# A Probabilistic Hadamard U-Net for MRI Bias Field Correction

arXiv: https://arxiv.org/abs/2403.05024 · HTML: https://mlsanity.com/paper/2403.05024/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 27 references that are in the map.
- First posted on arXiv 8 Mar 2024.

## Builds on (most cited first, 5 of 5)

- [The Medical Segmentation Decathlon](https://mlsanity.com/paper/2106.05735/) (2021)
- [A Probabilistic U-Net for Segmentation of Ambiguous Images](https://mlsanity.com/paper/1806.05034/) (2018)
- [A Weakly Supervised Consistency-based Learning Method for COVID-19 Segmentation in CT Images](https://mlsanity.com/paper/2007.02180/) (2020)
- [A Hybrid Quantum-Classical Approach based on the Hadamard Transform for the Convolutional Layer](https://mlsanity.com/paper/2305.17510/) (2023)
- [Discrete Cosine Transform Based Causal Convolutional Neural Network for Drift Compensation in Chemical Sensors](https://mlsanity.com/paper/2011.06681/) (2020)

## Built on by (most cited first, 4 of 4)

- [U-Net with Hadamard Transform and DCT Latent Spaces for Next-day Wildfire Spread Prediction](https://mlsanity.com/paper/2602.11672/) (2026)
- [HTMA-Net: Towards Multiplication-Avoiding Neural Networks via Hadamard Transform and In-Memory Computing](https://mlsanity.com/paper/2509.23103/) (2025)
- [Hybrid Multi-Dimensional MRI Prostate Cancer Detection via Hadamard Network-Based Bias Correction and Residual Networks](https://mlsanity.com/paper/2604.17107/) (2026)
- [ShearFuse-UNet: Hadamard, DCT, and Shearlet Transform Fusion for Next-Day Wildfire Spread Prediction](https://mlsanity.com/paper/2606.14071/) (2026)

## Abstract (arXiv)

Magnetic field inhomogeneity correction remains a challenging task in MRI analysis. Most established techniques are designed for brain MRI by supposing that image intensities in the identical tissue follow a uniform distribution. Such an assumption cannot be easily applied to other organs, especially those that are small in size and heterogeneous in texture (large variations in intensity), such as the prostate. To address this problem, this paper proposes a probabilistic Hadamard U-Net (PHU-Net) for prostate MRI bias field correction. First, a novel Hadamard U-Net (HU-Net) is introduced to extract the low-frequency scalar field, multiplied by the original input to obtain the prototypical corrected image. HU-Net converts the input image from the time domain into the frequency domain via Hadamard transform. In the frequency domain, high-frequency components are eliminated using the trainable filter (scaling layer), hard-thresholding layer, and sparsity penalty. Next, a conditional variational autoencoder is used to encode possible bias field-corrected variants into a low-dimensional latent space. Random samples drawn from latent space are then incorporated with a prototypical corrected image to generate multiple plausible images. Experimental results demonstrate the effectiveness of PHU-Net in correcting bias-field in prostate MRI with a fast inference speed. It has also been shown that prostate MRI segmentation accuracy improves with the high-quality corrected images from PHU-Net. The code will be available in the final version of this manuscript.