Partial Orthogonality Research Articles

Decomposition and compensation of fringe harmonic errors by use of their partial orthogonality in phase-shifting fringe projection profilometry

In phase-shifting fringe projection profilometry, fringe harmonics have been recognized as one of the main error-inducing factors. Generally, the response of a phase-shifting algorithm to fringe harmonics strongly depends on the used phase shifts and is usually unpredictable, especially when using nonuniform phase shifts. For this reason, it is difficult to eliminate the phase-measuring errors caused by fringe harmonics in a general case, for example, when the phase shifts are not uniform but arbitrarily valued. To overcome this problem, this paper analyzes the phase error function related to each fringe harmonic under the condition of using arbitrary phase shifts, reveals the partial orthogonality of these functions, and then derives an algorithm for decomposing the harmonics-induced phase errors. In the implementation, this algorithm calculates a coarse phase map first in the least squares sense from captured fringe patterns, and then estimates the coefficients of fringe harmonics from this calculated phase map by use of the partial orthogonality of the error functions. By using the estimated harmonic coefficients, the phase map is updated, thus having improved accuracies so that the harmonics-induced phase errors are compensated for through an iterative procedure. The simulation and experimental results demonstrate this method to be effective and efficient in measuring fringe phases.

Variable selection in heterogeneous panel data models with cross‐sectional dependence

AbstractThis paper studies the Bridge estimator for a high‐dimensional panel data model with heterogeneous varying coefficients, where the random errors are assumed to be serially correlated and cross‐sectionally dependent. We establish oracle efficiency and the asymptotic distribution of the Bridge estimator, when the number of covariates increases to infinity with the sample size in both dimensions. A BIC‐type criterion is also provided for tuning parameter selection. We further generalise the marginal Bridge estimator for our model to asymptotically correctly identify the covariates with zero coefficients even when the number of covariates is greater than the sample size under a partial orthogonality condition. The finite sample performance of the proposed estimator is demonstrated by simulated data examples, and an empirical application with the US stock dataset is also provided.

Compact agile Tchebycheff transform variant for temporal compression of neural signals on brain-implantable microsystems

This paper proposes a new approximate variant of the Tchebycheff transform for the compression of intra-cortically-recorded neural signals on brain-implantable microsystems. The proposed approximate transform is achieved using an innovative algorithm that guarantees ‘partial orthogonality’ of the basis functions corresponding to the coefficients conveying the majority of the signal energy. This significantly reduces inter-coefficient energy leakage in the proposed transform variant, which helps meaningfully enhance the data compression rate. While keeping the signal reconstruction accuracy within the acceptable range, the proposed Tchebycheff transform variant is maximally truncated in order to reduce the hardware cost of the associated implementation. Based on the resulting truncated approximate transform, a hardware-efficient 256-channel neural signal compressor is designed, prototyped, and tested using pre-recorded neural signals. Operated at a clock rate of 960 kHz, the prototyped neural signal compressor exhibits a true compression rate of 1880 with an RMS reconstruction error of 2.8%.

Co-consensus semi-supervised multi-view learning with orthogonal non-negative matrix factorization

Semi-supervised multi-view learning has recently achieved appealing performance with the consensus relation between samples. However, in addition to the relation between samples, the relation between samples and their assemble centroid is also important to the learning. In this paper, we propose a novel model based on orthogonal non-negative matrix factorization, which allows exploring both the consensus relations between samples and between samples and their assemble centroid. Since this model utilizes more consensus information to guide the multi-view learning, it can lead to better performance. Meanwhile, we theoretically derive a proposition about the equivalency between the partial orthogonality and the full orthogonality. Based on this proposition, the orthogonality constraint and the label constraint are simultaneously implemented in the proposed model. Experimental evaluations on five real-world datasets show that our approach outperforms the state-of-the-art methods, where the improvement is 6% average in terms of ARI index.

Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states {mathbf {P}}—i.e., the set of one-dimensional projections on a complex Hilbert space H– and the orthomodular lattice {mathbf {L}} of closed subspaces of H. These six groups are isomorphic when the dimension of H is ge 3. The latter hypothesis is absolutely necessary in this identification. For example, the automorphisms group of all bijections preserving orthogonality and the order on {mathbf {L}} identifies with the bijections on {mathbf {P}} preserving transition probabilities only if dim(H)ge 3. Despite of the difficulties caused by M_2({mathbb {C}}), rank two algebras are used for quantum mechanics description of the spin state of spin-frac{1}{2} particles. However, there is a counterexample for Uhlhorn’s version of Wigner’s theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, let M and N be two atomic JBW^*-triples not containing rank–one Cartan factors, and let {mathcal {U}} (M) and {mathcal {U}} (N) denote the set of all tripotents in M and N, respectively. We show that each bijection Phi : {mathcal {U}} (M)rightarrow {mathcal {U}} (N), preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Molnár to the wider setting of atomic JBW^*-triples not containing rank–one Cartan factors, and provides new models to present quantum behavior.

Representations of optical fibre communications in three‐ and four‐dimensional spaces

The authors study some electromagnetic field representations with a focus on optical communications, light polarisation and phase. Well-known three- and four-dimensional representations are compared with each other and with new ones that are presented and analysed from an analytical point of view. A symmetry emerges in the relationships between the various representations, showing that they are closely related formulas. In fact, different representations point out different characteristics or domains of the same phenomenon. This study results in a more accurate description of the electromagnetic field and especially in its capacity to carry information for telecommunications purposes. From a technological point of view, the partial orthogonality of the described spaces suggests the possibility of exploiting simultaneously more than one representation in order to carry more information or the possibility of using these representations as a natural codifying system with the aim of increasing reliability or confidentiality of communications.

Fast ADMM for Sum-of-Squares Programs Using Partial Orthogonality

When sum-of-squares (SOS) programs are recast as semidefinite programs (SDPs) using the standard monomial basis, the constraint matrices in the SDP possess a structural property that we call partial orthogonality. In this paper, we leverage partial orthogonality to develop a fast first-order method, based on the alternating direction method of multipliers (ADMM), for the solution of the homogeneous self-dual embedding of SDPs describing SOS programs. Precisely, we show how a “diagonal plus low rank” structure implied by partial orthogonality can be exploited to project efficiently the iterates of a recent ADMM algorithm for generic conic programs onto the set defined by the affine constraints of the SDP. The resulting algorithm, implemented as a new package in the solver CDCS, is tested on a range of large-scale SOS programs arising from constrained polynomial optimization problems and from Lyapunov stability analysis of polynomial dynamical systems. These numerical experiments demonstrate the effectiveness of our approach compared to common state-of-the-art solvers.

Inter-Class Orthogonal Main Effect Plans for Asymmetrical Experiments

In this paper we construct ‘inter-class orthogonal’ main effect plans (MEPs) for asymmetrical experiments. In such a plan, the factors are partitioned into classes so that any two factors from different classes are orthogonal. We have also defined the concept of “partial orthogonality” between a pair of factors. In many of our plans, partial orthogonality has been achieved when (total) orthogonality is not possible due to divisibility or any other restriction. We present a method of obtaining inter-class orthogonal MEPs. Using this method and also a method of ‘cut and paste’ we have obtained several series of inter-class orthogonal MEPs. One of them happens to be a series of orthogonal MEP (OMEPs) [see Theorem 3.6], which includes an OMEP for a 330 experiment on 64 runs. We have also obtained a series of MEPs which are almost orthogonal in the sense that every contrast is non-orthogonal to at most one more. A member of this series is an MEP for a 310210 experiment on 32 runs in which the only non-orthogonality is between the linear contrasts of pairs of three-level factors. Plans of small size (≤ 15 runs) are also constructed by ad-hoc methods. Among these plans there are MEPs for a 42.32.2 and a 35.2 experiment on 12 runs and a 52.32 experiment on 15 runs.

Power Allocation for Non-Orthogonal Millimeter Wave Systems With Mixed Traffic

We consider the problem of power allocation for the down-link of a 5G cellular system operating in the mm-wave band and serving two sets of users: fix-rate (FR) users (transmitting data at fixed rate), typically seen in device-to-device communications, and variable-rate (VR) users (that can change their transmission data rate), typically requiring high data rate services. The power allocation objective is the maximization of the spectral efficiency of VR users while ensuring that FR users get the required rate. In contrast with the existing literature on power allocation, we exploit the sparsity of the virtual mm-wave channel matrix, obtained by applying fixed discrete-Fourier transform beamformers at both the transmitter and the receiver. Exploiting the channel partial orthogonality, users are first grouped based on the mutual interference and then the power is allocated among and within groups.

Asymptotic Properties of Marginal Least-Square Estimator for Ultrahigh-Dimensional Linear Regression Models with Correlated Errors

ABSTRACTIn this article, we discuss asymptotic properties of marginal least-square estimator for ultrahigh-dimensional linear regression models. We are specifically interested in probabilistic consistency of the marginal least-square estimator in the presence of correlated errors. We show that under a partial orthogonality condition, the marginal least-square estimator can achieve variable selection consistency. In addition, we demonstrate that if a mutual orthogonality holds, the marginal least-square estimator satisfies estimation consistency. The discussed theories are exemplified through extensive simulation studies.

