Structure Tensor Total Variation

Abstract

Abstract. We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as structure tensor total variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale and vector-valued images. It generalizes several existing variational penalties, including the total variation seminorm and vectorial extensions of it. Meanwhile, thanks to the structure tensor's ability to capture first-order information around a local neighborhood, the STV functionals can provide more robust measures of image variation. Further, we prove that the STV regularizers are convex while they also satisfy several invariance properties w.r.t. image transformations. These properties qualify them as ideal candidates for imaging applications. In addition, for the discrete version of the STV functionals we derive an equivalent definition that is based on the patch-based Jacobian operator, a novel linear operator which extends the Jacobian matrix. This alternative definition allow us to derive a dual problem formulation. The duality of the problem paves the way for employing robust tools from convex optimization and enables us to design an efficient and parallelizable optimization algorithm. Finally, we present extensive experiments on various inverse imaging problems, where we compare our regularizers with other competing regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually.

Introducing diffusion tensor to high order variational model for image reconstruction

“…Roussous and Maragos [10] developed a functional that utilises only eigenvalues of the structure tensor. Similar work by Lefkimmiatis et al [13] used Schatten-E-mail address: zkpan@qdu.edu.cn (Z. Pan).…”

“…Anisotropic diffusion tensor can be used to describe the local geometry at an image pixel, thus making it appealing for various image processing tasks [6][7][8][9][10][11][12][13]. Variational methods allow easy integration of constraints and use of powerful modern optimisation techniques such as primal-dual [14][15][16], fast iterative shrinkagethresholding algorithm [17,18], and alternating direction method of multipliers [2][3][4][19][20][21][22][23][24].…”

Hyperspectral image restoration based on spatio-spectral structure tensor regularization

“…In [13], the correlation of the spatial structure in a local region of a multi/hyperspectral image is exploited by using structure tensor total variation (STV), and high-performance restoration can be achieved without computationally inefficient non-local search. This is because STV is defined as the nuclear norm of the structure tensor, a matrix consisting of gradient components in a local region, and thus it can evaluate the semi-local spatial correlation of images.…”

Image Denoising Using Directional Adaptive Variable Exponents Model

“…In the second step, this information is used to construct an energy expression whose minimization favours solutions respecting the local geometry obtained by the structure tensor analysis. In [21], the so-called structure tensor total variation is proposed to solve inverse problems in imaging. The regularizer is based on penalizing the eigenvalues of the structure tensor.…”

“…The regularizer is based on penalizing the eigenvalues of the structure tensor. The work [20] extends [21] in the sense that in [20] the regularizer is based on the eigenvalues of a non-local version of the structure tensor. In [22], solution adaptive variants of the total variation were considered where the adaptivity is modelled as a fixed point problem.…”