Tomas Sauer

Gitta Kutyniok, Tomas Sauer

In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of non-stationary bivariate subdivision schemes, which allow to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.

Tomas Sauer

The paper considers a symbolic approach to Prony's method in several variables and its close connection to multivariate polynomial interpolation. Based on the concept of universal interpolation that can be seen as a weak generalization of univariate Chebychev systems, we can give estimates on the minimal number of evaluations needed to solve Prony's problem.

Mariantonia Cotronei, Caroline Moosmüller, Tomas Sauer et al.

We study many properties of level-dependent Hermite subdivision, focusing on schemes preserving polynomial and exponential data. We specifically consider interpolatory schemes, which give rise to level-dependent multiresolution analyses through a prediction-correction approach. A result on the decay of the associated multiwavelet coefficients, corresponding to a uniformly continuous and differentiable function, is derived. It makes use of the approximation of any such function with a generalized Taylor formula expressed in terms of polynomials and exponentials.

Mariantonia Cotronei, Milvia Rossini, Tomas Sauer et al.

Like the continous shearlet transform and their relatives, discrete transformations based on the interplay between several filterbanks with anisotropic dilations provide a high potential to recover directed features in two and more dimensions. Due to simplicity, most of the directional systems constructed so far were using prediction--correction methods based on interpolatory subdivision schemes. In this paper, we give a simple but effective construction for QMF (quadrature mirror filter) filterbanks which are the discrete object between orthogonal wavelet analysis. We also characterize when the filterbank gives rise to the existence of refinable functions and hence wavelets and give a generalized shearlet construction for arbitrary dimensions and arbitrary scalings for which the filterbank construction ensures the existence of an orthogonal wavelet analysis.

Jesús Carnicer, Tomas Sauer

In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon-Berzolari and Chung-Yao. By means of proper H-bases for the vanishing ideal of the configuration, we derive properties of the matrix of first syzygies of this ideal which allow us to draw conclusions on the geometry of the point configuration.

Jean-Louis Merrien, Tomas Sauer

Hermite subdivision schemes act on vector valued data that is not only considered as functions values in $\mathbb{R}^r$, but as consecutive derivatives, which leads to a mild form of level dependence of the scheme. Previously, we have proved that a property called spectral condition or sum rule implies a factorization in terms of a generalized difference operator that gives rise to a "difference scheme" whose contractivity governs the convergence of the scheme. But many convergent Hermite schemes, for example, those based on cardinal splines, do not satisfy the spectral condition. In this paper, we generalize the property in a way that preserves all the above advantages: the associated factorizations and convergence theory. Based on these results, we can include the case of cardinal splines and also enables us to construct new types of convergent Hermite subdivision schemes.

Thomas Lang, Tomas Sauer

Segmentation, i.e., the partitioning of volumetric data into components, is a crucial task in many image processing applications ever since such data could be generated. Most existing applications nowadays, specifically CNNs, make use of voxelwise classification systems which need to be trained on a large number of annotated training volumes. However, in many practical applications such data sets are seldom available and the generation of annotations is time-consuming and cumbersome. In this paper, we introduce a novel voxelwise segmentation method based on active learning on geometric features. Our method uses interactively provided seed points to train a voxelwise classifier based entirely on local information. The combination of an ad hoc incorporation of domain knowledge and local processing results in a flexible yet efficient segmentation method that is applicable to three-dimensional volumes without size restrictions. We illustrate the potential and flexibility of our approach by applying it to selected computed tomography scans where we perform different segmentation tasks to scans from different domains and of different sizes.

Tomas Sauer

Prony's problem in several variables has attracted some attention recently and provides an interesting combination of polynomial ideal theory with analytic and numeric computations. This note points out further connections to Hankel operators of finite rank as they appear in multidimensional moment problems, shift invariance signal spaces, annihilating ideals of filters and factorization of the Hankel matrices and operators by means of Vandermonde matrices. In fact, it turns out that these concepts are essentially equivalent.

Tomas Sauer

Prony's method, in its various concrete algorithmic realizations, is concerned with the reconstruction of a sparse exponential sum from integer samples. In several variables, the reconstruction is based on finding the variety for a zero dimensional radical ideal. If one replaces the coefficients in the representation by polynomials, i.e., tries to recover sparse exponential polynomials, the zeros associated to the ideal have multiplicities attached to them . The precise relationship between the coefficients in the exponential polynomial and the multiplicity spaces are pointed out in this paper.

