International Journal of Wavelets, Multiresolution and Information Processing

The concept of $K$-frames was recently introduced by Găvruta⁷ in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let ${𝒜}$ be a unital $C^{\ast }$-algebra, ${\mathscr{H}},{𝒦}$ be finitely or countably generated Hilbert ${𝒜}$-modules, and $F,G$ be adjointable operators from ${\mathscr{H}}$ to ${𝒦}$. In this paper, we study a class of $(F,G)$-bounded operators and $(F,G)$-operator frames for ${ℒ}({\mathscr{H}},{𝒦})$. We also prove that the pseudo-inverse of $T\in {ℒ}({\mathscr{H}},{𝒦})$ exists if and only if $T$ has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert $C^{\ast }$-modules. Further, we also present some perturbation results for $(F,G)$-operator frames in ${ℒ}({\mathscr{H}},{𝒦})$.

We study the orthogonal rational functions with a given sequence of poles on the half plane. We give the weak-star convergence results for the rational measures which relate to the Poisson kernel and reproducing kernel. Those rational measures represent the same inner product as the normalized Nevanlinna measure in ${{ℒ}}_n$. Moreover, we consider the convergence properties of the interpolants which interpolate the Carathéodory function $F(z)$.

In a separable Hilbert space ${\mathscr{H}}$, two frames ${\{f_i\}}_{i\in I}$ and ${\{g_i\}}_{i\in I}$ are said to be woven if there are constants $0<A\le B$ so that for every $\sigma \subset I$, ${\{f_i\}}_{i\in \sigma }\cup {\{g_i\}}_{i\in {\sigma }^c}$ forms a frame for ${\mathscr{H}}$ with the universal bounds $A,B$. This paper provides methods of constructing woven frames. In particular, bounded linear operators are used to create woven frames from a given frame. Several examples are discussed to validate the results. Moreover, the notion of woven frame sequences is introduced and characterized.

In this paper, we obtain some new results on the construction of $K$-g-frames and meanwhile, we correct two corresponding results. We also give a Paley–Wiener type stability theorem for $K$-g-frames, which provides an improvement to one known result on this topic.

In this paper, we first introduce the notation of weaving continuous fusion frames in separable Hilbert spaces. After reviewing the conditions for maintaining the weaving $c$-fusion frames under the bounded linear operator and also, removing vectors from these frames, we will present a necessarily and sufficient condition about $c$-woven and $c$-fusion woven. Finally, perturbation of these frames will be introduced.

Hyperspectral imaging (HSI) has shown great potential in the use of paddy variety identification. However, the quality of HSI images taken by a hyperspectral camera under non-ideal illumination is vulnerable to environmental influences such as shadows and noises, leading to a degraded identification result. This problem is addressed in this study by a two-stage image processing method. First, to eliminate the influence of shadows, a grayscale image based on the reflectance slope is synthesized. The synthetic reflectance slope image (SRSI) is binarized for image segmentation and shape features extraction. Secondly, an HSI image de-noising technology based on weighted spatial filtering (WSF), which integrates both spatial and spectral information of the HSI image, is proposed to reduce the influence of noises. Finally, the extracted shape, spectral and texture features are combined and input into the support vector machine for paddy variety identification. Four varieties of paddy with different origins were tested in the experiments. The experiment results showed that compared with color images, the SRSIs could help obtain more accurate shape features. The results also showed that the WSF method can significantly reduce noises and improves the paddy variety identification accuracy.

The complex rock fracture and road pavement cracks are more difficult to extract than the other linear objects in an image. In rock engineering, the rock fracture is an important factor that might cause tunnel and bridge collapse, or rock slope or dam damage. In road construction, the crack is one of the main pavement diseases. To avoid the difficultly of extracting fractures/cracks in an image, a new algorithm for tracking the central lines of fractures or cracks is studied to alleviate the problem for image segmentation. It includes four aspects: (1) a new fractional differential template is established to enhance the blurring and weak fractures/cracks in an image, compared with the traditional fractional differential template Tiansi, the new template has no zero coefficient and can enhance the micro-fractures/cracks; (2) in order to decrease the difficulty level of fracture/crack extraction, an algorithm for extracting the feature points of the fracture/crack central line is proposed based on the idea of Steger algorithm; (3) after linking short gaps based on distance, the long gap linking is made according to the principle of hydrodynamics, it first makes judgment if the two neighboring feature points are in one crack or not, in which, the feature points are regarded as two spring resources, then in light on the idea of water gushed out of the spring, when the two water flows meet together, the two points are recognized in one crack, otherwise they are not in one crack and cannot be connected together and (4) if the two neighboring feature points are in one crack, then the distance and the curvature between the two line segments are calculated, if they are less than the given thresholds, the linking path is searched and the gap is filled. Compared with the four traditional algorithms by testing hundred images, the new algorithm can accurately and quickly extract the central lines in complex rock fracture and rough road pavement cracks, which can increase the accuracy of crack/fracture image segmentation compared to the other algorithms.

