# Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction

arXiv: https://arxiv.org/abs/2409.07797 · HTML: https://mlsanity.com/paper/2409.07797/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 12 Sep 2024.

## Builds on (most cited first, 12 of 12)

- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [DiffIR: Efficient Diffusion Model for Image Restoration](https://mlsanity.com/paper/2303.09472/) (2023)
- [Real-world Noisy Image Denoising: A New Benchmark](https://mlsanity.com/paper/1804.02603/) (2018)
- [Multi-channel Weighted Nuclear Norm Minimization for Real Color Image Denoising](https://mlsanity.com/paper/1705.09912/) (2017)
- [Multi-stage image denoising with the wavelet transform](https://mlsanity.com/paper/2209.12394/) (2022)
- [Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction](https://mlsanity.com/paper/1512.01003/) (2015)
- [Fast L1-L2 minimization via a proximal operator](https://mlsanity.com/paper/1609.09530/) (2016)
- [Color Image and Multispectral Image Denoising Using Block Diagonal Representation](https://mlsanity.com/paper/1902.03954/) (2019)
- [Multi-channel Nuclear Norm Minus Frobenius Norm Minimization for Color Image Denoising](https://mlsanity.com/paper/2209.08094/) (2022)
- [Weighted Truncated Nuclear Norm Regularization for Low-Rank Quaternion Matrix Completion](https://mlsanity.com/paper/2101.02443/) (2021)
- [Fast, nonlocal and neural: a lightweight high quality solution to image denoising](https://mlsanity.com/paper/2403.03488/) (2024)
- [How to Best Combine Demosaicing and Denoising?](https://mlsanity.com/paper/2408.06684/) (2024)

## Built on by (most cited first, 6 of 6)

- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)
- [Infrared-Assisted Single-Stage Framework for Joint Restoration and Fusion of Visible and Infrared Images under Hazy Conditions](https://mlsanity.com/paper/2411.12586/) (2024)
- [Quaternionic Reweighted Amplitude Flow for Phase Retrieval in Image Reconstruction](https://mlsanity.com/paper/2501.02180/) (2025)
- [A Novel Wasserstein Quaternion Generative Adversarial Network for Color Image Generation](https://mlsanity.com/paper/2512.08542/) (2025)
- [FSCM: Frequency-Enhanced Spatial-Spectral Coupled Mamba for Infrared Hyperspectral Image Colorization](https://mlsanity.com/paper/2605.15880/) (2026)
- [Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm](https://mlsanity.com/paper/2606.19046/) (2026)

## Abstract (arXiv)

Color image restoration methods typically represent images as vectors in Euclidean space or combinations of three monochrome channels. However, they often overlook the correlation between these channels, leading to color distortion and artifacts in the reconstructed image. To address this, we present Quaternion Nuclear Norm Minus Frobenius Norm Minimization (QNMF), a novel approach for color image reconstruction. QNMF utilizes quaternion algebra to capture the relationships among RGB channels comprehensively. By employing a regularization technique that involves nuclear norm minus Frobenius norm, QNMF approximates the underlying low-rank structure of quaternion-encoded color images. Theoretical proofs are provided to ensure the method's mathematical integrity. Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.