Restoring medical images with combined noise base on the nonlinear inverse scale space method

Chenwei Li; Donghong Zhao; Chenwei Li; Donghong Zhao

doi:10.3934/mmc.2025016

Mathematical Modelling and Control

2025, Volume 5, Issue 2: 216-235. doi: 10.3934/mmc.2025016

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Research article

Restoring medical images with combined noise base on the nonlinear inverse scale space method

Chenwei Li ,
Donghong Zhao ^,

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received: 10 September 2024 Revised: 24 October 2024 Accepted: 28 November 2024 Published: 09 June 2025

This paper proposes a model that eliminates combined additive and multiplicative noise by merging the Rudin, Osher, and Fatemi (ROF) and I-divergence data fidelity terms. Many important techniques for denoising and restoring medical images were included in this model. The addition of the I-divergence fidelity term increases the complexity of the model and difficulty of the solution in comparison with the ROF. To solve this model, we first proposed the generalized concept of the maximum common factor based on the inverse scale space algorithm. Different from general denoising algorithms, the inverse scale space method exploits the fact that $ u $ starts at some value $ c_0 $ and gradually approaches the noisy image $ f $ as time passes which better handles noise while preserving image details, resulting in sharper and more natural-looking images. Furthermore, a proof for the existence and uniqueness of the minimum solution of the model was provided. The experimental findings reveal that our proposed model has an excellent denoising effect on images destroyed by additive noise and multiplicative noise at the same time. Compared with general methods, numerical results demonstrate that the nonlinear inverse scale space method has better performance and faster running time on medical images especially including lesion images, with combined noises.
- inverse scale space method,
- total variation,
- medical image denoising,
- combined noises
Citation: Chenwei Li, Donghong Zhao. Restoring medical images with combined noise base on the nonlinear inverse scale space method[J]. Mathematical Modelling and Control, 2025, 5(2): 216-235. doi: 10.3934/mmc.2025016

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Abstract

This paper proposes a model that eliminates combined additive and multiplicative noise by merging the Rudin, Osher, and Fatemi (ROF) and I-divergence data fidelity terms. Many important techniques for denoising and restoring medical images were included in this model. The addition of the I-divergence fidelity term increases the complexity of the model and difficulty of the solution in comparison with the ROF. To solve this model, we first proposed the generalized concept of the maximum common factor based on the inverse scale space algorithm. Different from general denoising algorithms, the inverse scale space method exploits the fact that $ u $ starts at some value $ c_0 $ and gradually approaches the noisy image $ f $ as time passes which better handles noise while preserving image details, resulting in sharper and more natural-looking images. Furthermore, a proof for the existence and uniqueness of the minimum solution of the model was provided. The experimental findings reveal that our proposed model has an excellent denoising effect on images destroyed by additive noise and multiplicative noise at the same time. Compared with general methods, numerical results demonstrate that the nonlinear inverse scale space method has better performance and faster running time on medical images especially including lesion images, with combined noises.

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