International Journal of Wavelets, Multiresolution and Information Processing

In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data.

Continuous wavelet transform (CWT) is a linear convolution of signal and wavelet function for a fixed scale. This paper studies the algorithm of CWT with Morlet wavelet as mother wavelet by using nonzero-padded linear convolution. The time domain filter, which is a non-causal filter, is the sample of wavelet function. By making generalized discrete Fourier transform (GDFT) and inverse transform for this filter, we can get a geometrically weighted periodic extension of the filter when evaluated outside its original support. From this extension of the time domain filter, we can get a causal filter. In this paper, GDFT-based algorithm for CWT, which has a more concise form than that of linear convolution proposed by Jorge Martinez, is constructed by using this causal filter. The analytic expression of the GDFT of this filter, which is essential for GDFT-based algorithm for CWT, is deduced in this paper. The numerical experiments show that the calculation results of GDFT-based algorithm are stable and reliable; the running speed of GDFT-based algorithm is faster than that of the other two algorithms studied in our previous work.

In signal processing, several applications necessitate the efficient reprocessing and representation of data. Compression is the standard approach that is used for effectively representing the signal. In modern era, many new techniques are developed for compression at the sensing level. Compressed sensing (CS) is a rising domain that is on the basis of disclosure, which is a little gathering of a sparse signal’s linear projections including adequate information for reconstruction. The sampling of the signal is permitted by the CS at a rate underneath the Nyquist sampling rate while relying on the sparsity of the signals. Additionally, the reconstruction of the original signal from some compressive measurements can be authentically exploited using the varied reconstruction algorithms of CS. This paper intends to exploit a new compressive sensing algorithm for reconstructing the signal in bio-medical data. For this purpose, the signal can be compressed by undergoing three stages: designing of stable measurement matrix, signal compression and signal reconstruction. In this, the compression stage includes a new working model that precedes three operations. They are signal transformation, evaluation of $\mathrm{\Theta }$ and normalization. In order to evaluate the theta ($\mathrm{\Theta }$) value, this paper uses the Haar wavelet matrix function. Further, this paper ensures the betterment of the proposed work by influencing the optimization concept with the evaluation procedure. The vector coefficient of Haar wavelet function is optimally selected using a new optimization algorithm called Average Fitness-based Glowworm Swarm Optimization (AF-GSO) algorithm. Finally, the performance of the proposed model is compared over the traditional methods like Grey Wolf Optimizer (GWO), Particle Swarm Optimization (PSO), Firefly (FF), Crow Search (CS) and Glowworm Swarm Optimization (GSO) algorithms.

Face annotation, a modern research topic in the area of image processing, has useful real-life applications. It is a really difficult task to annotate the correct names of people to the corresponding faces because of the variations in facial appearance. Hence, there still is a need for a robust feature to improve the performance of the face annotation process. In this work, a novel approach called the Deep Gabor-Oriented Local Order Features (DGOLOF) for feature representation has been proposed, which extracts deep texture features from face images. Seven recently proposed face annotation methods are considered to evaluate the proposed deep texture feature under uncontrolled situations like occlusion, expression changes, illumination and pose variations. Experimental results on the LFW, IMFDB, Yahoo and PubFig databases show that the proposed deep texture feature provides efficient results with the Name Semantic Network (NSN)-based face annotation. Moreover, it is observed that the proposed deep texture feature improves the performance of face annotation, regardless of all the challenges involved.

Although traditional dictionary learning (DL) methods have made great success in pattern recognition and machine learning, it is extremely time-consuming, especially in the training stage. The projective dictionary pair learning (DPL) learned the synthesis dictionary and the analysis dictionary jointly to achieve a fast and accurate classifier. However, the dictionary pair is initialized as random matrices without using any data samples information, it required many iterations to ensure convergence. In this paper, we propose a novel compact DPL and refining method based on the observation that the eigenvalue curve of sample data covariance matrix usually decrease very fast, which means we can compact the synthesis dictionary and analysis dictionary. For each class of the data samples, we utilize the principal components analysis (PCA) to retain global important information and compact the row space of a synthesis dictionary and the column space of an analysis dictionary in the first stage. We further refine the learned dictionary pair to achieve a more accurate classifier during compact dictionary pair refining, which combines the orthogonality of PCA with the redundancy of DL. We solve this refining problem in closed-form completely, naturally reducing the computation complexity significantly. Experimental results on the Extended YaleB database and AR database show that the proposed method achieves competitive accuracy and low computational complexity compared with other state-of-the-art methods.

Due to the huge advancement in technology, digitizing the multimedia content like text, images and videos has become easier. Everyday huge amounts of multimedia content are shared through the social networks using internet. Sometimes this multimedia content can be hacked by the hackers. This will lead to the misuse of the data. On the other hand, the medical content needs high security and privacy. Motivated by this, joint secured medical image compression–encryption mechanisms are proposed in this paper using multiscale transforms and symmetric key encryption techniques. The multiscale transforms involved in this paper are wavelet transform, bandelet transform and curvelet transform. The encryption techniques involved in this paper are international data encryption algorithm (IDEA), Rivest Cipher (RC5) and Blowfish. The encoding technique used in this paper is embedded block coding with truncation (EBCOT). Experimental results are done for the proposed works and evaluated by using various parameters like Peak Signal-to-Noise Ratio (PSNR), Mean Square Error (MSE), Image Quality Index (IQI) and Structural Similarity Index (SSIM), Average Difference (AD), Normalized Cross-correlation (NK), Structural Content (SC), Maximum difference (MD), Laplacian Mean Squared Error (LMSE) and Normalized Absolute Error (NAE). It is justified that the proposed approaches in this paper yield good results.

