Controlling spatial correlation in k-space interpolation networks for MRI reconstruction: denoising versus apparent blurring

TL;DR

The methodology advances practical and theoretical aspects of ML-based MRI reconstruction by reinstating reconstruction noise characterization as a cornerstone for performance evaluation, eliminating the need for fully sampled references and introducing apparent blurring, quantifying nonlinear signal mixing without dependence on reference images.

Abstract

Purpose: Interpretability is essential for the clinical adoption of state-of-the-art machine learning (ML) methods in magnetic resonance imaging (MRI). Conventional evaluation of ML reconstructions relies heavily on aggregate image metrics that require fully sampled references. These metrics, inherited from classical image processing and natural image ML, often overlook the critical challenge of noise amplification specific to medical image reconstruction. This study aims to analyze the influence of nonlinear activations on spatial noise variance distribution of k-space interpolation networks (RAKI) and to provide a framework for incorporating variance maps during network training. Methods: We present an analytical framework that decomposes pixel-level noise variance into components reflecting linear and nonlinear characteristics of RAKI. By applying automatic differentiation on the image-space equivalent of the network, variance maps are computed during each training iteration, enabling runtime quality assessment beyond data consistency. We introduce apparent blurring, quantifying nonlinear signal mixing without dependence on reference images. By incorporating variance maps into the traning loss as regularizers, our self-informed RAKI architecture (G-factor-informed RAKI, GIF-RAKI) can directly integrate updated noise characteristics during runtime. Results: Experimental results demonstrate that variance components quantitatively explain network behavior. GIF-RAKI outperforms conventional RAKI variants in image fidelity and noise suppression. Conclusion: Our methodology advances practical and theoretical aspects of ML-based MRI reconstruction by reinstating reconstruction noise characterization as a cornerstone for performance evaluation, eliminating the need for fully sampled references. GIF-RAKI also enables optimization of the trade-off between denoising and apparent blurring.

Figures (10)

