International Journal of Wavelets, Multiresolution and Information Processing

We give the most general expression of orthonormal piecewise constant wavelet bases, in the case when the refinement process consists in dividing a domain into four subdomains having the same area. We show that the classical Haar bases are optimal with respect to the ∞-norm of the reconstruction error, but not with respect to the 1-norm. The construction applies not only to planar domains, but also to surfaces of revolution, like sphere, cone, paraboloid, or two-sheeted hyperboloid.

In this paper, a novel image fusion algorithm based on framelet transform is presented. The core idea is to decompose all the images to be fused into low and high-frequency bands using framelet transform. For fusion, two different selection strategies are developed and used for low and high-frequency bands. The first strategy is adaptive weighted average based on local energy and is applied to fuse the low-frequency bands. In order to fuse high-frequency bands, a new strategy is developed based on texture while exploiting the human visual system characteristics, which can preserve more details in source images and further improve the quality of fused image. Experimental results demonstrate the efficiency and better performance than existing image fusion methods both in visual inspection and objective evaluation criteria.

Let φ be a continuous function in L²(ℝ) such that the sequence {φ(t - n)}_n∈ℤ is a frame sequence in L²(ℝ) and assume that the shift-invariant space V(φ) generated by φ has a multi-banded spectrum σ(V). The main aim in this paper is to derive a multi-channel sampling theory for the shift-invariant space V(φ). By using a type of Fourier duality between the spaces V(φ) and L²[0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V(φ).

In recent years, considerable efforts have been made in the research of sparse representation for signals over overcomplete dictionaries. The dictionaries can be either pre-specified transforms or designed by learning from a set of training signals. In the paper, the dictionary learning problem was extended into a quadratic programming framework. A projected gradient with line search method was presented for solving this large-scale box-constrained quadratic program. The non-negative dictionary learned using this method was applied to image de-noising. Experimental results demonstrated that this learning-based method had better performance than the wavelet-based, the variation-based and the K-SVD methods.

We propose a stochastic gradient descent algorithm for the least square regression with coefficient regularization. An explicit expression of the solution via sampling operator and empirical integral operator is derived. Learning rates are given in terms of the suitable choices of the step sizes and regularization parameters.

In this paper, we present an explicit method to construct directly in the x-domain compactly supported scaling functions corresponding to the wavelets adapted to a sum of differential operators with constant coefficients. Here the adaptation to an operator is taken to mean that the wavelets give a diagonal form of the operator matrix. We show that the biorthogonal compactly supported wavelets adapted to a sum of differential operators with constant coefficients are closely connected with the representation of the null-space of the adjoint operator by the corresponding scaling functions. We consider the necessary and sufficient conditions (actually the Strang–Fix conditions) on integer shifts of a compactly supported function (distribution) f ∈ S'(ℝ) to represent exactly any function from the null-space of a sum of differential operators with constant coefficients.

The Easy Path Wavelet Transform (EPWT),²⁰ has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal N-term approximations for piecewise Hölder continuous functions with singularities along curves.

Spectral decompositions of translation and dilation operators are built in terms of suitable orthonormal bases of L²(ℝ), leading to spectral formulas for scaling functions and orthonormal wavelets associated with multiresolution analysis (MRA). The spectral formulas are useful to compute compactly supported scaling functions and wavelets. It is illustrated with a particular choice of the orthonormal bases, the so-called Haar bases, which yield a new algorithm related to the infinite product matrix representation of Daubechies and Lagarias.

The forward and inverse wavelet transform using the continuous Morlet basis may be symmetrized by using an appropriate normalization factor. The loss of response due to wavelet truncation is addressed through a renormalization of the wavelet based on power. The spectral density has physical units which may be related to the squared amplitude of the signal, as do its margins the mean wavelet power and the integrated instant power, giving a quantitative estimate of the power density with temporal resolution. Deconvolution with the wavelet response matrix reduces the spectral leakage and produces an enhanced wavelet spectrum providing maximum resolution of the harmonic content of a signal. Applications to data analysis are discussed.

In this paper, orthogonal multiwavelets on interval [0, 1] with arbitrary compact support γ and integer dilation factor a are studied. Firstly, the concept of multiwavelets on the interval [0, 1] is generalized to the arbitrary integer dilation factor a, a ≥ 2, a ∈ ℤ, and an algorithm for constructing orthogonal multiscaling function and multiwavelets on interval [0, 1] with dilation factor a is presented. Secondly, the decomposition and reconstruction formulas of multiwavelets on interval [0, 1] are deduced. Finally, the "balancing" concept of multiwavelets on the interval [0, 1] with dilation factor a is defined, and the algorithm of balancing the unbalanced multiwavelets on the interval [0, 1] is also given.