**Some general remarks**

Two ideas arise quite naturally at the end of this 4 years period: a lot of work has been done in field of interest of the SSG, but the way to a full exploitation of the application of wavelets and multiresolution analysis in geodesy is still quite long. It is also clear that the quite broad list of items in the program was very ambitious.

**Report about activities and results.**

Several of the members has developed an important activity in the field of the SSG or in fields that are strictly related. Some of the results are quite essentially reported here after.

*Spherical wavelets. *This
field has been mainly treated by the Geomathematics Group of Kaiserlautern.
Several papers have been published. It seams that, after several mathematical
developments, we are approaching the possibility of practical applications.

*Image analysis. *This
has been specially treated by Y. Pan and B. Schaffrin using a bayesian
approach.

*Wavelets and collocation. *This
is a field not originally mentioned in the terms of reference of the SSG.
Some authors (Kostakis, Keller) treated the problem under different points
of view. In fact the well known procedure of multi-step collocation is
in some way a kind of multiresolution analysis.

*Other treated fields* include DTM analysis and transformation,
the analysis of time series of INS measurements and the quite original
development of Slepian functions for the sphere.

**Publications**
Albertella A., Sneeuw N., Sanso F. (1998). The Slepian
Problem on the Sphere.

IV Hotine-Marussi Symp. on Math. Geodesy. Trento.

Bayer M., Beth S., Freeden W. (1998). Geophysical Field Modelling by Multiresolution Analysis. Acta Geodaetica et Geophysica Hungarica 33, pp. 289-319.

Bayer M., Beth S., Maier T. (1998). Wavelet Modelling of Vectorial Components in Gravitational Field Determination. In: Freeden W. (ed.). Progress in Geodetic Science at GW98, pp. 208-217.

Benciolini B. (1997). Wavelets: An Introduction. International Archives of Phtogrammetry and Remote Sensing, Vol. XXXII, Part 6w1 - Proceedings of the workshop: International Cooperation and Technology Transfer, Padova, pp. 31-34.

Benciolini B., Zatelli P. (1997). Wavelets Applications in Local Geoid Computation. Scientific Assembly of IAG, Rio de Janeiro.

Blais J.A.R. (1996). Spectrum Considerations Using Maximum Entropy and Multiresolution Considerations. Journal of Geodesy, Vol. 70, No. 3, pp. 349-356.

Blais, J.A.R., He K., Larouche C. (1995). Multiresolution Analysis in Geomatics Applications. Canadian Geomatics CD-ROM, National Defence and Geomatics Canada.

Blais, J.A.R. (1996). Multiresolution Spatial Data Representation and Processing in GISs. Final Report submitted in February, 1996, to GIS and (Services Division, Natural Resources Canada, Ottawa, 41 pp.)

Bruton A.M., Skaloud J., Schwarz K.P. (1999). The Use of Wavelets for the Analysis and Denoising of Kinematic Geodetic Measurements. Poster presented in Symposium G4 at IUGG99 in Birmingham.

Chao B. F., Naito I. (1995). Wavelet Analysis Provides a new Tool for Studying Earth’s Rotation. EOS, Trans. Amer. Geophys. Union 76, pp. 161-165.

Kotsakis C., Sideris M.G. (1999) The Long Road from Deterministic Collocation to Multiresolution Approximation (Technical Report).

Freeden W. (1998). The Uncertainty Principle and Its Role in Physical Geodesy. In: Progress in Geodetic Science at GW98, pp. 225-236.

Freeden W., Schneider F. (1998). Wavelet Approximation on Closed Surfaces and Their Application to Boundary-Value Problems of Potential Theory. Math. Meth. in the Appl. Sci. 21, pp. 129-165.

Freeden W., Windheuser U. (1996). Spherical Wavelet Transform and Its Discretization. Advances in Computational Mathematics 5, pp. 51-94.

Freeden W., Windheuser U. (1997). Combined Spherical Harmonics and Wavelet Expansion - A Future Concept in Earth’s Gravitational Determination. Applied and Computational Harmonic Analysis (ACHA) 4, pp.1-37.

Freeden W. (1999). Multiscale Modelling of Spaceborne Geodata. 351 pp., ISBN 3-519-02600-7.

Freeden W., Gervens T., Schreiner M. (1998). Constructive Approximation on the Sphere. Oxford Science Publication, Oxford University Press, 427 pp., ISBN 0-19-853682-8.

Freeden W., Schneider F. (1998). An Integrated Wavelet Concept of Physical Geodesy. Journal of Geodesy 72, pp. 259-281.

Keller W. (1995). From Fourier Transform to Wavelet Transform - Similarities and Differences. International Lectures Series, Dept. of Geomathematics Engineering, Univ. of Calgary.

Keller W. (1996). Kontinueierliche Wavelet Transformation. Zeitschrift fur Vermessungswesen 121, pp. 563 - 572.

Keller W. (1997a). Collocation in Reproducing Kernel Hilbert Spaces of a Multi-Scale Analysis. 1997 Meeting of EGS.

Keller W. (1997b). Geoid Computation by Collocation in Wavelet Spaces. Scientific Assembly of IAG, Rio de Janeiro.

Kotsakis C. (1999a). The Multiresolution Character of Collocation (Paper submitted to Journal of Geodesy).

Kotsakis C. (1999b). Wavelets and Collocation: An Interesting Similarity. IUGG99, Symposium G6.

Liu H.S., Chao B.F. (1998). Wavelet Spectral Analysis of the Earth’s Orbital Variations and Paleoclimatic Cycles. J. Atmos. Soc. 55, pp. 227-236.

Liu L.T., Hsu H.T., Gao B.X. (1998). A New Family of Orthonormal Wavelets Bases. Journal of Geodesy 72, pp. 294-303.

Michel V. (1998). Wavelet Based Method for the Gravimetry Problem. In: Freeden W. (ed.). Progress in Geodetic Science at GW98, pp. 283-298.

Michel V. (1999). A Multiscale Method for the Gravimetry Problem: Theoretical and Numerical Aspects of Harmonic and Anharmonic Modelling. PhD-thesis, University of Kaiserslautern.

Yanjun Pan, Schaffrin B. (1998a). Adaptive Wavelets for Multiscale DEM Transformation. AVN 105, pp. 101-104.

Yanjun Pan, Schaffrin B. (1998b). Bayesian Wavelet Analysis for Image De-noising and Image Fusion. 22nd. Europ. Meeting of Statisticians, Vilnius, Lithuania.

Yanjun Pan, Schaffrin B. (1999a). Bayesian Wavelet Analysis for Image De-noising. Photogr., Goinf. N2, pp. 95-100.

Yanjun Pan, Schaffrin B. (1999b). Image Fusion with Wavelets. (Accepted for publication on GIS August 1999).

Schreiner M. (1996). A Pyramid Scheme for Spherical Wavelets. Technical Report AGTM, Report No. 170, Geomathematics Group, University of Kaiserslautern.

Skaloud J., Bruton A.M., K.P. Schwarz K.P. (1999). Detection and Filtering of Short-Term (1/f) Noise in Inertial Sensors (Accepted for publication in ION Navigation.).

Windheuser U. (1995). Spherical Wavelet Packets and Its Application in Physical Geodesy. In: Neunzert H. (ed.). Progress in Industrial Mathematics - At the European Consortium of Mathematics in Industry’94.

Windheuser U. (1995). Spharische Wavelets: Theorie und
Anwendung in der Physikalischen Geodasie. PhD-thesis, University of Kaiserslautern,
Geomathematics Group.