Wavelet footprints: theory, algorithms, and applications

• Published in: IEEE transactions on signal processing Vol. 51; no. 5; pp. 1306 - 1323
• Main Authors: Dragotti, P.L., Vetterli, M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.05.2003; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Dictionaries; Discontinuity; Energy management; Exact sciences and technology; Footprints; Information analysis; Information, signal and communications theory; Mathematical analysis; Mathematical methods; Mathematical models; Noise reduction; Polynomials; Robustness; Signal analysis; Signal and communications theory; Signal processing; Signal processing algorithms; Signal representation. Spectral analysis; Signal, noise; Telecommunications and information theory; Theorems; Wavelet; Wavelet coefficients; Wavelet transforms; Signal compression; Discontinuity; Compression; nonlinear approximation; Noise reduction; Piecewise polynomial techniques; matching pursuit; Signal representation; Algorithm; denoising; Deconvolution; Wavelet transformation; wavelets; Signal processing
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2003.810296

Wavelet-based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients. Discontinuities often carry relevant signal information, and therefore, they represent a critical part to analyze. We study the dependency across scales of the wavelet coefficients generated by discontinuities. We start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual (Theorem 1). We then introduce the notion of footprints, which are scale space vectors that model discontinuities in piecewise polynomial signals exactly. We show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints. This also leads to efficient approximation of piecewise smooth functions. Finally, we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising, compression, and (nonblind) deconvolution. In the case of compression, we also prove that at high rates, footprint-based algorithms attain optimal performance (Theorem 3).