Signal Processing for MRI

Richard S. Spencer, M.D., Ph.D.

National Institutes of HealthNational Institute on Aging, Baltimore, MD

spencer@helix.nih.govspencer@helix.nih.gov

A Philosophical DebateA Philosophical Debate Do we live in a digital Do we live in a digital

world or an analog world?world or an analog world?

An Engineering RealityAn Engineering RealityWe live in a digitized world.We live in a digitized world.

Andrew Lipson, LEGO bricks, ca. 2000

Auguste Rodin, bronze, ca. 1880

Fundamental fact: Fundamental fact:

• • MRI data is acquired in k-spaceMRI data is acquired in k-space • • x-space is just a derived quantity x-space is just a derived quantity (which we happen to be interested in)(which we happen to be interested in) Therefore, we need to understand:Therefore, we need to understand:

kx

ky

x

y

DataData ImageImage

??

Plan: to demonstrate that Plan: to demonstrate that • The basic concepts of time/frequency signalprocessing can be carried over to MRI

• kkxx and and kkyy are the relevant sampling intervals are the relevant sampling intervals

• The imaging equation defines the transformation between conjugate variables--Fourier

• Sampling and other operations on data are performed in k-space; the convolution theorem supplies the resulting effect in the image

• Both DSP and physical effects must be considered

Digitization of a Time-Domain Analog SignalDigitization of a Time-Domain Analog Signal

g(t)g(t) ggsampsamp(t)(t)

( )sn

t nT

gsamp (t) g(t) (t nTs ) n

g(nTs ) (t nTs )n

Sampling Interval TSampling Interval Tss

Data spaced at intervals TData spaced at intervals Tss

ttTTss

==

rfrf

GGss

GGpepe

GGreadread

9090 180180

yk = yG

xk = xG sT

ADCADC

echo signalecho signal

Sampling during MRI Sampling during MRI signal acquisitionsignal acquisition

Sampling time, tSampling time, t

Sampling in k-spaceSampling in k-space

Read directionRead direction Phase directionPhase direction

For both dimensions: data is spaced at intervals For both dimensions: data is spaced at intervals kk

kx

kyxk = xG t

xk = xG sT

yk = yG

yk = yG

/ 2

Significance of this:Significance of this:

• • From a post-processing point of view, read and phase From a post-processing point of view, read and phase directions in MRI can be handled in an identical fashiondirections in MRI can be handled in an identical fashion

• • Much of what you already know about signal Much of what you already know about signal processing of sampled time-domain signals can processing of sampled time-domain signals can be immediately carried over to MRIbe immediately carried over to MRI

Data in k-space is (usually) regularly sampled on a grid.Data in k-space is (usually) regularly sampled on a grid.

This sampling is entirely analogous to sampling ofThis sampling is entirely analogous to sampling oftime-domain data: time-domain data:

intervals are intervals are kkxx and and kky y instead of interval instead of interval TTss

The k-space sampling function is written:The k-space sampling function is written:

N, number of sampledN, number of sampledpoints in kpoints in kxx or k or kyy

/ 2

/ 2

( ; ) ( )N

n N

Comb k k k n k

k-space data are numbers assigned to each grid point: k-space data are numbers assigned to each grid point:

These are the samplesThese are the samples sssampsamp(k(kxx, k, kyy))

Conceptually, we can consider Conceptually, we can consider sssampsamp(k(kxx, k, kyy) )

to be a sampled version of some continuous function, s(kto be a sampled version of some continuous function, s(kxx, k, kyy))

kkkk

•• •• ••••

••••

The above dealt with signal acquisitionThe above dealt with signal acquisition

To proceed: To proceed: consider the physicsconsider the physics

Relationship Between Signal and Relationship Between Signal and Precessing Spins During ReadPrecessing Spins During Read

ssxx=cos(2=cos(2t)t)

ssyy=sin(2=sin(2t)t)

Signal and Spins During ReadSignal and Spins During Read

Signal from dx dy:Signal from dx dy: s(t; x, y) dx dy = s(t; x, y) dx dy = (x, y) (x, y) eei 2i 2 t t dx dydx dy

rxG

xk rt Gxt k x

2( / ) ( , ) xi k xxs t k x x y e dxdy

Integrate over Integrate over all excited spinsall excited spins→→

Consideration of the phase encode gradientConsideration of the phase encode gradientleads to the celebrated leads to the celebrated imaging equationimaging equation

……relates k-space data, relates k-space data, s(ks(kxx, k, kyy) to the image, ) to the image, (x,y)(x,y)

