International Journal of Wavelets, Multiresolution and Information Processing

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every $(U,C)$-controlled Riesz basis ${\{f_k\}}_{k=1}^{\infty }$ is in the form ${\{U^{-1}CM{e}_k\}}_{k=1}^{\infty }$, where $M$ is a bijective operator on $H$. Furthermore, we propose an equivalent accessible condition to the sequence ${\{f_k\}}_{k=1}^{\infty }$ being a $(U,C)$-controlled Riesz basis.

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products $U{P}_{\lambda }$ and $U{P}_G$ that are similar to the Heisenberg uncertainty product. $U{P}_{\lambda }$ and $U{P}_G$ differ from each other by the metric used for the Vilenkin group $G$. We discuss analogs of a quantitative uncertainty principle. Representations for $U{P}_{\lambda }$ and $U{P}_G$ in terms of Walsh and Haar basis are given.

In order to improve fused image quality of multi-spectral (MS) image and panchromatic (PAN) image, a new remote sensing image fusion algorithm based on robust principal component analysis (RPCA) and non-subsampled shearlet transform (NSST) is proposed. First, the first principle component PC1 of MS image is extracted via principal component analysis (PCA). Then, the component PC1 and PAN image are decomposed by NSST to get the low and high frequency subbands, respectively. For the low frequency subband, the sparse matrix of PAN image by RPCA decomposition is used to guide the fusion rule; for the high frequency subbands, the fusion rule employed is based on adaptive PCNN model. Finally, the fusion image is obtained by inverse NSST transform and inverse PCA transform. The experimental results illustrate that the proposed fusion algorithm outperforms other classical fusion algorithms (PCA, Curvelet, NSCT, NSST and NSCT-PCNN) in terms of visual quality and objective evaluation in whole, and achieve better fusion performance.

By using biorthogonal wavelets, Reynaud-Bouret, Rivoirard and Tuleau-Malot provide the adaptive and optimal $L^2$-risk estimation for density functions (not necessarily having compact support) in a Besov space $B_{r,q}^s(ℝ)$ [P. Reynaud-Bouret, V. Rivoirard and C. Tuleau-Malot, Adaptive density estimation: A curse of support?, J. Stat. Plan. Inference 141(1) (2011) 115–139]. The authors pose an open problem: Can $L^p$-risk ($1\le p<\infty$) estimation be given in their setting? In this paper, we try to solve that problem for $p\in [2,+\infty )$ by using wavelet estimators.

Different graph kernels may correspond to using different notions of similarity or may be using information coming from multiple sources. In this paper, we develop a common method to construct combined graph kernel (CGK) which is based on a family of graph kernels. We define three kinds of CGK. The first one is called the weighted combined graph kernel and is a parametric CGK. The second one is called the accuracy ratio weighted combined graph kernel and is a non-parametric CGK. The third one is called the product combined graph kernel and also belongs to non-parametric CGK. The three kinds of definition of CGK can be applied for constructing CGK based on a family of graph kernels. This family of kernels is demonstrated based on the Weisfeiler–Lehman (WL) sequence of graphs in this paper, including a highly efficient subtree kernel, edge kernel, and shortest path kernel. Experiments demonstrate that our CGK based on WL graph kernels outperforms the corresponding single WL graph kernel on several classification benchmark data sets.

In this paper, we propose a consolidated framework for the automatic selection of the most discriminant subbands for the problem of face recognition. Essentially, the face images are transformed into textures using the linear binary pattern (LBP) approach, these texturized-faces undergo the wavelet packet decomposition resulting in several subband images. We propose to use the energy features to effectively represent these subband images. The underlying statistical patterns of the data are harnessed in form of information-theoretic metrics to select the most discriminant subbands. The proposed algorithms are extensively evaluated on several standard databases and are shown to always pick the most significant subbands resulting in better performance. The proposed algorithms are entirely generic and do not depend on the selection of features or/and classifiers.

The multiwindow discrete Gabor transform (M-DGT) is an important time-frequency analysis tool in many applications. In this paper, sparse time-frequency representation (TFR) based on M-DGT with sparse regularization theory is presented. The M-DGT is first formulated as a convex constrained optimization model by minimizing the objective function with a mixed ${\ell }_1$–${\ell }_2$ norm of the M-DGT coefficients. Then, an iterative algorithm based on the split Bergman method is utilized to compute the sparse Gabor time-frequency spectrum of the analyzed signal. According to the Heisenberg uncertainty principle, using an analysis window with good time resolution in M-DGT will lead to the Gabor TFR with high frequency resolution and vice versa. To obtain the sparse TFR with good time-frequency resolution (or concentration), the sparse spectra of M-DGT can be combined by the arithmetic average or the geometric average. Numerical experiments clearly show that the proposed method is an effective and powerful tool for analyzing nonstationary signals, by which the high time-frequency concentration (or resolution) of the Gabor time-frequency spectrum can be obtained as compared to traditional M-DGT.

Let $X=X\cup Z$ be a data set in ${ℝ}^D$, where $X$ is the training set and $Z$ is the test one. Many unsupervised learning algorithms based on kernel methods have been developed to provide dimensionality reduction (DR) embedding for a given training set $\mathrm{\Phi }:X\to {ℝ}^d$ ($d\ll D$) that maps the high-dimensional data $X$ to its low-dimensional feature representation $Y=\mathrm{\Phi }(X)$. However, these algorithms do not straightforwardly produce DR of the test set $Z$. An out-of-sample extension method provides DR of $Z$ using an extension of the existent embedding $\mathrm{\Phi }$, instead of re-computing the DR embedding for the whole set $X$. Among various out-of-sample DR extension methods, those based on Nyström approximation are very attractive. Many papers have developed such out-of-extension algorithms and shown their validity by numerical experiments. However, the mathematical theory for the DR extension still need further consideration. Utilizing the reproducing kernel Hilbert space (RKHS) theory, this paper develops a preliminary mathematical analysis on the out-of-sample DR extension operators. It treats an out-of-sample DR extension operator as an extension of the identity on the RKHS defined on $X$. Then the Nyström-type DR extension turns out to be an orthogonal projection. In the paper, we also present the conditions for the exact DR extension and give the estimate for the error of the extension.

Sliced inverse regression (SIR) is a powerful method to deal with a dimension reduction model. As well known, SIR is equivalent to a transformation-based projection pursuit, where the optimal directions are just the directions in SIR. In this paper, we consider simultaneous estimations of optimal directions for functional data and optimal transformations. We take a reproducing kernel Hilbert space approach. Both the directions and the transformations are chosen from reproducing kernel Hilbert spaces. A learning rate is established for the estimators.

In this paper, we consider the commutator

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

Partial differential equation (PDE) occurs in the mathematical modeling of many physical phenomena arising in science and engineering. In this paper, we present wavelet-based lifting scheme for the numerical solution of nonlinear PDEs using different wavelet filter coefficients. The numerical results obtained by this scheme are compared with the exact solution to demonstrate the accuracy and also speeds up convergence in lesser computational time as compared with the existing schemes. Some test problems are presented for the validity and applicability of the scheme.