Frank Bauer: Research and Development in Applied Mathematics (Publications)

Stephani, H., Hermann, M., Bauer, F. and Heise, B.

Wavelet-Based Channel Reduction and Clustering for Terahertz Hyperspectral Data Abstract:
Nowadays, terahertz time-domain spectroscopy is a sophisticated imaging technique of increasing importance in security control applications as well as non-destructive testing. In an hyperspectral image a full spectrum of the range of several terahertz (THz) is assigned to each pixel and creates an enormous amount of data. Therefore, advanced methods for automated data analysis are required. We present a wavelet-based approach for channel reduction and feature selection for a subsequent clustering leading to an image segmentation. The main focus of our method is set on the appropriate dimensionality reduction adapted to the THz spectral characteristics of the samples under investigation. A feature reduction to less than 5% is achieved, thereby enabling a channel-wise image processing on the reduced data set. Furthermore, unsupervised classification is chosen for an automatized segmentation including all channel information represented in the wavelet domain. Relevant characteristics of the THz spectra are preserved by our feature selection, in particular the distribution of the peak position and peak depth. The proposed method for channel reduction is verified by extensive simulations at first. Finally, it is demonstrated on various real-world measurements of chemical compounds. The improved performance of the analysis on the reduced feature set could be shown in comparison with the evaluation on the full data set. ,

Journal of Applied Spectroscopy (submitted 2010)

Survey:
Bauer, F. and Lukas, M.

Comparing parameter choice methods for regularization of ill-posed problems Abstract:
In the literature on regularization, many different parameter choice methods have been proposed in both deterministic and stochastic settings. However, based on the available information, it is not always easy to know how well a particular method will perform in a given situation, and how it compares to other methods. This paper reviews most of the existing parameter choice methods, and evaluates and compares them in a large simulation study for spectral cut-off and Tikhonov regularization. The test cases cover a wide range of linear inverse problems with both white and colored stochastic noise. The results show some marked differences between the methods, in particular, in their stability with respect to the noise and its type. We conclude with a table of properties of the methods and a summary of the simulation results, from which we identify the best methods. ,

Inverse Problems (submitted 2010)

(Preprint)

Bauer, F. and Mathé, P.

Parameter Choice Methods Using Minimization Schemes Abstract:
In this paper we establish a generalized framework, which allows to prove convergenence and optimality of parameter choice schemes for inverse problems based on minimization in a generic way. We show that the well known quasi-optimality criterion falls in this class. Furthermore we present a new parameter choice method and prove its convergence by using this newly established tool. ,

Journal of Complexity (submitted 2010)

(Preprint)

Bauer, F.

Parameter Choice by Fast Balancing Abstract:
Choosing the regularization parameter for inverse problems is of major importance for the performance of the regularization method.
We will introduce a fast version of the Lepskij balancing principle and show that it is a valid parameter choice method for Tikhonov regularization both in a deterministic and a stochastic noise regime as long as minor conditions on the solution are fulfilled. ,

(2010)

(Preprint)

Bauer, F.

Applying Lepskij-Balancing in Practice Abstract:
In practice, the Lepskij balancing principle for choosing the regularization parameter in inverse problems has exhibited a better behavior than one would have expected from just looking at the proofs.
We will present an explanation for this behavior by using a stochastic model for noise and initial data. Furthermore, we will prove that a small modification of the algorithm also improves the (theoretical) performance of the method, in both speed and accuracy. ,

(2010)

(Preprint)

Bauer, F., Schöner, H., Dorington, A., Heise, B., Wieser, V., Payne, A., Cree, M. and Moser, B.,

Image Processing for 3D-Scans Generated by TOF Range Cameras Abstract:
Time-of-flight full-field range cameras use a correlative imaging technique to generate three-dimensional measurements of the environment. Though reliable and cheap they have the disadvantage of high measurement noise and errors that limit the practical use of these cameras in industrial applications.
We show how some of these limitations can be overcome with specially adapted image processing techniques that can be used to reduce the measurement noise. We compare three different de-noising techniques using data acquired from two different off-the-shelf cameras. Best results were obtained with the Anisotropic Diffusion technique, and we demonstrate how this algorithm can be implemented in real time using a Graphics Processing Unit. ,

IEEE/OSA Journal of Display Technology (submitted 2009)

Bauer, F. and Gutting, M.

