Abstract
We propose a strictly convex functional in which the regular term consists of the total variation term and an adaptive logarithm based convex modification term. We prove the existence and uniqueness of the minimizer for the proposed variational problem. The existence, uniqueness, and long-time behavior of the solution of the associated evolution system is also established. Finally, we present experimental results to illustrate the effectiveness of the model in noise reduction, and a comparison is made in relation to the more classical methods of the traditional total variation (TV), the Perona-Malik (PM), and the more recent D-α-PM method. Additional distinction from the other methods is that the parameters, for manual manipulation, in the proposed algorithm are reduced to basically only one.
Highlights
Noise removal, edge detection, contrast enhancement, inpainting, and segmentation have been the subject of intense mathematical image analysis and processing research for nearly three decades
(e) D-α-PM model that, even though in performance metrics, especially in terms of peak-signal-tonoise ratio (PSNR) and mean absolute deviation/error (MAE), the D-α-PM method seems to perform better, the similarity curves attest that our method generates restored images that more closely match the original image than the results obtained by the D-α-PM method (see Figures 2(e) and 4(e))
We have proposed a modified total variation model based on the strictly convex modification for image denoising
Summary
Edge detection, contrast enhancement, inpainting, and segmentation have been the subject of intense mathematical image analysis and processing research for nearly three decades. A number of edge indicators have been proposed and logically grafted into the partial differential equation (PDE) based evolution equations [7, 11, 20] Some of these PDEs originate from variational problems. U∈BV(Ω) Ω where 0 ≤ α(x) ≤ 1 is a control factor which controls the speed of diffusion depending on whether the region is homogeneous or an edge Further literature surveys attest to the fact that research in effective regularization functionals which have the ability to generate diffusion processes that restore images, while simultaneously preserving critical images features, the analysis, and practical implementation of such models, is still an extremely active concern. In this paper, we propose a new adaptive total variation (TV) formulation for image denoising, which is strictly convex.
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