Signal Processing for MRI Richard S. Spencer, M.D., Ph.D. National Institutes of Health National Institute on Aging, Baltimore, MD spencer@helix.nih.gov

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

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

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

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

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

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

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

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)

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)

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 →

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

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

* Convolution of x(t) and h(t) = = ? Fix Result Fold, slide h(t-t) 1 Fold, slide * = ? ½ ½ 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) ½ (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)

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

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

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

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

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 • 

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

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

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

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

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

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

• 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)

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)} =

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

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)

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

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

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

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

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

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

Rewrite in terms of k: Gread TE t Therefore:

Note: Gread Therefore:

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

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

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

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