International Journal of Wavelets, Multiresolution and Information Processing

Banach frame systems in Banach spaces have been defined and studied. A sufficient condition under which a Banach space, having a Banach frame, has a Banach frame system has been given. Also, it has been proved that a Banach space E is separable if and only if E* has a Banach frame ({φ_n},T) with each φ_n weak*-continuous. Finally, a necessary and sufficient condition for a Banach Bessel sequence to be a Banach frame has been given.

The extension principles play an important role in characterizing and constructing of wavelet frames. The common extension principles, the unitary extension principle (UEP) or the oblique extension principle (OEP), are based on the unitarity of the modulation matrix. In this paper, we state the UEP and OEP for refinable function vectors in the polyphase representation. Finally, we apply our results to directional wavelets on triangles which we have constructed in a previous work. We will show that the wavelet system generates a tight frame for L²(ℝ²).

Long range dependent (LRD) stationary time series have historically served to model real time series with apparent changes in local mean level. A natural tool to study changes in local mean level is the unbalanced Haar wavelet transformation (UHT). In this work, UHT is used to study changes in local mean level in LRD models and several real and simulated time series exhibiting LRD. In particular, simulations for LRD models suggest that changes in local mean level occur at times essentially governed by a homogeneous Poisson arrival process, and only the local mean levels themselves inherit the LRD property of the original time series. These properties are compared with the analogous ones in several real and simulated time series. The results are mixed though generally in favor of LRD models. The approach based on UHT is also compared to several alternatives such as defining changes in local mean level through kernel smoothing. The interest throughout is mainly in very long time series such as those collected in the studies of data traffic over Internet.

Al-Cu thin films were deposited by DC magnetron sputtering. The films are characterized by atomic force microscopy and its surface morphologies are analyzed by wavelet technique. Multiresolution signal decomposition wavelet technique was employed to extract the surface roughness from the AFM images of Al-Cu thin films. It is observed that the Al-Cu thin films exhibit higher surface roughness value with increasing deposition time. The calculated surface roughness of the thin films, using wavelet technique, is comparable with that of its experimental values.

The Kernel Foley–Sammon Transform (KFST) performs well in solving nonlinear discriminant analysis problems. However, as a kernel method, KFST also faces with large kernel matrix calculation problems O(n³) with the sample size n. KFST will be very costly and even intractable to compute when n is large. In this paper, we propose a Fast Clustering-based Kernel Foley–Sammon Transform (FCKFST) approach to tackle this problem. FCKFST solves KFST over a reductive l × l matrix representing clustering data instead of an n × n matrix of the original data, where l is exceedingly smaller than n. This paper also proves that FCKFST improving the calculating efficiency does not decrease the classification precision with comparison to KFST. We apply our method to digit and image recognition problems, and we obtain good experimental results.

This paper focuses on the development of a new two-dimensional representation for images that can capture different features in images, ranging from highly directional ones to fully isotropic ones. We propose a multiselectivity analysis, defined by combining an isotropic multiscale and multidirection decomposition. The result is new half-continuous frame for each selectivity level. The angular selectivity of these frames grows with selectivity level. This selectivity level can be adapted locally to the content of the image; so it can be seen as an adaptive selectivity representation, which present adaptively isotropic, directional and intermediary features in images. The numerical experiments presented in this paper demonstrate that the adaptive selectivity approach is very competitive in image denoising and enhancement.

Coping with geometrical attacks in transform domain is crucial when we design a robust image watermarking scheme. In this paper, a novel contourlet-domain image watermarking scheme, which is robust to common signal processing and geometrical attacks, is proposed. First, the region with maximum energy in the directional subbands is considered for watermarking, i.e. the watermark can be embedded into the significant region as well as highly textured region of the host image. Then, for each coefficient of the selected subband, the strength factor was adaptively adjusted in terms of the energy of its parent and neighbor coefficients. Consequently, the tradeoff between the transparency and robustness of watermark can be achieved. Furthermore, the robust feature points, which can survive various signal processing and affine transformation, are extracted by using the Harris–Laplace detector. In watermark detection, the geometrical distortion of image is identified by using the feature template constructed during embedding phase, i.e. watermark resynchronization is performed. Experimental results show that the proposed watermark scheme is invisible and robust against common signal processing such as median filtering, sharpening, noise adding and JPEG compression, etc and geometrical attacks such as rotation, translation, scaling, row or column removal, shearing and local random bend, etc.

The redundancy of continuous wavelet transforms implies that the wavelet coefficients are not independent of each other. This interdependence allows the reconstruction or approximation of the wavelet transform, and of the original field, from a subset of the wavelet coefficients. Contrasting with lines of modulus maxima, known to provide useful partition functions and some data compaction, the reconstruction from single-scale coefficients is derived for the Hermitian family of wavelets. The formula is exact in the continuum for d-dimensional fields, and its limitations under discretization are illustrated.