Steen Moeller Center for Magnetic Resonance research University of Minnesota

Steen Moeller Center for Magnetic Resonance research University of Minnesota moeller@cmrr.umn.edu Lot of material is from a talk by Douglas C. Noll Department of Biomedical Engineering Functional MRI Laboratory University of Michigan

Image reconstruction what is it that we are trying to do Image reconstruction Some ways that we are doing it Image reconstruction When we know more, we can get more with less

The MR Signal Equation and Fourier Reconstruction What is the objective for Reconstruction Steps in the Gridding Reconstruction Optimization of Gridding Other Fourier Inversion Methods Conclusions

MR spin= nuclear spin angular momentum of atoms We measure the free induction decay of the spin density through the induced electromotive force. 1. Main field B 0 Alignment of nuclear spin (steady state/reference state) 2. Radiofrequency field B 1 Systematic flip the rotation vector (also slice selector) 3. Linear gradient G. Spatial encoded frequencies

MR spin= nuclear spin angular momentum of atoms We measure the free induction decay of the spin density through the induced electromotive force. 1. Main field B 0 Alignment of nuclear spin (steady state/reference state) 2. Radiofrequency field B 1 Systematic flip the rotation vector (also slice selector) 3. Linear gradient G. Spatial encoded frequencies

A net magnetization (M)parallel to the magnetic field

MR spin= nuclear spin angular momentum of atoms We measure the free induction decay of the spin density through the induced electromotive force. 1. Main field B 0 Alignment of nuclear spin (steady state/reference state) 2. Radiofrequency field B 1 Systematic flip the rotation vector (also slice selector) 3. Linear gradient G. Spatial encoded frequencies

1. The resonance (the Lamor frequency) of the nuclear spins are linearly dependent on magnetic field strength 2. Apply a varying field Figure: http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/lauterbur-paul.pdf

1. The resonance (the Lamor frequency) of the nuclear spins are linearly dependent on magnetic field strength 2. Apply a varying field 3. One-to-one correspondence of spatial location in the x direction with frequency of precession. 4. The spectrum of S(t) is the picture of the magnetization as a function of spatial location. 5. To get 2D repeat 1D with a second gradient

In 2D, the received signal in MRI is: is a Fourier integral, where k-space is defined as: FOV y k x k i y x dxdy e y x m k k s y x,,, ) 2 ( t y y t x x d G t k d G t k 0 0 2 2

A Fourier perspective (k-space) to repeat 1D exp. (Phase encoding) Time is proportional to k-space Sampling time of s(t) (Read out)

Gradient Waveforms k-space G x k y G y Time (ms) k x

RO PE Acq

A Fourier perspective (k-space) 2 challenges with EPI -spacing of samples is not equidistant -positive and negative gradients are not 100% balanced. Fix: - sinc interpolation to equidistant grid - re-align echoes with navigator information. Sampling time of s(t) (Read out spacing gradient area)

MR Signal Equation Fourier Transform m x, y s k x, k y

If the signal equation is a Fourier transform s k, k mx, y x y FOV i2 ( kx xkyy) dxdy Then the image reconstruction should simply be the inverse Fourier transform: mˆ x, y sk, k x kspace y e e Right? i2 ( kx xk yy) dk x dk y

There is the question of discretization K-space is sampled and of finite extent The image is also sampled One nice solution is the discretize the inverse FT: mˆ x, y sk, k x y e i2 ( kx xk yy) dk x dk y m n, m s, k j k l Fourier kernel k-space area of each sample

For Cartesian sampling, this is quite easy: The k-space area for each sample is a constant The Fourier kernel is uniformly sampled The image reconstruction is then: mˆ n, m C N 2 j l s k i, k l exp i2 nj N ml N which can be easily implemented using the inverse 2D fast Fourier transform (FFT):

Thus, the inverse 2D FFT is the standard image reconstruction for uniformly sampled MRI: Inverse Fourier Transform

Original Fourier Windowed Gibb s Ringing No, and sometimes it isn t even the best reconstruction. The Fourier reconstruction is best reconstruction in the least-squares sense.

4 different Sampling patterns FFT of signal with missing signals. Guess what the truth is? 75% missing 50% missing 25% missing

As we need to find an m that best matches the data, what happens in reality? In reality (s + noise) = Em, so which m to choose, how and what are the implications? What do we know about m?

The inverse FT: mˆ x, y sk, k no longer has uniform sample areas The discretized inverse using time-domain data must account for non-uniform sample density: x y mˆ x, y j s x j y e e i2 ( kx xk yy) i2 ( k ( j) xk dk x ( j) y) dk y area k-space area of each sample j

This is known as the density compensation function or DCF The corrects for nonuniform sampling, e.g. Many methods for calculating DCF: Jacobian of time/k-space transformation Local area density function Voronoi areas PSF optimization 4-shot spiral samples and sample area function (Voronoi diagram)

Single-shot spiral acquisition 4000 samples, 64x64 matrix

The most standard of the reconstruction methods for arbitrary sampled data? x y mˆ x, y Sometimes called: j s j Discrete FT or DFT reconstruction e i2 ( k ( j) xk ( j) y) Weighted correlation methods (Maeda) DCF Conjugate phase reconstruction (Macovski) j

The MR Signal Equation and Fourier Reconstruction Steps in the Gridding Reconstruction Optimization of Gridding Other Fourier Inversion Methods Conclusions

Problem: This DFT reconstruction is computationally inefficient: x y mˆ x, y j s j e i2 ( k ( j) xk ( j) y) DCF General Idea: Interpolate non-uniform samples onto a Cartesian grid so that the FFT can be used for image reconstruction j

Non-uniform Sampling

Cartesian Sampling

Non-uniform Sampling Cartesian Samples and Bi-Linear Interpolation Bi-linear interpolation works, but produces image artifacts.

