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Two-Parameter Selection Techniques forProjection-based Regularization Methods:
Application to Partial-Fourier pMRI
Misha E. Kilmer1
Scott Hoge2
1 Dept. of Mathematics, Tufts UniversityMedford, MA
2 Dept. of Radiology, Brigham and Women’s Hospital andHarvard Medical School, Boston, MA
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 1/29
Overview
• Hybrid Methods for single parameter Tikhonovregularization
• Generalization to specialized, two parameter case• Parallel MRI background• Numerical results• Conclusions and future work
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 2/29
MotivationForward model is a (real) linear system
Ax− η = bex,
where• A is
◦ m× n, (e.g. n ≥ 100, 000,m ≥ n)◦ not available explicitly (fast matvecs)◦ ill-conditioned
• Only b = bex + η measured (known)• η white (or close)
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 3/29
Regularized Problem
minx‖Ax− b‖22 + λ2‖L1x‖
22 + µ2‖L2x‖
22
where λ, µ are not known a priori.
Our Motivation: Image reconstruction in partial Fourier,parallel (i.e. multiple coil) MRI.
Assumption∗: L1, L2, n/2× n,[
L1
L2
]
invertible.
Issue: Choosing appropriate (λ, µ) without naive solutionover a grid of possible choices.
*can be relaxedTwo-Parameter Selection Techniques for Projection-based Regularization Methods – p. 4/29
Single Parameter CaseFirst consider
minx‖Ax− b‖22 + λ2‖Lx‖22,
where L is (cheaply) invertible. A change of variablesy = Lx gives
miny
∥
∥
∥
∥
[
AL−1
λI
]
y −
[
b
0
]∥
∥
∥
∥
2
2
.
Naive Approach: for fixed set of λ’s, repeatedly solve anduse a heuristic (e.g. L-curve, methods from previous talk) toapproximate the best one.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 5/29
Single Parameter Case, cont
Since cheap to compute Av and L−1v, use LSQR to solve.
Solve instead
miny∈Kk
∥
∥
∥
∥
[
AL−1
λI
]
−
[
b
0
]∥
∥
∥
∥
2
2
where, with C = AL−1,Kk = span{CT b, (CT C)CT b, (CTC)2CT b, . . . , (CT C)k−1CT b}
Soln. cost ≈ k times sum of matvec cost with A and cost ofL−1v. If we wanted to solve this “accurately” for eachspecific λ, k could change and be large. Too expensive!
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 6/29
Projected ProblemIn LSQR we have the relations
AL−1Vk = Uk+1Bk, u1 = Uk+1e1 = βb
where Vk is n× k, Uk is m× k each with orthogonal columnsand Bk is k + 1× k bidiagonal.
minyk∈Kk
∥
∥
∥
∥
[
AL−1
λI
]
yk −
[
b
0
]∥
∥
∥
∥
2
2
becomes, with xk = Vkyk:
minzk
∥
∥
∥
∥
[
Bk
λIk
]
zk − βe1
∥
∥
∥
∥
2
2
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 7/29
Regularized, Projected Problem
minzk
∥
∥
∥
∥
[
Bk
λIk
]
zk − βe1
∥
∥
∥
∥
2
2
This is a size k (small), Tikhonov-regularized, projectedproblem.
KEY: Choose λ “optimally” for this problem. Then, theregularized solution to the original equation is set as
y(λ∗)k = Vkz
(λ∗)k .
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 8/29
Benefits
minzk
∥
∥
∥
∥
[
Bk
λIk
]
zk − βe1
∥
∥
∥
∥
2
2
• Can compute y(λ)k = L−1x
(λ)k 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)k ‖ = ‖y
(λ)k ‖ = ‖z
(λ)k ‖ virtually free
◦ ‖Ax(λ)k − b‖ = ‖AL−1y
(λ)k − b‖ = ‖Bk+1z
(λ)k − βe1‖
virtually free• If k not too large, other options possible (e.g. WGCV,
[Chung, Nagy, O’Leary ‘08])
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 9/29
Two Parameter Case
minx‖Ax− b‖22 + λ2‖L1x‖
22 + µ2‖L2x‖
22
Recall the assumption L1, L2, n/2× n,[
L1
L2
]
invertible.
minx‖Ax− b‖22 + λ2
∥
∥
∥
∥
[
L1µλL2
]
x
∥
∥
∥
∥
2
2
Fix c = µ/λ, define Lc =
[
L1
cL2
]
and y(λ,µ) = Lcx(λ,µ):
miny
∥
∥
∥
∥
[
AL−1c
λI
]
y −
[
b
0
]∥
∥
∥
∥
2
2Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 10/29
Two Parameter Case
minzk
∥
∥
∥
∥
[
Bk,c
λI
]
zk,c − βe1
∥
∥
∥
∥
2
2
• For a fixed value of c, a different projected problem,regularized using Tikhonov.
• Question: Which values of λ need to be tested for fixedc?
• Question: What information about the projectedproblems do we retain to make a decision about both λand µ?
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 11/29
GridTypically, choose a set of `1 values for λ equally spaced inlog space. Likewise, `2 log-equispaced points for µ. Then“search” over the `1`2 possible pairs in the grid.
Thus, in logspace, each pair (λ, µ) lies on one of the`1 + `2 − 1 lines of slope 1 in this grid.
Using µ = cλ, each line corresponds to one value of c. Foreach fixed c, we need only take the λ values on this line.For each projected problem, at most min(`1, `2) λ values aretested, at best, 1. Work could be done in parallel.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 12/29
SummaryOriginal:
y(λ,µ) = arg miny
∥
∥
∥
∥
[
AL−1c
λI
]
y −
[
b
0
]∥
∥
∥
∥
2
2
,
x(λ,µ) = L−1c y(λ,µ), µ = cλ.
