Wavelet Transforms — A Quick Study

The wavelet transform has become a useful computational tool for a variety of signal and image processing applications. For example, the wavelet transform is useful for the compression of digital image files; smaller files are important for storing images using less memory and for transmitting images faster and more reliably. The FBI uses wavelet transforms for compressing digitally scanned fingerprint images. NASA’s Mars Rovers used wavelet transforms for compressing images acquired by their 18 cameras. The wavelet-based algorithm implemented in software onboard the Mars Rovers is designed to meet the special requirements of deep-space communication. In addition, JPEG2K (the newer JPEG image file format) is based on wavelet transforms. Wavelet transforms are also useful for ‘cleaning’ signals and images (reducing unwanted noise and blurring). Some algorithms for processing astronomical images, for example, are based on wavelet and wavelet-like transforms. This Quick Study describes the wavelet transform, illustrates why it is effective for noise reduction, and briefly describes several improvements of the basic wavelet transform and basic noise reduction method used in the illustration. We describe what the wavelet transform is, and we describe algorithms for processing a signal after its wavelet transform has been computed. First we should point out that there are two basic types of wavelet transform. One type of wavelet transform is designed to be easily reversible (invertible); that means the original signal can be easily recovered after it has been transformed. This kind of wavelet transform is used for image compression and cleaning (noise and blur reduction). Typically, the wavelet transform of the image is first computed, the wavelet representation is then modified appropriately, and then the wavelet transform is reversed (inverted) to obtain a new image. The second type of wavelet transform is designed for signal analysis; for example, to detect faults in machinery from sensor measurements, to study EEG or other biomedical signals, to determine how the frequency content of a signal evolves over time. In these cases, a modified form of the original signal is not needed and the wavelet transform need not be inverted (it can be done in principle, but requires a lot of computation time in comparison with the first type of wavelet transform). In this Quick Study we will focus on those wavelet transforms that are easily invertible. The most basic wavelet transform is the Haar transform described by Alfred Haar in 1910. It serves as the prototypical wavelet transform. We will describe the (discrete) Haar transform, as it