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A Tensor Variational Formulation of Gradient Energy Total Variation

Freddie Åström, George Baravdish, Michael Felsberg

arXiv:2608.29172v1cs.CV

TL;DR

Existing denoising formulations based on structure tensors do not support an explicit formal Euler–Lagrange derivation, while conventional smoothing can remove image structure. The paper introduces GETV by placing the gradient energy tensor in a total variation functional, proves convexity, and derives a tensor-based Euler–Lagrange equation. Experiments suggest competitive denoising for grayscale and colour images, particularly for fine structures and high-frequency information.

  • Problem

    Existing structure-tensor total variation formulations do not allow an explicit formal Euler–Lagrange derivation, while basic low-pass denoising removes image structure along with noise.

  • Method

    The paper introduces the gradient energy tensor into a total variation regularization functional and formulates the resulting tensor-based diffusion scheme variationally.

  • Results

    GETV is proved convex, and experiments suggest favourable denoising performance compared with EAD and TV for grayscale and colour images.

  • Takeaways & Limitations

    GETV is reported as suitable for denoising images containing high-frequency information such as fine structures.

  • Takeaways & Limitations

    The gradient energy tensor uses second- and third-order derivatives, making GET sensitive to noise, and it is generally not positive semidefinite.

Abstract

from arXiv · show

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.

1 Computer Vision Laboratory, Link¨oping University, Sweden

The paper lists an affiliation with the Center for Medical Image Science and Visualization and Linköping University in Sweden.

  • The Center for Medical Image Science and Visualization is listed as an affiliation.
  • Linköping University is also listed as an institutional affiliation.
  • The listed institutions are located in Sweden.

1 Introduction

The paper introduces GETV, a tensor-based total variation formulation using the gradient energy tensor, to address limitations of existing denoising approaches. It provides a formal variational framework and reports competitive results for grayscale and colour images.

  • Introduction: Low-pass filtering removes noise but also removes image structure, motivating adaptive diffusion and tensor-based regularization.Standard TV can produce cartoon-like images, while structure tensors may miss structures smaller than their averaging-filter width.
  • Introduction: The proposed method is reported to preserve fine, high-frequency structures better than competing approaches in denoising examples.GETV preserves sharp edges and details, while EAD tends to oversmooth images and TV fails on some auto-tuned colour cases.
  • Related works: Existing structure-tensor formulations do not permit an explicit formal Euler–Lagrange derivation, whereas the proposed framework does.
  • Contributions: GETV inserts the gradient energy tensor into a total variation regularization functional.The formulation considers both eigenvalues and eigenvectors of the gradient energy tensor.
  • Contributions: GETV is proved to be a convex functional.
  • Contributions: GETV combines EAD and TV characteristics and achieves highly competitive grayscale and colour image denoising results.

2 Variational approach to image enhancement

The section develops variational image diffusion using fidelity and regularization terms, then explains tensor-based anisotropic diffusion through the structure tensor and its diffusion tensor.

  • 2.1 Energy minimization: The variational framework minimizes an energy combining data fidelity with a regularization term weighted by λ.The regularization controls smoothness, while the fidelity term represents agreement with the observed image.
  • 2.1 Energy minimization: The Euler-Lagrange equation characterizes the stationary point minimizing the energy under homogeneous Neumann boundary conditions.The boundary condition is ∇u · n = 0 on the image boundary.
  • 2.1 Energy minimization: Total variation provides an alternative variational regularizer whose minimizer can be obtained through a parabolic initial value problem or primal-dual formulations.A constant offset can also be incorporated into the norm to avoid discontinuous solutions.
  • 2.2 Tensor-based anisotropic diffusion: Tensor-based anisotropic diffusion smooths images parallel to local image structure using a transformed structure tensor.The structure tensor estimates local orientation, and a nonlinear diffusivity function produces the diffusion tensor.
  • 2.2 Tensor-based anisotropic diffusion: The structure tensor is a Gaussian-windowed second-moment matrix whose dominant eigenvector is orthogonal to image structure.The diffusion tensor attenuates filtering across structures through an eigenvalue-dependent diffusivity function.

3 Gradient energy tensor

The gradient energy tensor describes directional gradient energy without the structure tensor’s post-convolution, while higher-order derivatives improve orientation sensitivity but increase noise sensitivity.

