Wavelets and Multiresolution Processing

– 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