Two-Parameter Selection Techniques for Projection-based Regularization Methods: Application to Partial-Fourier pmri

Transcription

1 Two-Parameter Selection Techniques for Projection-based Regularization Methods: Application to Partial-Fourier pmri Misha E. Kilmer 1 Scott Hoge 1 Dept. of Mathematics, Tufts University Medford, MA Dept. of Radiology, Brigham and Women s Hospital and Harvard Medical School, Boston, MA Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 1/9

2 Overview Hybrid Methods for single parameter Tihonov regularization Generalization to specialized, two parameter case Parallel MRI bacground Numerical results Conclusions and future wor Two-Parameter Selection Techniques for Projection-based Regularization Methods p. /9

3 Motivation Forward model is a (real) linear system where A is Ax η = b ex, m n, (e.g. n 100, 000,m n) not available explicitly (fast matvecs) ill-conditioned Only b = b ex + η measured (nown) η white (or close) Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 3/9

4 Regularized Problem min x Ax b + λ L 1 x + µ L x where λ,µ are not nown a priori. Our Motivation: Image reconstruction in partial Fourier, parallel (i.e. multiple coil) MRI. Assumption : L 1,L,n/ n, [ L1 L invertible. Issue: Choosing appropriate (λ, µ) without naive solution over a grid of possible choices. *can be relaxed Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 4/9

5 Single Parameter Case First consider min x Ax b + λ Lx, where L is (cheaply) invertible. A change of variables y = Lx gives min y [ AL 1 λi y [ b 0 Naive Approach: for fixed set of λ s, repeatedly solve and use a heuristic (e.g. L-curve, methods from previous tal) to approximate the best one.. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 5/9

6 Single Parameter Case, cont Since cheap to compute Av and L 1 v, use LSQR to solve. Solve instead min y K [ AL 1 λi [ b 0 where, with C = AL 1, K = span{c T b, (C T C)C T b, (C T C) C T b,...,(c T C) 1 C T b} Soln. cost times sum of matvec cost with A and cost of L 1 v. If we wanted to solve this accurately for each specific λ, could change and be large. Too expensive! Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 6/9

7 Projected Problem In LSQR we have the relations AL 1 V = U +1 B, u 1 = U +1 e 1 = βb where V is n, U is m each with orthogonal columns and B is + 1 bidiagonal. min y K [ AL 1 λi y [ b 0 becomes, with x = V y : min z [ B λi z βe 1 Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 7/9

8 Regularized, Projected Problem min z [ B λi z βe 1 This is a size (small), Tihonov-regularized, projected problem. KEY: Choose λ optimally for this problem. Then, the regularized solution to the original equation is set as y (λ ) = V z (λ ). Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 8/9

9 Benefits min z [ B λi z βe 1 Can compute y (λ) = L 1 x (λ) with short-term recurrences for multiple λ simultaneously. Try to choose optimal λ for projected problem using appropriate heuristic [K. and O Leary, 01 Lx (λ) = y(λ) = z(λ) virtually free b = AL 1 y (λ) virtually free Ax (λ) b = B +1 z (λ) βe 1 If not too large, other options possible (e.g. WGCV, [Chung, Nagy, O Leary 08) Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 9/9

10 Two Parameter Case min x Ax b + λ L 1 x + µ L x Recall the assumption L 1,L,n/ n, min x Ax b + λ Fix c = µ/λ, define L c = [ L1 cl [ L1 µ λ L [ L1 L x invertible. and y (λ,µ) = L c x (λ,µ) : min y [ AL 1 c λi y [ b 0 Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 10/9

11 Two Parameter Case min z [ B,c λi z,c βe 1 For a fixed value of c, a different projected problem, regularized using Tihonov. Question: Which values of λ need to be tested for fixed c? Question: What information about the projected problems do we retain to mae a decision about both λ and µ? Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 11/9

12 Grid Typically, choose a set of l 1 values for λ equally spaced in log space. Liewise, l log-equispaced points for µ. Then search over the l 1 l possible pairs in the grid. Thus, in logspace, each pair (λ,µ) lies on one of the l 1 + l 1 lines of slope 1 in this grid. Using µ = cλ, each line corresponds to one value of c. For each fixed c, we need only tae the λ values on this line. For each projected problem, at most min(l 1,l ) λ values are tested, at best, 1. Wor could be done in parallel. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 1/9

13 Summary Original: y (λ,µ) = arg min y [ AL 1 c λi y [ b 0, x (λ,µ) = L 1 c y (λ,µ), µ = cλ. Apply steps of LSQR to approximately it, equivalent to: z (λ),c = arg min [ B,c λi z βe 1 Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 13/9

