## Current Research Interests

My research interests are in the areas of signal theory, sampling and reconstruction of bandlimited signals, system approximation, quantization, and signal adapted processing. Further, I work on applications of signal processing concepts and methods in forensic DNA analysis.

Below current and past research topics are presented in more detail.

### Sampling Functionals / System Approximation

For pointwise sampling it is known that there exist stable LTI systems and functions in the the Paley-Wiener space \(\mathcal{PW}_{\pi}^{1}\) such that the approximation process diverges, regardless of the oversampling factor. Recently, it was shown that the divergence can be overcome by using more general measurement functionals that are based on a complete orthonormal system. However, this approach requires the approximation process to have an increased bandwidth.

However, there exists a two channel system approximation process that is uniformly convergent for all stable LTI systems and all functions in \(\mathcal{PW}_{\pi}^{1}\). An advantage of the two channel approach compared to the one channel approach is the reduction of the approximation bandwidth, which can be exactly the same as the input function bandwidth.

- Ullrich J. Mönich, Holger Boche,
*"A Two Channel System Approximation for Bandlimited Functions"*, IEEE Transactions on Information Theory, vol. 63, no. 9, pp. 5496-5505, September 2017. https://dx.doi.org/10.1109/TIT.2017.2724550 - Holger Boche, Ullrich J. Mönich,
*"Signal and System Approximation from General Measurements"*, in "New Perspectives on Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's 85th birthday", Applied and Numerical Harmonic Analysis, Birkhauser (Springer-Verlag), 2014. http://dspace.mit.edu/handle/1721.1/92323

### Bio Security: Forensic DNA Analysis

Recently a lot of effort has been put in studying fully continuous interpretation approaches for DNA mixtures, where not only the location of a peak is used but also the actual peak heights. Since the peak heights contains additional information, these methods are superior to previous methods, where thresholds have been applied.

An important step toward the development of these methods is a complete understanding of the statistical properties of the peak heights and the derivation of a signal model. Based on such a model, algorithms for finding the number of contributors in a DNA mixture and for determining the probability that a given suspect contributed to a DNA sample can be developed.

- Ullrich J. Mönich, Ken Duffy, Muriel Médard, Viveck Cadambe, Lauren E. Alfonse, Catherine Grgicak,
*"Probabilistic Characterisation of Baseline Noise in STR Profiles"*, Forensic Science International: Genetics, vol. 19, pp. 107-122, November 2015. http://dx.doi.org/10.1016/j.fsigen.2015.07.001 - Ullrich J. Mönich, Catherine Grgicak, Viveck Cadambe, Jason Y.
Wu, Genevieve. Wellner, Ken Duffy, Muriel Médard,
*"A signal model for forensic DNA mixtures"*, in 2014 Asilomar Conference on Signals, Systems and Computers, Nov. 2014. http://www.rle.mit.edu/ncrc/wp-content/uploads/2014/12/asilomar_2014.pdf

### Signal Theory: Signals Reconstruction and System Approximation

Today's information society is based on the power of digital signal processing, and the theoretical foundation for the digital processing of analog signals is the Shannon sampling theorem. Without any further information about the signal, except its bandlimitedness, Nyquist rate sampling is optimal. However, if the signal to exhibits some additional structure, it is possible to reduce the sampling rate below the Nyquist rate. Compressed sensing provides a framework for sampling and recovering of sparse signals, however, sparsity is only one of many possible structures a signal can have.

Since signal processing is concerned with the processing of signals, a fundamental question is: which analog systems can be stably approximated by digital systems? The answer depends on many parameters, like the signal space under consideration, the sampling rate, the kind of sampling, etc. Recently, it has been shown that by using suitable generalized measurement functionals, it is possible to have a uniformly convergent approximation of stable linear time-invariant systems for the space \(\mathcal{PW}_\pi^1\) if oversampling is applied.

- Holger Boche, Ullrich J. Mönich,
*"Signal and System Approximation from General Measurements"*, arXiv, 2014. http://arxiv.org/abs/1402.1092 - Ullrich J. Mönich, Holger Boche,
*"Non-equidistant sampling for bounded bandlimited signals"*, Signal Processing, vol. 90, no. 7, pp. 2212-2218, July 2010. http://dx.doi.org/10.1016/j.sigpro.2010.02.002

### Quantization in Signal Reconstruction and System Approximation

Quantization and thresholding are two important operations in practical systems. Although quantization is a purely deterministically process, it is usually analyzed stochastically to simplify the mathematical treatment. Quantization and thresholding of signal samples can lead to divergent reconstruction processes. A deterministic treatment of quantization reveals surprising facts. For example, for certain signals the reconstruction error increases unboundedly as the quantization step size is reduced to zero.

- Holger Boche, Ullrich J. Mönich,
*"Behavior of the Quantization Operator for Bandlimited, Nonoversampled Signals"*, IEEE Transactions on Information Theory, vol. 56, no. 5, pp. 2433-2440, May 2010, http://dx.doi.org/10.1109/TIT.2010.2044072

## Previous Research Interests

### Component-Based Face Detection

Robust face detection algorithms are important, especially in the computer-human interaction and in surveillance applications. In this work a component-based face detection algorithm is developed, which is more robust in particular in the presence of occlusions. The four face components (eyes, nose, mouth) are detected, using a cascade of boosted classifiers, where the classifiers work with Haar-like features. An online demonstrator is available at http://facedetection.nue.tu-berlin.de/.

- Online Demonstrator (http://facedetection.nue.tu-berlin.de)
- Lutz Goldmann, Ullrich J. Mönich, Thomas Sikora,
*"Components and Their Topology for Robust Face Detection in the Presence of Partial Occlusions"*, IEEE Transactions on Information Forensics and Security, vol. 2, no. 3, pp. 559-569, Sep. 2007, http://dx.doi.org/10.1109/TIFS.2007.902019 - Ullrich J. Mönich,
*"Objektdetektion basierend auf Komponenten und ihrer Topologie"*, (Diploma thesis)