Multiwavelet With Special Properties And Its Application In Image Processing

Multiwavelet is the new development of wavelet analysis.It is very attractive because it can possess orthogonality,short support,high order of smoothness,high order of vanishing moments,and symmetry/antisymmetry simultaneously.However,the further development of multiwavelet has confronted two problems.One is that the construction of multiwavelet is very difficult.There is no uniform and simple relationship between the highpass filters and lowpass filters in multiwavelet as that in scale-value wavelet.So it's not easy to compute multiwavelets even if we know the corresponding multiscaling functions.The other one is that the application of multiwavelet is not as good as expected though it has perfect theory support.And its application is inconvenient because of prefiltering.Lian found one way to tackle these problems by introducing a method of constructing Armlet orthogonal multiwavelet,whose filter meets flipping property and highpass annihilation.The paper is based on Lian's results. We first extend the notion of Armlet and flipping property of filters to biorthogonal system.Then we derive a concise relationship between the highpass and lowpasss filters in biorthogonal set.This relationship makes it easy to construct multiwavelet because multiwavelets and multiscaling functions are decided only by two polynomials.It also provides much flexible for construction because it contains parameters and the number of parameters can be pointed as needed.With the help of these relationships, we construct a set of Armlet multiwavelets with multiplicity r=2,such as odd-length orthogonal Armlet multiwavelets,biorthogonal Armlet multiwavelets and BBMA multiwavelets.In addition,we also prove that multiwavelet with Armlet property can guarantee wavelet decomposition with highpass output not being effected by polynomial perturbation of the input.This speciality makes it convenient for these multiwavelets to application,because they needn't prefiltering just like what balanced multiwavelets can do.Finally,we apply these examples to image processing. The simulation results show that multiwavelet with Armlet property has a promising future in application.