david atkinson philip batchelor david larkmancomputational aspects of mri is mri data band-limited?...

Computational Aspects of MRI


David AtkinsonPhilip BatchelorDavid Larkman


Programme

09:30 – 11:00 Fourier, sampling, gridding, interpolation. Matrices and Linear Algebra

11:30 – 13:00 MRI

Lunch (not provided)

14:00 – 15:30SVD, eigenvalues. Regularisation, Norms, Conjugate Gradient,

Compressed Sensing.

16:00 – 17:30Coordinate systems and geometrical transforms, DICOM, Jacobians


Fourier, Sampling and Gridding

David Atkinson

[email protected]

mailto:[email protected]


Resources

• References in lecture notes http://cmic.cs.ucl.ac.uk/david_atkinson/training

• Maths for Medical Imaging summer school http://www.maths4medicalimaging.co.uk/

• MathWorld: http://mathworld.wolfram.com/• MATLAB manual.• IEEE Trans Med Imag (1999) 18 1049-1075 Survey:

Interpolation Methods in Medical Image Processing. Lehman et al.

• IEEE Trans Med Imag (1991) 30 473-478. Selection of a Convolution Function for Fourier Inversion Using Gridding. Jackson et al.

http://cmic.cs.ucl.ac.uk/david_atkinson/training

http://www.maths4medicalimaging.co.uk/

http://mathworld.wolfram.com/


The Fourier Transform & Its Applications.Ronald Bracewell

Numerical Recipes 3rd Edition: The Art of Scientific Computing

William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery

Scientific Computing: An Introductory SurveyMichael T. Heath

http://www.cse.uiuc.edu/heath/scicomp/notes/

http://www.cse.uiuc.edu/heath/scicomp/notes/


Outline

• Fourier and MRI.• Continuous and Discrete FT• Pixels and FOV.• FT pairs and relations.• Convolution.• Filtering.• Aliasing.• Sampling Theory.• The FFT.• Interpolation• Gridding


Fourier and MRIAnalogue to Digital Converter (ADC)

Samples k-space

Host reconstruction computer

Discrete Fourier Transform

dxexhkH ikx∫∞

∞−= )()(


Fourier Relations in MRI

Time domain signal from ADC[s]

Temporal Frequency

[Hz or s-1]K-spaceSpatial frequency[m-1]

Image space

[m]

ADC Host


Continuous and Discrete, Forward and Reverse Fourier Transforms

dxexhfH ifx∫∞

∞−= π2)()( dfefHxh ifx

∫∞

∞−

−= π2)()(

fk πω 2==

∑=

−−−=N

j

NkjiejhkH1

/)1)(1(2)()( π∑=

−−+=N

k

NkjiekHN

jh1

/)1)(1(2)(1)( π

Definitions of forward and reverse vary – just be consistent.

Note 1/N scale factor in only one of the Discrete FTs.

Angular frequency


Discretely Sampled Data: Pixels and FOV

fΔ≈ 2

1Δ− 2

1 0

Δ pixel

ΔN Δ1

FOV

ΔN1

⇔


Fourier Transform Pairs

k-space constant offset



For a rect that just covers the image FOV, the sinc will go through 0 at the k-space sample points.



FT of a series of spikes is another set of spikes with reciprocal spacing



Kaiser-Bessel. Fourier behaviour has analytic expression


Fourier Transform Relations

[ ]ifbfHbxhafH

aaxh

fHxh

π2exp)()(

1)(

)()(

⇔+

⎟⎠⎞

⎜⎝⎛⇔

⇔

scaling

shifting

convolution)()( fHfG⇔∫∞

∞−−≡∗ τττ dxhghg )()( ⊗

Rotation in one domain is a rotation by the same angle in the Fourier domain


Motion During an MRI scanRotation example

Time

Linear profile order

Rotation mid-way through scan.

Ghosting in PE direction.


Convolutionrect spike

∫∞

∞−−≡∗ τττ dxhghg )()(

x

⊗

x x


Convolutionrect rect

∫∞


x

⊗


Convolutionrect gaussian

∫∞


x

⊗

x


Image Filtering

• Low pass filter passes low frequencies.– Commonly used for noise reduction.

• High pass filter passes high frequencies.– e.g. separate cardiac from respiratory signal.

• Filtering (k-space multiplication) is equivalent to convolution in the image domain.


Filtering and Convolution

⊗

x


Discrete Sampling

continuous object

FT of sampling pattern

periodic replication

⊗

continuous k-space

discrete sampling

sampledk-space×


Discrete Sampling: Aliasing

continuous object

FT of sampling pattern

periodic replicationaliasing

⊗

continuous k-space

wider discrete sampling

sampledk-space×


image sampled every 8th pixel


Aliasing cont.

• Too wide a sampling pattern in k-space leads to image aliasing: MR image wrap around.

• Too widely separated pixels in a digital camera leads to spatial frequency aliasing: Image is not wrapped but has features with wrong spatial frequencies.

• Anti-aliasing filters are effective but must be used BEFORE digitisation.


Original

Cubic interpolation to 1/8 size,with anti-aliasing

Cubic interpolation to 1/8 size,without anti-aliasing

imshow(imresize(x,0.125,'Method','cubic','Antialiasing',true))


Sampling Theory

• The values of a function between samples can be recovered exactly, if,– function is band-limited – sampled at or above the Nyquist rate.

