Nonlinear structure tensors

Abstract

In this article we introduce nonlinear versions of the popular structure tensor, also known as second moment matrix. These nonlinear structure tensors replace the Gaussian smoothing of the classical structure tensor by discontinuity-preserving nonlinear diffusions. While nonlinear diffusion is a well-established tool for scalar and vectorvalued data, it has not often been used for tensor images so far. Two types of nonlinear diffusion processes for tensor data are studied: an isotropic one with a scalar-valued diffusivity, and its anisotropic counterpart with a diffusion tensor. We prove that these schemes preserve the positive semidefiniteness of a matrix field and are therefore appropriate for smoothing structure tensor fields. The use of diffusivity functions of total variation (TV) type allows us to construct nonlinear structure tensors without specifying additional parameters compared to the conventional structure tensor. The performance of nonlinear structure tensors is demonstrated in three fields where the classic structure tensor is frequently used: orientation estimation, optic flow computation, and corner detection. In all these cases the nonlinear structure tensors demonstrate their superiority over the classical linear one. Our experiments also show that for corner detection based on nonlinear structure tensors, anisotropic nonlinear tensors give the most precise localisation.

Adaptive Structure Tensors and their Applications

“…The adaptation quality can be further improved by nonlinear operators, which use the updated data in a kind of feedback loop for the adaptation. Two such nonlinear operators have been proposed for the structure tensor, firstly the concept based on robust statistics by van den Boomgaard and van de Weijer [37], and secondly the techniques based on nonlinear diffusion, proposed by Weickert and Brox [41,7]. These methods will now be explained in more detail.…”

“…Consequently, a discontinuity in one matrix channel inhibits also smoothing in the others. There exists also an anisotropic counterpart to this scheme, which has been introduced in [41,7]. In the anisotropic case not only the amount of diffusion is adapted locally to the data but also the direction of smoothing.…”

“…A very similar approach is to use adaptive Gaussian windows [26,23] for choosing the local neighborhood. Also by nonlinear diffusion one can perform data-adaptive smoothing that avoids the integration of ambiguous data [41,7]. A second possibility to enhance local orientation estimation is to keep the non-adaptive window, but to clearly choose one of the ambiguous orientations by means of robust statistics [37].…”

Nonlinear Diffusion on the 2D Euclidean Motion Group

“…This richer structure is often overlooked, e.g. if one applies component-wise nonlinear diffusion on tensor images [7]. When processing tensor images or orientation scores, it is actually more natural to define the evolution equation on the Euclidean motion group, leading to scale spaces on the Euclidean motion group.…”

A Generic Approach to the Filtering of Matrix Fields with Singular PDEs

“…Extensions of curvature-based PDEs to matrix fields have been proposed in [11] and more recently in [16], based on generalisations of the so-called structure tensor for scalar images to matrix fields. The research on these structure-tensor concepts has been initiated by [19,7]. The approaches to matrix field regularisation suggested in [9] are based on differential geometric considerations.…”