ESPIRiT - Learning Coil Sensitivities from Data

Parallel MRI reconstruction relies heavily on knowing the sensitivity of each coil. ESPIRiT offers a data-driven way to learn these sensitivities, directly from acquired k-space data. Let’s break down the ideas step by step.

1. The Key Idea: Coupling Between Coils

In multi-coil MRI, the signals from different coils are correlated. ESPIRiT leverages these correlations to learn coil sensitivities.

• Each coil is a sensor providing a different view of the same underlying image.
• Instead of assuming a coil model, ESPIRiT observes the data to learn how coils are coupled.

2. Capturing Local Correlations: k-Space Patches

To extract these couplings:

1. Take small patches in $k$-space across all coils.
2. Flatten and stack them into a matrix $A$, where each row represents one patch.
3. Solve for the null-space vector $h$, which acts as a convolution kernel in k-space:

$$
(A∗h)≈0
$$

• $h$ also represents a constraint that all valid k-space patches satisfy.
• Intuitively, $h$ is the “language” of coil relationships—any valid $k$-space data should approximately satisfy it.

3. Moving to Image Space: The Operator $G$

• Apply an inverse Fourier transform to $h$ → becomes operator $G$ in image space.
• Applying $G$ to the multi-coil image $$u$$ should leave it unchanged if the data obeys the learned constraints:

$$
G u \approx u
$$

• $G$ is Hermitian; its eigenvectors with eigenvalue $≈ 1$ define the signal subspace.
• These eigenvectors correspond to the coil sensitivity maps we need.

4. Reconstructing the Image

Once we have sensitivity maps $s_i$, the multi-coil image can be written as:

$$
u = m \cdot s
$$

where $m$ is the underlying image to reconstruct.

We solve for $m$ with a regularized least squares problem:

$$
\min_m ∑_c ∥P\mathcal F(\hat s^cm)−y_c∥^2+λR(m)
$$

• $y$ = acquired k-space data
• $\mathcal F$ = Fourier transform
• $S$ = diagonal operator of coil sensitivities
• $R(m)$ = optional regularization
• $P$ = sampling mask

5. ESPIRiT Pipeline Diagram

$k$-space patches
│
▼
Null-space vector $h$
│
▼
Image-space operator $G$
│
▼
Eigenvectors (Sensitivity maps $s$)
│
▼
Final image $m$

ESPIRiT elegantly combines linear algebra, Fourier transforms, and eigen-decomposition to reconstruct high-quality images using multi-coil MRI data. Its strength lies in learning coil sensitivities directly from the data, making parallel imaging robust and flexible.