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# Project B26: Information extracting sensor networks

Duration: | April 2012 - December 2012 |

Project Heads: | Gitta Kutyniok |

Researchers: | Mijail Guillemard Friedrich Philipp |

Support: | DFG Research Center "Mathematics for Key Technologies"(Matheon) |

Official Page: | Matheon Project B26 |

# Project Description

**Background**

Sensor networks are customarily designed to measure and transmit data (messages) by aiming at a minimal MSE. However, in most practical situations not the complete message is required at the receiver or fusion center, but only particular information. Taking this novel viewpoint into account, sensor networks of significantly reduced complexity/cost are envisionable. As an exemplary situation, we mention a sensor network which shall measure the mean of sampled physical processes. By the central limit theorem, in this case an optimal reduction of the impact of noise is not necessary. This will be taken care of by the computation at the receiver side, thus enabling the utilization of cheaper sensors. The paradigm shift in the design of sensor networks - extracting information instead of transmitting the entire physical process as accurately as possible - was very recently pioneered by H. Boche (TU München). A mathematical foundation is still almost completely missing, with very first intriguing results by H. Boche and U.J. Mönich. In this project, we will consider different signal models and focus on the extraction of the maximal value as exemplarily seeked information of a physical process under the impact of tresholding and noise. The goal is to develop a general mathematical framework yielding optimal design criteria for sensor networks extracting such information, followed by testing practical realizations.

**Research Program**

In this project, we will consider the following model situation: A physical process will be described by functions belonging to a particular signal space, which could be a Sobolov space or a Paley-Wiener space of band-limited functions. This process will then be measured by a fixed number of sensors through (equidistant or nonequidistant) sampling, which locally process those measurements. In a last step the processed and thresholded measurements will be transmitted - and presumably be impacted by noise - to a fusion center, which performs a last processing step. The main performance criteria is to optimally compute or approximate the mean of the initial physical process. Our main goal is both to derive a deep mathematical understanding of this situation leading to optimality criteria and to provide an algorithmic realization followed by practical tests.

Initial results from H. Boche and U.J. Mönich (TU München) indicate that even for one sensor, the nonlinearity of the thresholding operator leads to completely new territory, requiring non-standard functional analysis methodologies. In fact, it was, for instance, shown that in the situation of a Paley-Wiener space as signal space and under the impact of thresholding and quantization operators, the classical Shannon sampling series does fail completely in approximating the initial function in the sense of an infinite worst case L^\infty-error for specifically chosen functions. This already indicates the delicacy of the task at hand. To achieve the previously detailed objectives in sensor network design, we aim to solve the following subgoals:

1. Analysis of the one-sensor-situation.

- Design of an extraction strategy of the maximal value.
- Characterization of the class of signals, which allows perfect extraction.
- L^\infty-error analysis for all signals in the considered signal space.
- Algorithmic realization and numerical tests.

2. Analysis of the multiple-sensors-situation.

- Design of an extraction strategy of the maximal value under transmission without local processing in the sensors.
- Characterization of the class of signals, which allows perfect extraction.
- Design of a local processing strategy to enlarge this class.
- L^\infty-error analysis for all signals in the considered signal space.
- Algorithmic realization and numerical tests.

3. Practical tests with external partners (see below).

# Project Output

**Publications**

- Scalable Frames, Gitta Kutyniok, Kasso A. Okoudjou, Friedrich Philipp, Elizabeth K. Tuley, , Preprint, arXiv:1204.1880v3

- Signal Analysis with Frame Theory and Persistent Homology M. Guillemard, H. Boche, G.Kutyniok, and F. Philipp. In Proc. Sampta13, 2013

- Preconditioning of frames, G. Kutyniok, K. A. Okoudjou, and F. Philipp. Submitted, 2013.

- Perfect preconditioning of frames by a diagonal operator.

G. Kutyniok, K. Okoudjou, and F. Philipp. In Proc. Sampta’13, 2013.

- Signal recovery from thresholded frame,

H. Boche, M. Guillemard, G. Kutyniok and F. Philipp. Submitted, 2013.

**Cooperations**

Within Matheon:

Matheon B3: Integrated Planning of Multi-layer Telecommunication Networks

Matheon B23: Robust optimization for network applications

External: