International Journal of Wavelets, Multiresolution and Information Processing

In practical examples, there are numerous signals that are sparse in a redundant frame rather than an orthonormal basis. This paper mainly focuses on such sparse recovery via $l_q$-analysis-based dual frame with Weibull matrices under the assumption that signals are sparse or compressible in a general frame. First, we give the $l_{2,q}$ robust null space property and show that it is weaker than the $D$-RIP when $D$ is a general frame. Second, we show that Weibull random matrices with a general dual frame satisfy the $l_{2,q}$ robust null space property with high probability. Finally, we investigate the stability estimate of $l_q$-analysis based dual frame with Weibull matrices. The result shows that it remains stable and can guarantee accurate recovery of signals with high probability.

In this paper, we introduce the concept of $d$-dilation (dual) pseudo-splines in $L^2(ℝ)$ with dilation $d\ge 2$, characterize the linear independence of the integer shifts, the subdivision schemes of polynomial reproduction and Hölder regularity for (dual) pseudo-splines with general dilation. We present a new method to determine the Hölder regularity of refinable functions with general dilation $d\ge 2$. Furthermore, we compare the regularities between pseudo-splines and dual ones.

The work is concentrated to get good despeckled performance under non-homomorphic framework by involving simple distributions to obtain closed form Maximum-a-Posteriori probability (MAP) solutions. To estimate noise-free wavelet coefficient, distributions with lesser number of parameters allied Laplacian/Gaussian probability density function (pdf) for signal reflectivity and Rayleigh/Gaussian pdf for noisy signal are employed. Thus, the four despeckling methods are formed, namely LRMAP, GRMAP, LGMAP and GGMAP to despeckle SAR images and the effectiveness of different distributions is studied. The despeckling method is made adaptive by estimating the local variance of high frequency image using wavelet sub-band coefficient statistics. Also, the parameter space used in the proposed methods does not involve initialization, iterative search and convergence problems. The performances of the proposed methods are evaluated in terms of Equivalent Number of Looks (ENL), Peak Signal to Noise Ratio (PSNR), Edge Preservation ($\beta )$ and Mean Structural Similarity Index Measure (MSSIM). Experimental results show that LRMAP yields good results over all methods and GGMAP does not perform good for all images in terms of all quality metrics. Also, the proposed methods yield good quality metrics in less computing time as compared with the method available in the literature.

For Hilbert spaces $H$ and $K_i(i\in 𝕁)$, we use the notations $B_H^g({\{K_j\}}_{j\in 𝕁})$, $F_H^g({\{K_j\}}_{j\in 𝕁})$ and $R_H^g({\{K_j\}}_{j\in 𝕁})$ to denote the sets of all $g$-Bessel sequences, $g$-frames and Riesz bases in $H$ with respect to ${\{K_j\}}_{j\in 𝕁}$, respectively. By defining a linear operation and a norm, we prove that $B_H^g({\{K_j\}}_{j\in 𝕁})$ becomes a Banach space and is isometrically isomorphic to the operator space $B(H,K)$, where $K={\oplus }_{j\in 𝕁}{K}_j$. In light of operator theory, it is proved that $F_H^g({\{K_j\}}_{j\in 𝕁})$ and $R_H^g({\{K_j\}}_{j\in 𝕁})$ are open sets in $B_H^g({\{K_j\}}_{j\in 𝕁})$. This implies that both $g$-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection $\pi$ from $B_H^g({\{K_j\}}_{j\in 𝕁})$ onto the $C^{\ast }$-algebra $B(H)$, a multiplication and an involution on the Banach space $B_H^g({\{K_j\}}_{j\in 𝕁})$ are defined so that $B_H^g({\{K_j\}}_{j\in 𝕁})$ becomes a unital $C^{\ast }$-algebra that is isometrically isomorphic to the $C^{\ast }$-algebra $B(H)$, provided that $H$ and $K$ are isomorphic.

Affine transform is to describe the same target at different viewpoints to obtain the relationship between images of approximate model. Affine invariant feature extraction plays an important role in object recognition and image registration. Firstly, the definition of polar radius integral transform (PRIT) is put forward by means of the characterization of affine transform mapping straight line into straight line, where PRIT computes the integral along the polar radius direction and converts images into closed curves which keep the same affine transform with original images. Secondly, in order to extract affine invariant feature, an affine invariant feature extraction algorithm is also given based on PRIT. The proposed algorithm can be used to combine contour-based methods with region-based methods. It has some advantages of fewer amounts of computations and feasibility of feature extraction for objects with several components. Finally, the capability of anti-noise (Gaussian noise, salt and pepper noise) of PRIT is discussed. The simulation experiment results show that PRIT can effectively extract the affine invariant features, but also the low order PRIT has very strong robustness to noise.

By applying the multiscale method to the Möbius transformation function, we construct the multiscale analytic sampling approximation (MASA) to any function in the Hardy space $H^2(𝕋)$. The approximation error is estimated, and it is proved that the MASA is robust to sample error. We prove that the MASA can be expressed by a Hankel matrix, making use of which, a fast algorithm is established to compute the MASA. Since what we acquire in practice may well be the samples on time domain instead of the analytic ones on the unit disc of the complex plane, we establish a fast algorithm for acquiring analytic samples. Numerical experiments are carried out to demonstrate the efficiency of the MASA.

In this paper, we study the performance of kernel-based regression learning with non-iid sampling. The non-iid samples are drawn from different probability distributions with the same conditional distribution. A more general marginal distribution assumption is proposed. Under this assumption, the consistency of the regularization kernel network (RKN) and the coefficient regularization kernel network (CRKN) are proved. Satisfactory capacity independently error bounds and learning rates are derived by the techniques of integral operator.

Active Shape Model (ASM) is a most effective method of facial landmarking. It employs two models, profile model and shape model, to match the position of facial landmark. In this paper, we introduce a new model based on relative position feature (RPF) in local region to improve ASM. We found the fact that landmarks with larger matching error have more shape matching displacement. So, in our method, RPF model is used to adjust the position of landmarks with more shape matching displacement in every matching iteration. STASM (Stacked ASM) is practical standard of ASM and is proved to be the best method of locating face landmarks. Our experiments on STASM show significant performance improving, especially on databases in which faces are partially blocked by glasses or artificial black square.