( ),( ) ( , ) ( , )x yi k x k yx ys k k x y e dxdy x y

( ) 1, ,( , ) ( ) ( )x yi k x k yx y x y x yx y s k k e dk dk s k k

Note: the Fourier transform arises from the physicsNote: the Fourier transform arises from the physics

Combine Fourier transforms with convolutionCombine Fourier transforms with convolutionto make use of the all-powerful to make use of the all-powerful Convolution TheoremConvolution Theorem

Convolution of x(t) and h(t)Convolution of x(t) and h(t)

( ) ( ) ( ) ( ) ( )y t x h t d x t h t

Arises naturally when considering:Arises naturally when considering:

the observable effects of intended or unintended actions on data the observable effects of intended or unintended actions on data

digital filtersdigital filters

fold fold slide slide multiply multiply integrate integrate

½½

y(t)y(t)

tt00-t-t11 4t4t11 5t5t11tt11 2t2t11 3t3t11

-t-t11

00

tt11

2t2t11

3t3t11

4t4t11

5t5t11(a)(a)

(b)(b)

(c)(c)

(d)(d)

(e)(e)

(f)(f)

(g)(g)

(h)(h)

h(-th(-t11--

x(x(

x(x(

x(x(

x(x(

x(x(

x(x(

x(x(

h(-h(-

h(th(t11--

h(2th(2t11--

h(3th(3t11--

h(4th(4t11--

h(5th(5t11--

11

11

x(x(

h(h(

½½

11

h(t-h(t-

½½

tt

Convolution of x(t) and h(t) = Convolution of x(t) and h(t) = ( ) ( ) ( ) ( ) ( )y t x h t d x t h t

Brigham, The FFT, 1988Brigham, The FFT, 1988

**Fold, slideFold, slide

= ?= ?

Slide, multiply, integrateSlide, multiply, integrateFixFix

ResultResult

Ingredients:Ingredients:h(t) and g(t), and their Fourier transforms H(h(t) and g(t), and their Fourier transforms H(), G(), G()) = Fourier transform= Fourier transform = Inverse Fourier transform= Inverse Fourier transform• • = multiplication= multiplication* = the convolution operator * = the convolution operator

4 ways of writing the convolution theorem:4 ways of writing the convolution theorem:

I. {f*g} = F • GI. {f*g} = F • G

II. {f • g} = F * GII. {f • g} = F * G

III. {F • G} = f * gIII. {F • G} = f * g

IV. {F * G} = f • gIV. {F * G} = f • g

The Convolution TheoremThe Convolution Theorem

1

1

1

Our application is based on the imaging equation:Our application is based on the imaging equation:

{{(x, y)} = s(k(x, y)} = s(kxx, k, kyy))

{s(k{s(kxx, k, kyy)} = )} = (x, y)(x, y)

Version III. {s • H} = Version III. {s • H} = * h * h

1

1

Ideal data in k spaceIdeal data in k space

Various non-idealities or filtersVarious non-idealities or filters

Visible effect on the imageVisible effect on the image

= the ideal image= the ideal image

With this, we can understand the effects that With this, we can understand the effects that sampling, truncation, and relaxation sampling, truncation, and relaxation

in k-space have on the imagein k-space have on the image

AliasingAliasingdirect sampling effectdirect sampling effect

The point spread functionThe point spread functiontruncation--signal processingtruncation--signal processing

relaxation--physicsrelaxation--physics

AliasingAliasingaka wrap-around, aka fold-overaka wrap-around, aka fold-over

??

??

Equally good digital choices!Equally good digital choices!

Thus, high frequency sinusoids, when sampled, Thus, high frequency sinusoids, when sampled, can be mis-assigned to a lower frequency!can be mis-assigned to a lower frequency!

To avoid this, sample at a rate To avoid this, sample at a rate S S = 1/T= 1/Tss which satisfies which satisfies

S S > 2 • > 2 • where where is the frequency of the sinusoid is the frequency of the sinusoid

This rate, 2 • This rate, 2 • is called the is called the NyquistNyquist rate, rate, NN

To avoid aliasing: To avoid aliasing: S S > > NN 2 • 2 •

1/1/

Fourier decomposition permits extension of this theorem Fourier decomposition permits extension of this theorem to a general bandlimited (-to a general bandlimited (-maxmax, , maxmax) signal, described as:) signal, described as:

g(t) G()e i2td max

max

Then aliasing is avoided by Then aliasing is avoided by ensuringensuring

S S > > NN 2 • 2 • maxmax

Note: for a non-bandlimited signal, apply an anti-aliasing prefilter:Note: for a non-bandlimited signal, apply an anti-aliasing prefilter:

