the fourier transform in mri/fmri

STAT692, Wisconsin Rowe, MCW

The Fourier Transform in MRI/fMRI

Daniel B. Rowe, [email protected]

Department of BiophysicsDivision of Biostatistics

Graduate School of Biomedical Sciences


Outline

➣ Introduction

➣ One Dimensional FT

• Time series constituents and Fourier spectrum.

➣ Two Dimensional FT

• An image constituents and Fourier spectrum.

➣ Two Dimensional MR Image Formation.

• k-space and MR Image Reconstruction.

➣ Two Dimensional MR Image Processing

• Smoothing/Edge detetection.

➣ One Dimensional fMRI Time Series

• fMRI time series Fourier spectrum and filtering.

➣ FMRI Time Series Statistics


Outline

➣ Introduction

➣ One Dimensional FT

• Time series constituents and Fourier spectrum.

➣ Two Dimensional FT

• An image constituents and Fourier spectrum.

➣ Two Dimensional MR Image Formation

• k-space and MR Image Reconstruction.

➣ Two Dimensional MR Image Processing

• Smoothing/Edge detetection.

➣ One Dimensional fMRI Time Series

• fMRI time series Fourier spectrum and filtering.

➣ FMRI Time Series Statistics


Introduction

Place volunteer/patient into MRI scanner.

MCW GE 3T Long Bore


Introduction

In MRI we image a real-valued 3D object, R(x, y, z).Lattice of volume elements, voxels.

! >Volume Slice

Consider a single slice R(x, y).


Introduction

In MRI we aim to image a real-valued object, R(x, y).Di!erent tissues have di!erent magnetic properties yielding contrast.

T !2 weighted image


One Dimensional FT

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

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No noise added!


One Dimensional FT-Continuous

The FT of a continuous function f(x) is

F (k) =

! +"

#"f(x)e#i2!kx dx

also denoted as F{f(x)} and its inverse to be

f(x) =

! +"

#"F (k)e+i2!kx dk

also denoted as F#1{F (k)}.

Don’t forget thatei" = cos(!) + i sin(!) .



F (k) =

! "

#"f(x)[cos(2"kx) ! i sin(2"kx)] dx

=

! "

#"f(x) cos(2"kx) dx ! i

! "

#"f(x) sin(2"kx) dx

= FC(k) ! iFS(k)

FC(k) =

! "

#"

"

#$

j

Aj cos(2"#jx) +$

j

Bj sin(2"#jx)

%

& cos(2"kx) dx

FS(k) =

! "

#"

"

#$

j

Aj cos(2"#jx) +$

j

Bj sin(2"#jx)

%

& sin(2"kx) dx

The cos() sin() and sin() cos() cross terms are zero.Nonzero values at constituent frequencies where Aj and Bj nonzero.



Fourier Transform properties.Property Function Transform

Linearity af(x) + bg(x) aF (k) + bG(k)

Similarity f(ax) 1|a|F (ka)

Shifting f(x ! a) e#i2!kaF (k)

Derivative d!f(x)dx! (i2"k)#F (k)



Convolution of functions f(x) and g(x) is defined as

f(x) " g(x) =

! +"

#"f(!) g(x ! !) d! .

Further

F {f(x) · g(x)} = F (k) " G(k) ,

and

F {f(x) " g(x)} = F (k) · G(k) .



Convolution properties.f(x) ! g(x) = g(x) ! f(x) commutative

f(x) ! [g(x) ! h(x)] = [f(x) ! g(x)] ! h(x) associative

f(x) ! [g1(x) + g2(x)] = f(x) ! g1(x) + f(x) ! g2(x) distributive

d f(x)∗g(x)dx = d f(x)

dx ! g(x) = f(x) ! d g(x)dx derivative

h(x # x0) = f(x # x0) ! g(x) = f(x) ! g(x # x0) shift

if h(x) = f(x) ! g(x)


One Dimensional FT-Discrete

The (finite) Discrete FT can be derived from the continuous FT(with assumptions).

F (p!k)' () *Complex

=n#1$

q=#n

f(q!x)' () *Complex

e#i2"pq

2n

for p = !n, . . . , n ! 1

f(q!x)' () *Complex

=1

2n

n#1$

p=#n

F (p!k)' () *Complex

ei2"pq

2n

for q = !n, . . . , n ! 1 .

There are some assumptions here.The constituent frequencies do not change in time.The constituent frequencies are # 1/(2!x).



