Stephan Didas

Highly Accurate Optic Flow Computation with Theoretically Justified Warping

In this paper, we suggest a variational model for optic flow computation based on non-linearised and higher order constancy assumptions. Besides the common grey value constancy assumption, also gradient constancy, as well as the constancy of the Hessian and the Laplacian are proposed. Since the model strictly refrains from a linearisation of these assumptions, it is also capable to deal with large displacements. For the minimisation of the rather complex energy functional, we present an efficient numerical scheme employing two nested fixed point iterations. Following a coarse-to-fine strategy it turns out that there is a theoretical foundation of so-called warping techniques hitherto justified only on an experimental basis. Since our algorithm consists of the integration of various concepts, ranging from different constancy assumptions to numerical implementation issues, a detailed account of the effect of each of these concepts is included in the experimental section. The superior performance of the proposed method shows up by significantly smaller estimation errors when compared to previous techniques. Further experiments also confirm excellent robustness under noise and insensitivity to parameter variations.

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.

Morphology for matrix data: Ordering versus PDE-based approach

Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: one is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering- or PDE-driven morphological operators for matrix-valued data.

Splines in Higher Order TV Regularization

Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m−1 solve also a minimization problem with quadratic data term and m-th order total variation (TV) regularization term. In contrast to problems with quadratic regularization terms involving m-th order derivatives, the spline knots are not known in advance but depend on the input data and the regularization parameter λ. More precisely, the spline knots are determined by the contact points of the m–th discrete antiderivative of the solution with the tube of width 2λ around the m-th discrete antiderivative of the input data. We point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the Sobolev space W2,0m. From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.

Regularity and scale-space properties of fractional high order linear filtering

We investigate the use of fractional powers of the Laplacian for signal and image simplification. We focus both on their corresponding variational techniques and parabolic pseudodifferential equations. We perform a detailed study of the regularisation properties of energy functionals, where the smoothness term consists of various linear combinations of fractional derivatives. The associated parabolic pseudodifferential equations with constant coefficients are providing the link to linear scale-space theory. These encompass the well-known α-scale-spaces, even those with parameter values α > 1 known to violate common maximum-minimum principles. Nevertheless, we show that it is possible to construct positivity-preserving combinations of high and low-order filters. Numerical experiments in this direction indicate that non-integral orders play an essential role in this construction. The paper reveals the close relation between continuous and semi-discrete filters, and by that helps to facilitate efficient implementations. In additional numerical experiments we compare the variance decay rates for white noise and edge signals through the action of different filter classes.

A generic approach to diffusion filtering of matrix-fields

SummaryDiffusion tensor magnetic resonance imaging, is a image acquisition method, that provides matrix- valued data, so-called matrix fields. Hence image processing tools for the filtering and analysis of these data types are in demand. In this article, we propose a generic framework that allows us to find the matrix-valued counterparts of the Perona–Malik PDEs with various diffusivity functions. To this end we extend the notion of derivatives and associated differential operators to matrix fields of symmetric matrices by adopting an operator-algebraic point of view. In order to solve these novel matrix-valued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting. Numerical experiments performed on both synthetic and real world data substantiate the effectiveness of our novel matrix-valued Perona–Malik diffusion filters.

Stability and local feature enhancement of higher order nonlinear diffusion filtering

This paper discusses the extension of nonlinear diffusion filters to higher derivative orders. While such processes can be useful in practice, their theoretical properties are only partly understood so far. We establish important results concerning L2-stability and forward-backward diffusion properties which are related to well-posedness questions. Stability in the L2-norm is proven for nonlinear diffusion filtering of arbitrary order. In the case of fourth order filtering, a qualitative description of the filtering behaviour in terms of forward and backward diffusion is given and compared to second order nonlinear diffusion. This description shows that curvature enhancement is possible with of fourth order nonlinear diffusion in contrast to second order filters where only edges can be enhanced.

Relations between higher order TV regularization and support vector regression

We study the connection between higher order total variation (TV) regularization and support vector regression (SVR) with spline kernels in a one-dimensional discrete setting. We prove that the contact problem arising in the tube formulation of the TV minimization problem is equivalent to the SVR problem. Since the SVR problem can be solved by standard quadratic programming methods this provides us with an algorithm for the solution of the contact problem even for higher order derivatives. Our numerical experiments illustrate the approach for various orders of derivatives and show its close relation to corresponding nonlinear diffusion and diffusion–reaction equations.

A General Structure Tensor Concept and Coherence-Enhancing Diffusion Filtering for Matrix Fields

Coherence-enhancing diffusion filtering is a striking application of the structure tensor concept in image processing. The technique deals with the problem of completion of interrupted lines and enhancement of flow-like features in images. The completion of line-like structures is also a major concern in diffusion tensor magnetic resonance imaging (DT-MRI). This medical image acquisition technique outputs a 3D matrix field of symmetric 3 × 3-matrices, and it helps to visualize, for example, the nerve fibers in brain tissue. As any physical measurement, DT-MRI is subjected to errors causing faulty representations of the tissue corrupted by noise and with visually interrupted lines or fibers.

Combined ℓ2 data and gradient fitting in conjunction with ℓ1 regularization

We are interested in minimizing functionals with ℓ2 data and gradient fitting term and ℓ1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ1 norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ2 gradient fitting term can be used to avoid both edge blurring and staircasing effects.