• The Nyquist rate is twice the highest frequency in the signal.– (The sampling needs to catch the up and down of a

sine wave.)• “Band-limited” means its Fourier Transform goes

to zero at the edges.


f Δ≈ 21

Δ− 21 0

Δ pixel

ΔN Δ1

FOVΔN

1

⇔

Band-limited: continuous frequencies assumed zero outside range shown

Nyquist rate is ΔN1

2/Δ− N 2/Δ+≈ N


Is MRI Data Band-Limited?

• In theory data cannot be band-limited in both image and k-space domains.

• For full FOV, raw, unchopped data the k-space is band-limited if there is no image wrap. The image is often effectively band-limited as the k-space signal falls into the noise.


Truncation Artefacts

When the object is not band-limited, we have to truncate the frequencies.– truncation– multiply frequencies by a rect– convolve image with a sinc– image ringing


Truncation

Gibbs effect. Ripples narrow but never disappear.


Use of Hermitian Symmetry

• The k-space of a real object (no imaginary component) is Hermitian symmetric.

• Used in the “half Fourier” MRI acquisitions. Note in reality object has non-zero phase and a phase correction is applied.

• Scan times ~5/8 of whole data achieved.

),(),( *yxyx kkSkkS −−=


The Fast Fourier Transform Algorithm

• Revolutionised signal processing.• Performs the FT on discrete data.• Requires data to be regularly sampled.• A number of practical issues…


FT of a Gaussian is …?


Discrete FT is periodic

“Expect” to seeRead The Manual


FFT Algorithm

• Pay attention to (i.e. read the manual):– Scaling.– Forward/inverse definitions.– Location of zero frequency (DC).

• MATLAB for N even: DC at N/2 + 1• MATLAB for N odd: DC at (N+1)/2

– Shifting• MATLAB: Apply ifftshift, then FFT or iFFT, then fftshift

– Dimensions over which to apply FT – through slice?


Complex nature of Fourier Coefficients

• Always use complex numbers when dealing with FFT.

• Pass through without special care.• Take modulus, phase etc at the end.


Interpolation

• Finding the value of a function between measured points.

x

?

a


Interpolation Approaches

1. Fit a polynomial-type function to all the data points. Function values between points can be computed.

• not well suited to images with many pixels.2. Repeatedly fit within local regions.3. Use sampling theory.


Local Interpolations

• Nearest neighbour• Linear• Cubic

x

?


Local Interpolations

• Nearest neighbour• Linear• Cubic

x

?

[schematic]


Interpolation and Convolution

x

?

Nearest neighbour interpolation is equivalent to convolution with a rectangle.

d

d


Interpolation and Convolution

x

Linear interpolation is equivalent to convolution with a triangle.

a b c

kernelion interpolatr triangula theis where kkfkff ccaab +≈

af

cf

bf


Effect of convolution

• Convolution in image domain is multiplication by FT of kernel in k-space i.e. a low pass filtering or blurring.


Link to Sampling Theory

• A sinc kernel has a k-space filter that is a rect• For a band-limited image, multiplication of k-

space by a rect does no damage to k-space or the image.

• A band-limited function can be interpolated at any point exactly by sinc interpolation if it was sampled at the Nyquist rate.

• But, a sinc kernel has infinite extent…


Sinc Interpolation Kernel

x

?

xxx sinsinc =


• Smaller kernels: faster interpolation, more blurry results.• Interpolation error can oscillate with a period of 1 pixel.• In iterative algorithms e.g. registration, sometimes use

linear interpolation during the algorithm and a larger kernel for final display.

• Sinc interpolation for a rigid shift can be implemented by applying a phase ramp in the Fourier domain.

• For a fixed kernel applied across the data, can perform a de-apodisation


Gridding

Gridding, or “re-gridding”, maps irregularly sampled data to a regular grid.

• The FFT requires regularly sampled data as input.• Motion during a single scan can put data off a regular grid.• Non-Cartesian k-space trajectory, e.g. radial, spiral

Gridding Methods• MATLAB function griddata• Convolution re-gridding (interpolation)• Non-uniform FFT (nuFFT)


MATLAB griddata

• Delauney triangulation of irregular points.• Triangular interpolation using the

measured values at the triangle vertices.


Convolution Re-gridding(recap on interpolation)

xa b c

af

cf

bf

Linear interpolation of regularly sampled data: convolution with a triangle kernel centred at the position b where we wish to evaluate the function.

ccaab kfkff +≈


Convolution Re-gridding(irregular samples)

x

Apply a sampling density correction.Centre the kernel at each regular grid point.Compute convolution.

De-apodise.


De-apodisation

• In convolution re-gridding, the k-space has been convolved with a kernel.

• Equivalently, the image has been multiplied by the FT of the kernel.– bright in image centre

• Divide image by FT of kernel to de-apodise.– beware of zeros.


Other gridding issues

• The kernel is often chosen to be a Kaiser Bessel.

• The grid may be “oversampled” – finer k-space resolution, chop doubled FOV after processing.

• Convolution is in 2D.• Sampling density correction is non-trivial

for general sampling pattern.


Discrete Fourier Transform of Non-Uniform Data

• The Discrete Fourier Transform can be computed by summation - data still needs to be sampling density corrected O(N2).

• Non-uniform Fast Fourier Transform uses oversampled grid and FFT to achieve speed O(mNlogN).


Summary

• FT is linear.• Discrete sampling raises issues of

aliasing, gridding etc.

• Useful to think about issues in both image and k-space domains.


Interpolation in MATLAB

• griddata – scattered data• imresize – image resizing with anti-

aliasing.• interp2, interpn – 2D and nD

interpolation.• imtransform – apply geometrical

transform• makeresampler – user specified

interpolation for imtransform

david atkinson philip batchelor david larkmancomputational aspects of mri is mri data band-limited?...

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