• Figure 1: Denoising vs. apparent blurring: a trade-off for nonlinear reconstructions. (A) Nonlinear RAKI reconstruction ($R=5$) with $\mathbb{C}\text{LReLU}$ factor $\alpha = 0.0$ demonstrates noticeable apparent blurring in the central region, revealing ventricular structures in the difference maps. Conversely, linear reconstructions with $\mathbb{C}\text{LReLU}$ factor $\alpha=1$ and GRAPPA show pronounced noise amplification. An intermediate nonlinearity level ($\alpha = 0.5$) achieves the best balance between effective denoising and acceptable apparent blurring. (B) Schematic representation of pixel-level variance analysis using backpropagation. The reconstruction serves as the forward model, while variance analysis exploits the backward gradient pass also used during optimization. The pixel-level variance of a single, real-valued output pixel is computed with respect to all pixels of the multichannel, complex input via built-in autodifferentiation, fully compatible with the network optimizer. This enables establishing an end-to-end connection between input and output pairs describing network noise characteristics.
• Figure 2: Structure of the pixel-level variance. Projected pixel-level variance maps for $\mathbb{C}\text{LReLU}$ factor $\alpha \in \{0, 0.3, 0.5, 0.7\}$ in a 5-fold acceleration setup are shown in 1D along the central line (top) and in 2D (bottom), respectively. Variance maps are reconstructed with dimensions of $127 \times 127$ after training. The examined pixel coordinate $[35;63]$, representing the eigenpixel, is outlined with a red square on all panels. Replica pixels, denoted by red circles, appear as the $(R-1)$ alias of the eigenpixel term along the PE direction, defining an effective FOV. Eigenpixel dominance is evident on the 1D plot as well as the reflected intensity pattern repeating over the effective FOV determined by the acceleration factor on the 2D images. The background magnitude increases as nonlinearity increases, enabling more signal mixing into the examined pixel. Effectively, the baseline level of the background determines the pixel contamination for the reconstruction. The widening of the eigenpixel and replica pixel peaks, which is now included in pixel contamination, is attributed to the network denoising capacity caused by introduced nonlinearity, as described by dawood_image_2025. The nomenclature for total variance (all pixels), maximum variance (eigenpixel contribution, red squares), replica pixels (red circles), and pixel contamination (all pixels except the eigenpixel) is visually motivated.
• Figure 3: Scalar- and vector-valued autodifferentiation and variance map nomenclature. Calculating variance maps of the reconstructed output with respect to the aliased input through scalar-valued autodifferentiation (pixel gradient) using two embedded loops (A), or through vector-valued autodifferentiation (Jacobian) in a single step (B). The numerator dimensions for autodifferentiation are highlighted in orange, while the denominator dimensions are highlighted in blue for both approaches. The returned variance map component, calculated in a given step, is highlighted in red. The vector-valued calculation is considerably faster on a GPU than the scalar-valued method, at the expense of an $n_Y \times n_x$ larger memory load, as indicated by the color code. Visualization of the Einstein summation convention, omitting the channel dimension, used for fast runtime variance map calculation (C). Schematic representation of how the introduced variance maps describing the complete reconstruction output are composed from pixel-level Jacobian (D,E,F). The total variance, also known as the generalized g-factor (D), the maximum variance, termed eigenpixel (E), and residual variance, termed pixel contamination (F), map calculations are shown using the Einstein convention for 4-fold undersampling. $\dag$: Hermitian adjoint; IMG-RAKI: image-space RAKI reconstruction.
• Figure 4: Workflow of g-factor-informed RAKI. The workflow of GIF-RAKI comprises two branches: the k-space data consistency branch (A), corresponding to conventional RAKI, and an additional variance minimization term in image space (B). The schematic illustration shows network training on a fully sampled, low-resolution central k-space region (A). Low-resolution images of size $n_y^{\text{var, low}} \times n_x^{\text{var, low}} = 32 \times 32$ are derived from the undersampled k-space for the image-space branch (B). Black arrows between the two branches indicate that the updated k-space convolution weights are transferred to image space at each iteration step. Autodifferentiation is employed to compute the Jacobian and derive the variance maps based on the updated convolution weights (B). The nonlinear $\mathbb{C}\text{LReLU}$ activations are independent across branches and are retained in k-space within the image-space loop due to memory constraints. (C) A schematic overview of both branches highlights the gradient flow, illustrating the number of backpropagations through the computational graph via autodifferentiation (orange boxes). The backward pass is executed twice for the variance regularization loop (bottom, C), but only once for the k-space data consistency branch (top, C). During optimization, only the parameters of the data consistency branch are updated based on the joint k-space data consistency and image-space regularization, since the image-space branch has no additional trainable parameters of its own. Channel dimensions are omitted for clarity and indicated symbolically by arrows with dimensions. $\circledast$: convolution; $\odot$: element-wise multiplication; $\Sigma : n^{(0)}$: summation in channel dimension $n^{(0)}$
• Figure 5: Effect of nonlinearity on variance components. RAKI and GRAPPA reconstructions, $\times 15$ magnified difference maps, generalized g-factor, eigenpixel, pixel contamination, and apparent blurring maps are shown for $\mathbb{C}\text{LReLU}$ factors $\alpha \in \{0.0, 0.1, 0.2, \ldots, 1.0\}$ with $R=5$ and $ACS=48$. Mean variances computed within the whole brain are displayed above each subfigure. The difference maps and aggregate quality metrics motivate the use of higher nonlinearity levels for improved denoising, as confirmed by the decreased g-factor values. The variance maps illustrate the composition of the g-factor: pixel contamination contributions remain negligible in the linear regime ($\alpha \approx 1.0$), but become dominant under strong nonlinearity ($\alpha \approx 0.0$), producing a noticeable blurring effect. This is reflected in the considerable drop of the eigenpixel contribution in the g-factor. An intermediate nonlinearity level ($\alpha \approx 0.5$) achieves the best balance between effective denoising and acceptable apparent blurring. The apparent blurring maps capture the visual effect of the dominant pixel contamination in the total variance, denoted by red contours for $\mathbf{g}_{\text{blur}} = 0.5$. The case $\alpha = 1$ corresponds to a RAKI network with linear activations, where pixel contamination is within numerical precision. In this purely linear case, the total variance equals the eigenpixel map, consistent with GRAPPA behavior. NMSE: normalized mean squared error ($\times 100$), PSNR: peak signal-to-noise ratio, SSIM: structural similarity index ($\times 100$).