1. Signal Processing for MRI
Richard S. Spencer, M.D., Ph.D.
National Institutes of Health
National Institute on Aging, Baltimore, MD

2. A Philosophical Debate Do we live in a digital
world or an analog world?
Auguste Rodin, bronze, ca. 1880
An Engineering Reality
We live in a digitized world.
Andrew Lipson, LEGO bricks, ca. 2000

3. ky y ? kx x Fundamental fact: Therefore, we need to understand:
• MRI data is acquired in k-space
• x-space is just a derived quantity
(which we happen to be interested in)
Therefore, we need to understand: kx ky x y ?
Data
Image

4. Plan: to demonstrate that
• The basic concepts of time/frequency signal
processing can be carried over to MRI
• kx and ky 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

5. Digitization of a Time-Domain Analog Signal
Sampling Interval Ts  =
gsamp(t)
g(t) t Ts
Data spaced at intervals Ts

6. Sampling during MRI signal acquisition
90
180
echo signal rf Gs 
Gpe
Gread
ADC
Sampling time, t

7. ky kx Sampling in k-space Phase direction Read direction
For both dimensions: data is spaced at intervals k

8. Data in k-space is (usually) regularly sampled on a grid.
This sampling is entirely analogous to sampling of
time-domain data:
intervals are kx and ky instead of interval Ts
Significance of this:
• From a post-processing point of view, read and phase directions in MRI can be handled in an identical fashion
• Much of what you already know about signal
processing of sampled time-domain signals can
be immediately carried over to MRI

9. The k-space sampling function is written:
N, number of sampled
points in kx or ky • • • • • • k k
k-space data are numbers assigned to each grid point:
These are the samples ssamp(kx, ky)
Conceptually, we can consider ssamp(kx, ky)
to be a sampled version of some continuous function, s(kx, ky)

10. The above dealt with signal acquisition
To proceed: consider the physics
Relationship Between Signal and
Precessing Spins During Read 
sx=cos(2t)
sy=sin(2t)

11. Signal and Spins During Read
Signal from dx dy: s(t; x, y) dx dy = (x, y) ei 2  t dx dy
Integrate over
all excited spins →

12. Consideration of the phase encode gradient
leads to the celebrated imaging equation
…relates k-space data, s(kx, ky) to the image, (x,y)
Note: the Fourier transform arises from the physics

13. Arises naturally when considering:
Combine Fourier transforms with convolution
to make use of the all-powerful Convolution Theorem
Convolution of x(t) and h(t)
fold  slide  multiply  integrate
Arises naturally when considering:
the observable effects of intended or unintended actions on data
digital filters

14. * Convolution of x(t) and h(t) = = ? Fix Result Fold, slide
h(t-t) 1
Fold, slide *
= ? 1 t 1 t t t
x(t)
Fix
h(2t1-t)
Slide, multiply, integrate
x(t)
x(t)
h(t1-t)
(d) t
2t1
h(3t1-t) t1 t t
(c)
(e)
3t1
x(t)
x(t)
h(-t)
y(t)
h(4t1-t) t
(b)
(f)
4t1 t
x(t)
x(t)
h(5t1-t)
h(-t1-t)
(g) t t
(a)
5t1
-t1
Result
-t1 t1
2t1
3t1
4t1
5t1 t
Brigham, The FFT, 1988
(h)

15. The Convolution Theorem
Ingredients:
h(t) and g(t), and their Fourier transforms H(), G()
= Fourier transform
= Inverse Fourier transform
• = multiplication
* = the convolution operator
4 ways of writing the convolution theorem:
I {f*g} = F • G
II. {f • g} = F * G
III {F • G} = f * g
IV. {F * G} = f • g

16. Our application is based on the imaging equation:
{(x, y)} = s(kx, ky)
{s(kx, ky)} = (x, y)
Visible effect on the image
Version III {s • H} =  * h
Various non-idealities or filters
Ideal data in k space
 = the ideal image

17. With this, we can understand the effects that
sampling, truncation, and relaxation
in k-space have on the image
Aliasing
direct sampling effect
The point spread function
truncation--signal processing
relaxation--physics

18. aka wrap-around, aka fold-over
Aliasing
aka wrap-around, aka fold-over ?
Equally good digital choices! ?

