International Journal of Applied Mathematics and Machine Learning

In order to achieve the multi-focus image fusion task, a sparse representation method based on quaternion for multi-focus image fusion is proposed in this paper. Firstly, the RGB color information of each pixel in the color image is represented by quaternion based on the relevant knowledge of computational mathematics, and the color image pixel is processed as a whole vector to maintain the relevant information between the three color channels. Secondly, the dictionary represented by quaternion and the sparse coefficient represented by quaternion are obtained by using the our proposed sparse representation model. Thirdly, the coefficient fusion is carried out by using the “max-L1” rule. Finally, the fused sparse coefficient and dictionary are used for image reconstruction to obtain the quaternion fused image, which is then converted into RGB color multi-focus fused image. Our method belongs to computational mathematics, and uses the relevant knowledge in the field of computational mathematics to help us carry out the experiment. The experimental results show that the method has achieved good results in visual quality and objective evaluation.

SAR Target Recognition Using Improved Sparse Representation with Local Reconstruction

In order to handle the problem of synthetic aperture radar (SAR) target recognition, an improved sparse representation-based classification (SRC) is proposed. According to the sparse coefficient vector resulting from the global dictionary, the largest coefficient in each class is taken as the reference. Then, the surrounding neighborhoods of the sample with the largest coefficient are selected to construct the optimal local dictionary in each training class. Afterwards, the samples in the local dictionary are used to reconstruct the test sample to be identified. Finally, the decision is made according to the comparison of the reconstruction errors from different classes. In the experiments, the proposed method is verified based on the moving and stationary target acquisition and recognition (MSTAR) dataset. The results show that the proposed method has performance advantages over existing methods, which demonstrates its effectiveness and robustness.

Compressed sensing based fingerprint imaging system using a chaotic model-based deterministic sensing matrix

A novel deterministic sensing matrix design approach applied to enable secured compressed sensing and transmission of fingerprint images. The performance of the sensing matrix was analyzed in detail by varying compressed sensing and security parameters. The number of sampling and sparse coefficient are the parameters taken under consideration from compressed sensing, whereas the encryption key is from the security scheme. The first stage in the performance study is acquiring the signal, and followed by sparsely modelling it using Orthogonal Matching Pursuit (OMP) algorithm. The sparse modelled data is multiplied by the proposed deterministic partial orthogonal Discrete Cosine Transform (DCT) sensing matrix to reduce its dimension and encrypt it. To introduce confusion on the DCT matrix rows, the pseudo-random permutation is applied to the DCT matrix rows before sensing matrix derivation. Additionally, recovering and decryption of the compressed image accomplished with the help of a convex optimization method. The results obtained from the simulation of the proposed system confirmed that a better Peak Signal to Noise Ratio (PSNR) than the recommended value for wireless transmission is archived using a sample below 25% without losing a significant number of fingerprint minutiae.

Pansharpening of WorldView-2 Data via Graph Regularized Sparse Coding and Adaptive Coupled Dictionary

The spectral mismatch between a multispectral (MS) image and its corresponding panchromatic (PAN) image affects the pansharpening quality, especially for WorldView-2 data. To handle this problem, a pansharpening method based on graph regularized sparse coding (GRSC) and adaptive coupled dictionary is proposed in this paper. Firstly, the pansharpening process is divided into three tasks according to the degree of correlation among the MS and PAN channels and the relative spectral response of WorldView-2 sensor. Then, for each task, the image patch set from the MS channels is clustered into several subsets, and the sparse representation of each subset is estimated through the GRSC algorithm. Besides, an adaptive coupled dictionary pair for each task is constructed to effectively represent the subsets. Finally, the high-resolution image subsets for each task are obtained by multiplying the estimated sparse coefficient matrix by the corresponding dictionary. A variety of experiments are conducted on the WorldView-2 data, and the experimental results demonstrate that the proposed method achieves better performance than the existing pansharpening algorithms in both subjective analysis and objective evaluation.

Big Data Business Actual Analysis: Stock Price Prediction Based on Time Series Model

This paper selects the daily closing price data of the Shanghai Composite Index from January 1, 2016 to December 31, 2017, excluding holidays, and preprocesses the data. After taking the logarithm and converting it into the rate of return data, the first-order difference is performed to make it into a stable time series, and then the ARMA(p,q) model is constructed. Through parameter significance test, residual test and characteristic root test, according to the minimum principle of AIC, the optimal model is finally determined to be ARMA(2,5) of sparse coefficient, and the expression of the model is obtained. The GARCH(1,1) model is established for the residual of ARMA(2,5), and the model expression is obtained. In order to directly predict the return rate of the Shanghai Composite Index, the ARIMA(2,1,5) model of the sparse coefficient is constructed for the return rate of the Shanghai Composite Index, and the model expression is obtained. By predicting the Shanghai Composite Index return data on January 2, 2018, it is found that the prediction error of the model is small, and it can be used for subsequent predictions.

Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method

Multi-term time-fractional partial differential equations (PDEs) have become a hot topic in the field of mathematical physics and are used to improve the modeling accuracy in the description of anomalous diffusion processes compared to the single-term PDE results. This research includes the numerical solutions of two-term time-fractional PDE models using an efficient and accurate local meshless method. Due to the advantages of the meshless nature and ease of applicability in higher dimensions, the demand for meshless techniques is increasing. This approach approximates the solution on a uniform or scattered set of nodes, resulting in well-conditioned and sparse coefficient matrices. Numerical tests are performed to demonstrate the efficacy and accuracy of the proposed local meshless technique.

Hyperspectral Imagery Classification Based on Multiscale Superpixel-Level Constraint Representation

Sparse representation (SR)-based models have been widely applied for hyperspectral image classification. In our previously established constraint representation (CR) model, we exploited the underlying significance of the sparse coefficient and proposed the participation degree (PD) to represent the contribution of the training sample in representing the testing pixel. However, the spatial variants of the original residual error-driven frameworks often suffer the obstacles to optimization due to the strong constraints. In this paper, based on the object-based image classification (OBIC) framework, we firstly propose a spectral–spatial classification method, called superpixel-level constraint representation (SPCR). Firstly, it uses the PD in respect to the sparse coefficient from CR model. Then, transforming the individual PD to a united activity degree (UAD)-driven mechanism via a spatial constraint generated by the superpixel segmentation algorithm. The final classification is determined based on the UAD-driven mechanism. Considering that the SPCR is susceptible to the segmentation scale, an improved multiscale superpixel-level constraint representation (MSPCR) is further proposed through the decision fusion process of SPCR at different scales. The SPCR method is firstly performed at each scale, and the final category of the testing pixel is determined by the maximum number of the predicated labels among the classification results at each scale. Experimental results on four real hyperspectral datasets including a GF-5 satellite data verified the efficiency and practicability of the two proposed methods.

A Multi-Scale Compressed Sensing Algorithm Based on Variational Mode

The compressed sensing algorithm based on the hybrid sparse base (TFWBST+wave atom) usually uses two kinds of image sparse transformations to realize the sparse representation of structure and texture respectively. However, due to the lack of constraints on image texture and structure and the lack of orthogonality of the two sparse bases, the sparse coefficient of structure and the sparse coefficient of texture after transformation are often not good enough to reflect their respective components, that is, the texture coefficient often loses the detail information of texture. To overcome this phenomenon, this paper combines the compressed sensing algorithm based on hybrid base with the layered variational image decomposition method to form the variational multi-scale compressed sensing, which is to establish the CS image reconstruction model with minimal energy functional. The layered variational image decomposition decomposes image into different feature components by minimizing energy functional. The reconstruction of each layer by compressed sensing algorithm is very suitable for texture and detail reconstruction. In this model, TFWBST transform and wave atom are combined as a joint sparse dictionary, and the image decomposition is carried out under the (BV, G, E) variational framework, which is introduced into multi-scale compressed sensing technology to reconstruct the original image. In this new functional, TFWBST transform and wave atom are used to represent structure and texture respectively, and multiscale (BV, G, E) decomposition which can decompose an image into a sequence of image structure, texture and noise is added for restricting image parts. Experiments show that the new model is very robust for noise, and that can keep edges and textures stably than other multi-scale restoration and reconstruction of images.

Robust Object Tracking via Reverse Low-Rank Sparse Learning and Fractional-Order Variation Regularization

Object tracking based on low-rank sparse learning usually makes the drift phenomenon occur when the target faces severe occlusion and fast motion. In this paper, we propose a novel tracking algorithm via reverse low-rank sparse learning and fractional-order variation regularization. Firstly, we utilize convex low-rank constraint to force the appearance similarity of the candidate particles, so as to prune the irrelevant particles. Secondly, fractional-order variation is introduced to constrain the sparse coefficient difference in the bounded variation space, which allows the difference between consecutive frames to exist, so as to adapt object fast motion. Meanwhile, fractional-order regularization can restrain severe occlusion by considering more adjacent frames information. Thirdly, we employ an inverse sparse representation method to model the relationship between target candidates and target template, which can reduce the computation complexity for online tracking. Finally, an online updating scheme based on alternating iteration is proposed for tracking computation. Experiments on benchmark sequences show that our algorithm outperforms several state-of-the-art methods, especially exhibiting better adaptability for fast motion and severe occlusion.