Remainder Matrix Research Articles

An Optimal Polarization SAR Three-Component Target Decomposition Based on Semi-Definite Programming

The model-based polarimetric synthetic aperture radar (PolSAR) target decomposition decodes the scattering mechanism of the target by analyzing the essential scattering components. This paper presents a new general three-component scattering power decomposition method by establishing optimization problems. It is known that the existing three-component decomposition method prioritizes the contribution of volume scattering, which often leads to volume scattering energy overestimation and may make double-bounce scattering and odd-bounce scattering component power negative. In this paper, a full parameter optimization method based on the remainder matrix is proposed, where all the elements of the coherency matrix will be taken into account including the remaining T13 component. The optimization is achieved with no priority order by solving the problem using semi-definite programming (SDP) based on the Schur complement theory. By doing so, the problem of volume scattering energy overestimation and negative powers will be avoided. The performance of the proposed approach is demonstrated and evaluated with AIRSAR and GF-3 PolSAR data sets. The experimental results show that by using the proposed method, the power contributions of volume scattering in two sets of data were reduced by at least 2.6% and 3.7% respectively, compared to traditional methods. And the appearance of negative power of double-bounce scattering and odd-bounce scattering are also avoided compared with those of the existing three-component decomposition.

What the Volume of Matrices Says?

We know matrices and their transposes and we also know flip matrices. In my previous paper Matrices-One Review, I introduced transprocal matrix. Flip matrices are transpose of transprocal matrices. Now I would like to introduce water image of four matrices said above and properties of such matrices. Also we know, determinant of sum of matrices is not equal to sum of determinant of matrices. Why can’t we get equal value on addition side and additive side of determinant of matrix addition and subtraction? This question triggered me to find the reason. The basic algebra of mensuration gave ideas to retreat determinant of matrix addition and subtraction. I extent that ideas for matrices sum. Further, in 1812, French mathematician Jacques Philippe Marie Binet described how to multiply matrices. Matrices are defined on addition, subtraction and multiplication but not in division. By the inspiration of Binet, I would like to describe how to do divisions on matrices. This idea is derived from division of fractions. In division of fraction, reciprocal of divisor fraction multiplies with dividend fraction. I do the same in division on matrices with some modifications. By this way, we could find quotient matrix and remainder matrix which satisfy division algorithm. So we could say, determinant of division of dividend matrix and divisor matrix is equal to division of determinant of dividend matrix and determinant of divisor matrix.

An Optimal Nonnegative Eigenvalue Decomposition for the Freeman and Durden Three-Component Scattering Model

Model-based decomposition allows the physical interpretation of polarimetric radar scattering in terms of various scattering mechanisms. A three-component decomposition proposed by Freeman and Durden has been popular, though significant shortcomings have been identified. In particular, it can result in negative eigenvalues for the component terms and the remainder matrix, hence violating fundamental requirements for physically meaningful decompositions. In addition, since the algorithm solves for the canopy term first, the contribution of the canopy is often over-estimated. In this paper, we show how to determine the parameters for the Freeman–Durden model in a way that minimizes the total power in the remainder matrix without favoring any individual component in the model, while simultaneously satisfying the constraints of nonnegative eigenvalues. We illustrate our analytical solution by comparison with the Freeman–Durden algorithm, as well as the nonnegative eigenvalue decomposition (NNED) proposed by van Zyl et al. The results show that this optimum algorithm generally assigns less power to the volume scattering than either the original Freeman–Durden or the NNED algorithms.

