International Journal of Wavelets, Multiresolution and Information Processing

In this paper, we develop the basic stability theory for g-frames and g-Riesz bases in Hilbert C*-module. Also, we study stability of the dual g-frames in Hilbert C*-module. We extend the Casazza–Christensen perturbation theorem to g-frames in Hilbert C*-module but this is not valid for g-Riesz bases. We prove some characterizations of g-frames and g-Riesz bases in Hilbert C*-module.

A two-dimensional point process, if considered as a random measure, can be expressed as a countable sum of Delta Dirac measures concentrated at some random points. Then a continuous wavelet transform can be applied to obtain information on some structural properties. We introduce the notions of wavelet-based isotropy, main anisotropy direction and anisotropy degree to characterize the implicit anisotropic structure of the point process. We propose several statistical hypothesis tests that are proved to be useful to test for the presence of anisotropy. An application to a real case is also included.

Images are often corrupted by noise due to the imperfection of image acquisition systems and transmission channels. Traditional algorithms perform image denoising in the pixel domain. However, the transform domain denoising methods have shown outstanding success over the last decades. There are many image denoising methods which are in existence over the last decades, originated from various disciplines such as probability theory, statistics, partial differential equations, linear and nonlinear filtering, spectral and multiresolution analysis due to the robustness of the systems. Recently, image denoising has been attracting much attention using the wavelet transform. Wavelet based approach provides a particularly useful method for image denoising when the preservation of contents in the scene is of importance because the local adaptivity is based explicitly on the values of the wavelet detail coefficients. In this paper, we have proposed a new thresholding technique based on local contrast and adaptive mean in the wavelet transform domain.

An explicit, simple characterization of those operators that arise as Weyl–Heisenberg frame operators on L²(ℝⁿ) is obtained in terms of the concept of tile vertices of an operator. Generalizations of these frames to abstract Hilbert spaces are introduced and these and their frame operators are characterized in terms of Weyl–Heisenberg frames and frame operators in Euclidean spaces. Appropriate "window" versions of these concepts and results are also presented.

In this cryptosystem, we have considered RGB images for two-dimensional (2D) data security. Security of RGB images during transmission is a major concern, discussed globally. This paper proposes a novel technique for color image security by random hill cipher (RHC) over SL_n(𝔽) domain associated with 2D discrete wavelet transform. Existing techniques have discussed the security of image data on the basis of the keys only (which provide only one layer of security for image data), but in the proposed cryptosystem, the keys and the arrangement of RHC parameters are imperative for correct decryption of color image data. Additionally, key multiplication side (pre or post) with the RGB image data should inevitably be known, to correctly decrypt the encrypted image data. So, the proposed cryptosystem provides three layers of security for RGB image data. In this approach, we have considered keys from the special linear group over a field 𝔽, which provides enormous key-space for the proposed cryptosystem. A computer simulation on standard examples and results is given to support the fixture of the scheme. Security analysis, and detailed comparison between formerly developed techniques and proposed cryptosystem are also discussed for the robustness of the technique. This method will have large potential usage in the digital RGB image processing and the security of image data.

In this paper, we introduce the notion of finite wave packet groups for positive prime integers, as groups of cyclic dilations, translations and modulations. We will present a unified group theoretical approach for the cyclic wave packet transform on finite Abelian groups of prime order. Then, we study basic properties of this windowed transform.

In this paper, we show that [½, 1) is not a fixed point set of any wavelet induced isomorphism. The same holds for [0, ½). This settles Problem 1 posed in the paper "On wavelet induced isomorphisms" by Singh in Int. J. Wavelets Multiresolut. Inf. Process.8 (2010) 359–371. Further, we obtain that every other subinterval of [0, 1) of measure ½ is a fixed point set of some wavelet induced isomorphism.

Compressive sensing (CS) has been proposed for images that are sparse under a certain transform domain. However, many natural images are not strictly sparse in the transform domain, causing a tail-folding effect that degrades the performance of the CS reconstruction. To decrease such effect, a sparse-filtering (SF) in Directional Lifting Wavelet Transform (DLWT) domain based Bayesian compressive sensing (BCS) algorithm (DLWT-SF-TSW-BCS) is proposed. At the encoder, DLWT, an efficient multi-scale geometrical analysis (MGA) tool, is used to produce the sparse representation for natural images. Then sparse-filtering is adopted to cut off the small DLWT coefficients before random measurement. At the decoder, the interscale tree-structure redundancy in DLWT domain is further exploited in Bayesian reconstruction. Experimental results show that the proposed DLWT-SF-TSW-BCS algorithm significantly outperforms other state-of-the-art CS reconstruction algorithms, for example, peak signal to noise ratio (PSNR) gain up to 10.00 dB over the tree structured wavelet compressive sensing (TSW-CS).

In this paper, we present an algorithm to automatically detect meaningful modes in a histogram. The proposed method is based on the behavior of local minima in a scale-space representation. We show that the detection of such meaningful modes is equivalent in a two classes clustering problem on the length of minima scale-space curves. The algorithm is easy to implement, fast and does not require any parameter. We present several results on histogram and spectrum segmentation, grayscale image segmentation and color image reduction.

Support vector machine regression (SVMR) is a regularized learning method in reproducing kernel Hilbert spaces with epsilon-insensitive loss function. Different from the previously known works on the generalization ability of SVMR with independent and identically distributed (i.i.d.) samples, in this paper, we consider the generalization ability of SVMR algorithm based on non-i.i.d. samples, uniformly ergodic Markov chain (u.e.M.c.) samples. We give an error analysis for SVMR algorithm based on u.e.M.c. samples and obtain the optimal learning rate for the SVMR algorithm based on u.e.M.c. samples.