••--maxmax

==maxmaxtt

Non-bandlimited signalNon-bandlimited signal prefilterprefilter bandlimited signalbandlimited signal

The convolution theorem defines the The convolution theorem defines the effect on the image of samplingeffect on the image of sampling

( ; ) ( )Comb k k k m k

A straightforward calculation shows:A straightforward calculation shows:

-1-1 {Comb(k; {Comb(k; k)} = 1/ k)} = 1/ k • Comb(x; 1/ k • Comb(x; 1/ k)k)

kk xx

-1-1

kk 1/1/kk

We can now calculate:We can now calculate:

sampsamp(x) = (x) = -1-1{s(k) • Comb(k; {s(k) • Comb(k; k)}k)}

= = -1-1{s(k)} {s(k)} * * -1-1{Comb(k; {Comb(k; k)} k)}

==

Obtain replicates, spaced at a distance Obtain replicates, spaced at a distance 1/ 1/ k apartk apart

( ) ( ;1/ )x Comb x k

Replication:Replication:LL

LL

1/ 1/ kk

Provided 1/ Provided 1/ k > L, there is no overlap and k > L, there is no overlap and correct reconstruction is possiblecorrect reconstruction is possible

xx1/1/kk

**

• • kkxx < 1/L < 1/Lxx

• • kkyy < 1/L < 1/Lyy

Using only the convolution theorem, we found thatUsing only the convolution theorem, we found thatwe can avoid aliasing by selectingwe can avoid aliasing by selecting

L / 2L / 2

= 0= 0

This is This is equivalent toequivalent to the Nyquist sampling theorem the Nyquist sampling theoremi) (definition)

ii) k < 1/L (the condition derived above)

i) and ii) iii)

which can be written:

iv)

using the value of max max , we obtainv) Ts < 1/(2 maxmax ) ) which can also be written: ss > 2 maxmax = The Nyquist condition= The Nyquist condition

max = L

2 xG

xk = xG sT

xG 1/sT L

1 sT xG L

This was derived for the read direction, butThis was derived for the read direction, butidentical considerations apply in the identical considerations apply in the

phase encode directionphase encode direction

Thus, to fit the entire object into the image, one needs to sampleThus, to fit the entire object into the image, one needs to samplein k-space such that in k-space such that k < 1 / L is satisfiedk < 1 / L is satisfied

k is called the FOVk is called the FOV

FOV = 7.5 cmFOV = 7.5 cmAliased in phase encodeAliased in phase encode

FOV = 15 cmFOV = 15 cmNon-aliasedNon-aliased

Actual:Actual:

••-k-kmaxmax kkmaxmaxkk

==

TruncatedTruncatedk-space datak-space data

s(k) s(k) • s(k) = s • s(k) = strunc trunc (k)(k)

Point Spread Function Due to Point Spread Function Due to Signal ProcessingSignal ProcessingActual data are samples from truncated k-space Actual data are samples from truncated k-space

The convolution theorem can help define the The convolution theorem can help define the result of this truncationresult of this truncation

The resulting 1-D image is given by:The resulting 1-D image is given by:

trunctrunc(x) = (x) = -1-1{s{strunctrunc(k)}(k)}

= = -1-1{ Rect(k) • s(k{ Rect(k) • s(kxx)})}

= = -1-1{ Rect(k)} * { Rect(k)} * -1-1{s(k)}{s(k)}

==

We will use: We will use: -1-1{{ (k)} =(k)} =Sin(x)/xSin(x)/x also known asalso known asSinc(x)Sinc(x)

max

max

sin(2 )2 * ( )k xk x x

Therefore, a delta function density distribution Therefore, a delta function density distribution in one dimension becomes: in one dimension becomes:

Ideal point objectIdeal point object Smearing from truncationSmearing from truncation Actual image:Actual image:blurredblurred

==**

(x)(x)sin(2 )

2kxkx

More truncation gives more blurringMore truncation gives more blurring

Point Spread Function Due to Truncation

x (arbitrary units)A

mp

litu

de

No

rmal

ize

d t

o M

axim

um

-k-kmaxmax kkmaxmax

-1024-1024 10241024

-512-512 512512

-256-256 256256

GGreadread

We can now calculate the combined We can now calculate the combined effects of sampling and truncation:effects of sampling and truncation:

samp, truncsamp, trunc(x) = (x) = -1-1{s(k) • Rect(k) • Comb(k; {s(k) • Rect(k) • Comb(k; k)}k)}

==-1-1{s(k)} {s(k)} ** -1-1{Rect(k)} {Rect(k)} **-1-1{Comb(k; {Comb(k; k)}k)}

==

Obtain: Obtain: •• replication, spaced at a distance replication, spaced at a distance 1/ 1/ k apartk apart

•• smearingsmearing

max

max

sin(2 )max 2

1( ) ( ;1/ ) 2 k x

k xx Comb x k kk

s(ks(kxx, k, kyy) ) • s(k • s(kxx, k, kyy) = s) = strunc trunc (k(kxx, k, kyy))

,max,max

,max ,max

sin(2 )sin(2 )

2 2 * ( , )yx

x y

k yk x

k x k y x y

In two dimensions: two dimensional truncation!In two dimensions: two dimensional truncation!

