International Journal of Wavelets, Multiresolution and Information Processing

Image deconvolution is an important problem, which has seen plenty of progress in the last decades. Due to its ill-posedness, a common approach is to formulate the reconstruction as an optimization problem$,$ regularized by an additional sparsity-enforcing term. This term is often modeled as an ${\ell }_1$ norm measured in the domain of a suitable signal transform. The resulting optimization problem can be solved by an iterative approach via Landweber iterations with soft thresholding of the transform coefficients. Previous approaches focused on thresholding in the wavelet-domain. In particular, recent work [C. Vonesch and M. Unser, A fast thresholded Landweber algorithm for wavelet-regularized multidimensional deconvolution, IEEE Trans. Image Process. 17(4) (2008) 539–549.] has shown that the use of Shannon wavelets results in particularly efficient reconstruction algorithms. The present paper extends this approach to Shannon shearlets, which we also introduce in this work. We show that for anisotropic blurring filters, such as the motion blur, the novel shearlet-based approach allows for further a improvement in efficiency. In particular, we observe that for such kernels using shearlets instead of wavelets improves the quality of image restoration and SERG, when compared after the same number of iterations.

High resolution digital cameras and state-of-the-art image editing software tools has given rise to large amount of manipulated images leaving no traces of being subjected to any manipulation. Passive or blind forgery detection algorithms are used in order to determine its authenticity. In this paper, an algorithm is proposed that blindly detects global rescaling operation using the statistical models computed based on quadrature mirror filter (QMF) decomposition. Fuzzy entropy measure is employed to choose the relevant features and to remove non-important features whereas artificial neural network classifier is used for forgery detection. Experimental results are presented on grayscale and $G$-component images of UCID database to prove the validity of the algorithm under different interpolation schemes. Results are provided for the detection of rescaled images with JPEG compression, arbitrary cropping and white Gaussian noise addition. Further, results are shown using USC-SIPI database to prove the robustness of the algorithm against the type of database.

For any symmetry group ${\mathscr{H}}$, any appropriate matrix dilation (compatible with ${\mathscr{H}}$) and any appropriate symmetry center $c$ we give an explicit method for the construction of ${\mathscr{H}}$-symmetric with respect to the center $c$ refinable masks which have sum rule of an arbitrary order $n$. Moreover, we give a description of all these masks. For any symmetry group ${\mathscr{H}}$, any appropriate matrix dilation (compatible with ${\mathscr{H}}$) and any appropriate row of symmetry centers $C$ we give two explicit methods for the construction of ${\mathscr{H}}$-symmetric with respect to the row of centers $C$ refinable matrix masks which have sum rule of an arbitrary order $n$. A description of all such matrix masks is also presented.

In this paper, a novel face descriptor, the Gabor-scale binary pattern (GSBP), is proposed to explore the neighboring relationship in spatial, frequency and orientation domains for the purpose of face recognition. In order to extract the GSBP feature, the Gabor-scale volume and the Gabor-scale vector are introduced by using a group of Gabor wavelet coefficients with a special orientation. Moreover, the Gabor-scale length pattern and the Gabor-scale ratio pattern are proposed. Compared with the existed methods, GSBP utilizes the deep relations between neighboring Gabor subimages instead of directly combining Gabor wavelet transform and local binary pattern. For estimating the performance of GSBP, we compare the proposed method with the related methods on several popular face databases, including LFW, FERET, AR, Yale and Extended YaleB databases. The experimental results show that the proposed method outperforms several popular face recognition methods.

In this paper, a numerical approximation based on Legendre wavelets has been developed to solve nonlinear fractional Volterra–Fredholm integro-differential equations. Legendre wavelets are generated by dilation and translation of Legendre polynomials. The properties of the Legendre wavelets are presented in the paper. The proposed wavelet method transforms the integral equations to a system of nonlinear algebraic equations and this algebraic system has been solved numerically by Newton’s method. Convergence analysis of the proposed method has been discussed in this paper. Some examples have been illustrated to show the applicability and accuracy of the present method.

Daubechies obtained the $n$-dimensional inversion formula for the continuous wavelet transform of spherically symmetric wavelets in $L^2({ℝ}^n)$ with convergence interpreted in the $L^2$-norm. From the wavelet $\psi \in L^2({ℝ}^n)$, Daubechies generated a doubly indexed family of wavelets ${\psi }_{a,b}$ by restricting the dilation parameter $a$ to be a real number greater than zero and the translation parameter $b$ belonging to ${ℝ}^n$. We show that $a$ can be chosen to be in ${ℝ}^n$ with none of the components $a_j$ vanishing. Further, we prove that if $f$ and ${\psi }_{a,b}$ are continuous in ${ℝ}^n$, then the convergence besides being in $L^2({ℝ}^n)$ is also pointwise in ${ℝ}^n$. We advance our theory further to the case when $f$ and $\psi$ both belong to $L^2({ℝ}^n)$ then convergence of the wavelet inversion formula is pointwise at all points of continuity of $f$. This result significantly enhances the applicability of the wavelet inversion formula to the image processing.

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

We consider the coefficient-based least squares regularized regression learning algorithm for the strongly and uniformly mixing samples. We obtain the capacity independent error bounds of the algorithm by means of the integral operator techniques. A standard assumption in theoretical study of learning algorithms for regression is the uniform boundedness of output sample values. We abandon this boundedness assumption and carry out the error analysis with output sample values satisfying a generalized moment hypothesis.

In this paper, we construct two new cyclotomic binary sequences of order $2d$ and length $2p^m$ which include the sequences constructed by Ke et al. in 2012. We also compute their linear complexity. The results show that these sequences have good linear complexity.

An improved glowworm swarm optimization algorithm with generalized opposition-based learning is proposed in this paper and is used in segmentation for magnetic resonance images. Noises are removed and intensity inhomogeneities are corrected in the MR images. Next, a clustering technique with glowworm swarm optimization algorithm with generalized opposition based learning is used. Finally, lesions are separated from the normal tissues of the brain in the post-processing step. The performance of the proposed methodology based on both numerical and visual results are compared with K-means and particle swarm optimization based methodologies over two sets of MR images. The experimental results demonstrate that the proposed methodology statistically outperforms other methodologies.

In this paper, we establish laws of the iterated logarithm for intersection times and volumes of Wiener sausages in critical dimensions by the high moment method.