Biorthogonal Wavelet Systems Research Articles

Tight frame: an efficient way for high-resolution image reconstruction

High-resolution image reconstruction arise in many applications, such as remote sensing, surveillance, and medical imaging. The model proposed by Bose and Boo [Int. J. Imaging Syst. Technol. 9 (1998) 294–304] can be viewed as passing the high-resolution image through a blurring kernel, which is the tensor product of a univariate low-pass filter of the form [1/2+ ε ,1,…,1,1/2− ε ], where ε is the displacement error. Using a wavelet approach, bi-orthogonal wavelet systems from this low-pass filter were constructed in [R. Chan et al., SIAM J. Sci. Comput. 24 (4) (2003) 1408–1432; R. Chan et al., Linear Algebra Appl. 366 (2003) 139–155] to build an algorithm. The algorithm is very efficient for the case without displacement errors, i.e., when all ε =0. However, there are several drawbacks when some ε ≠0. First, the scaling function associated with the dual low-pass filter has low regularity. Second, only periodic boundary conditions can be imposed, and third, the wavelet filters so constructed change when some ε change. In this paper, we design tight-frame symmetric wavelet filters by using the unitary extension principle of [A. Ron, Z. Shen, J. Funct. Anal. 148 (1997) 408–447]. The wavelet filters do not depend on ε , and hence our algorithm essentially reduces to that of the case where ε =0. This greatly simplifies the algorithm and resolves the drawbacks of the bi-orthogonal approach.

Effective Value Calculation in Wavelet Domain

This paper discusses the methods for calculating effective values of voltage and current in the wavelet domain. It is shown that when a voltage or current signal is decomposed and analyzed using discrete wavelets in a power-quality study, the effective values can be evaluated using the sets of coefficients at each decomposition level. Two formulas are mathematically proved in the paper. One is for orthogonal wavelet analysis; another one is for biorthogonal wavelet systems. Applications of these formulas are numerically illustrated based on two voltage waveforms, and the numerical issues related to using the formulas are also discussed. The formulas presented can be easily programmed and incorporated into power-quality analyzing devices.

The solution of ill-conditioned symmetric toeplitz systems via two-grid and wavelet methods

In this paper, we present a generalized multigrid method combined with wavelet filters for solving ill-conditioned symmetric Toeplitz systems T n x = b, where T n ϵ R n× n is generated by nonnegative functions with zeros. First, we propose the construction of general Cohen, Daubechies, and Feauveau (CDF) 9/7 biorthogonal wavelet systems, so that a new class of compactly supported biorthogonal wavelet systems GCDF are achieved with specified vanishing moments for scaling functions. In order to solve ill-conditioned Toeplitz systems by using the two-grid method (TGM), we use the constructed GCDF wavelets to get prolongation and restriction operators. As a result, the proposed TGM is proved by numerical experiment to be more efficient than the classic TGM, especially when T n is seriously ill-conditioned. For the vanishing moments being N, experimental tests illustrate that the TGM with damped-Jacobi smoother converges when the generating function has zeros of order less than or equal to 2N ( N ≤ 8). Besides, we prove in theory that the proposed method converges for Toeplitz systems that are generated by functions with zeros of order less than or equal to four.

Design of Silicon IP Cores for Biorthogonal Wavelet Transforms

A methodology for rapid silicon design of biorthogonal wavelet transform systems has been developed. This is based on generic, scalable architectures for the forward and inverse wavelet filters. These architectures offer efficient hardware utilisation by combining the linear phase property of biorthogonal filters with decimation and interpolation. The resulting designs have been parameterised in terms of types of wavelet and wordlengths for data and coefficients. Control circuitry is embedded within these cores that allows them to be cascaded for any desired level of decomposition without any interface logic. The time to produce silicon designs for a biorthogonal wavelet system is only the time required to run synthesis and layout tools with no further design effort required. The resulting silicon cores produced are comparable in area and performance to hand-crafted designs. These designs are also portable across a range of foundries and are suitable for FPGA and PLD implementations.

