On the reconstruction accuracy of multi-coil MRI with orthogonal projections

TL;DR

The paper treats multi-coil MRI coil combination as a linear compression problem in image space, using orthogonal projections to model coil dimensionality reduction. By leveraging random projections and PCA within the Grassmannian framework, it analyzes how reconstruction error, variance, SNR, and visual quality relate under different projection dimensions. Key finding: applying PCA-based compression before the final rSOS coil combination consistently yields higher SNR and better visual quality than uncompressed approaches, effectively providing denoising benefits. The results advocate for image-space PCA preprocessing in coil-compressed MRI pipelines and motivate extending the framework to k-space preprocessing to reduce computational burden in PI reconstructions.

Abstract

MRI signal acquisition with multiple coils in a phased array is nowadays commonplace. The use of multiple receiver coils increases the signal-to-noise ratio (SNR) and enables accelerated parallel imaging methods. Some of these methods, like GRAPPA or SPIRiT, yield individual coil images in the k-space domain which need to be combined to form a final image. Coil combination is often the last step of the image reconstruction, where the root sum of squares (rSOS) is frequently used. This straightforward method works well for coil images with high SNR, but can yield problems in images with artifacts or low SNR in all individual coils. We aim to analyze the final coil combination step in the framework of linear compression, including principal component analysis (PCA). With two data sets, a simulated and an in-vivo, we use random projections as a representation of the whole space of orthogonal projections. This allows us to study the impact of linear compression in the image space with diverse measures of reconstruction accuracy. In particular, the $L_2$ error, variance, SNR, and visual results serve as performance measures to describe the final image quality. We study their relationships and observe that the $L_2$ error and variance strongly correlate, but as expected minimal $L_2$ error does not necessarily correspond to the best visual results. In terms of visual evaluation and SNR, the compression with PCA outperforms all other methods, including rSOS on the uncompressed image space data.

Figures (3)

• Figure 1: Scatter plots for the simulated data set with different noise levels, showing the $L_2$ reconstruction error \ref{['err']} and variance \ref{['var']} of the final images obtained by combining the coils with rSOS and compression by random projections and PCA in $\mathop{\mathrm{\mathcal{G}}}\nolimits_{k,32}$. A varying amount of Gaussian noise was added in each coil assuming no correlation noise. The SNR value corresponds to the mean over all voxels in the reconstructed rSOS image volume without the linear compression.
• Figure 2: Cross-sectional image corresponding to the scatter plots in Figure \ref{['simplots']} for the simulated data set with different noise levels. The left column shows the first channel of the simulated noisy $32$-dim coil array before the coil combination step. The second column shows the final image obtained by rSOS including compression with the random projection $p \in \mathop{\mathrm{\mathcal{G}}}\nolimits_{28,32}$ that yields the minimal $L_2$ reconstruction error \ref{['err']}. The third column shows the image provided by rSOS without compression. The second last columns show the final images when using PCA in $\mathop{\mathrm{\mathcal{G}}}\nolimits_{1,32}$ and $\mathop{\mathrm{\mathcal{G}}}\nolimits_{4,32}$ for compression, yielding the highest mean SNR. We can directly see that the lowest $L_2$ error does not directly correspond to the highest SNR. Moreover, the visual evaluation suggests that the SNR describes the image quality more accurately.
• Figure 3: Left - Scatter plot for the in-vivo data set showing the reconstruction error \ref{['err']} and variance \ref{['var']} obtained by combining the coils with rSOS and compressing with random projections and PCA in $\mathop{\mathrm{\mathcal{G}}}\nolimits_{k,32}$. Right - Final cross-sectional images corresponding to the compression methods in the scatter plot. The SNR value corresponds to the mean over all voxels in the reconstructed image volume, see \ref{['SNR']}.

Theorems & Definitions (2)

• Remark 2.1
• Definition 2.2