Existence and Uniqueness of Weak Solutions for a New Class of Convex Optimization Problems Related to Image Analysis

Anas Tiarimti Alaoui,Mostafa Jourhmane,Jen-Chih Yao

doi:10.1155/2021/6691795

Anas Tiarimti Alaoui, Mostafa Jourhmane + Show 1 more

Open Access

https://doi.org/10.1155/2021/6691795

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Abstract

This paper proposes a new anisotropic diffusion model in image restoration that is understood from a variational optimization of an energy functional. Initially, a family of new diffusion functions based on cubic Hermite spline is provided for optimal image denoising. After that, the existence and uniqueness of weak solutions for the corresponding Euler–Lagrange equation are proven in an appropriate functional space, and a consistent and stable numerical model is also shown. We complement this work by illustrating some experiments on different actual brain Magnetic Resonance Imaging (MRI) scans, showing the proposed model’s impressive results.

Highlights

During the last few decades, the digital image has been significantly utilized as a noninvasive medical diagnosis or treatment tool
Since a digital image contains relevant information that should be carefully extracted from it, many engineers and mathematicians have established numerous techniques and theories [2,3,4] to collect quantitative or qualitative data
Along with the meaningful details, digital images carry false information as noise and other undesirable artifacts due to various reasons. is erroneous information can occur during image formation, transmission, or recording processes. erefore, one needs good models to identify and eliminate these noises while preserving relevant information and structures

Summary

Introduction

During the last few decades, the digital image has been significantly utilized as a noninvasive medical diagnosis or treatment tool. Many approaches have been developed to process images and showed great interest in using the variational approach [5,6,7,8,9,10,11,12,13,14,15], which attempts to establish mathematical models strictly related to the physical world by defining a diffusion equation from a particular optimization problem, which generally has the following form: minu􏼚F(u). Perona and Malik (Perona–Malik) [16] were the first to propose such a model in image processing, which corresponds to nonconvex energy functional This property leads to multiple solutions with staircase effects observed in practice [6, 17].

Novel Anisotropic Diffusion Model for Image Restoration

Mathematical Investigation of the Proposed Model

Numerical Implementation and Experimental Results

Experimental Procedures and Results