intrinsic voxel correlation in fmri daniel b. rowe, ph.d ...daniel/pubs/rowetalkjsm06.pdf · jsm...

JSM 06’ Rowe, MCW

Intrinsic voxel correlation in fMRI

Daniel B. Rowe, [email protected]

Joint with R.G. Hoffmann

Department of BiophysicsDivision of Biostatistics

Graduate School of Biomedical Sciences

8701 Watertown Plank Rd.Milwaukee, WI 53226

JSM 06’ Rowe, MCW

Outline

Complex Image Reconstruction Part I

• Complex voxels from complex k-space measurements

Complex Statistical Activation Method Part I

• Real-Valued: Magnitude-Only, Phase-Only

• Complex-Valued: Magnitude & Phase

Complex Image Reconstruction Part II

• Relating voxels to k-space measurements

• Statistical Implications

Complex Statistical Activation Methods Part II

• fMRI activation directly from k-space measurements

Discussion

JSM 06’ Rowe, MCW


The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)

32 64 96 128 160 192 224 2569

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11

32 64 96 128 160 192 224 256−3

−2

−1

0

1

2

3

10, cos 0/512 Hz 3 ∗ sin 8/512 Hz

32 64 96 128 160 192 224 256−1

−0.8

−0.6

−0.4

−0.2

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32 64 96 128 160 192 224 256−1

−0.8

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sin 32/512 Hz cos 4/512 Hz

32 64 96 128 160 192 224 256

6

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Sum

JSM 06’ Rowe, MCW


The Complex-Valued (Discrete) Fourier Transform (n=256, TR=2s)The DFT process is to make the data a vector y = (y1, ..., yn)′

Form the forward Fourier matrix Ω = ΩR + iΩI

f1...

fn

=

(ΩR + i ΩI

)

y1...

yn

n× 1 n× n n× 1

Complex Complex Real

(fR + ifI) =(ΩR + i ΩI

)(yR + iyI)

JSM 06’ Rowe, MCW


The Complex-Valued (Discrete) Fourier Transform (TR=2s)

(ΩR +i ΩI) *(yR +i yI) = (fR +i fI)

64 128 192 256

64

128

192

256 + i 64 128 192 256

64

128

192

256 *

32

64

96

128

160

192

224

256 + i

32

64

96

128

160

192

224

256 =

−.25

−.125

0

.125

.25 + i

−.25

−.125

0

.125

.25

JSM 06’ Rowe, MCW


The Complex-Valued (Discrete) Fourier Transform (TR=2s)

−.25

−.125

0

.125

.25

−.25

−.125

0

.125

.25

Cosines Sines

−0.25 −0.1875 −0.125 −0.0625 0 0.0625 0.125 0.18750

128

256

384

512

−0.25 −0.1875 −0.125 −0.0625 0 0.0625 0.125 0.1875

−128

−64

0

64

128

Cosines Sines

32 64 96 128 160 192 224 2569

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11

32 64 96 128 160 192 224 256−3

−2

−1

0

1

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3

32 64 96 128 160 192 224 256−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

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1

32 64 96 128 160 192 224 256−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

10, cos 0/512 Hz 3 ∗ sin 8/512 Hz sin 32/512 Hz cos 4/512 Hz

JSM 06’ Rowe, MCW


The Complex-Valued 2D (Discrete) Fourier Transform

24 48 72 96

24

48

72

9624 48 72 96

24

48

72

96

24 48 72 96

24

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72

9624 48 72 96

24

48

72

96

24 48 72 96

24

48

72

96

∑= cos + cos + sin + cos

FOV=192 mm, mat=96×96, vox=2 mm3

JSM 06’ Rowe, MCW


The Complex-Valued 2D (Discrete) Fourier Transform(ΩyR + iΩyI) * (RR + iRI) * (ΩxR + iΩxI)T = (SR + iSI)

32 64 96 128

32

64

96

12824 48 72 96

24

48

72

9632 64 96 128

32

64

96

128−.25 −.17 0 .17 .25

.25

.17

0

−.17

−.25

+ i * + i * + i = + i

32 64 96 128

32

64

96

12832 64 96 128

32

64

96

12832 64 96 128

32

64

96

128−.25 −.17 0 .17 .25

.25

.17

0

−.17

−.25


JSM 06’ Rowe, MCW


The Complex-Valued 2D (Discrete) Fourier Transform

−.25 −.17 0 .17 .25

.25

.17

0

−.17

−.25−.25 −.17 0 .17 .25

.25

.17

0

−.17

−.25

Real k-space Imaginary k-space

Note: Rotate bottom half up to get top!


JSM 06’ Rowe, MCW


Gx & Gy to encode, measure the complex-valued FT of the object.

S(kx, ky) =∫

R(x, y)e−i2π(xkx+yky)dx dy,

(a) EPI Pulse Sequence

−4 −3 −2 −1 0 1 2 3

−4

−3

−2

−1

0

1

2

3

kx

k y

(b) k-Space Trajectory

Kumar, Welti and Ernst: NMR Fourier Zeugmatography, J. Magn. Reson. 1975

Haacke et al.: Magnetic Resonance Imaging: Physical Principles and Sequence Design, 1999.

JSM 06’ Rowe, MCW


S(kx, ky) = SR(kx, ky) + iSI(kx, ky), complex-valued FT of the object.

(a) real: 96 × 96 (b) imaginary: 96 × 96


JSM 06’ Rowe, MCW


complex-valued 2D IFT(ΩyR + iΩyI) * (SR + iSI) * (ΩxR + iΩxI)T = (RR + iRI)

+ i * + i * + i = + i

JSM 06’ Rowe, MCW


Due to the imperfect Fourier encoding, the IFT reconstructedobject is complex-valued, R(x, y) = RR(x, y) + iRI(x, y).

(a) Real image, yRt (b) Imaginary image, yIt


JSM 06’ Rowe, MCW


This occurs over time in fMRI and results in complex-valued imagesand voxel time course observations, yt = yRt + iyIt.

JSM 06’ Rowe, MCW


Most fMRI studies transform from real-imaginary rectangular coordinatesto magnitude-phase polar coordinates, ρ(x, y) = m(x, y)eiφ(x,y).

(a) Magnitude, mt =√

y2Rt + y2

It(b) Phase, φt = atan4(yIt/yRt)


JSM 06’ Rowe, MCW


Collect a sequence of these reconstructed images over time.Form voxel time courses, yt = rte

iφt.

JSM 06’ Rowe, MCW

Complex Statistical Activation Method Part ITime series are complex-valued or bivariate with phase coupled means.

0

0.2

0.4

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1

1.2

00.5

1

032

6496

128160

192224

256

Imaginary

Realtim

e

yR

t

yI

Magnitude

r

Phase φ

03264961281601922242560

0.2

0.4

0.6

0.8

1

1.2

1.4

time

Real

Real

0 32 64 96 128 160 192 224 2560

0.2

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1

time

Imag

inary

Imaginary

The yR and yI time courses have related vector length info!This is a time series from a actual human experimental data!

JSM 06’ Rowe, MCW

Complex Statistical Activation Method Part ITime series are complex-valued or bivariate with phase coupled means.

0

0.2

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1.2

00.5

1

032

6496

128160

192224

256

Imaginary

Realtim

e

yR

t

yI

Magnitude

r

Phase φ

0 32 64 96 128 160 192 224 2561.5

1.55

1.6

1.65

1.7

1.75

time

Magn

itude

Magnitude

32 64 96 128 160 192 224 256

−3

−2

−1

0

1

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time

Phas

e

Phase

MO time courses only have vector length info!PO time courses only has vector angle info!Real-Imaginary or Magnitude-Phase time courses have all info!

JSM 06’ Rowe, MCW


Block-designed experiment: Off-On-Off-...-On-Off task

- Real

6

Imaginary

*

*

- Real

6

Imaginary

*

- Real

6

Imaginary

*

Real Magnitude-Only (MO/UP) Activation1,2,3

Real Phase-Only (PO) Activation4

Complex Magnitude w/ Constant Phase (CP) Activation5

Complex Magnitude &/or Phase (CM) Activation6

Complex Magnitude w/ Phase Regressor Activation7,8

1Bandettini et al.: MRM, 30:161-173, 1993. 2Friston et al.: HBM, 2:189-210, 1995.3Rowe and Logan: NeuroImage, 24:603-606, 2005. 4Rowe, Meller, and Hoffmann: Submitted, 20065Rowe and Logan: NeuroImage, 23:1078-1092, 2004. 6Rowe: NeuroImage, 25:1310-1324, 2005b.7Menon, MRM, 47:1-9, 2002. 8Nencka and Rowe: Submitted, 2006.

JSM 06’ Rowe, MCW



- Real

6

Imaginary

*

**

- Real

6

Imaginary

*

- Real

6

Imaginary

*







JSM 06’ Rowe, MCW



- Real

6

Imaginary

*

**

- Real

6

Imaginary

*

@@

@@@I

- Real

6

Imaginary

*







JSM 06’ Rowe, MCW



- Real

6

Imaginary

*

**

- Real

6

Imaginary

*

@@

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- Real

6

Imaginary

*

CCCCCCCCCO







JSM 06’ Rowe, MCW


complex-valued 2D F FT(ΩyR + iΩyI) * (RR + iRI) * (ΩxR + iΩxI)

T = (SR + iSI)

2 4 6 8

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2 4 6 8

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2 4 6 8

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+ i * + i * + i = + i

2 4 6 8

2

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2 4 6 8

2

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8

2 4 6 8

2

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8

2 4 6 8

2

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8

JSM 06’ Rowe, MCW


complex-valued 2D I FT(ΩyR + iΩyI) * (SR + iSI) * (ΩxR + iΩxI)

T = (RR + iRI)