The locust slice images have all the features such as strong self-similarity, piecewise smoothness and nonlinear texture structure. Multi-scale interpolation operator is an effective tool to describe such structures, but it cannot overcome the influence of noise on images. Therefore, this research designed the Shannon–Cosine wavelet which possesses all the excellent properties such as interpolation, smoothness, compact support and normalization, then constructing multi-scale wavelet interpolative operator, the operator can be applied to decompose and reconstruct the images adaptively. Combining the operator with the local filter operator (mean and median), a multi-scale Shannon–Cosine wavelet denoising algorithm based on cell filtering is constructed in this research. The algorithm overcomes the disadvantages of multi-scale interpolation wavelet, which is only suitable for describing smooth signals, and realizes multi-scale noise reduction of locust slice images. The experimental results show that the proposed method can keep all kinds of texture structures in the slice image of locust. In the experiments, the locust slice images with mixture noise of Gaussian and salt–pepper are taken as examples to compare the performances of the proposed method and other typical denoising methods. The experimental results show that the Peak Signal-To-Noise Ratio (PSNR) of the denoised images obtained by the proposed method is greater 27.3%, 24.6%, 2.94%, 22.9% than Weiner filter, wavelet transform method, median and average filtering, respectively; and the Structural Similarity Index (SSIM) for measuring image quality is greater 31.1%, 31.3%, 15.5%, 10.2% than other four methods, respectively. As the variance of Gaussian white noise increases from 0.02 to 0.1, the values of PSNR and SSIM obtained by the proposed method only decrease by 11.94% and 13.33%, respectively, which are much less than other 4 methods. This shows that the proposed method possesses stronger adaptability.

The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Then, Kutyniok and Strohmer generalized the notion of Wilson system to the lattices whose generator matrix is in Hermite normal form.

In the present publication, we introduce Wilson systems where the time-frequency shifts are combined symmetrically with respect to the origin. Using the arguments from previous papers that work also in this case, we give a full characterization of atoms for which the obtained Wilson systems are orthonormal bases in $L^2(R)$ and we generalize these results to different sign sequences and to symplectic lattices of density $1/2$.

In this paper, the problem of preview control for continuous-time singular stochastic systems is studied by the augmented error system approach. First, a static output feedback is introduced to make the closed-loop system impulse-free. And through the second restricted equivalent transformation, the continuous-time singular stochastic system is transformed into a normal stochastic system and some algebra equations. Then, an assistant system is designed to overcome the difficulty of trying to make derivation to a stochastic system like in deterministic system condition. And the stochastic system after the transformation is translated to the assistant system. In order to make the output of the singular system track the reference signal as accurately as possible without static error, an integrator is introduced. Based on the system after the translation and the error equation, an augmented error system is constructed with the state vector, the error vector and the reference signal. Finally, the tracking problem for the singular stochastic system is transformed into the optimal regulating problem for the augmented error system. By the dynamic programming method, the optimal preview controller for the original singular stochastic system is obtained. The existence and uniqueness solution of the Riccati equation is discussed and the numerical analysis shows the effectiveness for the preview controller designed in this paper.

Biomedical signal/image processing and analysis are always fascinating tasks in scientific researches, both theoretical and practical. One of the powerful tools in such topics is wavelet theory which has been proved to be challenging since its discovery. One of the best measures of the optimality of reconstruction of signals/images is the well-known Shannon’s entropy. In wavelet theory, this is very well known and researchers are familiar with it. In the present work, a step forward is proposed based on more general wavelet tools. New approach is proposed for the reconstruction of signals/images provided with multiwavelets Shannon-type entropy to evaluate the order/disorder of the reconstructed signals/images. Efficiency and accuracy of the approach is confirmed by a simulation study on several models such as ECG, EEG and DNA/Proteins’ signals.

Hyperspectral images (HSIs) have hundreds of narrow and adjacent spectral bands, which will result in feature redundancy, decreasing the classification accuracy. Feature (band) selection helps to remove the noisy or redundant features. Most traditional feature selection algorithms can be only performed on a single HSI scene. However, appearance of massive HSIs has placed a need for joint feature selection across different HSI scenes. Cross-scene feature selection is not a simple problem, since spectral shift exists between different HSI scenes, even though the scenes are captured by the same sensor. The spectral shift makes traditional single-dataset-based feature selection algorithms no longer applicable. To solve this problem, we extend the traditional ReliefF to a cross-domain version, namely, cross-domain ReliefF (CDRF). The proposed method can make full use of both source and target domains and increase the similarity of samples belonging to the same class in both domains. In the cross-scene classification problem, it is necessary to consider the class-separability of spectral features and the consistency of features between different scenes. The CDRF takes into account these two factors using a cross-domain updating rule of the feature weights. Experimental results on two cross-scene HSI datasets show the superiority of the proposed CDRF in cross-scene feature selection problems.

In this paper, we use a new general sequence corresponding to a $g$-Bessel sequence $\{{\mathrm{\Lambda }}_j:j\in J\}$ to characterize that $\{{\mathrm{\Lambda }}_j:j\in J\}$ are $l^2({\{{{𝒱}}_j\}}_{j\in J})$-linear independent, $g$-complete and a $g$-frame. We also use $\{{\mathrm{\Lambda }}_j:j\in J\}$ and the refinement $\{{\mathrm{\Gamma }}_{ij}{\mathrm{\Lambda }}_i:i\in I,j\in J_i\}$ to characterize each other to be (near) exact $g$-frames or $g$-Riesz bases. Finally, we give several constructions and an equivalent characterization of Besselian $g$-frames and near exact $g$-frames.