Wavelets, a modern tool for signal processing

transforms to compress digitally scanned fingerprints. NASA’s Mars rovers use them to compress images acquired by their 18 cameras. The new JPEG 2000 format is based on wavelet transforms; the smaller files allow one to store images using less memory and to transmit those images faster and more reliably. Researchers also use wavelet transforms to clean signals and images—that is, to reduce unwanted noise and blurring. As illustrated in the figure, some algorithms for processing astronomical images are based on wavelets and wavelet-like transforms. A wavelet transform, like a Fourier transform, involves integrating a product of a signal and an oscillating function. But unlike the everlasting sines and cosines of Fourier analysis, the oscillating functions in a wavelet transform are usually stretched and translated versions of a single oscillating function of short duration; indeed that localized function is the “wavelet.” This tutorial describes an elementary wavelet transform, illustrates why it is effective for compression and noise reduction, and briefly describes how the basic wavelet and noise reduction methods can be improved. The focus is on wavelet transforms used for image compression and reduction of noise and blur; such transforms must be invertible. But a second type of wavelet transform is worth noting briefly. It is designed for signal analysis—for example, to detect faults in machinery from sensor measurements, to study electroencephalograms or other biomedical signals, or to determine how the frequency content of a signal evolves over time. In those cases, the wavelet transform need not be invertible.