Simplified adaptive volume scattering model and scattering analysis of crops over agricultural fields using the RADARSAT-2 polarimetric synthetic aperture radar imagery

A simplified adaptive volume scattering model (SAVSM) for RADARSAT-2 polarimetric synthetic aperture radar (PolSAR), based on the n ’th power sine and cosine functions, is developed to characterize the changes in crop phenology within the growing season. A three-component model-based decomposition (TCMD) with SAVSM is also implemented with the non-negative eigenvalue decomposition. Multitemporal RADARSAT-2 data acquired in southwestern Ontario, Canada, are used for validating the proposed TCMD-SAVSM. The TCMD-SAVSM is first compared with the adaptive model-based decomposition proposed by Arii et al. in which ∇ f v is set to 0.01, demonstrating that TCMD-SAVSM not only consumes much less computing time but also the decomposed surface and double-bounce components are also more representative of the land cover reality. Using the percentage of power of the remainder matrix 0.001 as the evaluation criterion, the SAVSM achieved the best performance in corn, soybean and wheat fields compared with other popular volume scattering models (VSMs), with the highest mean and lowest standard deviation. It suggests that the SAVSM is a suitable VSM for describing seasonal changes of crop fields. Last, the decomposed surface, double-bounce and volume scattering components of wheat, soybean and corn at various growth stages are analyzed at the individual crop level, and the results demonstrate that their variations are in concert with the crop’s phenological development.

Estimation of connectivity measures in gappy time series

A new method is proposed to compute connectivity measures on multivariate time series with gaps. Rather than removing or filling the gaps, the rows of the joint data matrix containing empty entries are removed and the calculations are done on the remainder matrix. The method, called measure adapted gap removal (MAGR), can be applied to any connectivity measure that uses a joint data matrix, such as cross correlation, cross mutual information and transfer entropy. MAGR is favorably compared using these three measures to a number of known gap-filling techniques, as well as the gap closure. The superiority of MAGR is illustrated on time series from synthetic systems and financial time series.

Adaptive polarimetric decomposition using incoherent ground scattering models without reflection symmetry assumption

Most of the current incoherent polarimetric decompositions employ coherent models to describe ground scattering; however, this cannot truly reflect the fact especially in natural ground surfaces. This paper proposes a highly adaptive decomposition with incoherent ground scattering models (ADIGSM). In ADIGSM, Neumann’s adaptive model is employed to describe volume scattering, and to explain cross-polarized power in remainder matrix, so that we can obtain orientation angle randomness for both volume scattering and the dominant ground scattering. The computation of volume scattering parameters is strictly constrained for non-negative eigenvalues, while the volume scattering parameters that explain the most cross-polarized power are selected. When applying ADIGSM to NASA’s UAVSAR data, the negative component powers were obtained in quite a few forest pixels. Compared with several newest decompositions, the volume scattering power is obviously lowered, especially in areas dominated by surface scattering or double bounce scattering. The orientation angle randomness of each component is reasonable as well. ADIGSM has potential to be applied in the fields such as PolSAR image classification, land cover mapping, speckle filtering, soil moisture and roughness estimation, etc.

Improved van Zyl Polarimetric Decomposition Lessening the Overestimation of Volume Scattering Power

This paper improves van Zyl’s Nonnegative Eigenvalue Decomposition (NNED). Orientation angle compensation and helix scattering are introduced to the decomposition. The volume scattering parameters that explain the most cross-polarized power are selected. If volume scattering and helix scattering explain all cross-polarized power in the measured coherency matrix, then simply perform van Zyl decomposition to the remainder matrix; otherwise, the measured coherency matrix is decomposed into three components, i.e., helix scattering, volume scattering, and one ground scattering. The latter two scattering are all modeled by Neumann’s adaptive depolarizing model, according to which some cross-polarized power is attributed to ground scattering hence the orientation angle randomness of volume scattering and the dominant ground scattering are obtained. In this way, all cross-polarized power could be well explained. Experiments using UAVSAR data showed that more than 99.8% of total pixels are well fitted. Negative power is avoided. Compared with van Zyl decomposition, volume scattering power is reduced by up to 8.73% on average. The given volume scattering power is often lower than that by three latest NNED.

