Optimization transfer approach to joint registration / reconstruction for motion-compensated image reconstruction

Transcription

1 Optimization transfer approach to joint registration / reconstruction for motion-compensated image reconstruction Jeffrey A. Fessler EECS Dept. University of Michigan ISBI Apr. 5, 00 0

2 Introduction Image reconstruction of moving objects with unknown motion Joint estimation of motion parameters and object Jacobson et al., IEEE NSS 003 Taguchi et al. SPIE 007 Odille et al. MRM Jul. 008 Also super-resolution problems with unknown motion... (cf starting with low-resolution images vs starting with sinograms or k-space) Computational challenge: motion operator in forward model. We use optimization transfer to put motion estimation step in image domain

3 M frames (defined generally) Measurement model y m = A m x m + ε m, m =,...,M y m measured data for mth frame A m system matrix for mth frame x m unknown image for mth frame ε m measurement noise for mth frame Nominal goal: reconstruct image frames {x m } from measured data {y m }.

4 Object model Assume each frame is a spatial transformation of one base image: x m = T(α m ) c α m motion parameters for mth frame T( ) nonrigid warp operator c base image coefficient vector (e.g., for B-splines) x 0 = T(0) c Motion-compensated image reconstruction goal: reconstruct base image coefficients c and motion parameters {α m } from measured data {y m }. 3

5 Joint registration/reconstruction Combined measurement model / object model: y m = A m T(α m ) c+ ε m, m =,...,M Stacking, where α (α,..., α M ): y y. y M A A... y = A T(α) c+ ε A M T(α) T(α ). T(α M ) ε ε. ε M Penalized weighted least-squares (PWLS) estimation: (ĉc, ˆα) = argmin c,α Ψ(c, α) Ψ(c, α) = y A T(α) c W + R (c)+r (α) 4

6 Optimization by alternation Initialize base image c 0 and motion parameters α 0. Alternating updates: c n+ = argmin c α n+ = argmin α Ψ(α n, c) Ψ ( α, c n+) Update c using standard image reconstruction methods: c n+ = argmin c y A T(αn ) c W + R (c) Updating motion parameters α is challenging: α n+ = argmin α ψ(α), ψ(α) = y AT(α)c n+ W + R (α) 5

7 Optimization transfer / Majorize-minimize Alternative to minimizing cost function ψ(α) directly: S-step: find a surrogate function (majorizer) φ(α; α n ) s.t. φ(α; α n ) ψ(α), φ(α n ; α n ) = ψ(α n ) M-step: minimize surrogate function: α n+ = argmin α φ(α; α n ). α Guaranteed to decrease cost function ψ(α n ) monotonically. 6

8 Optimization Transfer Illustrated Ψ(x) and φ(x, x n ) Surrogate function Cost function x n x n+ 7

9 Optimization Transfer in d 8

10 Separable quadratic surrogate for WLS L(x) = y Ax W = i w i (y i [Ax] i ) = L(x n )+(x x n ) g n + (x xn ) A W A(x x n ) L(x n )+(x x n ) g n + (x xn ) D(x x n ) D /[ x ( x n + D g n)] Q(x; x n ), when D is any matrix that satisfies D A W A, where g n L(x n ) = A W (y Ax n ). Useful choice (Erdoğan and Fessler, PMB 999): ( D = diag{d j }, d j i w i a i j k a ik ). 9

11 Surrogate for motion parameters Recall: ψ(α) = y A } T(α)c {{ n+ } + R (α) W x Using result for WLS, the following majorizes ψ: φ(α, α n ) Q(T(α) c n, T(α n ) c n )+R (α) = D /[ T(α) c n ( T(α n ) c n + D g n)] + R (α), where the gradient depends on previous estimates: g n L(x n ) = A W (y A T(α n ) c n ). M-step is entirely image-domain operations: α n+ = argmin α φ(α, α n ). (Guobao Wang & Jinyi Qi did something similar for PET kinetics in ISBI 008) 0

12 Simulation Example MRI with randomly ordered k y,k z phase encodes True Object

13 Simulation Example K space data for 3 frame (log scale) Zero fill IFFT reconstructions, and their average True motion: horizontal translation: -, 4, 0

14 Joint motion estimation / reconstruction Joint estimate: Oracle reconstruction (3 alternations, 9 CG) ( known motion) Joint estimate using Optimization Transfer Oracle estimate using true motion Estimated horizontal translation: -.4, 4.4, 0.4 True horizontal translation: -, 4, 0 3

15 Discussion Proposed optimization transfer approach to joint image reconstruction and motion estimation Puts motion estimation step in image domain akin to image registration avoids expensive forward projections in registration step promotes software modularity (e.g., importance sampling, Bhagalia et al., IEEE T-MI, Aug. 009) Optimization transfer approach may require more iterations Usual nonconvexity issues (multi-resolution...) Generalizes readily to non-quadratic log-likelihood functions (e.g., PET, polyenergetic CT,...) Awaits evaluation with real data... Awaits comparisons with nonlinear CG Open problem: using ordered-subsets approaches efficiently 4