For the analysis and representation of stationary signals and images the conventional Fourier- and wavelet-based methods are particularly appropriate. However, in many areas in science and engineering we are faced with the problem to interpret digital signals and images which are not band-limited and have a non-stationary behaviour. Frequently, there are even further obstacles. The acquisition of signal or image measurements may be very expensive and therefore limited. In other applications, measurement sets are huge but contaminated by noise. Examples of the above are encountered in magnetic resonance imaging, infrared microscopy, fluorescence-lifetime imaging microscopy (FLIM), the analysis of seismic signals in geophysics, radar imaging (SAR/ISAR), tissue ageing models, vibration analysis, direction of arrival (DOA) detection, texture classification, radio frequency identification (RFID), non-destructive testing, satellite navigation, time series analysis, echolocation, induction motor diagnostics (MCSA), to name just a few.

Within the last few years, research on Prony-based methods has been intensified, as they offer an alternative to the compressed sensing approach. One essential advantage of the Prony method is that it does not need randomly collected measurements but works with deterministic sampling based on a sampling scheme which is adapted to the nonlinear signal model. At the same time, the Prony approach does not suffer the well-known curse of dimensionality in the multivariate case.

This Dagstuhl Seminar brought together a number of researchers from different areas in mathematics, engineering and industry. Topics included new mathematical insights and efficient numerical algorithms for problems based on exponential analysis as well as applications of exponential analysis models in engineering and life sciences. During this Seminar, the participants presented their newest results and discussed several open problems and applications from different perspectives.

The talks in the workshop partially had the character of survey talks and were arranged in different main research topics, as new mathematical theory (Day 1 and Day 5), new computational approaches (Day 2) and new results in applications in engineering (Day 3) and life sciences (Day 4).

On the first day, the talks focussed on the mathematical theory of exponential analysis. B. Beckermann and A. Matos emphasized the close connection between exponential analysis and rational approximation, which leads to new application areas as density reconstruction in an equilibrium problem in logarithmic potential theory and improved rational approximation of Markov functions. As shown in the talk by T. Sauer, there is also a close connection between multi-exponential analysis and continued fractions.

In the afternoon, the participants started four smaller thematic discussion groups to discuss new challenges and open problems throughout the week. These discussion groups particularly focussed on theory, more efficient computational algorithms and applications in engineering and biosciences. The discussions included general observations on the further development of exponential analysis and its ties to other subjects as well as further specific approaches for application of the theory to application problems in radar imaging or MRI. Still, we are far away from a full understanding of the relations between the different methods for the stable reconstruction of a parametric signal model, as well as from a systematic construction of Fourier analytic methods for the improved analysis of non-stationary signals.

In practice, Prony-based methods often suffer from a bad conditioning of the involved structured matrices and some extra effort is required to reliably execute the corresponding algorithms. Among the more successful implementations, the ESPRIT method, the Matrix-Pencil method, the approximate Prony method, and validated exponential analysis are established. The problem statement is also closely related to rational approximation theory and the structured low-rank approximation of structured matrices. These connections are still not completely understood and may lead to strongly improved reconstruction algorithms, able to treat more general sampling sets and deliver super-resolution results.

The second day was devoted to the problem of efficient computations for reconstruction or approximation of functions using multi-exponential analysis. First numerical methods like MUSIC are already implemented in the computer algebra system Maplesoft, as presented in the talk by J. Gerhard. The advantage of the application of Maplesoft is that it can process the data within a desired very high precision and therefore successfully handle these reconstruction problems which are known to be ill-conditioned. Another way to improve the numerical stability is the application of more sophisticated numerical approaches to structured matrices and the connection to rational approximation, such that the existing stable algorithms for rational approximation can be used, see the talk of G. Plonka. The survey presentation by H. Mhaskar turned the attention to the application of neural networks to approximation problems, and D. Potts presented new efficient Fourier methods to compute the so-called ANOVA decomposition for approximation of multivariate functions.

In the afternoon a mini-course on newest features of Maplesoft was presented by J. Gerhard, with a special focus on the problem of how to connect MAPLESOFT with other Software as MATLAB etc. The remaining time was used for further scientific discussions in smaller groups.

On the third day, the talks surveyed different new applications of exponential analysis in engineering. The presentation of F. Knaepkens showed new approaches for direction of arrival (DOA) estimation, image denoising and inverse synthetic aperture radar. Chromatic aberration in large antenna systems have been studied by D. de Villiers. R.-M Weideman showed her results on antenna position estimation through sub-sampled exponential analysis of signals in the near-field. The talks by R. Beinert and J. Prestin showed applications concerning phase retrieval in optical diffraction tomography and detection of directional jumps in images.

The survey presentations on the fourth day focussed on exponential models in life sciences. J. Gielis showed in his talk, how generalized Möbius-Listing bodies (GML) can be employed for better modelling and understanding of certain dynamical processes in the natural sciences. D. Li explained recent progress of advanced time-resolved imaging techniques based on exponential analysis models and their applications in life sciences, for example to reveal biological processes at the molecular level. In a further talk, R.G. Spencer reported on newest developments in Magnetic Resonance Relaxometry and Macromolecular Mapping to achieve more accurate myelin quantification in the brain that permits the establishment of physiological correlations. The underlying reconstruction problem is a seriously ill-posed inverse problem.

The last day of the workshop was again devoted to further results in mathematical theory of exponential analysis and connections to other areas of mathematics. In the talk by D. Batenkow, the degree of ill-posedness of the parameter reconstruction problem based on the exponential sum model was studied in more detail. The degree of condition of the problem essentially depends on the distribution of the frequency parameters. H. Vesolovska discussed the problem of recovering an atomic measure on the unit 2-sphere S\sup2; given finitely many moments with respect to spherical harmonics. A connection of exponential analysis to computer algebra problems was brought to our attention by M. Ishteva. She showed how the joint decomposition of a set of non-homogeneous polynomials can be computed using the canonical polyadic decomposition, and how this decomposition can be applied in nonlinear system identification.

This Dagstuhl meeting has been an important milestone for improved understanding of the large impact of exponential analysis tools in both theory and practice. During the thematic discussions in small groups in this meeting, several new aspects have been considered and several collaborations have been initiated or continued. Examples include new approaches for an improved modelling of antenna system frequency responses in radio frequency (Cuyt, De Villiers, Weideman) and for stabilised parameter estimation in exponential models using iterative factorizations of matrix pencils of Loewner matrices (Beckermann, Plonka-Hoch).

We mention that this seminar is related to Dagstuhl seminar 15251 on “Sparse modelling and multi-exponential analysis„ that took place in 2015. The discussions at the latter have led to many interesting collaborative projects, among which a funded Horizon-2020 RISE project (Research and Innovation Staff Exchange) with the acronym EXPOWER, standing for "Exponential analysis Empowering innovation" (grant agreement No 101008231).

It is our experience that these Dagstuhl seminars are timely and seminal. Through the meetings new collaborations and new potential are unlocked. There is a clear need to further connect stakeholders from the new theoretical developments and the identified industrial applications, as is our objective here and in the future.

Creative Commons BY 4.0