The DFT can be represented as

+

,f1...

fn

-

. = "̄

+

,y1...

yn

-

.

n $ 1 n $ n n $ 1

Complex Complex Complex (Real)

(fR + ifI) =/"̄R + i "̄I

0(yR + iyI)




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No noise added!




("̄R +i "̄I) *(yR +i yI) = (fR fI)

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Represent complex-valued time series as an image.




("̄R +i "̄I) *(yR +i yI) = (fR +i fI)

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Pre-multiply by complex-valued forward Fourier Matrix as an image.




("̄R +i "̄I) *(yR +i yI) = (fR +i fI)

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There are lines at the frequency locations.Real part (image) represents constituent cosine frequencies.Imaginary part (image) represents constituent sine frequencies.The intensity of the lines represents amplitude of that frequency.



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

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Two Dimensional FT-Discrete

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Cos Cos

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1= cos + cos + sin + cos

FOV=192 mm, mat=96$96, vox=2 mm3



The Complex-Valued 2D (Discrete) Fourier Transform("̄yR + i"̄yI) * (YR + iYI) * ("̄xR + i"̄xI)

T = (FR + iFI)

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+ i

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The Complex-Valued 2D (Discrete) Fourier Transform("̄yR + i"̄yI) * (YR + iYI) * ("̄xR + i"̄xI)

T = (FR + iFI)

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+ i * + i * + i =

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The Complex-Valued 2D (Discrete) Fourier Transform("̄yR + i"̄yI) * (YR + iYI) * ("̄xR + i"̄xI)T = (FR + iFI)

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+ i * + i * + i = + i

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Real k-space (Cosines) Imaginary k-space (Sines)

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cos cos sin cos



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Note: Rotate bottom half (complex conjugate) up to get top!Hermetian symmetry (property).


Two Dimensional MR Image Formation

So why do we need Fourier Transforms?




In MRI/fMRI our measurements are not voxel values!




In MRI/fMRI our measurements are not voxel values!

Our measurements are spatial frequencies!

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How do we get spatial frequencies?




We apply Gx & Gy magnetic field gradients to encodethen we measure the complex-valued DFT of the object.




We apply Gx & Gy magnetic field gradients to encodethen we measure the complex-valued DFT of the object.

Images are formed (Reconstructed) by a 2D IFT



(a) Gradient Echo-EPI Pulse Sequence

−4 −3 −2 −1 0 1 2 3

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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.



F (kx, ky) = FR(kx, ky) + iFI(kx, ky), the complex-valued DFT of object

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




complex-valued 2D IFT("yR + i"yI) * (FR + iFI) * ("xR + i"xI)T = (YR + iYI)

+ i * + i * + i = + i




Due to the imperfect Fourier encoding, the IFT reconstructedobject is complex-valued, Y (x, y) = YR(x, y) + iYI(x, y).

(a) Real image, yR (b) Imaginary image, yI




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

(a) Magnitude, m =!

y2R + y2

I(b) Phase, ! = atan4(yI/yR)



Two Dimensional MR Image Processing

There are two basic image (filtering) processing categories.

1) Image smoothing

2) Image sharpening

Both can be performed with the DFT.



For image processing, we first define a kernel (AKA mask).A 3 $ 3 kernel with weights denoted by w’s.

w1 w2 w3w4 w5 w6w7 w8 w9

We take this kernel and move it around the image.

A new image is made by summing the product of the kernel weightswith the pixel intensity values under the kernel.

The kernel weights typically sum to unity.



u(x, y)1/16 1/8 1/161/8 1/4 1/81/16 1/8 1/16

Y (x, y)· · · · · · · · ·· · · · · · · · ·· · · · · · · · ·· · · 202 198 207 · · ·qy · · 195 186 201 · · ·· · · 211 189 208 · · ·· · · · · · · · ·· · · · · · · · ·· · · · qx · · · ·

Make new image Z with value at (qx, qy) that is

Z(qx, qy) = 1/16 " 202 + 1/8 " 198 + ... + 1/8 " 189 + 1/16 " 208



The previous procedure of moving the kernel around andmaking new voxel values is the definition of convolution!

Z(qx, qy) =m$

s=#m

n$

r=#n

Y (qx ! r, qy ! s)u(r, s)

where n and m are the x and y dimensions of Y .u is zero padded to be of the same dimension as Y .

Image filtering (Smoothing) is computationally faster in freq space.


