% setup MRI-education-resources path and requirements
cd ../
startup

warning: using the gnuplot graphics toolkit is discouraged

The gnuplot graphics toolkit is not actively maintained and has a number
of limitations that are ulikely to be fixed.  Communication with gnuplot
uses a one-directional pipe and limited information is passed back to the
Octave interpreter so most changes made interactively in the plot window
will not be reflected in the graphics properties managed by Octave.  For
example, if the plot window is closed with a mouse click, Octave will not
be notified and will not update it's internal list of open figure windows.
We recommend using the qt toolkit instead.

loading image
loading signal

Image Reconstruction

Image reconstruction in MRI means converting the raw signal collected in k-space into images. For a fully-sampled acquisition using Cartesian sampling (e.g. on a grid), this can be done with a discrete Fourier Transform. In anticipation of advanced reconstruction methods, image reconstruction for MRI is formulated as a linear system. The technique of Partial Fourier imaging is also introduced.

Learning Goals

1. Describe how images are formed

   • Describe how MRI raw data is reconstructed into an image

2. Manipulate MRI sequence parameters to improve performance

   • Describe how the Partial Fourier method can be used

3. Manipulate and analyze MRI data

   • Reconstruct an image from raw data

Sorting the MRI Data into k-space

Recall that the MRI signal is proportional to the Fourier Transform of the net magnetization of our object, evaluated at the k-space location defined by the gradients:

\[ s_i(t) = \mathcal{F} \{ M_{XY}(\vec{r},0) \} |_{\vec{k} = \vec{k}_i(t)} \]

Note that this includes a dimension for the readout, \(t\), as well as different readouts for each TR, denoted by the subscript for the \(i^\mathrm{th}\) TR.

The data acquired during the readout will be sorted into data based on the k-space trajectory. The following example shows the Cartesian k-space trajectory, which acquired data in a raster fashion in k-space:

From this knowledge, the MR signal over time is stored in a data matrix with known k-space locations to create \(M(\vec{k})\).

Fourier Transform Reconstruction

Once the data has been sorted into the corresponding lines in k-space, an inverse Fourier Transform is applied to reconstruct an image of the transverse magnetization

\[\mathcal{F}^{-1}\{ M(\vec{k} )\} = m(\vec{r})\]

For typical Cartesian sampling, this can be done very simply with the 2D or 3D Fast Fourier Transform (FFT) algorithm

The following animation illustrates typical k-space data as it would be acquired for different k-space lines (left) and the resulting image as more and more lines of k-space are accumulated

We can also acquire our k-space lines in a “center-out” or random ordering, shown below

Partial Fourier

Partial Fourier methods, also known as fractional NEX, are a way to sample a reduced amount of k-space while still reconstructing an image.

These methods note one of the Fourier Transform properties which states that the positive k-space data equals the complex conjugate of the negative k-space data when the object is real-valued:

\[M(\vec{k}) = M^*(-\vec{k}), \ \mathrm{if} \ \mathcal{Im}(m(\vec{r})) = 0\]

Our object is the transverse magnetization, so if our transverse magnetization is all aligned along \(M_X\) then our object is real-valued and we can apply this relationship.

\[m(\vec{r}) = M_{XY}(\vec{r},0) = M_{X}(\vec{r},0) + i M_{Y}(\vec{r},0) \]

Under ideal conditions, the RF pulse can excite the net magnetization just into \(M_X\), meaning this requirement can be satisfied. Taken together, this means only half of the k-space data (e.g positive data \(\vec{k} \geq 0\)) can be acquired and it can be used to fill in the negative k-space data. This means 2 times faster scanning!

In practice, there are sources of phase that result in violation of the real-valued object condition. First, there can be a global phase offset (e.g. transverse magnetization aligned along \(M_Y\)), but this is not a problem since a global phase correction can be applied to correct for this. More problematic are other sources of phase, which come from the RF coil profiles, off-resonance, and chemical shift.

However, this additional phase is typically contained in the low spatial frequencies, in other words they can be measured by our central k-space values. Thus, practical Partial Fourier methods acquire slightly more than half of k-space, and then use specialized algorithms such as projection onto convex sets (POCS) or homodyne detection reconstructions that include correction of the low spatial frequency phase.

% Partial Fourier examples

dataname = 'Data/se_t1_sag_data';
load(dataname)
kdata = data;
S = size(kdata);

im_original = ifft2c(kdata);

subplot(2,1,1)
imagesc(real(im_original))
title('Real(image)')
colormap(gray), axis equal tight off
subplot(2,1,2)
imagesc(imag(im_original))
title('Imag(image)')
colormap(gray), axis equal tight off


% is this image suitable for Partial Fourier reconstruction?