Thomas Lang, Tomas Sauer

Thresholding is the most widely used segmentation method in volumetric image processing, and its pointwise nature makes it attractive for the fast handling of large three-dimensional samples. However, global thresholds often do not properly extract components in the presence of artifacts, measurement noise or grayscale value fluctuations. This paper introduces Feature-Adaptive Interactive Thresholding (FAITH), a thresholding technique that incorporates (geometric) features, local processing and interactive user input to overcome these limitations. Given a global threshold suitable for most regions, FAITH uses interactively selected seed voxels to identify critical regions in which that threshold will be adapted locally on the basis of features computed from local environments around these voxels. The combination of domain expert knowledge and a rigorous mathematical model thus enables a very exible way of local thresholding with intuitive user interaction. A qualitative analysis shows that the proposed model is able to overcome limitations typically occuring in plain thresholding while staying efficient enough to also allow the segmentation of big volumes.

Rosanna Campagna, Salvatore Mondrone, Tomas Sauer

Offset curves for planar trajectories are interesting in the generation of tool paths for numerically controlled industrial machines and in trajectory planning methods for autonomous driving systems. Theoretical offset curves may exhibit peculiar singularities, including self-intersections, which limit their use in practical applications. Existing approaches address these issue through geometric filtering techniques to detect and remove undesirable features but the computation of accurate and well-behaved offset curves remains a challenging task. We assume a first stage of functional approximation of trajectories by penalized Hermite spline regression enabling the simultaneous fitting of positions and tangents. The regularization is imposed on the second derivatives, effectively mitigating the jerk effect, which is particularly relevant in motion planning and path smoothing applications. Then, taking into account the geometrical pointwise properties of the resulting curve, we design two offset curves through the simultaneous approximation of function values and derivatives. Then, a mathematical model to obtain the so-called bi-offset as most fitting as with the original generator curve is proposed, also relating the offset range and pointwise curvature values. The adaptive reconstruction of the center line from the external boundaries is a topic of interest and is the main focus of our work. Numerical experiments confirm the reliability of our approach at every stage of the resolution process.

Günther Reitberger, Tomas Sauer

An important task when processing sensor data is to distinguish relevant from irrelevant data. This paper describes a method for an iterative singular value decomposition that maintains a model of the background via singular vectors spanning a subspace of the image space, thus providing a way to determine the amount of new information contained in an incoming frame. We update the singular vectors spanning the background space in a computationally efficient manner and provide the ability to perform block-wise updates, leading to a fast and robust adaptive SVD computation. The effects of those two properties and the success of the overall method to perform a state of the art background subtraction are shown in both qualitative and quantitative evaluations.

Juan Manuel Peña, Tomas Sauer

We consider the problem of updating the SVD when augmenting a "tall thin" matrix, i.e., a rectangular matrix $A \in \RR^{m \times n}$ with $m \gg n$. Supposing that an SVD of $A$ is already known, and given a matrix $B \in \RR^{m \times n'}$, we derive an efficient method to compute and efficiently store the SVD of the augmented matrix $[ A B ] \in \RR^{m \times (n+n')}$. This is an important tool for two types of applications: in the context of principal component analysis, the dominant left singular vectors provided by this decomposition form an orthonormal basis for the best linear subspace of a given dimension, while from the right singular vectors one can extract an orthonormal basis of the kernel of the matrix. We also describe two concrete applications of these concepts which motivated the development of our method and to which it is very well adapted.

Florian Boßmann, Tomas Sauer, Nada Sissouno

Mathematical methods of image inpainting involve the discretization of given continuous models. We present a method that avoids the standard pointwise discretization by modeling known variational approaches, in particular total variation (TV), using a finite dimensional spline space. Besides the analysis of the resulting model, we present a numerical implementation based on the alternating method of multipliers. We compare the results numerically with classical TV inpainting and give examples of applications.

Tomas Sauer

The paper gives an extension of Prony's method to the multivariate case which is based on the relationship between polynomial interpolation, normal forms modulo ideals and H--bases.