, C.: Measuring geodesic distances on the space of bounded diffeomorphisms. In: British Machine Vision Conference (BMVC). (22) 2. Modersitzki, J.: Numerical methods for image registration. Oxford University Press, New York (24) 3. Camion, V., Younes, L.: Geodesic interpolating splines. Volume 2134 of Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture notes in Computer Science. (21) Marsland, S., Twining, C.J.: Constructing diffeomorphic representations for the groupwise analysis of nonrigid registrations of medical images. IEEE Trans. Med. Imaging 23 (24) Cheney, W., Light, W.: A course in approximation theory. Brooks/Cole (1999) 6. Boggio, T.: Sulle funzioni di green d ordine m. Circolo Matematico di Palermo 2 (195) Gould, N.I.M., Orban, D., Toint, P.L.: Galahad, a library of thread-safe fortran 9 packages for large-scale nonlinear optimization. ACM Trans. Math. Softw. 29 (23) Arnold, V.I.: Mathematical methods of classical mechanics. 2 edn. Volume 6 of Graduate Texts in Mathematics. Springer Verlag, New York (1989) 58 pages. 9. Numerical Algorithms Group (NAG Manual) 1. Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Advances in Computational Mathematics 3 (1995)