1( , ) ( , )trunc trunc x yx y s k k

The image is given by:The image is given by:

1 1( , ) * ( , )x y x yrect k k s k k Conv Thm →Conv Thm →

••

What does this point spread function look like?What does this point spread function look like?

-1-1

PSF in 2D x-spacePSF in 2D x-spaceTruncation pattern in 2D k-spaceTruncation pattern in 2D k-space

-1-1

PSF in 2D x-spacePSF in 2D x-spaceTruncation pattern in 2D k-spaceTruncation pattern in 2D k-space

As in 1 dimension, width of point spread As in 1 dimension, width of point spread function is inverse to width of truncationfunction is inverse to width of truncation

-1-1

1/T1/T22* * = 1/T= 1/T22 + 1/T+ 1/T22´́

Net decayNet decay Thermodynamic decayThermodynamic decay Reversible dephasingReversible dephasing

In the gradient echo experiment, both TIn the gradient echo experiment, both T22 and Tand T22´́decay start from the beginning of each k-space line, at decay start from the beginning of each k-space line, at kkmaxmax

kkxx

*2/t Te

Next example: Point Spread Function Due to Next example: Point Spread Function Due to PhysicsPhysicsThe effect of TThe effect of T22* decay* decay

••kkkk

==

kkkk kkkk

Ideal dataIdeal data k-space filterk-space filter Actual data Actual data

*2/( ) ( ) ( ) ( ) t Ts k rect k s k rect k e

Gradient echo sequenceGradient echo sequence

*2/t Te

* *2 2/ / /t T TE T ke e e

*2GT

Rewrite in terms of k:Rewrite in terms of k:

k ( )rG t TE GGreadread

TETE

tt

*2/ /( ) ( ) ( ) ( ) TE T ks k rect k s k rect k e e

*2GT

Therefore:Therefore:

*2/ /1( ) ( ) TE T kPSF x rect k e e *

2GT

*2/ /2TE T kikxe e e

*2

m

m

k GT

kdk

Note:Note:2 m

s

kT

readG GGreadread

mk mk

sTTherefore: Therefore:

*2( ; / )sPSF PSF x T T

*2( ; / )sPSF PSF x T T

TT22* negligible:* negligible:

PSF as for truncationPSF as for truncation

TTs s = 5= 5TT22*;*;

RelaxationRelaxationbroadeningbroadening

TTs s = 10 = 10 TT22*;*;

PSF dominated PSF dominated by relaxationby relaxation

Broadening occurs as Broadening occurs as data acquisition time data acquisition time

lengthens on the lengthens on the time scale of Ttime scale of T22

** relaxation relaxation

Spread in Units of Reciprocal k-space Truncation Interval

Am

pli

tud

e N

orm

aliz

ed

to

Max

imu

m

Point Spread Function Due to Point Spread Function Due to TT22 decay decay

1/T1/T22

* * = 1/T= 1/T22 + 1/T+ 1/T22´́

In the spin echo experiment In the spin echo experiment • • TT22 decay starts from the beginning of each decay starts from the beginning of each

k-space line, at k-space line, at kkmaxmax

• • TT22´ effectively “starts” at k = 0, in the middle of acquisition´ effectively “starts” at k = 0, in the middle of acquisition

kkxx

2/t Te

2'/t Te

This PSF can be described by its full This PSF can be described by its full width at half-maximum (FWHM)width at half-maximum (FWHM)

T2´/Ts (arbitrary units)

FW

HM

of

PS

F D

ue

to T

2´ d

ecay

Conclusions:Conclusions:

• The basic concepts of time/frequency signalprocessing can be carried over to x-space/k-space in MRI

• The imaging equation defines the relevant Fourier conjugate variables

• kkxx and and kkyy are the sampling intervals, analogous to T are the sampling intervals, analogous to Tss

• Sampling and other operations on data are performed in k-space; the convolution theorem supplies the resulting effects on the image