Soft wavelet shrinkage, total variation (TV) diffusion, total variation regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the 1-D case. First we prove that Haar wavelet shrinkage on a single scale is equivalent to a single step of space-discrete TV diffusion or regularization of two-pixel pairs. In the translationally invariant case we show that applying cycle spinning to Haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of TV diffusion. We prove that space-discrete TV diffusion and TV regularization are identical, and that they are also equivalent to the SIDEs system when a specific force function is chosen. Afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the Laplacian pyramid of the signal. We analyse possibilities to avoid Gibbs-like artifacts for multiscale Haar wavelet shrinkage by scaling the thesholds. Finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and PDE / variational approaches. These methods are based on iterated shift-invariant wavelet shrinkage at multiple scales with scaled thresholds.
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.
This paper deals with establishing relations between a number of widely-used nonlinear filters for digital image processing. We cover robust statistical estimation with (local) M-estimators, local mode filtering in image or histogram space, bilateral filtering, nonlinear diffusion, and regularisation approaches. Although these methods originate in different mathematical theories, we show that their implementation reveals a highly similar structure. We demonstrate that all these methods can be cast into a unified framework of functional minimisation combining nonlocal data and nonlocal smoothness terms. This unification contributes to a better understanding of the individual methods, and it opens the way to new techniques combining the advantages of known filters.
The structure tensor, also known as second moment matrix or Förstner interest operator, is a very popular tool in image processing. Its purpose is the estimation of orientation and the local analysis of structure in general. It is based on the integration of data from a local neighborhood. Normally, this neighborhood is defined by a Gaussian window function and the structure tensor is computed by the weighted sum within this window. Some recently proposed methods, however, adapt the computation of the structure tensor to the image data. There are several ways how to do that. This article wants to give an overview of the different approaches, whereas the focus lies on the methods based on robust statistics and nonlinear diffusion. Furthermore, the dataadaptive structure tensors are evaluated in some applications. Here the main focus lies on optic flow estimation, but also texture analysis and corner detection are considered.
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