International Journal of Wavelets, Multiresolution and Information Processing

It is well-known that the different kinds of multiresolution analysis (MRA) structures generate different wavelets. In this paper, we give two uniform formulas on the number of mother functions for various wavelets associated with MRA structures. These formulas show that the number of mother functions of wavelets is determined by the support of the Fourier transform of the scaling function in MRA structure.

In this paper, the coefficient regularized regression algorithm with random projection is proposed. The excess error of the proposed algorithm associated with the reproducing kernel Hilbert space is bounded. Theoretical analysis shows that it is possible to learn directly in the projected domain and that random projection, as a preprocessing step in supervised learning, leads to empirical risk minimization on the projected data computationally simple.

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces.

We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

In this paper, we first present an adaptive intra-scale noise removal scheme, and estimate clean wavelet coefficients using new prior information with Bayesian estimation techniques. A new model using the non-informative improper Jeffreys' prior is given under the supposed Gaussian distribution for orthogonal wavelet transformation. Then, we propose a computationally feasible adaptive noise smoothing algorithm that considers the dependency characteristics of images. The wavelet coefficients are assumed to be non-Gaussian random variables for non-orthogonal redundancy transformation. The variances of the wavelet coefficients are estimated locally by a centered square-shaped window for every pixel within each subband. The experimental results show that the orthogonal wavelet transformation provides better results at the Gaussian assumption, while the non-orthogonal redundancy wavelet transformation performance tends to increase when the non-Gaussian bivariate distribution is used.

Texture classification is a challenging and important problem in image analysis. graphical models (GM) are promising tools for texture analysis. In this paper, we address the problem of learning the structure of Gaussian graphical models (GGM) for texture models. GGM can be considered as regression problems due to the connection between the local Markov properties and conditional regression of a Gaussian random variable. We utilize L₁-penalty regularization technique for appropriate neighborhood selection and parameter estimation simultaneously. The proposed algorithms are applied in texture synthesis and classification. Experimental results on Brodatz textures demonstrate that the proposed algorithms have good performance and prospects.

Transmission of block-coded images through error-prone radio mobile channel often results in lost blocks. Error concealment (EC) techniques exploit inherent redundancy and reduce visual artifacts through post-processing at the decoder side. In this paper, we propose an efficient quantization index modulation (QIM)-based data hiding scheme using dual-tree complex wavelet transform (DTCWT) for the application of image error concealment. The goal is achieved by embedding important information (image digest) as watermark signal that is extracted from the original image itself and is used to introduce sufficient redundancy in the transmitted image. At the decoder side, the extracted image digest is used to correct the damaged regions. DTCWT offers three-fold advantages viz. (1) high embedding capacity due to inherent redundancy that leads to the better reconstruction of high volume missing data, (2) better imperceptibly after data embedding since it most closely captures human visual system (HVS) characteristics than conventional DWT and (3) better watermark decoding reliability. Simulation results duly support the claims and relative performance improvement with respect to the existing results.

The lane information is essential to the highway intelligent vehicle applications. The direct description of the lanes is lane markings. Many vision methods have been proposed for lane markings detection. But in practice there are some problems to be solved by previous lane tracking systems such as shadows on the road, lighting changes, characters on the road and discontinuous changes in road types. Direction kernel function is proposed for robust detection of the lanes. This method focuses on selecting points on the markings edge by classification. During the classifying, the vanishing point is selected and the parts of the lane marking could form the lanes. The algorithm presented in this paper is proved to be both robust and fast by a large amount of experiments in variable occasions, besides, the algorithm can extract the lanes even in some parts of lane markings missing occasions.

Gabardo and Nashed have studied nonuniform multiresolution analysis based on the theory of spectral pairs in a series of papers, see Refs. 4 and 5. Farkov,³ has extended the notion of multiresolution analysis on locally compact Abelian groups and constructed the compactly supported orthogonal p-wavelets on L²(ℝ₊). We have considered the nonuniform multiresolution analysis on positive half-line. The associated subspace V₀ of L²(ℝ₊) has an orthonormal basis, a collection of translates of the scaling function φ of the form {φ(x ⊖ λ)}_λ∈Λ+ where Λ₊ = {0, r/N} + ℤ₊, N > 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ₊ is the set of non-negative integers. We find the necessary and sufficient condition for the existence of associated wavelets and derive the analogue of Cohen's condition for the nonuniform multiresolution analysis on the positive half-line.

Unitary Equivalence and Translation Representation play a key role in the Lax–Phillips Scattering Theory. In this paper we show that Translation Representation also plays an important role in Wavelet Theory, for Discrete Multi-Resolution Approximation as well as for Continuous Multi-Translation Approximation — to be defined in the paper.

Non-harmonic Fourier transform is useful for the analysis of transient signals, where the integral kernel is from the boundary value of Möbius transform. In this note, we study the Paley–Wiener type extension theorems for the non-harmonic Fourier transform. Two extension theorems are established by using real variable techniques.