Apply k steps of LSQR to approximately it, equivalent to:
z(λ)k,c = arg min
∥
∥
∥
∥
[
Bk,c
λIk
]
z − βe1
∥
∥
∥
∥
2
2
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 13/29
Summary, ContWe are able to compute the following with short-termrecurrences, for all appropriate values of c, λ1 byconsidering multiple projected problems:
• Solutions∗ y(λ,µ)k = Lcx
(λ,µ)k
• ‖y(λ,µ)k ‖ = ‖z
(λ)k,c ‖
• ‖r(λ,µ)k ‖ = ‖Ax
(λ,µ)k − b‖ = ‖Bk+1,cz
(λ)k,c − βe1‖
* Not needed to obtain items 2 and 3.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 14/29
GoalCompute near “optimal” values of λ, µ. Would like to do thisusing only information that was cheaply computed for eachprojected problem.
Following single-parameter case logic, knowing an “optimal”value of λ for each fixed-c line might be useful.
Added difficulty: each projected problem depends on afixed choice of c, but need whole picture. In particular,
‖Lcx(λ,µ)k ‖ is what is returned, not ‖L1x
(λ,µ)k ‖, ‖L2x
(λ,µ)k ‖.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 15/29
Regularization Parameter Selection
Scheme somewhat problem specific, main idea but may beuseful in other applications as well.
For each c, select regularization parameter λ for thecorresponding projected problem. Using µ = cλ, gives us`1 + `2 − 1 possible choices. Next use other (problemdependent) a priori information to select from among these.
For our application, enough to monitor sharp transitions inresidual norms (cheap, available).
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 16/29
Background: pMRI (2D)MRI uses magnetic field gradients and RF signals toencode field-of-view
← F →
Encoding typically corresponds to DFT, both sides−→ data is acquired in k space domain
k 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. 17/29
2D pMRISub-sampling k-space produces aliasing in spatial domain.
← F →
Use multiple receiver coilsand each coil subsamples in parallel. 4coils, each subsampling by 4→ 16 min.scan now takes 4 min.
����
����
W2
W34
W1
W
Reconstruct image one column at a time (regularized soln.to Wρ = s).Similarly, 3D, reconstruct volume one 2D image slice at atime. Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 18/29
Fast imaging using partial-Fourier encoding
Strategy: • Acquire one half of k-space (top 1/2)• Use conjugate-symmetry assumption to
reconstruct the other half
Issues: • Conjugate-symmetry implies a real-valuedimage
• Field inhomogeneity and gradient fielderrors prevent exact conjugate-symmetryin k-space encoding.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 19/29
Partial-Fourier Problem FormulationWant to constrain solution to be ‘nearly’ real.
We use a two-parameter minimization, to constrain real andimaginary components separately.
minρ
{
‖Wρ− s‖22 + λre‖<{ρ}‖22 + λim‖={ρ}‖
22
}
which is equivalent to solving
minx‖Ax− b‖22 + λ2‖L1x‖
22 + µ2‖L2x‖
22
with
A =
[
<{W} −={W}
={W} <{W}
]
, x =
[
<{ρ}
={ρ}
]
, L1 = [In, 0], L2 = [0, In]
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 20/29
Parameter SelectionSTEP 1: For each c (line) do:
• Compute the residual norms, as function of λ1, for theprojected problem (cheap!). Note these are the sameas residual norms corresponding to the large problem.
• Compute relative difference between neighboringterms on that line.
• Record λ value corresponding the sharpest transition.
STEP 2:
• If haven’t already, compute the xλ,µk ’s for these pairs.
• Throw out any “non-physical” solutions (e.g. ratio ofimaginary part to real part too large).
• Choose the remaining term with smallest residualnorm.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 21/29
Numerical ResultsHigh resolution phantom, 8 coil GE Scanner at BWH, single256x256 slice of 3D data set
• Sampled (partial Fourier) in ky-space 73 lines (at orabove 128); Sampled in kx-space 114 lines,nonuniformly.
• Acceleration factor ≈ 8• A is 133,152 x 131,072• λ1 = logspace(-5,2,10); λ2 = logspace(-3,4,10)• k fixed at 30• Simple thresholding on aliased image to throw out
non-physical solutions.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 22/29
Full Data Reconstruction
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400
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 23/29
Plot of Residual Norms
λim
λre
(log) Residual norm as function of parameters
2 4 6 8 10
1
2
3
4
5
6
7
8
9
102.8
3
3.2
3.4
3.6
3.8
4
4.2
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 24/29
Plot of Relative Error
λim
λre
(log) error norm as function of parameters
2 4 6 8 10
1
2
3
4
5
6
7
8
9
10
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 25/29
Reconstructionλ1(6) = 7.7e−2, λ2(5) = 1.29
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300
350
400
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 26/29
Reconstruction, zoom
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100
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200
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300
350
400
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 27/29
Phantom 2 results
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500
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550
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 28/29
Conclusions and Future Work• Projection approaches can be very computationally
efficient – choose the regularization parameter for thesmaller, projected problem (cheaper).
• For 2D, we select first for individual projectedproblems, then over the whole.
• Best selection methods may be problem dependent.• Basic idea valid when L1, L2 not this special: Transform
to standard form or use hybrid approach of [K.,Hansen, Espanol, ‘07].
• Issue of choosing k [Chung, Nagy, O’Leary ‘08]. Not afactor for our application.
Two-Parameter Selection Techniques for Projection-based Regularization Methods – p. 29/29
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