2012/10/31 by Ildar Khalidov, Michael Unser, Michaël Unser +1 · 6 citations
Computer Science · Engineering · Mathematics · #Algorithm #Artificial intelligence #Computer science #Connection (principal bundle) #Differential operator #Generality #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Mathematics #Multiplier (economics) #Operator (biology) #Pure mathematics #Realization (probability) #Sparse and Compressive Sensing Techniques #Statistics #Wavelet #White noise #math.CA #msc:42B15 #msc:42C40 #msc:60H15
paper · pdf · doi:10.1007/s00041-013-9306-1
published in Journal of Fourier Analysis and Applications 19(6), 1294-1322 (Birkhäuser) · 34 pages
arxiv created 2013/09/04 · openalex publication_date 2013/11/13 · arxiv updated 2014/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.