Locally Compact Orthogonal Wavelets on the Sphere Abstract:
Using a stereographical projection to the plane we construct an O(N log(N)) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 minutes on a standard computer using our MATLAB implementation. ,

Mathematical Methods in the Applied Sciences (submitted 2009)

(Preprint)

Bauer, F. and Clason, C.

On Theoretical Limits in Parallel Magnetic Resonance Imaging Abstract:
Based on a Fourier series expression of true image, receiver sensitivities, and measurements, it is possible to give theoretical limits for the perfect reconstructability of image and sensitivities in parallel magnetic resonance imaging. These limits depend on the smoothness of the sensitivities, number of receiver coils, and size of the acquired k-space measurement window. Different types of a priori information can be incorporated in the determination of these limits. Furthermore, the method employed is constructive and can serve as the basis for a nonlinear reconstruction scheme, as is shown using data from a simulated phantom. ,

IEEE Transactions on Image Processing (submitted 2008)

(Preprint)

Bauer, F. and Kindermann, S.

Recent Results on the Quasi-Optimality Principle Abstract:
We give a survey of results concerning the convergence and convergence rates and oracle inequalities for the quasi-optimality parameter choice in the most common regularization methods. ,

Inverse and Ill-posed Problems Vol. 17, Iss. 1, pp. 5-18 (2009)

Bauer, F., Hohage, T. and Munk, A.

Iteratively Regularized Gauss-Newton Method for Nonlinear Inverse Problems with Random Noise Abstract:
We study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. It is shown that the expected square error is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a-priori knowledge of the smoothness of the solution. For unknown smoothness the stopping index can be chosen adaptively based on Lepskij's balancing principle. For this stopping rule we establish an oracle inequality, which implies order optimal rates for deterministic errors, and optimal rates up to a logarithmic factor for random noise. The performance and the statistical properties of the proposed method are illustrated by Monte-Carlo simulations. ,

SIAM Journal on Numerical Analysis Vol. 47, Nr. 3, pp. 1827 - 1846 (2009)

(Preprint)

Moser, B., Schöner, H., Dorington, A., Payne, A., Cree, M., Bauer, F. and Heise, B.

A Clustering Based Denoising Technique for Range Images of Time of Flight Cameras Abstract:
A relatively new technique for measuring the 3D structure of visual scenes is provided by time of flight (TOF) cameras. Reflections of modulated light waves are recorded by a parallel pixel array structure. The time series at each pixel of the resulting image stream is used to estimate travelling time and thus range information. This measuring technique results in noise levels with variances changing over several orders of magnitude dependent on the illumination and material parameters.
This makes application of traditional global denoising techniques suboptimal. Using free aditional information from the camera we can get local information by clustering which allows for locally adapted smoothing. To illustrate the success of this method, we compare it with pure time averaging and edge preserving smoothing.
We show that this mathematical technique works without individual adaptations on two camera systems with highly different noise characteristics. ,

Proceedings CIMCA08, pp. 989 - 994 (2008)

Anzengruber, S., Bauer, F., Leitao, A. and Ramlau, R.

New Algorithms for Parallel MRI Abstract:
Magnetic Resonance Imaging with parallel data acquisition requires algorithms for reconstructing the patient’s image from a small number of measured lines of the Fourier domain (k-space). In contrast to well-known algorithms like SENSE and GRAPPA and its flavors we consider the problem as a non-linear inverse problem. However, in order to avoid cost intensive derivatives we will use Landweber-Kaczmarz iteration and in order to improve the overall results some additional sparsity constraints. ,

Journal of Physics, Conference Series Vol. 135, 012009 (8pp) (2008)

Moser, B., Bauer, F., Elbau, P., Heise, B. and Schöner, H.