Wavelets are closely related to Schrödinger’s wave functions and the interpretation of Born. Similar to the appearance of atomic orbital, it is proposed to combine anti-symmetric wavelets into orbital wavelets. The proposed approach allows the increase of the dimension of wavelets through this process. New orbital 2D-wavelets are introduced for the decomposition of still images, showing that it is possible to perform an analysis simultaneously in two distinct scales. An example of such an image analysis is shown.

Multi-view learning is a hot research direction in the field of machine learning and pattern recognition, which is attracting more and more attention recently. In the real world, the available data commonly include a small number of labeled samples and a large number of unlabeled samples. In this paper, we propose a latent multi-view semi-supervised classification method by using graph learning. This work recovers a latent intact representation to utilize the complementary information of the multi-view data. In addition, an adaptive graph learning technique is adopted to explore the local structure of this latent intact representation. To fully use this latent intact representation to discover the label information of the unlabeled data, we consider to unify the procedures of computing the latent intact representation and the labels of unlabeled data as a whole. An alternating optimization algorithm is designed to effectively solve the optimization of the proposed method. Extensive experimental results demonstrate the effectiveness of our proposed method.

In this paper, wavelet transform on a regression curve is investigated by using length-preserving projection and its application in financial data is also discussed. First, properties of wavelet filters on the regression trend curves are studied and two-scale equation of wavelet function is deduced on the regression trend curves. Second, the decomposition and reconstruction algorithm of discrete wavelet transform on regression trend curves is derived. Finally, two examples in financial data are given for discussion, based on decomposition and reconstruction algorithms on regression trend curves. Some new research interpretations are presented in dealing with financial data such as “volatility on regression growth trend”, “error on regression growth trend”, and so on.

We propose a new outline for adaptive dictionary learning methods for sparse encoding based on a hierarchical clustering of the training data. Through recursive application of a clustering method, the data is organized into a binary partition tree representing a multiscale structure. The dictionary atoms are defined adaptively based on the data clusters in the partition tree. This approach can be interpreted as a generalization of a discrete Haar wavelet transform. Furthermore, any prior knowledge on the wanted structure of the dictionary elements can be simply incorporated. The computational complexity of our proposed algorithm depends on the employed clustering method and on the chosen similarity measure between data points. Thanks to the multiscale properties of the partition tree, our dictionary is structured: when using Orthogonal Matching Pursuit to reconstruct patches from a natural image, dictionary atoms corresponding to nodes being closer to the root node in the tree have a tendency to be used with greater coefficients.

In this paper, an efficient novel dual permutation–substitution structure-based color image encryption algorithm is proposed. Initially, the Secure Hash Algorithm-512 (SHA-512) is applied to the input image to generate the initial values for the Lü system dynamically. In the first stage of permutation, inter-color-component pixel shuffling is carried out with a circular pixel-swapping mechanism. In the second stage, intra-color-component pixel shuffling is executed, based on pseudorandom positions generated by the Lü chaotic system. Pixel values are changed in the substitution stage, based on float-valued chaotic sequences generated by the Lü system. The performance of the proposed algorithm is evaluated with metrics such as key space, key sensitivity, histograms, correlation coefficients (vertical, horizontal and diagonal), information entropy, number of pixel changes rate (NPCR), unified average changing intensity (UACI), peak signal-to-noise ratio (PSNR), mean absolute error (MAE), contrast analysis, encryption time and the National Institute of Standards and Technology Special Publication 800-22 (NIST SP 800-22) statistical test. The experimental results obtained and performance assessment show that the proposed scheme has produced good results.