K-band: Self-supervised MRI Reconstruction via Stochastic Gradient Descent over K-space Subsets

TL;DR

This work introduces a practical, easy-to-implement, self-supervised training framework, which involves fast acquisition and self-supervised reconstruction and offers theoretical guarantees, and introduces training with stochastic gradient descent (SGD) over k-space subsets.

Abstract

Although deep learning (DL) methods are powerful for solving inverse problems, their reliance on high-quality training data is a major hurdle. This is significant in high-dimensional (dynamic/volumetric) magnetic resonance imaging (MRI), where acquisition of high-resolution fully sampled k-space data is impractical. We introduce a novel mathematical framework, dubbed k-band, that enables training DL models using only partial, limited-resolution k-space data. Specifically, we introduce training with stochastic gradient descent (SGD) over k-space subsets. In each training iteration, rather than using the fully sampled k-space for computing gradients, we use only a small k-space portion. This concept is compatible with different sampling strategies; here we demonstrate the method for k-space "bands", which have limited resolution in one dimension and can hence be acquired rapidly. We prove analytically that our method stochastically approximates the gradients computed in a fully-supervised setup, when two simple conditions are met: (i) the limited-resolution axis is chosen randomly-uniformly for every new scan, hence k-space is fully covered across the entire training set, and (ii) the loss function is weighed with a mask, derived here analytically, which facilitates accurate reconstruction of high-resolution details. Numerical experiments with raw MRI data indicate that k-band outperforms two other methods trained on limited-resolution data and performs comparably to state-of-the-art (SoTA) methods trained on high-resolution data. k-band hence obtains SoTA performance, with the advantage of training using only limited-resolution data. This work hence introduces a practical, easy-to-implement, self-supervised training framework, which involves fast acquisition and self-supervised reconstruction and offers theoretical guarantees.

Figures (9)

• Figure 1: The proposed k-band strategy. (a) Training data consists of k-space bands, with different orientations across training examples. (b) During training, the band is undersampled with a variable-density mask and the network learns to reconstruct images in a self-supervised manner. (c) During inference, the network is not limited to limited-resolution data. It receives variable-density undersampled data from the entire k-space, and reconstructs high-resolution images even though it never saw such examples during training.
• Figure 2: Examples of loss-weighting masks ($\mW$), where the number of distinct bands was (a) $k=10$, (b) $k=30$, (c) $k=180$. (d) Horizontal profile of the $\mW$ mask shown in (c), along the line between the two arrows. Notice that the loss-weighting mask inhibits the loss values in the k-space center and enhances them in the k-space periphery. Therefore, this mask facilitates equal learning across the entire k-space.
• Figure 3: (Top) Visualization of prospective sampling masks for training data acquisition and loss supervision. (Middle) Retrospective 1D Poisson disc undersampling mask used for input to the network during training and inference. (Bottom) Retrospective 2D Poisson disc undersampling mask used for input to the network during training and inference.
• Figure 4: Sampling masks used for input, for the five methods discussed in this work. In these examples $R_{band}=R_{vd}=4$.
• Figure 5: Evaluation of k-band for a range of band widths ($R_{band}$) and fixed inference-time acceleration ($R_{vd}=4$) using the fastMRI knee (green) and brain (blue) datasets. Note that k-band exhibits highly stable performance across a wide range of band widths for the training data (left-to-right).

Theorems & Definitions (8)

• proposition 1
• proof
• corollary 1
• proof
• corollary 2
• proposition 2
• proof
• corollary 3