A Tensor Variational Formulation of Gradient Energy Total Variation

Freddie Åström; George Baravdish; Michael Felsberg

cs.CVAug 29, 2026

A Tensor Variational Formulation of Gradient Energy Total Variation

Authors: Freddie Åström, George Baravdish, Michael Felsberg

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

Abstract

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.

Figures & tables

Figure 1: In this work we derive a gradient energy tensor-total variation (GETV) scheme. In this example, our approach clearly boosts the visual impression, PSNR and SSIM performance over EAD and TV in a colour image auto-denoising application.

Panel of Figure 2 — Figure 2: (a) Illustration of eigenvector basis at coordinate $(x,y)$ , the dashed (red) arrows indicate the eigenvectors of $W(\nabla u)$ and $S(\nabla u)$ and thick (black) arrows $\nabla u\nabla^{t}u$ . (b) Illustration of the paraboloid ( 15 ) where we set $c=\tau_{1}q_{1}^{2}+\tau_{2}q_{2}^{2}$ .

Panel of Figure 3 — Figure 3: A test image corrupted with $20$ standard deviation additive Gaussian noise and the corresponding denoised results. Observe that the errors larger than $10\%$ (magenta) are considerably less concentrated at high frequencies for the GETV than the other methods.

Panel of Figure 4 — Figure 4: Error measures, a higher W-PSNR indicates better recovery of high-frequency regions. The visual appearance for selected images is shown in figure 6 .

Panel of Figure 5 — Figure 5: Example from the grey-scale dataset for 15 standard deviation of noise. GETV shows an improvement over EAD and TV, both in PSNR and preservation of fine image structures such as the camera handle. Also, note that the images obtained with GETV appear less blurry than EAD and TV.

Panel of Figure 6 — Figure 6: Results from the Berkeley colour-image dataset with 15 standard deviation of noise where GETV excels. Consider particularly the text on the document and the grass behind the horse on the last row. Note that GETV in general preserves more fine details than EAD and TV, which both tends to oversmooth the images.

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