Denoising techniques for raw 3D data of TOF cameras based on clustering and wavelets Abstract:
In order to measure the 3D structure of a number of objects a comparably new technique in computer vision exists, namely time of ight (TOF) cameras. The overall principle is rather easy and has been applied using sound or light for a long time in all kind of sonar and lidar systems. However in this approach one uses modulated light waves and receives the signals by a parallel pixel array structure. Out of the travelling time at each pixel one can estimate the depth structure of a distant object. The technique requires measuring the intensity differences and ratios of several pictures with extremely high accuracy; therefore one faces in practice rather high noise levels. Object features as re ectance and roughness in uence the measurement results. This leads to partly high noise levels with variances dependent on the illumination and material parameters. It can be shown that a reciprocal relation between the variance of the phase and the squared amplitude of the signals exists. On the other hand, objects can be distinguished using these dependencies on surface characteristics. It is shown that based on local variances assigned to separated objects appropriate denoising can be performed based on Wavelets and edge-preserving smoothing methods. ,

Proceedings SPIE conference Vol. 6805, Nr. 1, pp. 68050E (2008)

Bauer, F. and Kindermann, S.

The Quasi-Optimality Criterion for Classical Inverse Problems Abstract:
The quasi-optimality criterion chooses the regularization parameter in inverse problems without requiring knowledge about the noise level. It it well-known that this cannot yield convergence for ill-posed problems in the worst case. In this paper we establish conditions providing lower bounds on the approximation error and the propagated noise error, such that these terms can be estimated from above and below by a geometric series. Using these we can show convergence and optimal order error bounds for Tikhonov regularization with the quasi-optimality criterion both in the case of deterministic problems as well as for stochastic noise. ,

Inverse Problems Vol. 24, 035002 (20 pp) (2008)

Bauer, F., Hohage, T. and Munk, A.
Newton Methods for Nonlinear Inverse Problems with Random Noise,
PAMM Vol. 7; Nr. 1, pp. 2060007-2060008 (2008)

Bauer, F. and Reiss, M.

Regularisation independent of the noise level: an analysis of quasi-optimality Abstract:
The quasi-optimality criterion chooses the regularization parameter in inverse problems without taking into account the noise level. This rule works remarkably well in practice, although Bakushinskii has shown that there are always counterexamples with very poor performance. We propose an average case analysis of quasi-optimality for spectral cut-off estimators (also known as truncated singular value decomposition, TSVD) and we prove that the quasi-optimality criterion determines estimators which are rate-optimal on average. Its practical performance is illustrated with a calibration problem from mathematical finance. ,

Inverse Problems Vol. 24, 055009 (16pp) (2008)

(Preprint)

Bauer, F.,

Split Operators for Oblique Boundary Value Problems Abstract:
In the field of gravity determination a special kind of boundary value problem respectively ill-posed satellite problem occurs; the data and hence side condition of our PDE are oblique second order derivatives of the gravitational potential. For this computationally demanding problem with millions of data points one classically one just uses the derivatives in radial direction because they commute with the Laplace operator $\Delta$. We will investigate in what extent this approach can also be used for other derivatives. We classify all first and purely second order operators $D$ which fulfill $\Delta D v = 0$ if $\Delta v = 0$. This allows us to solve the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of $D$, i.e. integration. ,

Applicable Analysis, Vol. 87; Nr. 1, pp. 45-57 (2008)

(Preprint)

Bauer, F. and Kannengiesser, S.

An Alternative Approach to the Image Reconstruction for Parallel Data Acquisition in MRI Abstract:
Magnetic Resonance Imaging with parallel data acquisition requires algorithms for reconstructing the patient's image from a small number of measured $k$-space lines.
In contrast to well-known algorithms like SENSE and GRAPPA and its flavors we consider the problem as a non-linear inverse problem. Fast computation algorithms for the necessary Fréchet derivative and reconstruction algorithms are given. ,

Mathematical Methods in the Applied Sciences Vol. 30, pp. 1437-1451 (2007)

(Preprint)

Bauer, F. and Munk, A.

