On Tensor-Based PDEs and their Corresponding Variational Formulations with Application to Color Image Denoising

Freddie Åström; George Baravdish; Michael Felsberg

cs.CVAug 23, 2026

On Tensor-Based PDEs and their Corresponding Variational Formulations with Application to Color Image Denoising

Authors: Freddie Åström, George Baravdish, Michael Felsberg

Organizations: Computer Vision Laboratory, Department of E.E., Linköping University · Center for Medical Image Science and Visualization (CMIV), Linköping University · Department of Science and Technology, Linköping University

Abstract

The case when a partial differential equation (PDE) can be considered as an Euler-Lagrange (E-L) equation of an energy functional, consisting of a data term and a smoothness term is investigated. We show the necessary conditions for a PDE to be the E-L equation for a corresponding functional. This energy functional is applied to a color image denoising problem and it is shown that the method compares favorably to current state-of-the-art color image denoising techniques.

Figures & tables

		SSIM				PSNR
Noise	Est	TR	Proposed	RGB	BM3D	TR	Proposed	RGB	BM3D
Lizard
$5$	$17.0$	$\mathbf{0.99}$	$\mathbf{0.99}$	$0.98$	$0.97$	$\mathbf{38.0}$	$\mathbf{38.0}$	$35.2$	$34.1$
$10$	$19.2$	$\mathbf{0.96}$	$\mathbf{0.96}$	$0.94$	$0.95$	$32.7$	$\mathbf{32.8}$	$30.6$	$32.3$
$20$	$24.2$	$0.90$	$0.91$	$0.87$	$\mathbf{0.92}$	$27.3$	$27.8$	$26.7$	$\mathbf{29.3}$
$50$	$57.1$	$0.75$	$\mathbf{0.77}$	$0.68$	$0.73$	$22.6$	$\mathbf{23.5}$	$22.0$	$23.2$

Table 1: SSIM and PSNR values

Panel of Figure 1 — Figure 1: Results. Even rows display some subregions of the image from the respectively previous row. The first four rows are illustrations using additive Gaussian noise of $50$ std, the next two rows of $20$ std, and the last two rows of 70 std noise. Best viewed in color.

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A Tensor Variational Formulation of Gradient Energy Total Variation

Freddie Åström, George Baravdish, Michael Felsberg

Computer Vision Laboratory, Linköping University, Sweden · Center for Medical Image Science and Visualization (CMIV), Linköping University · Department of Science and Technology, Linköping University, Sweden

Image Denoising

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We present a novel variational approach to a tensor-based total variation formulation which is called gradient energy total variation, GETV. We introduce the gradient energy tensor [6] into the GETV and show that the corresponding Euler-Lagrange (E-L) equation is a tensor-based partial differential equation of total variation type. Furthermore, we give a proof which shows that GETV is a convex functional. This approach, in contrast to the commonly used structure tensor, enables a formal derivation of the corresponding E-L equation. Experimental results suggest that GETV compares favourably to other state of the art variational denoising methods such as extended anisotropic diffusion (EAD)[1] and total variation (TV) [18] for gray-scale and colour images.

Abstract

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