Quadrature prefilters for the discrete wavelet transform

Discrepancies between the discrete wavelet transform and the coefficients of the wavelet series are known to be reducible by initialization of input data. Prefilters based on Lagrange interpolants are derived here for biorthogonal compact support wavelet systems, providing exact subspace projection in cases of local polynomial smoothness. The resulting divergence acceleration in a nonpolynomial test case is examined. Irregular sampling rates are also accommodated.

Rapid design of biorthogonal wavelet transforms

A rapid design methodology for biorthogonal wavelet transform cores has been developed based on a generic, scaleable architecture for wavelet filters. The architecture offers efficient hardware utilisation by combining the linear phase property of biorthogonal filters with decimation in a MAC-based implementation. The design has been captured in VHDL and parameterised in terms of wavelet type, data word length and coefficient word length. The control circuit is embedded within the cores and allows them to be cascaded without any interface glue logic for any desired level of decomposition. The design time to produce silicon layout of a biorthogonal wavelet system is typically less than a day. The resulting silicon cores produced are comparable in area and performance to hand-crafted designs. The designs are portable across a range of foundries and are also applicable to FPGA and PLD implementations.

Construction of nearly orthogonal interpolating wavelets

This paper first studies the measure of redundancy in biorthogonal wavelet systems and gives the explicit parameter representations of the general interpolating filters for lifting Donoho wavelets. Based on the measure and the parameter representations, a novel method is proposed to optimize co-redundancy degree which is then used to construct a family of nearly semi-orthogonal interpolating wavelets. Furthermore, by performing the local orthogonalization on the obtained wavelet functions we design a family of nearly orthogonal interpolating wavelets. Finally, some examples are given which show that our scheme is efficient in designing practical wavelet systems with good performance indexes.

The Wavelet Element Method: Part I. Construction and Analysis

The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some Rn. The domain of interest is split into subdomains which are mapped to a simple reference domain, heren-dimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the reference domain such that mapping them to each subdomain and matching along the interfaces leads to a wavelet system on the domain. In this paper we use adapted biorthogonal wavelet systems on the interval in such a way that tensor products of these functions can be used for the construction of wavelet bases on the reference domain. We describe the matching procedure in any dimensionnin order to impose continuity and prove that it leads to a construction of a biorthogonal wavelet system on the domain. These wavelet systems characterize Sobolev spaces measuring both piecewise and global regularity. The construction is detailed for a bivariate example and an application to the numerical solution of second order partial differential equations is given.

A new class of biorthogonal wavelet systems for image transform coding

We construct general biorthogonal Coifman wavelet systems, a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair. A time-domain design method is employed and closed-form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived. The resulting filter coefficients are all dyadic fractions, which is an attractive feature in the realization of multiplication-free discrete wavelet transform. Even-ordered systems in this family are symmetric, which correspond to linear-phase dual filters. In particular, three filterbanks (FBs) in this family are systematically verified to have competitive compression potential to the 9-7 tap biorthogonal wavelet FB by Cohen et al., which is currently the most widely used one in the field of wavelet transform coding. In addition, the proposed FB's have much smaller computational complexity in terms of floating-point operations required in transformation, and therefore indicate a better tradeoff between compression performance and computational complexity.

On asymptotic convergence of the dual filters associated with two families of biorthogonal wavelets

The asymptotic behavior of the dual filters associated with biorthogonal spline wavelet (BSW) systems and general biorthogonal Coifman wavelet (GBCW) systems are studied. As the order of the wavelet systems approaches infinity, the magnitude responses of the dual filters in the BSW systems either diverge or converge to some nonideal frequency responses. However, the synthesis filters in the GBCW systems converge to an ideal halfband lowpass filter without exhibiting any Gibbs-like phenomenon, and a subclass of the analysis filters also converge to an ideal halfband lowpass filter but with a one-sided Gibbs-like behavior. The two approximations of the ideal lowpass filter by the filter associated with a Daubechies orthonormal wavelet and by the synthesis filter in a GBCW system of the same order are compared. Such a study of the asymptotic behaviors of wavelet systems provides insightful characterization of these systems and systematic assessment and global comparison of different wavelet systems.