Optimal Regularization for Ill-Posed Problems in Metric SpacesAbstract:
We present a strategy for choosing the regularization parameter (Lepskij-type balancing principle) for ill-posed problems in metric spaces with deterministic or stochastic noise. Additionally we improve the strategy in comparison to the previously used version for Hilbert spaces in some ways. ,

Inverse and Ill-posed Problems Vol. 15, Iss. 2, pp. 137-148 (2007)

Bauer, F. and Ivanyshyn, O.

Optimal Regularization with Two Interdependent Regularization Parameters Abstract:
We will consider the situation of ill-posed problems with regularization methods which demand the choice of interdependent regularization parameters. We show that one can adapt the Lepskij-type balancing principle to this situation. ,

Inverse Problems Vol. 23, pp. 331 - 342 (2007)

(Preprint)

Bauer, F.

Some Considerations Concerning Regularization and Parameter Choice Algorithms Abstract:
Using a Bayesian type approach to inverse problems many phenomenons occurring in practice can be explained in a consistent way. In particular we can prove a discrete version of the quasi-optimality criterion to choose the regularization parameter just imposing minor additional a priori assumptions about the solution and the measurement noise. ,

Inverse Problems Vol. 23, pp. 837 - 858 (2007)

(Preprint)

Bauer, F., Pereverzev, S. and Rosasco, L.,

On Regularization Algorithms in Learning Theory Abstract:
In this paper we discuss a relation between Learning Theory and Regularization of linear ill-posed inverse problems. It is well known that Tikhonov regularization can be profitably used in the context of supervised learning, where it usually goes under the name of regularized least-squares algorithm. Moreover, the gradient descent algorithm was studied recently, which is an analog of Landweber regularization scheme. In this paper we show that a notion of regularization defined according to what is usually done for ill-posed inverse problems allows to derive learning algorithms which are consistent and provide a fast convergence rate. It turns out that for priors expressed in term of variable Hilbert scales in reproducing kernel Hilbert spaces our results for Tikhonov regularization match those in Smale and Zhou and improve the results for Landweber iterations obtained in Yao et al.
The remarkable fact is that our analysis shows that the same properties are shared by a large class of learning algorithms which are essentially all the linear regularization schemes. The concept of operator monotone functions turns out to be an important tool for the analysis. ,

Journal of Complexity Vol. 23 (1), pp. 52 - 72 (2007)

Bauer, F., Mathé, P. and Pereverzev, S.,

Local Solutions to Inverse Problems in Geodesy: The Impact of the Noise Covariance Structure upon the Accuracy of Estimation Abstract:
In many geoscientific applications, one needs to recover the quantities of interest from indirect observations blurred by colored noise. Such quantities often correspond to the values of bounded linear functionals acting on the solution of some observation equation. For example, various quantities are derived from harmonic coefficients of the Earth’s gravity potential. Each such coefficient is the value of the corresponding linear functional. The goal of this paper is to discuss new means to use information about the noise covariance structure, which allows order-optimal estimation of the functionals of interest and does not involve a covariance operator directly in the estimation process. It is done on the basis of a balancing principle for the choice of the regularization parameter, which is new in geoscientific applications. A number of tests demonstrate its applicability. In particular, we could find appropriate regularization parameters by knowing a small part of the gravitational field on the Earth’s surface with high precision and reconstructing the rest globally by downward continuation from satellite data. ,

Journal of Geodesy Vol 81, Nr 1, pp. 39 - 51 (2007)

(Preprint)

Bauer, F. and Pereverzev, S.,

An utilization of a rough approximation of a noise covariance within the framework of multi-parameter regularization Abstract:
Regularization under the assumption of badly known noise covariance operators is a demanding subject. In order to increase stability we propose a multi parameter regularization scheme and a parameter choice rule accordingly. We can show that the new scheme fulfills the same error bounds as the classically known ones.
Numerically we see that the new scheme is considerably more stable towards misestimations of the covariance operator than the old one-parameter one. ,

International Journal of Tomography & Statistics, Vol. 4, pp. 1 - 12 (2006)

(Preprint)

Bauer, F., Freeden, W. and Schreiner, M.

