VHU-Net: Variational hadamard U-Net for body MRI bias field correction

Bias field artifacts in magnetic resonance imaging (MRI) scans introduce spatially smooth intensity inhomogeneities that degrade image quality and hinder downstream analysis. To address this challenge, we propose a novel variational Hadamard U-Net (VHU-Net) for effective body MRI bias field correction. The encoder comprises multiple convolutional Hadamard transform blocks (ConvHTBlocks), each integrating convolutional layers with a Hadamard transform (HT) layer. Specifically, the HT layer performs channel-wise frequency decomposition to isolate low-frequency components, while a subsequent scaling layer and semi-soft thresholding mechanism suppress redundant high-frequency noise. To compensate for the HT layer's inability to model inter-channel dependencies, the decoder incorporates an inverse HT-reconstructed transformer block, enabling global, frequency-aware attention for the recovery of spatially consistent bias fields. The stacked decoder ConvHTBlocks further enhance the capacity to reconstruct the underlying ground-truth bias field. Building on the principles of variational inference, we formulate a new evidence lower bound (ELBO) as the training objective, promoting sparsity in the latent space while ensuring accurate bias field estimation. Comprehensive experiments on body MRI datasets demonstrate the superiority of VHU-Net over existing state-of-the-art methods in terms of intensity uniformity. Moreover, the corrected images yield substantial downstream improvements in segmentation accuracy. Our framework offers computational efficiency, interpretability, and robust performance across multi-center datasets, making it suitable for clinical deployment. The codes are available at https://github.com/Holmes696/Probabilistic-Hadamard-U-Net.