  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.
