Kevin Amaratunga

A Discrete Wavelet Transform without edge effects using wavelet extrapolation

The Discrete Wavelet Transform (DWT) is of considerable practical use in image and signal processing applications. For example, significant compression can be achieved through the use of the DWT. A fundamental problem with the DWT, however, is the treatment of finite length data sequences. Commonly used techniques such as circular convolution and symmetric extension can produce undesirable edge effects which propagate into the interior of the transformed data as the number of DWT iterations increases.In this paper, we develop a DWT applicable to Daubechies’ orthogonal wavelets which does not exhibit edge effects. The underlying idea is to extrapolate the data at the boundaries by determining the coefficients of a best fit polynomial through data points in the vicinity of the boundary. This approach can be regarded as a solution to the problem of orthogonal wavelets on an interval. However, it has the advantage that it does not involve the explicit construction of boundary wavelets. The extrapolated DWT is designed to be well conditioned and to produce a critically sampled output. The methods we describe are equally applicable to biorthogonal wavelet bases.

Multiplierless approximation of transforms with adder constraint

This letter describes an algorithm for systematically finding a multiplierless approximation of transforms by replacing floating-point multipliers with VLSI-friendly binary coefficients of the form k/2/sup n/. Assuming the cost of hardware binary shifters is negligible, the total number of binary adders employed to approximate the transform can be regarded as an index of complexity. Because the new algorithm is more systematic and faster than trial-and-error binary approximations with adder constraint, it is a much more efficient design tool. Furthermore, the algorithm is not limited to a specific transform; various approximations of the discrete cosine transform are presented as examples of its versatility.

Wavelet-Galerkin solution of boundary value problems

SummaryIn this paper we review the application of wavelets to the solution of partial differential equations. We consider in detail both the single scale and the multiscale Wavelet Galerkin method. The theory of wavelets is described here using the language and mathematics of signal processing. We show a method of adapting wavelets to an interval using an extrapolation technique called Wavelet Extrapolation. Wavelets on an interval allow boundary conditions to be enforced in partial differential equations and image boundary problems to be overcome in image processing. Finally, we discuss the fast inversion of matrices arising from differential operators by preconditioning the multiscale wavelet matrix. Wavelet preconditioning is shown to limit the growth of the matrix condition number, such that Krylov subspace iteration methods can accomplish fast inversion of large matrices.

Dyadic-based factorizations for regular paraunitary filterbanks and M-band orthogonal wavelets with structural vanishing moments

Paraunitary filterbanks (PUFBs) can be designed and implemented using either degree-one or order-one dyadic-based factorization. This work discusses how regularity of a desired degree is structurally imposed on such factorizations for any number of channels M /spl ges/ 2, without necessarily constraining the phase responses. The regular linear-phase PUFBs become a special case under the proposed framework. We show that the regularity conditions are conveniently expressed in terms of recently reported M-channel lifting structures, which allow for fast, reversible, and possibly multiplierless implementations in addition to improved design efficiency, as suggested by numerical experience. M-band orthonormal wavelets with structural vanishing moments are obtained by iterating the resulting regular PUFBs on the lowpass channel. Design examples are presented and evaluated using a transform-based image coder, and they are found to outperform previously reported designs.

Theory and factorization for a class of structurally regular biorthogonal filter banks

Regularity is a fundamental and desirable property of wavelets and perfect reconstruction filter banks (PRFBs). Among others, it dictates the smoothness of the wavelet basis and the rate of decay of the wavelet coefficients. This paper considers how regularity of a desired degree can be structurally imposed onto biorthogonal filter banks (BOFBs) so that they can be designed with exact regularity and fast convergence via unconstrained optimization. The considered design space is a useful class of M-channel causal finite-impulse response (FIR) BOFBs (having anticausal FIR inverses) that are characterized by the dyadic-based structure W(z)=I-UV/sup /spl dagger//+z/sup -1/UV/sup /spl dagger// for which U and V are M/spl times//spl gamma/ parameter matrices satisfying V/sup /spl dagger//U=I/sub /spl gamma//, 1/spl les//spl gamma//spl les/M, for any M/spl ges/2. Structural conditions for regularity are derived, where the Householder transform is found convenient. As a special case, a class of regular linear-phase BOFBs is considered by further imposing linear phase (LP) on the dyadic-based structure. In this way, an alternative and simplified parameterization of the biorthogonal linear-phase filter banks (GLBTs) is obtained, and the general theory of structural regularity is shown to simplify significantly. Regular BOFBs are designed according to the proposed theory and are evaluated using a transform-based image codec. They are found to provide better objective performance and improved perceptual quality of the decompressed images. Specifically, the blocking artifacts are reduced, and texture details are better preserved. For fingerprint images, the proposed biorthogonal transform codec outperforms the FBI scheme by 1-1.6 dB in PSNR.

Generalized hierarchical bases: a Wavelet‐Ritz‐Galerkin framework for Lagrangian FEM

Purpose – This paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains.Design/methodology/approach – Second‐generation wavelet construction gives rise to a powerful generalization of the traditional hierarchical basis (HB) finite element method (FEM). A framework based on piecewise polynomial Lagrangian multiwavelets is used to generate customized multiresolution bases that have not only HB properties but also additional qualities.Findings – For the 1D Poisson problem, we propose – for any given order of approximation – a compact closed‐form wavelet basis that block‐diagonalizes the stiffness matrix. With this wavelet choice, all coupling between the coarse scale and detail scales in the matrix is eliminated. In contrast, traditional higher‐order (n>1) HB do not exhibit this property. We also achieve full scale‐decoupling for the 2D Poisson problem on an irregular mesh. No traditional HB has this quality in 2D.Research l...

Time integration using wavelets

In this work, we describe how wavelets may be used for the temporal discretization of ODEs and PDEs. A major problem associated with the use of wavelets in time is that initial conditions are difficult to impose. A second problem is that a wavelet-based time integration scheme should be stable. We address both of these problems. Firstly, we describe a general method of imposing initial conditions, which follows on from some of our recent work on initial and boundary value problems. Secondly, we use wavelets of the Daubechies family as a starting point for the development of stable time integration schemes. By combining these two ideas we are able to develop schemes with a high order of accuracy. More specifically, the global error is O(hp-1), where p is the number of vanishing moments of the original wavelet. Furthermore, these time integration schemes are characterized by large regions of absolute stability, comparable to increasingly high order BDF methods. In particular, Daubechies D4 and D6 wavelets give rise to A-stable time-stepping schemes. In the present work we deal with single scale formulations. We note, however, that the standard multiresolution analysis for orthogonal wavelets on L2(R) applies here. This opens up interesting possibilities for treating BVPs and IVPs at multiple scales.

Spatially Adapted Multiwavelets and Sparse Representation of Integral Equations on General Geometries

In this paper we develop irregular wavelet representations for complex domains with the goal of demonstrating their potential for three-dimensional (3D) scientific and engineering computing applications. We show existence and construction of a large class of continuous spatially adapted multiwavelets in

Closed-form design of maximally flat IIR half-band filters

R^{\eta}

Multiplierless approximation of transforms using lifting scheme and coordinate descent with adder constraint

with vanishing moments over complex geometries. These wavelets share all of the major advantages of conventional wavelets in that they provide an analytical tool for studying data, functions, and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, spatially adapted multiwavelets allow fast transform, localization, and decorrelation on complex meshes, such as those encountered in finite element modeling. We show how these new constructions can be applied to partial differential equations cast in the integral form. We implement the wavelet approach for several model two-dimensional (2D) and 3D potential problems. It is shown that the optimal convergence rate is achieved, with only