MRI Reconstruction via Fourier Frames on Interleaving ... · Mid-Year Presentation ... Applied Mathematics & Statistics, and Scientiﬁc Computing (AMSC) ... October 2015: Code the

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


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)


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 .


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)


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.


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 .


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)


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.


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 .


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 .


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


Validation

Figure: Left: High-resolution image, 128x128. Right: Idealreconstruction, 16x16.


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.


Deliverables

Synthetic data setFourier frame sampling routineDownsampling routineRoutine to generate spectral dataTranspose reduction routineConjugate gradient routineFinal report and error analysis


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.


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.


Thank you!

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MRI Reconstruction via Fourier Frames on Interleaving ... · Mid-Year Presentation ... Applied Mathematics & Statistics, and Scientiﬁc Computing (AMSC) ... October 2015: Code the