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Generalized $L_p$-Norm Based Non-Local Means Denoising
Research Article | Published: 25 January 2026
ICCK Journal of Image Analysis and Processing Volume 2, Issue 1, 2026: 17-26
Volume 2, Issue 1, ICCK Journal of Image Analysis and Processing
Volume 2, Issue 1, 2026
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ICCK Journal of Image Analysis and Processing, Volume 2, Issue 1, 2026: 17-26

Open Access | Research Article | 25 January 2026
Generalized $L_p$-Norm Based Non-Local Means Denoising
by
Yash Gupta Yash Gupta 1
Jagan Mohan Jonnalagadda Jagan Mohan Jonnalagadda 1 *
1 Department of Mathematics, Birla Institute of Technology and Science, Pilani – Hyderabad Campus, Hyderabad 500078, India
* Corresponding Author: Jagan Mohan Jonnalagadda, [email protected]
DOI: 10.62762/JIAP.2025.744487
ARK: ark:/57805/jiap.2025.744487
Received: 19 December 2025, Accepted: 22 January 2026, Published: 25 January 2026  
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Abstract
Non-local means (NL-means) is a state-of-the-art image denoising algorithm that leverages self-similarity by averaging similar patches weighted by the classic $L_2$-norm distance. In this work, we extend the similarity measure to arbitrary $L_p$-norms ($1 \le p \le \infty$) and investigate their impact on denoising performance. We implement and evaluate NL-means with $p = 1, 2, 3, 4, \infty$ and compare via quantitative metrics (MSE, MAE, PSNR, SSIM), residual analysis, and visual inspection. Experiments on the \emph{Lena} image corrupted with AWGN ($\sigma = 20$), a widely used benchmark setting in the denoising literature, show that while $L_2$-norm remains optimal overall, other norms offer nuanced trade-offs in edge preservation and robustness. Our analysis demonstrates that $L_1$-norm offers superior impulse noise resilience, while higher norms like $L_3$ and $L_4$ exhibit enhanced structure preservation in gradient-rich regions. Additionally, we present a parameter sensitivity study showing how the optimal filtering parameter $h$ varies across different $L_p$-norms, and analyze computational complexity trade-offs. These findings, which are consistent with the general theoretical properties of $L_p$-norms, provide insights into optimizing the NL-means algorithm for specific image characteristics and noise distributions.
Graphical Abstract
Generalized $L_p$-Norm Based Non-Local Means Denoising
Keywords
non-local means
image denoising
norm
quantitative metrics
residual analysis
Data Availability Statement
The data and code supporting the findings of this study are available at the following repository: https://github.com/YashGupta3003/nonlocal-means-lp
Funding
This work was supported without any funding.
Conflicts of Interest
The authors declare no conflicts of interest.
AI Use Statement
The authors declare that no generative AI was used in the preparation of this manuscript.
Ethical Approval and Consent to Participate
Not applicable.
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Cite This Article
APA Style
Gupta, Y., & Jonnalagadda, J. M. (2026). Generalized Lp-Norm Based Non-Local Means Denoising. ICCK Journal of Image Analysis and Processing, 2(1), 17–26. https://doi.org/10.62762/JIAP.2025.744487
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TY  - JOUR
AU  - Gupta, Yash
AU  - Jonnalagadda, Jagan Mohan
PY  - 2026
DA  - 2026/01/25
TI  - Generalized $L_p$-Norm Based Non-Local Means Denoising
JO  - ICCK Journal of Image Analysis and Processing
T2  - ICCK Journal of Image Analysis and Processing
JF  - ICCK Journal of Image Analysis and Processing
VL  - 2
IS  - 1
SP  - 17
EP  - 26
DO  - 10.62762/JIAP.2025.744487
UR  - https://www.icck.org/article/abs/JIAP.2025.744487
KW  - non-local means
KW  - image denoising
KW  - norm
KW  - quantitative metrics
KW  - residual analysis
AB  - Non-local means (NL-means) is a state-of-the-art image denoising algorithm that leverages self-similarity by averaging similar patches weighted by the classic $L_2$-norm distance. In this work, we extend the similarity measure to arbitrary $L_p$-norms ($1 \le p \le \infty$) and investigate their impact on denoising performance. We implement and evaluate NL-means with $p = 1, 2, 3, 4, \infty$ and compare via quantitative metrics (MSE, MAE, PSNR, SSIM), residual analysis, and visual inspection. Experiments on the \emph{Lena} image corrupted with AWGN ($\sigma = 20$), a widely used benchmark setting in the denoising literature, show that while $L_2$-norm remains optimal overall, other norms offer nuanced trade-offs in edge preservation and robustness. Our analysis demonstrates that $L_1$-norm offers superior impulse noise resilience, while higher norms like $L_3$ and $L_4$ exhibit enhanced structure preservation in gradient-rich regions. Additionally, we present a parameter sensitivity study showing how the optimal filtering parameter $h$ varies across different $L_p$-norms, and analyze computational complexity trade-offs. These findings, which are consistent with the general theoretical properties of $L_p$-norms, provide insights into optimizing the NL-means algorithm for specific image characteristics and noise distributions.
SN  - 3068-6679
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  -
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@article{Gupta2026Generalize,
  author = {Yash Gupta and Jagan Mohan Jonnalagadda},
  title = {Generalized \$L\_p\$-Norm Based Non-Local Means Denoising},
  journal = {ICCK Journal of Image Analysis and Processing},
  year = {2026},
  volume = {2},
  number = {1},
  pages = {17-26},
  doi = {10.62762/JIAP.2025.744487},
  url = {https://www.icck.org/article/abs/JIAP.2025.744487},
  abstract = {Non-local means (NL-means) is a state-of-the-art image denoising algorithm that leverages self-similarity by averaging similar patches weighted by the classic \$L\_2\$-norm distance. In this work, we extend the similarity measure to arbitrary \$L\_p\$-norms (\$1 \le p \le \infty\$) and investigate their impact on denoising performance. We implement and evaluate NL-means with \$p = 1, 2, 3, 4, \infty\$ and compare via quantitative metrics (MSE, MAE, PSNR, SSIM), residual analysis, and visual inspection. Experiments on the \emph{Lena} image corrupted with AWGN (\$\sigma = 20\$), a widely used benchmark setting in the denoising literature, show that while \$L\_2\$-norm remains optimal overall, other norms offer nuanced trade-offs in edge preservation and robustness. Our analysis demonstrates that \$L\_1\$-norm offers superior impulse noise resilience, while higher norms like \$L\_3\$ and \$L\_4\$ exhibit enhanced structure preservation in gradient-rich regions. Additionally, we present a parameter sensitivity study showing how the optimal filtering parameter \$h\$ varies across different \$L\_p\$-norms, and analyze computational complexity trade-offs. These findings, which are consistent with the general theoretical properties of \$L\_p\$-norms, provide insights into optimizing the NL-means algorithm for specific image characteristics and noise distributions.},
  keywords = {non-local means, image denoising, norm, quantitative metrics, residual analysis},
  issn = {3068-6679},
  publisher = {Institute of Central Computation and Knowledge}
}
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CC BY Copyright © 2026 by the Author(s). Published by Institute of Central Computation and Knowledge. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
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