A semi-blind source separation method for differential optical absorption spectroscopy of atmospheric gas mixtures

Differential optical absorption spectroscopy (DOAS) is a powerful tool for detecting and quantifying trace gases in atmospheric chemistry [22]. DOAS spectra consist of a linear combination of complex multi-peak multi-scale structures. Most DOAS analysis routines in use today are based on least squares techniques, for example, the approach developed in the 1970s [18,19,20,21] uses polynomial fits to remove a slowly varying background (broad spectral structures in the data), and known reference spectra to retrieve the identity and concentrations of reference gases [23]. An open problem [22] is that fitting residuals for complex atmospheric mixtures often still exhibit structure that indicates the presence of unknown absorbers. &nbsp In this work, we develop a novel three step semi-blind source separation method. The first step uses a multi-resolution analysis called empirical mode decomposition (EMD) to remove the slow-varying and fast-varying components in the DOAS spectral data matrix ${\bf X}$. This has the advantage of avoiding user bias in fitting the slow varying signal. The second step decomposes the preprocessed data $\hat{{\bf X}}$ in the first step into a linear combination of the reference spectra plus a remainder, or $\hat{{\bf X}} = {\bf A}\,{\bf S} + {\bf R}$, where columns of matrix ${\bf A}$ are known reference spectra, and the matrix ${\bf S}$ contains the unknown non-negative coefficients that are proportional to concentration. The second step is realized by a convex minimization problem ${\bf S} = \mathrm{arg} \min \mathrm{norm}\,(\hat{{\bf X}} - {\bf A}\,{\bf S})$, where the norm is a hybrid $\ell_1/\ell_2$ norm (Huber estimator) that helps to maintain the non-negativity of ${\bf S}$. Non-negative coefficients are necessary in order for the derived proportional concentrations to make physical sense. The third step performs a blind independent component analysis of the remainder matrix ${\bf R}$ to extract remnant gas components. This step demonstrates the ability of the new fitting method to extract orthogonal components without the use of reference spectra. &nbsp We illustrate utility of the proposed method in processing a set of DOAS experimental data by a satisfactory blind extraction of an a-priori unknown trace gas (ozone) from the remainder matrix. Numerical results also show that the method can identify trace gases from the residuals.

Scaling LAPACK panel operations using parallel cache assignment

In LAPACK many matrix operations are cast as block algorithms which iteratively process a panel using an unblocked algorithm and then update a remainder matrix using the high performance Level 3 BLAS. The Level 3 BLAS have excellent scaling, but panel processing tends to be bus bound, and thus scales with bus speed rather than the number of processors ( p ). Amdahl's law therefore ensures that as p grows, the panel computation will become the dominant cost of these LAPACK routines. Our contribution is a novel parallel cache assignment approach to the panel factorization which we show scales well with p . We apply this general approach to the QR, QL, RQ, LQ and LU panel factorizations. We show results for two commodity platforms: an 8-core Intel platform and a 32-core AMD platform. For both platforms and all twenty implementations (five factorizations each of which is available in 4 types), we present results that demonstrate that our approach yields significant speedup over the existing state of the art.

Automatic Determination of Acceleration Factor Based on Residual and Functional in Shifted ICCG Method for 3-D Electromagnetic Field Analyses

This paper describes new strategy for determining an optimal acceleration factor in the shifted incomplete Cholesky conjugate gradient (ICCG) method. Although a useful method focused on the maximum diagonal entry in the incomplete Cholesky (IC) preconditioner was already proposed, it cannot always provide the optimal acceleration factor. In this paper, we propose new automatic determination methods based on the minimization of the norms of the remainder matrix, the residual, and the functional in the shifted ICCG method and discuss its effectiveness in the 3-D electromagnetic field analyses using the edge-based finite element method. It is concluded that the proposed methods based on the minimization of the norm of the residual or the functional can determine an appropriate acceleration factor within the acceptable computational cost and improve the convergence property of the shifted ICCG method.

A Novel Digital Watermarking Algorithm Based on Wavelet Lifting Scheme and Linear Regression

A novel digital watermarking algorithm is proposed in this paper, which is based on Linear Regression Constrains and Wavelet Lifting Scheme. Firstly we define a fixed matrix Q and use matrix Q MOD (Modulus after Division) the low-frequency data. The data is extracted from the 1-level wavelet decomposition of each sub-image. According to the remainder matrix, we can embed and extract watermark. The simulation results show that watermarked image can obtain a large value of PSNR and demonstrate that the algorithm has strong robustness to resist noise and clipping attacks, it also can greatly reduce carrier image distortion.