14 Summary, Cont We are able to compute the following with short-term recurrences, for all appropriate values of c, λ 1 by considering multiple projected problems: Solutions y (λ,µ) y (λ,µ) = z (λ),c r (λ,µ) = Ax (λ,µ) = L c x (λ,µ) b = B +1,c z (λ),c βe 1 * Not needed to obtain items and 3. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 14/9

15 Goal Compute near optimal values of λ,µ. Would lie to do this using only information that was cheaply computed for each projected problem. Following single-parameter case logic, nowing an optimal value of λ for each fixed-c line might be useful. Added difficulty: each projected problem depends on a fixed choice of c, but need whole picture. In particular, L c x (λ,µ) is what is returned, not L 1 x (λ,µ), L x (λ,µ). Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 15/9

16 Regularization Parameter Selection Scheme somewhat problem specific, main idea but may be useful in other applications as well. For each c, select regularization parameter λ for the corresponding projected problem. Using µ = cλ, gives us l 1 + l 1 possible choices. Next use other (problem dependent) a priori information to select from among these. For our application, enough to monitor sharp transitions in residual norms (cheap, available). Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 16/9

17 Bacground: pmri (D) MRI uses magnetic field gradients and RF signals to encode field-of-view F Encoding typically corresponds to DFT, both sides data is acquired in space domain space is sampled in line-by-line fashion. Reduce number of lines Reduce acquisition time Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 17/9

18 D pmri Sub-sampling -space produces aliasing in spatial domain. F Use multiple receiver coils and each coil subsamples in parallel. 4 coils, each subsampling by 4 16 min. scan now taes 4 min. W 1 W W W 3 Reconstruct image one column at a time (regularized soln. to Wρ = s). Similarly, 3D, reconstruct volume one D image slice at a time. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 18/9

19 Fast imaging using partial-fourier encoding Strategy: Acquire one half of -space (top 1/) Use conjugate-symmetry assumption to reconstruct the other half Issues: Conjugate-symmetry implies a real-valued image Field inhomogeneity and gradient field errors prevent exact conjugate-symmetry in -space encoding. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 19/9

20 Partial-Fourier Problem Formulation Want to constrain solution to be nearly real. We use a two-parameter minimization, to constrain real and imaginary components separately. { min Wρ s ρ + λ re R{ρ} + λ im I{ρ} } which is equivalent to solving min x Ax b + λ L 1 x + µ L x with A = [ R{W } I{W } I{W } R{W },x = [ R{ρ} I{ρ},L 1 = [I n, 0,L = [0,I n Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 0/9

21 Parameter Selection STEP 1: For each c (line) do: Compute the residual norms, as function of λ 1, for the projected problem (cheap!). Note these are the same as residual norms corresponding to the large problem. Compute relative difference between neighboring terms on that line. Record λ value corresponding the sharpest transition. STEP : If haven t already, compute the x λ,µ s for these pairs. Throw out any non-physical solutions (e.g. ratio of imaginary part to real part too large). Choose the remaining term with smallest residual norm. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 1/9

22 Numerical Results High resolution phantom, 8 coil GE Scanner at BWH, single 56x56 slice of 3D data set Sampled (partial Fourier) in y-space 73 lines (at or above 18); Sampled in x-space 114 lines, nonuniformly. Acceleration factor 8 A is 133,15 x 131,07 λ 1 = logspace(-5,,10); λ = logspace(-3,4,10) fixed at 30 Simple thresholding on aliased image to throw out non-physical solutions. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. /9

23 Full Data Reconstruction Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 3/9

24 Plot of Residual Norms (log) Residual norm as function of parameters λ re λ im Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 4/9

25 Plot of Relative Error (log) error norm as function of parameters λ re λ im 1 Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 5/9

26 Reconstruction λ 1 (6) = 7.7e, λ (5) = Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 6/9

27 Reconstruction, zoom Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 7/9

28 Phantom results Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 8/9

29 Conclusions and Future Wor Projection approaches can be very computationally efficient choose the regularization parameter for the smaller, projected problem (cheaper). For D, we select first for individual projected problems, then over the whole. Best selection methods may be problem dependent. Basic idea valid when L 1,L not this special: Transform to standard form or use hybrid approach of [K., Hansen, Espanol, 07. Issue of choosing [Chung, Nagy, O Leary 08. Not a factor for our application. Two-Parameter Selection Techniques for Projection-based Regularization Methods p. 9/9