  • 3 Gradient energy tensor: The gradient energy tensor is a real-valued symmetric tensor that determines the directional energy distribution of the signal gradient.Its symmetry ensures real eigenvalues and supports analysis through eigenvalue decomposition.
  • 3 Gradient energy tensor: Unlike the structure tensor, the gradient energy tensor does not require post-convolution of tensor components to form a rank-2 tensor.The paper expresses it equivalently using the image gradient, Hessian H, and ∇∆u.
  • 3 Gradient energy tensor: The gradient energy tensor uses second- and third-order derivatives, making it sensitive to noise while capturing orientations unavailable to the structure tensor.This derivative-based sensitivity is the principal trade-off identified for GET.
  • 3 Gradient energy tensor: The gradient energy tensor is not generally positive semi-definite; in two dimensions, positivity is reflected in the signs of its eigenvalues.The paper analyzes sufficient conditions through its component representation and trace-based eigenvalue argument.
  • 3 Gradient energy tensor: Because GET is not necessarily positive semi-definite, the paper defines GET+ as a positive semi-definite tensor.This construction provides the tensor used when positive semi-definiteness is required.
  • 3 Gradient energy tensor: The formulation considers both eigenvalues and eigenvectors of GET.The eigenvectors are denoted by V and the eigenvalues by ι1,2.

4 Introducing Gradient Energy Total Variation

GETV incorporates the gradient energy tensor into a tensor-based total variation functional, yielding a convex regularizer and a formally derived Euler–Lagrange equation. Its tensor formulation adapts filtering to image structures and is evaluated against EAD and TV.

  • GETV is introduced as a gradient energy tensor-based regularization functional for total variation denoising.
  • The eigendecomposition of ∇u∇tu is well-defined, including the zero-gradient case, using p = ∇u/|∇u| when |∇u| ≠ 0 and P = I otherwise.
  • The GETV functional R(u) is convex with respect to u.The proof represents the integrand as a quadratic form whose associated paraboloid has positive curvature everywhere.
  • With W(∇u) = I, GETV reduces to the standard total variation integrand |∇u|.
  • The formal minimization uses a tensor-based Euler–Lagrange theorem and computes derivatives of W through an explicit eigendecomposition.The resulting PDE is solved numerically with forward Euler and regularized finite differences.
  • The gradient energy tensor determines eigenvalues and eigenvectors that adapt filtering parallel to image structures.The exponential weighting makes λ1 small and λ2 larger near an image structure.

5 Application to image enhancement

GETV is evaluated against EAD and TV on grayscale and Berkeley color images corrupted by Gaussian noise, using multiple error measures and visual inspection. The results indicate that GETV better preserves fine and high-frequency structures, while EAD and TV tend to oversmooth or perform poorly in some settings.

  • Experiments’ setup: The evaluation uses standard grayscale images and 50 Berkeley color images corrupted with Gaussian noise at standard deviations 5, 10, 15, and 20.Grayscale images are 256 × 256, while Berkeley color images are 481 × 321; color processing uses the decorrelation CIELAB transform.
  • Experiments’ setup: GETV, EAD, and TV are compared using PSNR, SSIM, W-PSNR, A-IQA, and visual image quality.W-PSNR weights edge regions to assess preservation of high-frequency structures, while A-IQA supports parameter selection without using the noise-free image.
  • Results: For grayscale images, TV performs worse than EAD and GETV, while GETV preserves sharper details such as the cameraman’s camera handle.The reported A-IQA values are similar across methods, but visual inspection shows more detail preservation with GETV.
  • Results: For color images at lower noise levels, GETV and EAD differ marginally in PSNR and SSIM, but GETV is reported as more robust to variance.Visual comparisons particularly show improved clarity in document text, waves, and grass details for GETV.
  • Results: EAD and TV tend to oversmooth images, whereas GETV generally preserves more fine details in high-frequency regions.Manual parameter adjustment can improve error measures for some images, but the paper states that this is infeasible for large image collections.

6 Conclusion

The paper presents GETV as a tensor-based total variation approach using the gradient energy tensor. Its results suggest suitability for images containing high-frequency information and favorable denoising behavior compared with EAD and TV.

  • The paper presents a gradient energy total variation functional as a variational approach to tensor-based total variation.
  • GETV is reported as suitable for images containing high-frequency information such as fine structures.
  • Using A-IQA, the diffusion formulation is reported to perform well in denoising applications compared with EAD and TV.
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