Pavel Mrázek

On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs

Soft wavelet shrinkage, total variation (TV) diffusion, TV 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 one-dimensional 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 analyze possibilities to avoid Gibbs-like artifacts for multiscale Haar wavelet shrinkage by scaling the thresholds. 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.

Nonlinear structure tensors

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 vector-valued 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.

ON ROBUST ESTIMATION AND SMOOTHING WITH SPATIAL AND TONAL KERNELS

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 diusion, and regularisation approaches. Although these methods originate in dierent 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.

Adaptive Structure Tensors and their Applications

The structure tensor, also known as second moment matrix or Forstner 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 chapter 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 data-adaptive 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.

Correspondences between wavelet shrinkage and nonlinear diffusion

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet shrinkage denoising.

Diffusion-Inspired Shrinkage Functions and Stability Results for Wavelet Denoising

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of a (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet denoising. Moreover, by transferring stability notions from diffusion filtering to wavelet shrinkage, we derive conditions on the shrinkage function that ensure that shift invariant single-level Haar wavelet shrinkage is maximum–minimum stable, monotonicity preserving, and variation diminishing.

Rotationally Invariant Wavelet Shrinkage

Most two-dimensional methods for wavelet shrinkage are efficient for edge-preserving image denoising, but they suffer from poor rotation invariance. We address this problem by designing novel shrinkage rules that are derived from rotationally invariant nonlinear diffusion filters. The resulting Haar wavelet shrinkage methods are computationally inexpensive and they offer substantially improved rotation invariance.

Diffusion Filters and Wavelets: What Can They Learn from Each Other?

Nonlinear diffusion filtering and wavelet shrinkage are two methods that serve the same purpose, namely discontinuity-preserving denoising. In this chapter we give a survey on relations between both paradigms when space-discrete or fully discrete versions of nonlinear diffusion filters are considered. For the case of space-discrete diffusion, we show equivalence between soft Haar wavelet shrinkage and total variation (TV) diffusion for 2-pixel signals. For the general case of N-pixel signals, this leads us to a numerical scheme for TV diffusion with many favourable properties. Both considerations are then extended to 2-D images, where an analytical solution for 2 × 2 pixel images serves as building block for a wavelet-inspired numerical scheme for TV diffusion. When replacing space-discrete diffusion by fully discrete one with an explicit time discretisation, we obtain a general relation between the shrinkage function of a shift-invariant Haar wavelet shrinkage on a single scale and the diffusivity of a nonlinear diffusion filter. This allows to study novel, diffusion-inspired shrinkage functions with competitive performance, to suggest now shrinkage rules for 2-D images with better rotation invariance, and to propose coupled shrinkage rules for colour images where a desynchronisation of the colour channels is avoided. Finally we present a new result which shows that one is not restricted to shrinkage with Haar wavelets: By using wavelets with a higher number of vanishing moments, equivalences to higher-order diffusion-like PDEs are discovered.

From two-dimensional nonlinear diffusion to coupled Haar wavelet shrinkage

This paper studies the connections between discrete two-dimensional schemes for shift-invariant Haar wavelet shrinkage on one hand, and nonlinear diffusion on the other. We show that using a single iteration on a single scale, the two methods can be made equivalent by the choice of the nonlinearity which controls each method: the shrinkage function, or the diffusivity function, respectively. In the two-dimensional setting, this diffusion-wavelet connection shows an important novelty compared to the one-dimensional framework or compared to classical 2-D wavelet shrinkage: The structure of two-dimensional diffusion filters suggests to use a coupled, synchronised shrinkage of the individual wavelet coefficient channels. This coupling enables to design Haar wavelet filters with good rotation invariance at a low computational cost. Furthermore, by transferring the channel coupling of vector- and matrix-valued nonlinear diffusion filters to the Haar wavelet setting, we obtain well-synchronised shrinkage methods for colour and tensor images. Our experiments show that these filters perform significantly better than conventional shrinkage methods that process all wavelets independently.

Structural Adaptive Smoothing Procedures

An important problem in image and signal analysis is denoising. Given data y j at locations x j , j = 1, ..., N, in space or time, the goal is to recover the original image or signal m j , j = 1, ..., N, from the noisy observations y j , j = 1, ..., N. Denoising is a special case of a function estimation problem: If m j = m(x j ) for some function m(x), we may model the data y j as real-valued random variables Y j satisfying the regression relation