CHAOTIC COMPRESSIVE SENSING OF TV –UHF BAND IN IRAQ USING CHEBYSHEV GRAM SCHMIDT SENSING MATRIX

Cognitive radio (CR) is a promising technology for solving spectrum sacristy problem. Spectrum sensing is the main step of CR. Sensing the wideband spectrum produces more challenges. Compressive sensing (CS) is a technology used as spectrum sening in CR to solve these challenges. CS consists of three stages: sparse representation, encoding and decoding. In encoding stage sensing matrix are required, and it plays an important role for performance of CS. The design of efficient sensing matrix requires achieving low mutual coherence . In decoding stage the recovery algorithm is applied to reconstruct a sparse signal. İn this paper a new chaotic matrix is proposed based on Chebyshev map and modified gram Schmidt (MGS). The CS based proposed matrix is applied for sensing real TV signal as a PU. The proposed system is tested under two types of recovery algorithms. The performance of CS based proposed matrix is measured using recovery error (Re), mean square error (MSE), and probability of detection (Pd) and evaluated by comparing it with Gaussian, Bernoulli and chaotic matrix in the literature. The simulation results show that the proposed system has low Re and high Pd under low SNR values and has low MSE with high compression.

Recovery of continuous functions of two variables from their Fourier coefficients known with error

In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p < \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes $W^{\psi}_p$ of univariate functions. As to available information, we are given the noisy Fourier coefficients $y^{\delta}_{i,j} = y_{i,j} + \delta \xi_{i,j}$, $\delta \in (0,1)$, $i,j = 1,2, \dots$, of functions with respect to certain orthonormal system $\{ \varphi_{i,j} \}_{i,j=1}^{\infty}$, where the noise level is small in the sense of the norm of the space $l_p$, $1 \leq p < \infty$, of double sequences $\xi=( \xi_{i,j} )_{i,j=1}^{\infty}$ of real numbers. As a recovery method, we use the so-called $\Lambda$-method of summation given by certain two-dimensional triangular numerical matrix $\Lambda = \{ \lambda_{i,j}^n \}_{i,j=1}^n$, where $n$ is a natural number associated with the sequence $\psi$ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space $C ([0,1]^2)$ of continuous on $[0,1]^2$ functions. We showed, that for $1\leq p < \infty$, under the respective assumptions on the smoothness parameter $\psi$ and the elements of the matrix $\Lambda$, it holds \[ \Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\| \xi \|_{l_p} \leq 1} \Big\| y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\|_{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.\]

Non-Markovian memory strength bounds quantum process recoverability

Generic non-Markovian quantum processes have infinitely long memory, implying an exact description that grows exponentially in complexity with observation time. Here, we present a finite memory ansatz that approximates (or recovers) the true process with errors bounded by the strength of the non-Markovian memory. The introduced memory strength is an operational quantity and depends on the way the process is probed. Remarkably, the recovery error is bounded by the smallest memory strength over all possible probing methods. This allows for an unambiguous and efficient description of non-Markovian phenomena, enabling compression and recovery techniques pivotal to near-term technologies. We highlight the implications of our results by analyzing an exactly solvable model to show that memory truncation is possible even in a highly non-Markovian regime.

A method for optimal linear super-resolution image restoration

In this paper, we propose a super-resolution (pixel grid refinement) method for digital images. It is based on the linear filtering of a zero-padded discrete signal. We introduce a continuous-discrete observation model to create a reconstruction system. The proposed observation model is typical of real-world imaging systems - an initially continuous signal first undergoes linear (dynamic) distortions and then is subjected to sampling and the effect of additive noise. The proposed method is optimal in the sense of mean square recovery error minimization. In the theoretical part of the article, a general scheme of the linear super-resolution of the signal is presented and expressions for the impulse and frequency responses of the optimal reconstruction system are derived. An expression for the error of such restoration is also derived. For the sake of brevity, the entire description is presented for one-dimensional signals, but the obtained results can easily be generalized for the case of two-dimensional images. The experimental section of the paper is devoted to the analysis of the super-resolution reconstruction error depending on the parameters of the observation model. The significant superiority of the proposed method in terms of the reconstruction error is demonstrated in comparison with linear interpolation, which is usually used to refine the grid of image pixels.

Robust Matrix Completion By Exploiting Dynamic Low-Dimensional Structures

This paper studies the robust matrix completion problem for time-varying models. Leveraging the low-rank property and the temporal information of the data, we develop novel methods to recover the original data from partially observed and corrupted measurements. We show that the reconstruction performance can be improved if one further leverages the information of the sparse corruptions in addition to the temporal correlations among a sequence of matrices. The dynamic robust matrix completion problem is formulated as a nonconvex optimization problem, and the recovery error is quantified analytically and proved to decay in the same order as that of the state-of-the-art method when there is no corruption. A fast iterative algorithm with convergence guarantee to the stationary point is proposed to solve the nonconvex problem. Experiments on synthetic data and real video dataset demonstrate the effectiveness of our method.

