Algebraic geometric codes: basic notions

Crami??r-Rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists. The frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail. The set of subspaces, called the Grassmann manifold, and the set of covariance (positive-definite Hermitian) matrices have no fixed coordinate system associated with them and do not possess a vector space structure, both of which are required for deriving classical Crami??r-Rao bounds. Intrinsic versions of the Crami??r-Rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators. In the example of covariance matrix estimation, closed-form expressions for both the intrinsic and flat bounds are derived and compared with the root-mean-square error (RMSE) of the sample covariance matrix (SCM) estimator for varying sample support K. The accuracy bound on unbiased covariance matrix estimators is shown to be about (10/log 10)n/K 1/2 dB, where n is the matrix order. Remarkably, it is shown that from an intrinsic perspective, the SCM is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice. The RMSE of the standard method of estimating subspaces using the singular value decomposition (SVD)is compared with the intrinsic subspace Crami??r-Rao bound derived in closed form by varying both the signal-to-noise ratio (SNR) of the unknown p-dimensional subspace and the sample support. In the simplest case, the Crami??r-Rao bound on subspace estimation accuracy is shown to be about (p(n - p)1/2 K-1/2SN-1/2 rad for p-dimensional subspaces. It is seen that the SVD-based method yields accuracies very close to the Crami??r-Rao bound, esta blishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace. The analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere, such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation.

Covariance, Subspace, and Intrinsic CramrRao Bounds

In this paper, complex deterministic sensing matrices are explored to sample the signals in the single pixel imaging (SPI). A new analysissynthesis scheme is proposed to realize the complex deterministic sensing matrix for the DMD-based SPI. The analysis process divides the complex sensing matrix into real sensing matrix and imaginary sensing matrix, and multiple imaging is performed with these sensing matrices. After synthesizing the real and imaginary measurements, the final image of complex deterministic sensing matrix is reconstructed. The performance of deterministic sensing matrix is investigated through simulation and experiment. Compared with the random sensing matrix, the deterministic sensing matrix gives more favorable reconstructed images.

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Single-Pixel Imaging with Deterministic Complex-Valued Sensing Matrices

This paper studies the performance of different Reed-Solomon (RS) error control codes. Also, it proposes using efficient data randomize tool as an interleaver and comparing it with traditional one. The presented scenarios are evaluated over wireless channel. In the computer simulation experiments the performance degrading with the different data packet length is taken in considerations. The simulation results reveal that the block codes performance can be enhanced over wireless channel using efficient interleaver. Also, the proposed interleaved RS codes perform better than the traditional ones. The complexity of block codes can be decreased with effective real switching rules between the different code rate codes according to the channel conditions. The simulation results have revealed that the proposed technique is efficient for the wireless communication system.

Interleaved Reed-Solomon Codes With Code Rate Switching over Wireless Communications Channels

Nowadays, there are still difficulties in the implementation of Compressed Sensing (CS) sensors due to the nature of the measurement matrix. A binary measurement matrix can simplify the CS procedure significantly. However, due to the singularity of this class of measurement matrices, the convergence of some of existing CS reconstruction algorithms, such as the well-known block-based CS with smoothed-projected Landweber reconstruction (BCS-SPL) algorithm, is not guaranteed and can lead to an inaccurate recovery. In this paper we propose a simple, fast and efficient CS recovery algorithm that is able to recover the original image from compressed samples which are obtained using a binary measurement. Singular value decomposition (SVD) is coupled with the BCS-SPL algorithm in order to improve its recovery capability when a binary matrix is employed. The experimental results show that the proposed recovery algorithm has a better performance in terms of reconstruction quality when compared with existing reconstruction algorithm and yields images with quality that matches or exceeds those produced by the BCS-SPL algorithm. Additionally, the proposed algorithm is the most efficient in terms of recovery time, especially at high subrates.

Robust Image Reconstruction for Block-Based Compressed Sensing Using a Binary Measurement Matrix

In this letter, we construct a class of deterministic measurement matrices which are asymptotically optimal. For this purpose, we first apply the tensor product over the Reed Solomon (R-S) generator matrix to produce a new one; then employing this generator matrix, we construct a measurement matrix. If the R-S code is defined on \BBFq, then the resulting measurement matrices are of dimensions q2 ×q3, where q is an arbitrary prime power and its coherence would be 1/q, which is desirable for compressive sampling. We also illustrate the effectiveness of our proposed matrices in compressed sensing with some simulation examples.

A Reed-Solomon Code Based Measurement Matrix with Small Coherence

Recent matrix completion based methods have not been able to properly model the haplotype assembly problem for noisy observations. To deal with such cases, we propose a new minimum error correction (MEC) based matrix completion problem over the manifold of rank-one matrices. We then prove the convergence of a specific iterative algorithm to solve this problem. From the simulation results, the proposed method not only outperforms some well-known matrix completion based methods, but also shows a more accurate result compared to a most recent MEC-based algorithm for haplotype estimation.

Haplotype Assembly Using Manifold Optimization and Error Correction Mechanism

The authors solve the matrix completion (MC) problem based on manifold optimisation by incorporating the side information under which the columns of the intended matrix are drawn from a union of low-dimensional subspaces. It is proved that this side information leads us to construct new manifolds, as an embedded sub-manifold of the manifold of constant rank matrices, using which the MC problem is solved more accurately. The required geometrical properties of the aforementioned manifold are then presented for MC. Simulation results over both synthetic and real-world data show that the proposed method outperforms some recent techniques either based on side information or not.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-spr.2019.0377

Matrix completion with side information using manifold optimisation

Recent matrix completion based methods have not been able to properly model the Haplotype Assembly Problem (HAP) for noisy observations. To cope with such a case, in this letter we propose a new Minimum Error Correction (MEC) based matrix completion optimization problem over the manifold of rank-one matrices. The convergence of a specific iterative algorithm for solving this problem is proved. Simulation results illustrate that the proposed method not only outperforms some well-known matrix completion based methods, but also presents a more accurate result compared to a most recent MEC based algorithm for haplotype estimation.

Mohamad Mahdi Mohades

Papers

A Reed-Solomon Code Based Measurement Matrix with Small Coherence

Haplotype Assembly Using Manifold Optimization and Error Correction Mechanism

Matrix completion with side information using manifold optimisation

Haplotype Assembly Using Manifold Optimization and Error Correction Mechanism

Matrix completion with weighted constraint for haplotype estimation