Wavelets and Multiresolution Processing

Wavelets • Fourier transform has its basis functions in sinusoids • Wavelets based on small waves of varying frequency and limited duration • In addition to frequency, wavelets capture temporal information – Bound in both frequency and time domains – Localized wave and decays to zero instead of oscillating forever • Form the basis of an approach to signal processing and analysis known as multiresolution theory – Concerned with the representation and analysis of images at different resolutions – Features that may not be prominent at one level can be easily detected at another level • Comparison with Fourier transform – Fourier transform used to analyze signals by converting signals into a continuous series of sine and cosine functions, each with a constant frequency and amplitude, and of infinite duration – Real world signals (images) have a finite duration and exhibit abrupt changes in frequency – Wavelet transform converts a signal into a series of wavelets – In theory, signals processed by wavelets can be stored more efficiently compared to Fourier transform – Wavelets can be constructed with rough edges, to better approximate real-world signals – Wavelets do not remove information but move it around, separating out the noise and averaging the signal – Noise (or detail) and average are expressed as sum and difference of signal, sampled at different points * In a picture, the signal is given by pixels * Average and detail are represented by sum and difference of pixels * Implemented with a low-pass filter for average and high-pass filter for detail • Provide foundation for a new approach to signal processing and analysis called multiresolution – Concerned with the representation and analysis of images at more than one resolution – May be able to detect features at different resolutions – At the finest scale, average and detail are computed by sum and difference of neighboring pixels – We move to a coarser level by taking sum and difference of the previous levels in a recursive/iterative manner Background