A Tree Algorithm for Isotropic Finite Elements on the Sphere Abstract:
The earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0.4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modeling of geoscientifically relevant data on the sphere plays an important role. In this paper, we deal with isotropic kernel functions showing a local support (briefly called isotropic finite elements) for reconstructing square-integrable functions on the sphere. An essential tool is the concept of multiresolution analysis by virtue of the spherical up function. Because the up function is built by an infinite convolution product, we do not know an explicit representation of it. However, the tree algorithm for the multiresolution analysis based on the up functions can be formulated by convolutions of isotropic kernels of low-order polynomial structure. For these kernels, we are able to find an explicit representation, so that the tree algorithm can be implemented efficiently. ,

Numerical Functional Analysis and Optimization Vol. 27; Nr. 1 pp. 1 - 24 (2006)

(Preprint)

Bauer, F. and Hohage, T.

A Lepskij-type stopping rule for regularized Newton methods Abstract:
We investigate an a posteriori stopping rule of Lepskij-type for a class of regularized Newton methods and show that it leads to order optimal convergence rates for Hölder and logarithmic source conditions without a priori knowledge of the smoothness of the solution. Numerical experiments show that this stopping rule yields results at least as good as, and in some situations significantly better than, Morozov's discrepancy principle. ,

Inverse Problems, Vol. 21, pp. 1975 - 1991 (2005)

(Preprint)

Bauer, F. and Hohage, T.
On Lepskij's Stopping-Rule for Newton-type Methods with Random Noise,
PAMM Vol. 5; Nr. 1, pp. 15-18 (2005)

Bauer, F.,
Automatic Optimal Regularization for Ill-Posed Problems with Stochastical Noise without Imposing Smoothness Assumptions,
PAMM Vol. 5; Nr. 1, pp. 641-642 (2005)

Bauer, F.,
Automatic Regularization for Ill-Posed Problems with Stochastical Noise Estimate,
Proceedings of the 5^th International Conference on Inverse Problems in Engineering: Theory and Practice, UK, 11-15th July 2005
(2005)

Bauer, F. and Pereverzev, S.,

Regularization without Preliminary Knowledge of Smoothness and Error Behavior Abstract:
The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required the knowledge of smoothness of the solution or the overall error behaviour. We will show that we can guarantee an asymptotically almost optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behaviour. An application to the determination of the gravitational field out of satellite data will be shown. ,

European Journal of Applied Mathematics, Vol. 16; Nr. 3, pp. 303 - 317 (2005)

(Preprint)

Bauer, F.

Easy Differentiation and Integration of Homogeneous Harmonic Polynomials Abstract:
We will give explicit differentiation and integration rules for homogeneous harmonic polynomial polynomials and spherical harmonics in IR^3 with respect to the following differential operators: partial_1, partial_2, partial_3, x_3 partial_2 - x_2 partial_3, x_3 partial_1 - x_1 partial_3, x_2 partial_1 - x_1 partial_2 and x_1 partial_1 + x_2 partial_2 + x_3 partial_3. A numerical application to the problem of determining the geopotential field will be shown. ,

Publication not intended

Bauer, F.,
An Alternative Approach to the Oblique Derivative Problem in Potential Theory,
Berichte aus der Mathematik, Shaker Verlag, Aachen,
Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group (2004)

Bauer, F.,
Determination of the Geopotential Field out of Oblique Derivatives, an Alternate Approach,
Oberwolfach Reports, EMS Publishing House, Vol. 1, pp. 1435 - 1438 (2004)

(Preprint)

Bauer, F.,
Differential Equations of Lie Type,
Master's thesis, University of Warwick, Department of Mathematics (2001)