Random Models for Sparse Signals Expansion on Unions of Bases With Application to Audio Signals

• Published in: IEEE transactions on signal processing Vol. 56; no. 8; pp. 3468 - 3481
• Main Authors: Kowalski, M., Torresani, B.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive thresholding; Applied sciences; Audio signals; Bayesian methods; Categories; Classification; Coding, codes; Computer Science; denoising; Detection, estimation, filtering, equalization, prediction; Dictionaries; Engineering Sciences; Exact sciences and technology; Feasibility; Greedy algorithms; Image coding; Image processing; Information, signal and communications theory; Mathematical models; Mathematics; Miscellaneous; Noise reduction; nonlinear signal approximation; Probability; Signal analysis; Signal and communications theory; Signal and Image processing; Signal processing; Signal processing algorithms; Signal synthesis; Signal, noise; sparse representations; Telecommunications and information theory; Time frequency analysis; time-frequency decompositions; Unions; Waveforms; Dictionaries; Probabilistic approach; Acoustic signal; Image processing; Threshold detection; Noise reduction; sparse representations; Modeling; denoising; Audio signal; Linear combination; Algorithm performance; Coding; nonlinear signal approximation; Adaptive thresholding; Signal processing; Sparse representation; Feasibility; Discrete cosine transforms; time-frequency decompositions; Non-linear signal approximation; Sparse Representations; Time-frequency decompositions; Denoising
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2008.920144

A new approach for signal expansion with respect to hybrid dictionaries, based upon probabilistic modeling is proposed and studied. The signal is modeled as a sparse linear combination of waveforms, taken from the union of two orthonormal bases, with random coefficients. The behavior of the analysis coefficients, namely inner products of the signal with all basis functions, is studied in details, which shows that these coefficients may generally be classified in two categories: significant coefficients versus insignificant coefficients. Conditions ensuring the feasibility of such a classification are given. When the classification is possible, it leads to efficient estimation algorithms, that may in turn be used for denoising or coding purposes. The proposed approach is illustrated by numerical experiments on audio signals, using MDCT bases. However, it is general enough to be applied without much modifications in different contexts, for example in image processing.