Low-rank modeling of local k-space neighborhoods (LORAKS) for constrained MRI

Abstract

Recent theoretical results on low-rank matrix reconstruction have inspired significant interest in low-rank modeling of MRI images. Existing approaches have focused on higher-dimensional scenarios with data available from multiple channels, timepoints, or image contrasts. The present work demonstrates that single-channel, single-contrast, single-timepoint k-space data can also be mapped to low-rank matrices when the image has limited spatial support or slowly varying phase. Based on this, we develop a novel and flexible framework for constrained image reconstruction that uses low-rank matrix modeling of local k-space neighborhoods (LORAKS). A new regularization penalty and corresponding algorithm for promoting low-rank are also introduced. The potential of LORAKS is demonstrated with simulated and experimental data for a range of denoising and sparse-sampling applications. LORAKS is also compared against state-of-the-art methods like homodyne reconstruction, l1-norm minimization, and total variation minimization, and is demonstrated to have distinct features and advantages. In addition, while calibration-based support and phase constraints are commonly used in existing methods, the LORAKS framework enables calibrationless use of these constraints.

Fig. 1

Images reconstructed using simulated 256 × 256 Cartesian Fourier data acquired with different Δk values.

Fig. 2

Neighborhood systems for different k-space neighborhood radii R, with the centers of the neighborhood systems marked in red.

Fig. 3

Effects of the spatial support of an image on the singular value spectrum of the C matrix for different neighborhood sizes R.

Fig. 4

Images generated after rank-ℓ approximation of C with R = 5. (top row) Δk = 1. (middle row) Δk = 1/2. (bottom row) Δk = 1/3.

Fig. 5

Magnitudes of the Fourier transform reconstructions of the 8 linearly independent nullspace vectors f̃ for which Ĉ₇₄f̃ = 0, corresponding to the images displayed in the top row of Fig. 4 with R = 5 and Δk = 1. Image edge locations are superimposed in red for visual reference.

Fig. 6

Phase profiles generated with different k-space truncation radii Γ, which are used in simulations evaluating the rank characteristics of G and S. The gray scale ranges from a phase of −π (black) to π (white).

Fig. 7

Effects of the Fourier-domain phase truncation radius Γ on the singular value spectrum of the G matrix for different neighborhood sizes R.

Fig. 8

Effects of the Fourier-domain phase truncation radius Γ on the singular value spectrum of the S matrix for different neighborhood sizes R.

Fig. 9

Images generated after rank-ℓ approximation of the matrix S with R = 5. (top row) Γ = 0. (middle row) Γ = 2. (bottom row) Γ = 10.

Fig. 10

LORAKS denoising of the kiwi fruit dataset. (a) High-SNR magnitude image. (b) High-SNR phase image. (c) Magnitude image generated from the low-SNR simulation with conventional Fourier reconstruction. (d) LORAKS reconstruction using r_C = 31, r_G = 21, and r_S = 70. Also shown are mean-squared error images corresponding to (e) conventional Fourier reconstruction and (f) LORAKS reconstruction with r_C = 31, r_G = 21, and r_S = 70. The colorscale has been normalized such that the grayscale ranges from 0 to 1 for the images in (a–d).

Fig. 11

Plots of normalized ℓ₂-norm reconstruction error as a function of the rank constraint parameters r_C, r_G, and r_S. When particular rank constraint values are not given for specific curves, it indicates that the corresponding rank constraints were not imposed. Note that when G-based constraints are not used, the LORAKS reconstruction achieves its optimal value when r_C=27 (cyan curve). When S-based constraints are not used, the LORAKS reconstruction achieves its optimal value when r_C=22 (purple curve). When all constraints are used, the LORAKS reconstruction achieves its optimal value when r_C=31, r_G=21, and r_S=70 (dark yellow dash-dotted curve).

Fig. 12

LORAKS reconstruction of an undersampled T₂-weighted spin-echo brain image of a healthy subject. (a) Magnitude of the fully sampled reference image. (b) Phase of the fully sampled reference image. (c) Random k-space sampling mask. (d) Zero-filled Fourier reconstruction. (e) LORAKS reconstruction with r_C = 50. (f) LORAKS reconstruction with r_G = 120. (g) LORAKS reconstruction with r_S = 70. Note that the reconstructions shown in (f) and (g) still contain small residual aliasing artifacts, which are visible upon close inspection. (h–k) Error images corresponding to the reconstructions shown in (d–g). The colorscale has been normalized such that the grayscale ranges from 0 to 1 for the images in (d–g). For comparison, ℓ₁-norm and TV minimization results are shown in (l) and (n), respectively, with corresponding error images shown in (m) and (o).

Fig. 13

LORAKS reconstruction of the image from Fig. 12 with a “half k-space” sampling pattern. (a) Sampling mask. (b) Zero-filled Fourier reconstruction. (c) LORAKS reconstruction with r_C = 50. (d) LORAKS reconstruction with r_G = 120. (e) LORAKS reconstruction with r_S = 70. (f) Homodyne reconstruction [20]. Note that close inspection might be necessary to visualize the differences between the images shown in (b–f). (g–k) Error images corresponding to the reconstructions shown in (b–f). The colorscale has been normalized such that the grayscale ranges from 0 to 1 for the images in (b–f).

Fig. 14

LORAKS reconstruction of an undersampled T₁-weighted MPRAGE brain image of a patient with a stroke lesion. (a) Magnitude of the fully sampled reference image. (b) Phase of the fully sampled reference image. (c) Random sampling mask. (d) Zero-filled Fourier reconstruction. (e) LORAKS reconstruction with r_C = 50. (f) LORAKS reconstruction with r_G = 105. (g) LORAKS reconstruction with r_S = 75. (h–k) Error images corresponding to the reconstructions shown in (d–g). The colorscale has been normalized such that the grayscale ranges from 0 to 1 for the images in (d–g). For comparison, ℓ₁-norm and TV minimization results are shown in (l) and (n), respectively, with corresponding error images shown in (m) and (o).