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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

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

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Fundamental fact: MRI data is acquired in k-space MRI 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: kxkx kyky x y DataImage ?

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Plan: to demonstrate that The basic concepts of time/frequency signal processing can be carried over to MRI k x and k y are the relevant sampling intervals k x and k y 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

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Digitization of a Time-Domain Analog Signal g(t) g samp (t) Sampling Interval T s Data spaced at intervals T s t TsTsTsTs =

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rf GsGsGsGs G pe G read 90 90 180 180 ADC echo signal Sampling during MRI signal acquisition Sampling time, t

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Sampling in k-space Read direction Phase direction For both dimensions: data is spaced at intervals k kxkx kyky

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

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

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The above dealt with signal acquisition To proceed: consider the physics Relationship Between Signal and Precessing Spins During Read s x =cos(2 t) s y =sin(2 t)

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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

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

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Combine Fourier transforms with convolution to make use of the all-powerful Convolution Theorem Convolution of x(t) and h(t) 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 filters digital filters fold slide multiply integrate

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½ y(t) t 0 -t 1 4t 1 5t 1 t1t1t1t1 2t 1 3t 1 -t 1 0 t1t1t1t1 2t 1 3t 1 4t 1 5t 1 (a) (b) (c) (d) (e) (f) (g) (h) h(-t 1 - h(-t 1 - x( x( h(- h(- h(t 1 - h(t 1 - h(2t 1 - h(2t 1 - h(3t 1 - h(3t 1 - h(4t 1 - h(4t 1 - h(5t 1 - h(5t 1 - 1 1 x( x( h( h( ½ 1 h(t- h(t- ½ t Convolution of x(t) and h(t) = Brigham, The FFT, 1988 * Fold, slide = ? Slide, multiply, integrate Fix Result

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Ingredients: h(t) and g(t), and their Fourier transforms H( ), G( ) = Fourier transform = Fourier transform = Inverse Fourier transform = Inverse Fourier transform = multiplication = 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 The Convolution Theorem

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

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

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

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

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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 S > N 2 max Note: for a non-bandlimited signal, apply an anti-aliasing prefilter: - max = max max t Non-bandlimited signal prefilter bandlimited signal

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

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

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

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k x < 1/L x k x < 1/L x k y < 1/L y k y < 1/L y Using only the convolution theorem, we found that we can avoid aliasing by selecting L / 2 = 0 = 0 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) max using the value of max, we obtain max ) s max = The Nyquist condition v) T s 2 max = The Nyquist condition

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This was derived for the read direction, but identical considerations apply in the phase encode direction 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 k is called the FOV FOV = 7.5 cm Aliased in phase encode FOV = 15 cm Non-aliased

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Actual: -k max k max k = Truncated k-space data s(k) s(k) = s trunc (k) Point Spread Function Due to Signal Processing Actual data are samples from truncated k-space

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

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

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More truncation gives more blurring Point Spread Function Due to Truncation x (arbitrary units) Amplitude Normalized to Maximum -k max k max -1024 1024 -512 512 -256 256 G read

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

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s(k x, k y ) s(k x, k y ) = s trunc (k x, k y ) s(k x, k y ) s(k x, k y ) = s trunc (k x, k y ) In two dimensions: two dimensional truncation! The image is given by: Conv Thm Conv Thm

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

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

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1/T 2 * = 1/T 2 + 1/T 2 ´ 1/T 2 * = 1/T 2 + 1/T 2 ´ Net decay Thermodynamic decay Reversible dephasing In the gradient echo experiment, both T 2 and T 2 ´ decay start from the beginning of each k-space line, at k max kxkxkxkx Next example: Point Spread Function Due to Physics The effect of T 2 * decay

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

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Rewrite in terms of k: G read TE t Therefore:

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Note: Therefore:

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T 2 * negligible: PSF as for truncation T s = 5 T 2 *; Relaxationbroadening T s = 10 T 2 *; PSF dominated by relaxation Broadening occurs as data acquisition time lengthens on the time scale of T 2 * relaxation Spread in Units of Reciprocal k-space Truncation Interval Amplitude Normalized to Maximum

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Point Spread Function Due to T 2 decay 1/T 2 * = 1/T 2 + 1/T 2 ´ In the spin echo experiment T 2 decay starts from the beginning of each T 2 decay starts from the beginning of each k-space line, at k max T 2 ´ effectively starts at k = 0, in the middle of acquisition T 2 ´ effectively starts at k = 0, in the middle of acquisition kxkxkxkx

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This PSF can be described by its full width at half-maximum (FWHM) T 2 ´/T s (arbitrary units) FWHM of PSF Due to T 2 ´ decay

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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 k x and k y are the sampling intervals, analogous to T s k x and k y are the sampling intervals, analogous to T s Sampling and other operations on data are performed in k-space; the convolution theorem supplies the resulting effects on the image

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