Universal Measurement Matrix Design for Sparse and Co-Sparse Signal Recovery

Compressed Sensing (CS) avails mutual coherence metric to choose the measurement matrix that is incoherent with dictionary matrix. Random measurement matrices are incoherent with any dictionary, but their highly uncertain elements necessitate large storage and make hardware realization difficult. In this paper deterministic matrices are employed which greatly reduce memory space and computational complexity. To avoid the randomness completely, deterministic sub-sampling is done by choosing rows deterministically rather than randomly, so that matrix can be regenerated during reconstruction without storing it. Also matrices are generated by orthonormalization, which makes them highly incoherent with any dictionary basis. Random matrices like Gaussian, Bernoulli, semi-deterministic matrices like Toeplitz, Circulant and full-deterministic matrices like DFT, DCT, FZC-Circulant are compared. DFT matrix is found to be effective in terms of recovery error and recovery time for all the cases of signal sparsity and is applicable for signals that are sparse in any basis, hence universal.

Uniform approximation property of frames with applications to erasure recovery

<p style='text-indent:20px;'>In this paper, we introduce frames with the uniform approximation property (UAP). We show that with a UAP frame, it is efficient to recover a signal from its frame coefficients with one erasure while the recovery error is smaller than that with the ordinary recovery algorithm. In fact, our approach gives a balance between the recovery accuracy and the computational complexity.</p>

Tensor recovery from noisy and multi-level quantized measurements

Higher-order tensors can represent scores in a rating system, frames in a video, and images of the same subject. In practice, the measurements are often highly quantized due to the sampling strategies or the quality of devices. Existing works on tensor recovery have focused on data losses and random noises. Only a few works consider tensor recovery from quantized measurements but are restricted to binary measurements. This paper, for the first time, addresses the problem of tensor recovery from multi-level quantized measurements by leveraging the low CANDECOMP/PARAFAC (CP) rank property. We study the recovery of both general low-rank tensors and tensors that have tensor singular value decomposition (TSVD) by solving nonconvex optimization problems. We provide the theoretical upper bounds of the recovery error, which diminish to zero when the sizes of dimensions increase to infinity. We further characterize the fundamental limit of any recovery algorithm and show that our recovery error is nearly order-wise optimal. A tensor-based alternating proximal gradient descent algorithm with a convergence guarantee and a TSVD-based projected gradient descent algorithm are proposed to solve the nonconvex problems. Our recovery methods can also handle data losses and do not necessarily need the information of the quantization rule. The methods are validated on synthetic data, image datasets, and music recommender datasets.

Wavelet–Fourier CORSING techniques for multidimensional advection–diffusion–reaction equations

We present and analyze a novel wavelet–Fourier technique for the numerical treatment of multidimensional advection–diffusion–reaction equations based on the COmpRessed SolvING (CORSING) paradigm. Combining the Petrov–Galerkin technique with the compressed sensing approach the proposed method is able to approximate the largest coefficients of the solution with respect to a biorthogonal wavelet basis. Namely, we assemble a compressed discretization based on randomized subsampling of the Fourier test space and we employ sparse recovery techniques to approximate the solution to the partial differential equation (PDE). In this paper we provide the first rigorous recovery error bounds and effective recipes for the implementation of the CORSING technique in the multidimensional setting. Our theoretical analysis relies on new estimates for the local $a$-coherence, which measures interferences between wavelet and Fourier basis functions with respect to the metric induced by the PDE operator. The stability and robustness of the proposed scheme are shown by numerical illustrations in the one-, two- and three-dimensional cases.

Tensor Recovery from Noisy and Multi-Level Quantized Measurements

Higher-order tensors can represent scores in a rating system, frames in a video, and images of the same subject. In practice, the measurements are often highly quantized due to the sampling strategies or the quality of devices. Existing works on tensor recovery have focused on data losses and random noises. Only a few works consider tensor recovery from quantized measurements but are restricted to binary measurements. This paper, for the first time, addresses the problem of tensor recovery from multi-level quantized measurements by leveraging the low CANDECOMP/PARAFAC (CP) rank property. We study the recovery of both general low-rank tensors and tensors that have tensor singular value decomposition (TSVD) by solving nonconvex optimization problems. We provide the theoretical upper bounds of the recovery error, which diminish to zero when the sizes of dimensions increase to infinity. We further characterize the fundamental limit of any recovery algorithm and show that our recovery error is nearly order-wise optimal. A tensor-based alternating proximal gradient descent algorithm with a convergence guarantee and a TSVD based projected gradient descent algorithm are proposed to solve the nonconvex problems. Our recovery methods can also handle data losses and do not necessarily need the information of the quantization rule. The methods are validated on synthetic data, image datasets, and music recommender datasets.