2 4 6 8

2

4

6

8

2 4 6 8

2

4

6

8

2 4 6 8

2

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2 4 6 8

2

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+ i * + i * + i = + i

2 4 6 8

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2 4 6 8

2

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2 4 6 8

2

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2 4 6 8

2

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8

JSM 06’ Rowe, MCW


s =vec(STR, ST

I ) Ω =

[ΩR −ΩIΩI ΩR

]

s =

16

32

48

64

80

96

112

128

Ω =16 32 48 64

16

32

48

64 , − 16 32 48 64

16

32

48

64

16 32 48 64

16

32

48

64 , 16 32 48 64

16

32

48

64

s =(

sRsI

)ΩR = [(ΩyR ⊗ ΩxR)− (ΩyI ⊗ ΩxI)]

ΩI = [(ΩyR ⊗ ΩxI) + (ΩyI ⊗ ΩxR)]

⊗ is Kronecker product that multiplies each element of 1st matrix argument with entire 2nd matrix argument

JSM 06’ Rowe, MCW


r = Ω ∗ s

16

32

48

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112

128 = 16 32 48 64 80 96 112 128

16

32

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112

128 ∗

16

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128

JSM 06’ Rowe, MCW


r =

(rRrI

)→ RT = vec(r) = (RT

R, RTI )

16

32

48

64

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96

112

128 →

2

4

6

8

2

4

6

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2

4

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8

2

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6

8

2

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6

8

2

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8

2

4

6

8 ,

2

4

6

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2

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8

2

4

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2

4

6

8

2

4

6

8

2

4

6

8

2

4

6

8

2

4

6

8

↓← ←↓ RC = RR + i RI

→ 2 4 6 8

2

4

6

8

+ i 2 4 6 8

2

4

6

8

JSM 06’ Rowe, MCW

Statistical Implications

Showed that r = Ωs.

If E(s) = s0 and var(s) = Γ

then E(r) = Ωs0 and var(r) = ΩΓΩT

And vise versa.

We know we have spatially correlated voxels (the r’s).

This can only result from correlated k-space measurements (the s’s)

JSM 06’ Rowe, MCW


Could part of the correlation between voxels be fromtemporally correlated k-space measurements along the EPI trajectory?

−4 −3 −2 −1 0 1 2 3

−4

−3

−2

−1

0

1

2

3

kx

k y

k-Space Trajectory

Is (−3,−4) correlated with (−4,−4)?Is (−2,−4) correlated with (−3,−4)?

JSM 06’ Rowe, MCW


Could part of the correlation between voxels be fromk-space adjustments?

−4 −3 −2 −1 0 1 2 3

−4

−3

−2

−1

0

1

2

3

kx

k y

k-Space Trajectory

Shifting of k-space odd/even lines?

Evens left and odds right through timeinduces k-space correlation!

Effect on voxel spatial correlation?

JSM 06’ Rowe, MCW


Could part of the correlation between voxels be frompartial k-space acquisition?

−4 −3 −2 −1 0 1 2 3

−4

−3

−2

−1

0

1

2

3

kx

k y

Partial k-Space Acquisition

Acquistion of k-space botom 1/2 lines?

Make up top 1/2 from bottom 1/2!

Maybe k-space correlation correlatedsince same numbers.Effect on voxel spatial correlation?

JSM 06’ Rowe, MCW


What about Apodization, AKA k-space smoothing?

k-Space Apodization

Effect on voxel spatialcorrelation?

Increase correlation ofhigher spatial frequencies?

Bernstein, King, Zhou: Handbook of MRI Pulse Sequences, Academic Press, 2004.

”an outstanding reference source that covers all the important aspects of pulse sequence design and implementation.”–G.H. Glover, in NMR IN BIOMEDICINE (2005)

JSM 06’ Rowe, MCW


Sample k-space Correlation (along EPI trajectory) 8× 8 imagevar(s) var(r) var(m)

16 32 48 64 80 96 112 128

16

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12816 32 48 64 80 96 112 128

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12816 32 48 64

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64−1

−0.8

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1

Real-Imaginary k-space Real-Imaginary Voxels Magnitude Voxels

Some good correlations ∼ .3 in magnitude data.Maybe mising out on correlations seen in complex?Should decompose covariance into constituent parts.var(s) = Γ = ΓP + ΓN + ΓI

var(r) = ΩΓΩT = ΩΓPΩT + ΩΓNΩT + ΩΓIΩT

JSM 06’ Rowe, MCW

Discussion

Described Complex Image Reconstruction.

Related voxels to k-space measurements.

Not shown fMRI activation directly from k-space measurements!

Further research is needed.

To be continued ...

JSM 06’ Rowe, MCW

Thank You.

Could complex-valued activation be the future of fMRI analysis?Compute activation directly from original k-space measurements!

Collaborators:Mr. Andy NenckaDr. Brent LoganDr. Ray Hoffmann

Colleagues:Dr. Jim HydeDr. Shi-Jiang LiDr. Andrzej Jesmanowicz

intrinsic voxel correlation in fmri daniel b. rowe, ph.d ...daniel/pubs/rowetalkjsm06.pdf · jsm...

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