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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:

Cartesian k-space trajectory and pulse sequence

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

Image formation from sequential Cartesian k-space data

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

Image formation from center-out Cartesian k-space data Image formation from random Cartesian k-space data