19. Thus, high frequency sinusoids, when sampled,
can be mis-assigned to a lower frequency!
To avoid this, sample at a rate S = 1/Ts which satisfies
S > 2 • 
where  is the frequency of the sinusoid
1/
This rate, 2 •  is called the Nyquist rate, N
To avoid aliasing: S > N  2 • 

20. Then aliasing is avoided by ensuring
Fourier decomposition permits extension of this theorem
to a general bandlimited (-max, max) signal, described as:
Then aliasing is avoided by ensuring
S > N  2 • max
Note: for a non-bandlimited signal, apply an anti-aliasing prefilter: •
-max =
max t
Non-bandlimited signal
prefilter
bandlimited signal

21. The convolution theorem defines the effect on the image of sampling
A straightforward calculation shows:
-1 {Comb(k; k)} = 1/ k • Comb(x; 1/ k) k x
-1 k
1/k

22. We can now calculate:
samp(x) = -1{s(k) • Comb(k; k)}
= -1{s(k)} * -1{Comb(k; k)} =
Obtain replicates, spaced at a distance 1/ k apart

23. * Replication: x 1/k L L 1/ k
Provided 1/ k > L, there is no overlap and correct reconstruction is possible

24. Using only the convolution theorem, we found that
we can avoid aliasing by selecting
• kx < 1/Lx
• ky < 1/Ly
L / 2
This is equivalent to the Nyquist sampling theorem
i) (definition)
ii) k < 1/L (the condition derived above)
i) and ii) 
iii)
which can be written:
iv)
using the value of max , we obtain
v) Ts < 1/(2 max ) which can also be written: s > 2 max = The Nyquist condition
 = 0

25. Thus, to fit the entire object into the image, one needs to sample
in k-space such that k < 1 / L is satisfied
k is called the FOV
This was derived for the read direction, but
identical considerations apply in the
phase encode direction
FOV = 7.5 cm
Aliased in phase encode
FOV = 15 cm
Non-aliased

26. • Point Spread Function Due to Signal Processing
Actual data are samples from truncated k-space • = k
-kmax
kmax
Truncated
k-space data
Actual:
s(k)  • s(k) = strunc (k)

27. The convolution theorem can help define the result of this truncation
Sin(x)/x
also known as
Sinc(x)
We will use: -1{
(k)} =
The resulting 1-D image is given by:
trunc(x) = -1{strunc(k)}
= -1{ Rect(k) • s(kx)}
= -1{ Rect(k)} * -1{s(k)} =

28. * Therefore, a delta function density distribution
in one dimension becomes: = *
(x)
Ideal point object
Smearing from truncation
Actual image:
blurred

29. More truncation gives more blurring
-kmax
kmax
Point Spread Function Due to Truncation
-1024
Gread
1024
Amplitude Normalized to Maximum
-512
512
-256
256
x (arbitrary units)

30. We can now calculate the combined effects of sampling and truncation:
samp, trunc(x) = -1{s(k) • Rect(k) • Comb(k; k)}
=-1{s(k)} * -1{Rect(k)} *-1{Comb(k; k)} =
Obtain:
• replication, spaced at a distance 1/ k apart
• smearing

31. In two dimensions: two dimensional truncation!
s(kx, ky)  • s(kx, ky) = strunc (kx, ky)
The image is given by:
Conv Thm → •

32. What does this point spread function look like?
-1
Truncation pattern in 2D k-space
PSF in 2D x-space

33. As in 1 dimension, width of point spread function is inverse to width of truncation
-1
Truncation pattern in 2D k-space
PSF in 2D x-space
-1

34. Next example: Point Spread Function Due to Physics
The effect of T2* decay
1/T2* = 1/T2 + 1/T2´
Net decay
Thermodynamic decay
Reversible dephasing
In the gradient echo experiment, both T2 and T2´
decay start from the beginning of each k-space line, at kmax kx

35. • Gradient echo sequence = k k k k k k Ideal data k-space filter
Actual data

36. Rewrite in terms of k:
Gread TE t
Therefore:

37. Note:
Gread
Therefore:

38. time scale of T2* relaxation
Broadening occurs as
data acquisition time
lengthens on the
time scale of T2* relaxation
Amplitude Normalized to Maximum
Ts = 10 T2*;
PSF dominated
by relaxation
Ts = 5T2*;
Relaxation
broadening
Spread in Units of Reciprocal k-space Truncation Interval
T2* negligible:
PSF as for truncation

39. Point Spread Function Due to T2 decay
1/T2* = 1/T2 + 1/T2´
In the spin echo experiment
• T2 decay starts from the beginning of each
k-space line, at kmax
• T2´ effectively “starts” at k = 0, in the middle of acquisition kx

40. This PSF can be described by its full width at half-maximum (FWHM)
FWHM of PSF Due to T2´ decay
T2´/Ts (arbitrary units)

41. Conclusions: • The basic concepts of time/frequency signal
processing can be carried over to x-space/k-space in MRI
• The imaging equation defines the relevant
Fourier conjugate variables
• kx and ky are the sampling intervals, analogous to Ts
• Sampling and other operations on data are performed
in k-space; the convolution theorem supplies the resulting
effects on the image