This paper studies the problem of Simultaneous Sparse approximation (SSA). This problem arises in many applications that work with multiple signals maintaining some degree of dependency, e. g., radar and sensor networks. We introduce a new method towards joint recovery of several independent Sparse signals with the same support. We provide an analytical discussion of the convergence of our method, called Simultaneous Iterative Method (SIM). In this study, we compared our method with other group-Sparse reconstruction techniques, namely Simultaneous Orthogonal Matching Pursuit (SOMP) and Block Iterative Method with Adaptive Thresholding (BIMAT), through numerical experiments. The simulation results demonstrated that SIM outperformed these algorithms in terms of the metrics Signal to Noise Ratio (SNR) and Success Rate (SR). Moreover, SIM is considerably less complicated than BIMAT, which makes it feasible for practical applications such as implementation in MIMO radar systems.

Summary In this paper, the task is to return from a set of multiplicities from a model to obtain an approximation of that model using Sparse approximation. The term ‘ approximation’ indicate the sufficiency of an interpretation that is close enough to the true mode, i. e. reality. In geosciences, the multiplicities are provided by multiple-point statistical (MPS) methods. Realistic modeling of the earth interior demands for more sophisticated geostatistical methods based on true available images, i. e. the training images. Among the available MPS methods, the DisPat algorithm is a distance-based MPS method, which generates appealing realizations for stationary and non-stationary training images by classifying the patterns based on distance functions using kernel methods. Advances in non-stationary image modeling is an advantage of the DisPat method. Realizations generated by the MPS methods form the training set for the Sparse approximation. The Sparse approximation is comprising of two steps, Sparse coding and dictionary update, which are alternately used to optimize the trained dictionary. Model selection algorithms like LARS are used for Sparse coding. LARS optimizes the regression model sequentially by choosing a proper number of variables and adding the best variable to the active set in each iteration. The ILS-DLA dictionary learning algorithm addresses the internal structure of the dictionary by considering the overlapping or non-overlapping blocks and the inversion task according to the internal structure of the trained dictionary. The ILS-DLA is fast in the sense that it inverts smaller blocks constructing the trained dictionary rather than inverting the entire dictionary. The trained dictionary is sequentially updated by alternating between Sparse coding and dictionary training steps. According to the experiments, the compressed sparsity-based image model is superior to 90% of the generated realizations by 90% probability...

Gravity-field recovery of the Earth using reconstruction of spherical harmonic coefficients up to specified degree and order requires proper data sampling based on Shannon-Nyquist rate. Since, these coefficients are globally significant, the sampling must be done uniformly on the Earth, which it takes much time and expense to collect and process data. Many studies have been done in the field of sampling analysis of spherical harmonics [1, 2]. Sneeuw [2] showed a lack of Nyquist sampling rate can cause aliasing of second type in gravity-field modeling. Recently based on Compressive Sensing (CS) theorem the sampling rate can be substantially reduced and a signal can be approximated in Sparse sense with fewer sampled data that has main role in reconstruction. In this case, the desired signal can be reconstructed, using only some base functions, which are most strongly correlated with the problem. Therefore, based on this strategy, the base functions posing the best solution to the problem will be selected and the sampling rate for regional gravity field modeling will be decreased significantly. When we say a signal is m-Sparse, it means that there are at most nonzero components in the signal. In this case, only m coefficients of the signal have large magnitude, and others are zero, or have very small values. Here, the desired signal can be reconstructed with its large components without loss of more information. The zero-norm of a vector which is defined as, specifies the sparsity-level of a signal. Sparse approximation has been discussed in many studies [4, 5, 6, 7, 8, 9]. The basic idea proposed by Mallat and Zhang [4] is called matching pursuit (MP), which is an iterative Sparse approximation method to reconstruct a signal under specified conditions by replacing a complex Sparse problem with a simple optimized solution. Pati et al. [6] modified this algorithm into orthogonal matching pursuit (OMP), which is used for non-orthogonal dictionaries and converges faster than MP. The regularized orthogonal matching pursuit (ROMP) algorithm popularized by Needell and Vershynin [8] is an iterative Sparse approximation method where at each iteration m nonzero components of unknown parameters that most closely resemble the properties of the desired signal are selected. Needell and Tropp [9] refined the ROMP algorithm with compressive sampling matching pursuit (CoSaMP), which identifies locations of the large energy of a signal at each iteration. All these algorithms try to find column vectors in the design matrix that most strongly correlate with the desired signal. It is also assumed that the design matrix is well-posed and prior knowledge of the sparsity-level of the signal is clear. Usually, in practical application an ill-posed problem may be encountered, also the sparsity-level of the signal is not exactly clear, which make it difficult to use conventional iterative methods of CS. In this paper we present a new dynamic algorithm called Stabilized Orthogonal Matching Pursuit (SOMP) for gravity-field recovery of the earth using Sparse approximation of geopotential spherical harmonic coefficients, which is compatible with the ill-posed problem and can determine the sparsity-level of the signal, properly. Numerical result of the calculated spherical harmonics coefficients up to degree and order 36 shows that the algorithm is able to reconstruct the Earth's gravity-field with precision in mind the number of samples is 50% lower than the Nyquist rate.

The difficulties in the measurement of rainfall interception in forests confirm the necessity of presenting models. The widely used models for estimating rainfall interception are physical-based models, among which the Sparse Gash is the most commonly used. We evaluated the Sparse Gash model for estimating the rainfall interception of five forest stands (two chestnut-leaved oak stands, two oriental beech stands, and one velvet maple stand) in the Hyrcanian region. In each stand, the gross rainfall and throughfall were measured using 5 and 20 rainfall collectors, respectively, and rainfall interception was calculated by subtracting the throughfall from gross rainfall. To evaluate the performance of the model, we used statistical metrics: Error percentage (Error), Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and the Model Efficiency coefficient (CE). Based on the Pearson correlation coefficient, the correlation between the values estimated by the model and the observed values was statistically significant at a 95% confidence interval. In all forests, the values of the CE were higher than 0. 5, indicating the appropriate efficiency of the model. Based on the Error, the model showed good capability in estimating the rainfall interception of four forest stands (i. e., oriental beech in Lajim, chestnut-leaved oak in Kohmiyan and Sari, and velvet maple in Sari Error metric were found to be-10. 3%, +12. 7%, +10. 8%, and +15. 4%, respectively). Studying the performance of physically-based models in forests with different species and different allometric, climatic and rainfall characteristics completes the information gap about the efficiency of models to estimate rainfall interception.