Scaling LAPACK panel operations using parallel cache assignment

In LAPACK many matrix operations are cast as block algorithms which iteratively process a panel using an unblocked algorithm and then update a remainder matrix using the high performance Level 3 BLAS. The Level~3 BLAS have excellent weak scaling, but panel processing tends to be bus bound, and thus scales with bus speed rather than the number of processors ( p ). Amdahl's law therefore ensures that as p grows, the panel computation will become the dominant cost of these LAPACK routines. Our contribution is a novel parallel cache assignment approach which we show scales well with p . We apply this general approach to the QR and LU panel factorizations on two commodity 8-core platforms with very different cache structures, and demonstrate superlinear panel factorization speedups on both machines. Other approaches to this problem demand complicated reformulations of the computational approach, new kernels to be tuned, new mathematics, an inflation of the high-order flop count, and do not perform as well. By demonstrating a straight-forward alternative that avoids all of these contortions and scales with p , we address a critical stumbling block for dense linear algebra in the age of massive parallelism.

Comparison Criteria for Parallel Orderings in ILU Preconditioning

This paper introduces block red-black ordering in a general three-dimensional form for parallel incomplete LU (ILU) preconditioning. This parallel ordering method is designed to attain fast convergence with reduced synchronization among processors in parallelized forward and backward substitutions. In this method, the grid-nodes are divided into blocks and red-black ordering is applied to them. Since blocks with identical colors never have a data-dependency, the blocks in each color can be processed in parallel. Moreover, in order to compare parallel orderings, we propose a new tool for investigating orderings on convergence. The analytic index for convergence, which is based on the remainder matrix, is easily computed and has a unique value for a fixed ordering. Nodes are classified into seven groups, and the effect of each group on convergence is estimated. Numerical tests using a three-dimensional problem confirm the validity of both proposed parallel ordering and analytic convergence comparison methods.

Orderings for Parallel Conjugate Gradient Preconditioners

The efficient solution of the linear system of equations $A\mathbf{x} = \mathbf{b}$ on parallel computers sometimes requires a reordering of the unknowns. We investigate the effects of various orderings on the rate of convergence and the computation rate of the preconditioned conjugate method with SSOR as the preconditioner. As a model problem, we use a generalized Poisson equation $\nabla (K \nabla u ) = f$, discretized with finite differencing in two and three dimensions. The natural ordering has the highest rate of convergence yet results in poor to modest parallelism, whereas the classical red/black (RB) ordering fully exploits vector/parallel properties but has a deleterious effect on the rate of convergence. Intermediate between the RB and natural orderings are the ``many-color'' orderings studied by Harrar and Ortega, but we show that these have rather poor communication properties. Alternatives are domain decomposition orderings, which we consider with and without separator sets. We investigate the effectiveness of these orderings on parallel machines, the Intel iPSC/860 hypercube in particular. We also examine the effect of the remainder matrix on the rate of convergence.

A modified decoupled linear programming technique for power system state estimation

Decoupled linear programming (LP) with constant Jacobian matrices is an effective tool for state estimation. The use of LP in different iterations is computationally time consuming for large power systems. This paper presents a modified decoupled linear programming technique to solve the state estimation problem. This technique is based on the sensitivity of the optimal solution to changes in the input data without the need to re-solve the problem for each new value. This technique uses LP to solve the subproblems of active and reactive powers only once and forms the inverse of the basis matrix of each subproblern from the final simplex-extended tableau. In other iterations, the right hand side of the LP which depends on the active and reactive power mismatches is changed only when the remainder matrix is not changed. The solution can be generated by direct multiplication of the inverse of the basis matrix of each sub-problem by the corresponding new power mismatch vector in a sequential manner until the optimum solution is reached. In the proposed method, both the advantages of decoupling and the bad data rejection property of LP are provided with a considerable improvement in computational efficiency.