Chapter 18: Fast Imaging & Compressed Sensing

MRI's weakness is scan time. In standard 2DFT, every phase-encode line costs one TR of waiting, so scan time is roughly TR × number of phase-encode lines × averages — often several minutes. The obvious way to go faster is to skip lines of k-space: halve the lines, halve the time. The catch is that naive skipping breaks the image — and how it breaks points directly to the reconstruction trick.

Undersampling and artifacts

The k-space sample spacing $\Delta k$ sets the field of view (Chapter 16). Skip every other line and $\Delta k$ doubles, folding the field of view in half. Signal that belonged outside the body wraps to the other side — aliasing. The more uniformly you skip, the more this fold-over stacks up as regular replicas.

Skip lines randomly instead — dense in the center, sparse at the edges — and the corruption changes character entirely. Instead of regular replicas, you get noise-like artifacts scattered across the whole image. This structureless corruption is called incoherent.

Simulation: undersampling artifacts

Change how k-space is skipped and see how the zero-fill reconstruction (treating unacquired samples as zero) breaks. Uniform R=2 and R=4 stack fold-over replicas, worsening as the reduction factor grows. Switch to variable-density random and the corruption scatters into noise. It is exactly this incoherent, noise-like artifact that compressed sensing can remove.

Compressed sensing MRI

Recall the sparse-view CT of Chapter 10. The keys to recovering an image from few projections were that the image is sparse in the gradient domain, that the measurements are incoherent, and that the reconstruction is nonlinear. MRI is a natural stage for all three: MR images are piecewise smooth with sparse gradients, variable-density random sampling is incoherent, and all that remains is to run a nonlinear reconstruction. This is CS-MRI, the application where compressed sensing first flourished (Lustig et al. 2007).

Reconstruction is the optimization: of all images consistent with the measured k-space samples, choose the one with the smallest total variation.

$$\min_x \; \mathrm{TV}(x) \quad \text{s.t.} \quad \mathcal{F}_\Omega x = y$$

Here $\mathcal{F}_\Omega$ is the Fourier transform at the sampled k-space locations and $y$ are the measurements. The solver has exactly the same skeleton as ASD-POCS in Chapter 10, alternating a data-consistency step and a TV-descent step. CT's forward and back projections are simply replaced by the 2D FFT and inverse FFT.

Simulation: CS-MRI reconstruction

From 25% variable-density random sampling, compare zero-fill (left) with TV-regularized iterative reconstruction (right). As the iterations proceed, the noise-like artifacts on the right vanish and the RMSE drops well below zero-fill. Raising the TV step size $\lambda$ strengthens the smoothing; too large and fine detail collapses into staircases. Set $\lambda = 0$ for data consistency alone and confirm that it never improves on zero-fill.

Three routes to acceleration

The approaches to going faster by undersampling k-space fall into three families. Parallel imaging (SENSE, GRAPPA) uses the differing sensitivities of multiple receive coils to unfold aliasing — notably, it works even with uniform undersampling. Compressed sensing uses incoherent sampling and sparsity to erase incoherent artifacts. And deep learning learns the recovery itself from data; the image restoration and unrolled networks of Chapter 12 are active research in MRI reconstruction too. Real scanners increasingly combine all three.

Key points

MRI scan time is proportional to the number of phase-encode lines, so skipping k-space lines speeds it up. Uniform skipping produces regular fold-over aliasing; variable-density random skipping produces incoherent, noise-like artifacts. Compressed sensing exploits that incoherence and sparsity, recovering the image with the same TV iteration as Chapter 10 (operator swapped for the FFT). Combined with parallel imaging and deep learning, it underpins fast imaging on real scanners. The next chapter leaves MRI for nuclear-medicine tomography (PET/SPECT), which measures radiation emitted from the body — and where the iterative reconstruction of Chapter 8 shows its true form.

References

• Lustig M, Donoho D, Pauly JM. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance in Medicine 58, 1182–1195 (2007).
• Pruessmann KP et al. SENSE: Sensitivity encoding for fast MRI. Magnetic Resonance in Medicine 42, 952–962 (1999).
• Griswold MA et al. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magnetic Resonance in Medicine 47, 1202–1210 (2002).
• Hammernik K et al. Learning a variational network for reconstruction of accelerated MRI data. Magnetic Resonance in Medicine 79, 3055–3071 (2018).