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MRI Reconstruction via Fourier Frames onInterleaving SpiralsMid-Year Presentation
Christiana SabettApplied Mathematics & Statistics, and Scientific Computing
(AMSC)University of Maryland, College Park
Advisors: John Benedetto, Alfredo Nava-TudelaMathematics, IPST
[email protected], [email protected]
December 3, 2015
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Talk Outline
Problem OverviewComputational Approach
DiscretizationSpectral RepresentationGenerating Spectral DataSpectral Equivalence Along FrameMatrix Representation
ImplementationTranspose Reduction
ValidationTimelineDeliverables
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Problem Overview
Goal: Recover an image f : E → R where E ⊂ R2 usingspectral data from an MRI machine.
Images courtesy of U.S. Patent US5485086 A and “Carolyn’s MRI”, by ClintJCL (Flickr)
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Discretization
High ResolutionLet χ1 = 1
k ,pk ∈ 1k ∀k
B−1k=0 be a refined tagged partition
of E .We approximate f using the piecewise constant function fχ1
due to lack of access to real MRI data.Low Resolution
Let χ2 = 2j ,qj ∈ 2
j ∀jN1N2−1j=0 be a coarse tagged
partition of E such that fχ2 is piecewise constant.For each qj ∈ χ2, there is a corresponding pk ∈ χ1.We will approximate fχ2 from fχ1 .
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Partition Representation
Under the partition χ1, we have the representation
fχ1 =B−1∑k=0
f (pk )11k
(1)
and the equivalent spectral representation
fχ1 =B−1∑k=0
f (pk )11k. (2)
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Generating Spectral Data
Choose M ≥ N1N2 points αi = (λi , µi) ∈ Ω ⊂ R2 on theinterleaving spirals to get
fχ1(αi) =B−1∑k=0
f (pk )11k(αi). (3)
Figure: Points in the frame for 10 interleaving spirals. Limited domainΩ will induce error in the reconstruction.
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Spectral Equivalence
Let
g =
N1N2−1∑j=0
cj12j
(4)
be an image formed over the coarse partition χ2. Given thespectral data fχ1(αi), we want to find cj that solve
minc
M−1∑i=0
|fχ1(αi)− g(αi)|2 (5)
We compare the actual recovered image g to the idealrecovered image fχ2 formed by local averaging over fχ1 .
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Matrix Representation
LetF = [fχ1(α0) fχ1(α1) ... fχ1(αM−1)]T
andF = [c0 c1 ... cN1N2−1]T.
Define H such that [H]i,j = Hj(αi), where Hj(αi) = 12j(αi).
We form the overdetermined system
F = HF. (6)
(M × 1) = (M × N1N2)(N1N2 × 1)
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Implementation
We will solve the least-squares problem
F = (H∗H)−1H∗F, (7)
(N1N2 × 1) = (N1N2 × N1N2)(N1N2 × M)(M × 1)
whereH is the Bessel map`2(0,1, ...,N1N2 − 1)→ `2(0,1, ...,M − 1)H∗ is its adjointH∗H is the frame operator
We begin with transpose reduction.
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Transpose Reduction
Let A = H∗H and b = H∗ f . Define Vi = (H0(αi ), ...,HN1N2−1(αi ))∗
such that
H =
H0(α0) · · · HN1N2−1(α0)H0(α1) · · · HN1N2−1(α1)
......
...H0(αM−1) · · · HN1N2−1(αM−1)
=
V ∗
0V ∗
1...
V ∗M−1
.
Then,
b = H∗F =
∑M−1
i=0 H0(αi )Fi...∑M−1
i=0 HN1N2−1(αi )Fi
=M−1∑i=0
FiVi .
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Transpose Reduction
Similarly,
H∗H =
∑M−1
i=0 H0(αi )H0(αi ) · · ·∑M−1
i=0 H0(αi )HN1N2−1(αi )
.
.
.∑M−1i=0 HN1N2−1(αi )H0(αi ) · · ·
∑M−1i=0 HN1N2−1(αi )HN1N2−1(αi )
=
M−1∑i=0
H0(αi )H0(αi ) · · · H0(αi )HN1N2−1(αi )
.
.
.HN1N2−1(αi )H0(αi ) · · · HN1N2−1(αi )HN1N2−1(αi )
=
M−1∑i=0
H0(αi )
H1(αi )
.
.
.HN1N2−1(αi )
(H0(αi ) H1(αi ) · · · HN1N2−1(αi ))
=
M−1∑i=0
Vi V∗i .
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Transpose Reduction
To construct A = H∗H and b = H∗ f :1. Let Vj = (H0(α0), ...,HN1N2−1(α0))∗
2. Set A = VjV ∗j and b = f0Vj
3. For j = 0 : M − 1Set Vj = (H0(αj ), ...,HN1N2−1(αj ))∗
A = A + VjV ∗j
b = b + fjVj
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Validation
Figure: Left: High-resolution image, 128x128. Right: Idealreconstruction, 16x16.
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Timeline
October 2015: Code the sampling routine to form theFourier frame. [Complete]November 2015: Validation on small problems. [Ongoing]December 2015: Code the transpose reductionalgorithm [Complete] and begin testing.January 2016: Code the conjugate gradient algorithm.February - March 2016: Error analysis/testing.April 2016: Finalize results.
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
Deliverables
Synthetic data setFourier frame sampling routineDownsampling routineRoutine to generate spectral dataTranspose reduction routineConjugate gradient routineFinal report and error analysis
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
References
Au-Yeung, Enrico, and John J. Benedetto. “GeneralizedFourier Frames in Terms of Balayage.” Journal of FourierAnalysis and Applications 21.3 (2014): 472-508.Benedetto, John J., and Hui C. Wu. “Nonuniform Samplingand Spiral MRI Reconstruction.” Wavelet Applications inSignal and Image Processing VIII (2000).Bourgeois, Marc, Frank T. A. W. Wajer, Dirk Van Ormondt,and Danielle Graveron-Demilly. “Reconstruction of MRIImages from Non-Uniform Sampling and Its Application toIntrascan Motion Correction in Functional MRI.” ModernSampling Theory (2001): 343-63.Goldstein, Thomas, Gavin Taylor, Kawika Barabin, andKent Sayre. “Unwrapping ADMM: Efficient DistributedComputing via Transpose Reduction.” Sub. 8 April 2015.
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals
References
I. Daubechies. Ten Lectures on Wavelets. Society forIndustrial and Applied Mathematics, Philadelphia, PA,1992.Wang, Z., A.C. Bovik, H.R. Sheikh, and E.P. Simoncelli.“Image Quality Assessment: From Error Visibility toStructural Similarity.” IEEE Transactions on ImageProcessing IEEE Trans. on Image Process. 13.4 (2004):600-12.Benedetto, John J., Alfredo Nava-Tudela, Alex Powell, andYang Wang. MRI Signal Reconstruction by Fourier Frameson Interleaving Spirals. Technical report. 2008.
Christiana Sabett (AMSC) MRI Reconstruction via Fourier Frames on Interleaving Spirals