Singularity detection and processing with wavelets

• Published in: IEEE transactions on information theory Vol. 38; no. 2; pp. 617 - 643
• Main Authors: Mallat, S., Hwang, W.L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-03-1992; Institute of Electrical and Electronics Engineers
• Subjects: Applied sciences; Exact sciences and technology; Fourier transforms; Frequency measurement; Image processing; Information, signal and communications theory; Mathematics; Noise reduction; Signal analysis; Signal processing; Signal processing algorithms; Telecommunications and information theory; Wavelet analysis; Wavelet transforms; White noise; Image quality; Singularity; Two dimensional signal; Image processing; Wavelet transformation; Signal reconstruction; Signal analysis; Edge detection; Signal detection
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/18.119727

The mathematical characterization of singularities with Lipschitz exponents is reviewed. Theorems that estimate local Lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed. It is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately. The local frequency of such oscillations is measured from the wavelet transform modulus maxima. It has been shown numerically that one- and two-dimensional signals can be reconstructed, with a good approximation, from the local maxima of their wavelet transform modulus. As an application, an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales. In two dimensions, the wavelet transform maxima indicate the location of edges in images.< >