Two Dimensional MR Image Processing-Smoothing

Image Real Kernel Real Image FT Real Kernel FT Real

Image Imag. Kernel Imag. Image FT Imag. Kernel FT Imag.

kernel=

.0000 .0000 .0000 .0001 .0001 .0001 .0000 .0000 .0000

.0000 .0000 .0004 .0017 .0026 .0017 .0004 .0000 .0000

.0000 .0004 .0041 .0154 .0239 .0154 .0041 .0004 .0000

.0001 .0017 .0154 .0581 .0906 .0581 .0154 .0017 .0001

.0001 .0026 .0239 .0906 .1412 .0906 .0239 .0026 .0001

.0001 .0017 .0154 .0581 .0906 .0581 .0154 .0017 .0001

.0000 .0004 .0041 .0154 .0239 .0154 .0041 .0004 .0000

.0000 .0000 .0004 .0017 .0026 .0017 .0004 .0000 .0000

.0000 .0000 .0000 .0001 .0001 .0001 .0000 .0000 .0000



Image FT Real Kernel FT Real Prod. FT Real IFT Prod. Real

Image FT Imag. Kernel FT Imag. Prod. FT Imag. IFT Prod. Imag.



Original Image Smoothed Image




Smothing Caveats:1) Increases/Induces local voxel correlation

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Original Image Corr Smoothed Image Corr2) t-statistics need to be renormalized

K =21

wj under independence


One Dimensional fMRI Time Series

In fMRI we get complex-valued images over timeand voxel time course observations, yt = yRt + iyIt.



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

i$t.


One Dimensional fMRI Time SeriesTime series are complex-valued or bivariate with phase coupled means.

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Real: Task related changes!

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Imaginary: Task related changes!

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



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Magnitude: Task related magnitude changes!

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Phase: Often relatively constant temporally.

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!



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Magnitude: Task related magnitude changes!

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Phase: Task related phase changes!

Real-Imaginary or Magnitude-Phase time courses have all info!Recent work indicates that phase time courses may exhibit TRPCsMenon, 2002; Hoogenrad et al., 1998; Borduka et al., 1999; Chow et al., 2006;



Block-designed experiment: O!-On-O!-...-On-O! task

! Real

"

Imaginary

#############$

#########$#####$

! Real

"

Imaginary

#########$

%%%%%%%%%%%%%&

'''''''''(

! Real

"

Imaginary

#########$

%%%%%%%%%&

))

)))*

➣ Complex Magnitude w/ Constant Phase (CP) Activation1,2

➣ Complex Magnitude &/or Phase (CM) Activation3

➣ Real Magnitude-Only (MO/UP) Activation4,5

➣ Real Phase-Only (PO) Activation6

1Rowe and Logan: NeuroImage, 23:1078-1092, 2004. 2Rowe: NeuroImage 25:1124-1132, 2005a.3Rowe: NeuroImage, 25:1310-1324, 2005b. 4Bandettini et al.: Magn Reson Med, 30:161-173, 1993.5Friston et al.: Hum Brain Mapp, 2:189-210, 1995. 6Rowe, Meller, & Ho!mann: J Neuro Meth, in press, 2006.



Let’s consider the magnitude of the time series and its FT.

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TS Phase TS FT Imag TS FT Phase



Let’s filter some FT frequencies of the time series.

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TS FT Mag. TS FT Mag. filt

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TS FT Phase TS FT Phase101,102,110,118,152,160,168,169



Let’s filter some FT frequencies of the time series.

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TS FT Mag. TS FT real TS Mag.

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Original Time Series Filtered Time Series

This filtering will reduce your residual variance!A smaller variance means larger activation statistics!But you have changed the temporal autocorrelation!


FMRI time series statistics

Activation statistic (measure of association) is computed in every voxel.

Many ways to compute activation statistics. Magnitude vs. Complex

Activation is another topic all to itself! Happy to return.

Need to separate activation signal from noise!

Thresholding and the multiple comparisons problem.

Thresholding is another topic all to itself!Unthresh Activation

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PCE Activation

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FDR Activation

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FWE Activation

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Summary

➣ Introduction

➣ One Dimensional FT• Time series constituents and Fourier spectrum.

➣ Two Dimensional FT• An image constituents and Fourier spectrum.

➣ Two Dimensional MR Image Formation• k-space and MR Image Reconstruction.

➣ One Dimensional fMRI time series• fMRI time series Fourier spectrum and filtering.

➣ FMRI time series statistics

Thank You.

the fourier transform in mri/fmri

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