Bernhard Burgeth

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.

Properties of Higher Order Nonlinear Diffusion Filtering

This paper provides a mathematical analysis of higher order variational methods and nonlinear diffusion filtering for image denoising. Besides the average grey value, it is shown that higher order diffusion filters preserve higher moments of the initial data. While a maximum-minimum principle in general does not hold for higher order filters, we derive stability in the 2-norm in the continuous and discrete setting. Considering the filters in terms of forward and backward diffusion, one can explain how not only the preservation, but also the enhancement of certain features in the given data is possible. Numerical results show the improved denoising capabilities of higher order filtering compared to the classical methods.

Curvature-Driven PDE Methods for Matrix-Valued Images

Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edge-like structures in tensor fields, we first generalise Di Zenzo’s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts.

Median and related local filters for tensor-valued images

We develop a concept for the median filtering of tensor data. The main part of this concept is the definition of median for symmetric matrices. This definition is based on the minimisation of a geometrically motivated objective function which measures the sum of distances of a variable matrix to the given data matrices. This theoretically well-founded concept fits into a context of similarly defined median filters for other multivariate data. Unlike some other approaches, we do not require by definition that the median has to be one of the given data values. Nevertheless, it happens so in many cases, equipping the matrix-valued median even with root signals similar to the scalar-valued situation. Like their scalar-valued counterparts, matrix-valued median filters show excellent capabilities for structure-preserving denoising. Experiments on diffusion tensor imaging, fluid dynamics and orientation estimation data are shown to demonstrate this. The orientation estimation examples give rise to a new variant of a robust adaptive structure tensor which can be compared to existing concepts. For the efficient computation of matrix medians, we present a convex programming framework. By generalising the idea of the matrix median filters, we design a variety of other local matrix filters. These include matrix-valued mid-range filters and, more generally, M-smoothers but also weighted medians and @a-quantiles. Mid-range filters and quantiles allow also interesting cross-links to fundamental concepts of matrix morphology.

How to Choose Interpolation Data in Images

We introduce and discuss shape-based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of

Median Filtering of Tensor-Valued Images

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Morphology for matrix data: Ordering versus PDE-based approach

-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the “importance” of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting into the continuous model based on fat pixels (balls with positive radius) and study by

Morphological Operations on Matrix-Valued Images

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Anisotropic Continuous-Scale Morphology

-convergence the asymptotics when the radius vanishes. In this way, we obtain relevant information about the optimal distribution of the best interpolation pixels. We show that the resulting optimal data sets are identical to sets that can also be motivated using level set ideas and approximation theoretic considerations. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

Mathematical morphology for matrix fields induced by the Loewner ordering in higher dimensions

Novel matrix-valued imaging techniques such as diffusion tensor magnetic resonance imaging require the development of edge-preserving nonlinear filters. In this paper we introduce a median filter for such tensor-valued data. We show that it inherits a number of favourable properties from scalar-valued median filtering, and we present experiments on synthetic as well as on real-world images that illustrate its performance.