Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data. (English) Zbl 0786.62094

Donoho, David L.

Daubechies, Ingrid (ed.), Different perspectives on wavelets. American Mathematical Society short course on wavelets and applications, held in San Antonio, TX (USA), January 11-12, 1993. Providence, RI: American Mathematical Society. Proc. Symp. Appl. Math. 47, 173-205 (1993).

Summary: We describe wavelet methods for recovery of objects from noisy and incomplete data. The common themes: (a) the new methods utilize nonlinear operations in the wavelet domain; (b) they accomplish tasks which are not possible by traditional linear/Fourier approaches to such problems. We attempt to indicate the heuristic principles, theoretical foundations, and possible application areas for these methods. Areas covered: (1) Wavelet De-Noising. (2) Wavelet Approaches to Linear Inverse Problems. (4) Wavelet Packet De-Noising. (5) Segmented Multi-Resolutions. (6) Nonlinear Multi-Resolutions.
For the entire collection see [Zbl 0782.00059].

MSC:

62M99	Inference from stochastic processes
62N99	Survival analysis and censored data
65C99	Probabilistic methods, stochastic differential equations
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M20	Inference from stochastic processes and prediction
62P99	Applications of statistics
46N30	Applications of functional analysis in probability theory and statistics

Keywords:

wavelet de-noising; linear inverse problems; wavelet packet de-noising; segmented multi-resolutions; nonlinear multi-resolutions; data analysis; signal processing; wavelet methods; recovery of objects; noisy and incomplete data; nonlinear operations; heuristic principles; theoretical foundations