0.7 Generalizations of the basic multiresolution wavelet system  (Page 15/28)

Geronimo-hardin-massopust multiwavelets

A set of multiscaling filters based on fractal interpolation functions were developed in [link] , and the corresponding multiwavelets were constructed in [link] . As shown in [link] , they

are both symmetrical and orthogonal—a combination which is impossible for two-band orthogonal scalarwavelets. They also have short support, and can exactly reproduce the hat function. These interesting properties make multiwavelet a promisingexpansion system.

Spline multiwavelets

Spline bases have a maximal approximation order with respect to their length, however spline uniwavelets are only semiorthogonal [link] . A family of spline multiwavelets that are symmetric and orthogonal is developed in [link] .

Other constructions

Other types of multiwavelets are constructed using Hermite interpolating conditions [link] , matrix spectral factorization [link] , finite elements [link] , and oblique projections [link] . Similar to multiwavelets, vector-valued wavelets and vector filter banks are also developed [link] .

Applications

Multiwavelets have been used in data compression [link] , [link] , [link] , noise reduction [link] , [link] , and solution of integral equations [link] . Because multiwavelets are able to offer a combination of orthogonality,symmetry, higher order of approximation and short support, methods using multiwavelets frequently outperform those using the comparable scalewavelets. However, it is found that prefiltering is very important, and should be chosen carefully for the applications [link] , [link] , [link] . Also, since discrete multiwavelettransforms operate on size- $R$ blocks of data and generate blocks of wavelet coefficients, the correlation within each block of coefficients needs to beexploited. For image compression, predictions rules are proposed to exploit the correlation in order to reduce the bit rate [link] . For noise reduction, joint thresholding coefficients within each block improve the performance [link] .

Overcomplete representations, frames, redundant transforms, and adaptive bases

In this chapter, we apply the ideas of frames and tight frames introduced in Chapter: Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Right Frames, and unconditional Bases as well as bases to obtain a more efficient representation of many interesting signal classes. It might be helpful to review the material onbases and frames in that chapter while reading this section.

Traditional basis systems such as Fourier, Gabor, wavelet, and wave packets are efficient representations for certain classes of signals, butthere are many cases where a single system is not effective. For example, the Fourier basis is an efficient system for sinusoidal or smooth periodicsignals, but poor for transient or chirp-like signals. Each system seems to be best for a rather well-defined but narrow class of signals.Recent research indicates that significant improvements in efficiency can be achieved by combining several basis systems. One can intuitivelyimagine removing Fourier components until the expansion coefficients quit dropping off rapidly, then switching to a different basis system to expandthe residual and, after that expansion quits dropping off rapidly, switching to still another. Clearly, this is not a unique expansionbecause the order of expansion system used would give different results. This is because the total expansion system is a linear combination of theindividual basis systems and is, therefore, not a basis itself but a frame. It is an overcomplete expansion system and a variety of criteriahave been developed to use the freedom of the nonuniqueness of the expansion to advantage. The collection of basis systems from which asubset of expansion vectors is chosen is sometimes called a dictionary.