Computational Aspects of MRI

Transcription

1 David Atkinson Philip Batchelor David Larkman

2 Programme 09:30 11:00 Fourier, sampling, gridding, interpolation. Matrices and Linear Algebra 11:30 13:00 MRI Lunch (not provided) 14:00 15:30 SVD, eigenvalues. Regularisation, Norms, Conjugate Gradient, Compressed Sensing. 16:00 17:30 Coordinate systems and geometrical transforms, DICOM, Jacobians

3 Fourier, Sampling and Gridding David Atkinson

4 Resources References in lecture notes Maths for Medical Imaging summer school MathWorld: MATLAB manual. IEEE Trans Med Imag (1999) Survey: Interpolation Methods in Medical Image Processing. Lehman et al. IEEE Trans Med Imag (1991) Selection of a Convolution Function for Fourier Inversion Using Gridding. Jackson et al.

5 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 Survey Michael T. Heath

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

7 Fourier and MRI Analogue to Digital Converter (ADC) Samples k-space Host reconstruction computer Discrete Fourier Transform H ( k) = h( x) e ikx dx

8 Fourier Relations in MRI ADC Host Time domain signal from ADC [s] K-space Spatial frequency [m -1 ] Temporal Frequency [Hz or s -1 ] Image space [m]

9 H Continuous and Discrete, Forward and Reverse Fourier Transforms 2πifx ( f ) = h( x) e dx h( x) = H ( f ) e df 2πifx H ( k) = N j= 1 h( j) e 2πi( j 1)( k 1)/ N h( j) = 1 N N k= 1 H ( k) e + 2πi( j 1)( k 1) / N Definitions of forward and reverse vary just be consistent. Note 1/N scale factor in only one of the Discrete FTs. Angular frequency ω = k = 2πf

10 Discretely Sampled Data: Pixels and FOV f 1 0 2Δ 1 2Δ Δ NΔ pixel FOV 1 NΔ 1 Δ

11 Fourier Transform Pairs k-space constant offset

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

13 Fourier Transform Pairs FT of a series of spikes is another set of spikes with reciprocal spacing

14 Fourier Transform Pairs Kaiser-Bessel. Fourier behaviour has analytic expression

15 Fourier Transform Relations h( x) H ( 1 h( ax) a h( x + b) f ) H H ( f a f )exp [ 2πifb] scaling shifting Rotation in one domain is a rotation by the same angle in the Fourier domain g h g( τ ) h( x τ ) dτ G( f ) H ( f ) convolution

16 Motion During an MRI scan Rotation example Time Linear profile order Rotation mid-way through scan. Ghosting in PE direction.

17 Convolution rect spike g h g( τ ) h( x τ ) dτ x x x

18 Convolution rect rect g h g( τ ) h( x τ ) dτ x

19 Convolution rect gaussian g h g( τ ) h( x τ ) dτ x x

20 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.

21 Filtering and Convolution x

22 Discrete Sampling sampled k-space continuous k-space discrete sampling continuous object FT of sampling pattern periodic replication

23 Discrete Sampling: Aliasing sampled k-space continuous k-space wider discrete sampling continuous object FT of sampling pattern periodic replication aliasing

24 image sampled every 8 th pixel

25 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.

26 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))

27 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.

28 NΔ / 2 + NΔ / 2 f 1 2Δ Δ Δ NΔ pixel FOV 1 NΔ 1 Δ Band-limited: continuous frequencies assumed zero outside range shown Nyquist rate is 1 NΔ

29 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 bandlimited as the k-space signal falls into the noise.

30 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

31 Truncation Gibbs effect. Ripples narrow but never disappear.

32 Use of Hermitian Symmetry The k-space of a real object (no imaginary component) is Hermitian symmetric. S( k x, k y ) = S * ( k x, k y ) 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.

33 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

34 FT of a Gaussian is?

35 Discrete FT is periodic Expect to see Read The Manual

36 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?

37 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.

38 Interpolation Finding the value of a function between measured points.? a x

39 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.

40 Local Interpolations Nearest neighbour Linear Cubic? x

41 Local Interpolations Nearest neighbour Linear Cubic? x

42 Local Interpolations Nearest neighbour Linear Cubic [schematic]? x

43 Interpolation and Convolution Nearest neighbour interpolation is equivalent to convolution with a rectangle. d? d x

44 Interpolation and Convolution Linear interpolation is equivalent to convolution with a triangle. f f where k k b a a + is f c k c the triangular interpolation kernel f c f b f a a b c x

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

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47 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

48 sinc x = Sinc Interpolation Kernel sin x x? x

49 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

50 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)

51 MATLAB griddata Delauney triangulation of irregular points. Triangular interpolation using the measured values at the triangle vertices.

52 Convolution Re-gridding (recap on interpolation) f c f f k + b a a f c k c f b f a a b c x Linear interpolation of regularly sampled data: convolution with a triangle kernel centred at the position b where we wish to evaluate the function.

53 Convolution Re-gridding (irregular samples) Apply a sampling density correction. Centre the kernel at each regular grid point. Compute convolution. De-apodise. x

54 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 deapodise. beware of zeros.

55 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.

56 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(N 2 ). Non-uniform Fast Fourier Transform uses oversampled grid and FFT to achieve speed O(mNlogN).

57 Summary FT is linear. Discrete sampling raises issues of aliasing, gridding etc. Useful to think about issues in both image and k-space domains.

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59 Interpolation in MATLAB griddata scattered data imresize image resizing with antialiasing. interp2, interpn 2D and nd interpolation. imtransform apply geometrical transform makeresampler user specified interpolation for imtransform