Non-uniform Sampling One Cartesian Sample and Interpolation Region Gridding interpolates from a k-space region.

1. Density compensation of k-space data 2. Convolution with a blurring kernel S blur S C 3. Resampling blurred data at uniform locations 4. Inverse FFT mˆ 1 blur F S blur 5. Elimination of blurring effect ( deapodization ) mˆ mˆ blur / F 1 C F 1 1 S F 1 S F C

1D Example Original Object K-space Data

1D Example Highest density of samples DCF is small to compensate Density compensated data

1D Example Compensated Data Convolution The convolution result is a blurred version of the data it is applied to (still in k-space)

1D Example Uniform resampling of blurred data

Inv. FFT Observe the effect of the blurring function: Apodization or shading Ideal Image

After deapodization the final image is nearly perfect Ideal Image

The MR Signal Equation and Fourier Reconstruction Steps in the Gridding Reconstruction Optimization of Gridding Other Fourier Inversion Methods Conclusions

The devil in the details The DCF is a source of error and must be carefully chosen Sidelobe aliasing can be minimized by Oversampling of Cartesian grid (increase FOV) Optimizing the interpolation kernel

Density compensation must be performed before gridding Voronoi areas Jacobian of time/k-space transformation Need to find a continuous variable transformation The k r weighting in projection imaging is an example of this DCF is not obvious for cases where k-trajectories cross: Rosettes PROPELLOR

A suboptimal DCF will lead to a point spread function (PSF) that is not a d-function. Iterative refine weightings so that the PSF approaches a d-function (Pipe, MRM, 41:179-186, 1999) Comparison of initial and optimized DCF for spiral trajectory

2x oversampling of blurred k-space data (it is a continues function) Original FOV 2x FOV of reconstructed object

Consider the gridding reconstruction with an oversampled Cartesian grid: This is aliased energy that can fold onto the object with standard sampling

Fourier summation Gridding Gridding with oversampling Oversampling reduces aliasing of sidelobes, and thus, reduces reconstruction error, Particularly at the edges of the image

Sidelobe aliasing can be minimized by oversampling of the Cartesian grid This keeps the most intense sidelobe energy from folding (aliasing) onto object The convolution kernel can be optimized by minimizing sidelobe energy The Kaiser-Bessel function is known to be nearly optimal (best ratio of mainlobe to sidelobe energy) Larger kernels are better Optimal K-B parameters (Beatty IEEE-TMI 24:799-080, 2005). Both approaches require more computation

The main reason to do the gridding reconstruction is computation speed-up Method Operations* Operations - Example (M=16,384; N=128; W=3; V=2) mˆ x j DCF js j Gridding with Oversampling and FFT e i2 k j x MN 2 2.7 x 10 8 MW 2 +(VN) 2 log 2 (VN) 7.6 x 10 5 Cartesian Sampling N 2 log 2 N 1.1 x 10 5 *M = total number of samples, NxN = image matrix size W = convolution kernel width, V = oversampling factor

The MR Signal Equation and Fourier Reconstruction Steps in the Gridding Reconstruction Optimization of Gridding Other Fourier Inversion Methods Examples of different approaches to solving the same problem. Conclusions

Uses highly oversampled grids (4x, 8x) Very simple interpolation (next neighbor) Density compensation is easy Projections onto Convex Sets (POCS) approach FOV Constraint from Moriguchi, MRM, 51:343-352, 2004. Data Constraint

Idea: find optimal estimate of Cartesian sampled k-space data Uniform ReSampling (URS), Block URS (BURS) (Rosenfeld, Moriguchi) For space limited object, sinc interpolation is optimal Still have problem of density compensation Solution: structure the problem backwards

Expression interpolating from uniform samples (x) to non-uniform samples (b) Ax b ~ a sinc k k ij i j Find pseudo-inverse or regularized inverse # b A x Density compensation built in! Block version operates on smaller regions in k-space Non-uniform Samples k ~ j Uniform Samples ki

Idea: find the image that best fits the Fourier data by simulating the MR signal equation Different methods Harshbarger, Twieg. IEEE Trans Med Imaging 1999; 18(3):196-205. Sutton, Noll, Fessler. IEEE Trans Med Imaging 2003; 22(2):178-188. Others Density compensation can be used, but is not necessary Solution often found by a fast,iterative method, such as the conjugate gradient method

Expression interpolating from image samples (x) to nonuniform samples (y) Ax a ij y exp i 2 ( k Typically, we solve for the image, x, by finding the minimum of some cost function: i x j ) ( x) 2 1 y Ax R( x ) 2 Algorithm involves going back and forth between image and k-domain. Needs gridding and reverse gridding, e.g. NU-FFT and NU-IFFT

Gridding (Fourier) Reconstruction Acquired K-Space Data K-Space Trajectory Sample Density Estimated Image Gridding, IFFT Estimated K-Space Data ISE Reconstruction K-Space Trajectory Estimated Image Acquired K-Space Data MR Simulator NU-FFT Update Rule NU-IFFT

The gridding reconstruction Allows image reconstruction from non-uniformly spaced k-space samples Can be highly accurate Is computationally efficient Requires knowledge of k-space trajectory and and a density compensation function (DCF) Other approaches can: Eliminate the need for density compensation Correct for field inhomogeneity and other artifacts

ISMRM unbound http://www.ismrm.org/mri_unbound/sequence.htm NUFFT http://www.cims.nyu.edu/cmcl/nufft/nufft.html