Automatic 3D reconstruction of a polyhedral object from a single line drawing under perspective projection

A novel method is proposed to automatically recover a 3D polyhedral object from a single 2D line drawing in perspective projection. In the algorithm, a camera calibration based on partial orthogonality is performed in order to retrieve multiple pinhole cameras. Then, 3D orientations of edge lines are estimated by a factor graph inference. Finally, 3D reconstruction is conducted through hybrid optimization by combining beam search with simulated annealing. Parameterization based on inverse depths is proposed to reduce the dimension of variables in optimization. Experimental results show that the algorithm proposed in the study can reconstruct reasonable 3D objects from perspective-projected line drawings, which cannot be handled by existing approaches. Qualitative and quantitative comparisons are provided to illustrate the advantages of the proposed method.

A feasible method for sensor network localization

A nonlinear programming (NLP) model with partial orthogonality constraints, relaxed from the polynomial optimization problem, is proposed and analysed for solving sensor network localization. The NLP model is difficult to solve as the orthogonality constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with this difficulty, we apply the Cayley transform (a Crank–Nicolson-like update scheme) to preserve it. Combining with the gradient descent method, we develop a curvilinear search algorithm, and analyse its convergence. In practice, we accelerate our method by taking nonlinear conjugate gradient method and Barzilai–Borwein steps. Numerical experiments are given to demonstrate the efficiency of the proposed method, for the problems with the number of sensors up to 15,000 and the distance constraints up to 135,817.

Analytic mode-matching for acoustic scattering in three dimensional waveguides with flexible walls: Application to a triangular duct

An analytic mode-matching method suitable for the solution of problems involving scattering in three-dimensional waveguides with flexible walls is presented. Prerequisite to the development of such methods is knowledge of closed form analytic expressions for the natural fluid–structure coupled waveforms that propagate in each duct section and the corresponding orthogonality relations. In this article recent theory [J.B. Lawrie, Orthogonality relations for fluid–structural waves in a 3-D rectangular duct with flexible walls, Proc. R. Soc. A. 465 (2009) 2347–2367] is extended to construct the non-separable eigenfunctions for acoustic propagation in a three-dimensional rectangular duct with four flexible walls. For the special case in which the duct cross-section is square, the symmetrical nature of the eigenfunctions enables the eigenmodes for a right-angled, isosceles triangular duct with flexible hypotenuse to be deduced. The partial orthogonality relation together with other important properties of the triangular modes are discussed. A mode-matching solution to the scattering of a fluid–structure coupled wave at the junction of two identical semi-infinite ducts of triangular cross-section is demonstrated for two different sets of “junction” conditions.

Electromagnetic scattering from both finite semi-circular channels and bosses in a conducting plane: TM case

The behavior of TM-wave scattering from both semi-circular channels and bosses in a conducting plane is investigated. The scattered field is represented in terms of an infinite series of radial modes with unknown coefficients. By applying separation of variables and employing the partial orthogonality of sine functions, the unknown coefficients are obtained. The boundary conditions are checked for semi-circular channels and bosses to verify the algorithm proposed. Plotted results for the radar cross sections reveal how the scattering properties may be varied by changing several physical and geometrical parameters of the structures.

ORACLE EFFICIENT VARIABLE SELECTION IN RANDOM AND FIXED EFFECTS PANEL DATA MODELS

This paper generalizes the results for the Bridge estimator of Huang, Horowitz, and Ma (2008) to linear random and fixed effects panel data models which are allowed to grow in both dimensions. In particular, we show that the Bridge estimator isoracle efficient. It can correctly distinguish between relevant and irrelevant variables and the asymptotic distribution of the estimators of the coefficients of the relevant variables is the same as if only these had been included in the model, i.e. as if an oracle had revealed the true model prior to estimation.In the case of more explanatory variables than observations we prove that the Marginal Bridge estimator can asymptotically correctly distinguish between relevant and irrelevant explanatory variables if the error terms are Gaussian. Furthermore, a partial orthogonality condition of the same type as in Huang et al. (2008) is needed to restrict the dependence between relevant and irrelevant variables.

A Need for Speed: Genetic Encoding of Rapid Cycloaddition Chemistries for Protein Labelling in Living Cells

Rapid reactions: Several reactants for strain-promoted cycloaddition reactions have been genetically encoded as the side chains of noncanonical amino acids. This results in decisive improvements for the fluorescent labelling of intracellular proteins such as quantitative turnover, completion of labelling reactions within minutes, fluorogenic effects and even partial orthogonality for multicolour labelling.

Orthogonality relations for fluid-structural waves in a three-dimensional, rectangular duct with flexible walls

An exact expression for the fluid-coupled structural waves that propagate in a three-dimensional, rectangular waveguide with elastic walls is presented in terms of the non-separable eigenfunctions ψ n ( y , z ). It is proved that these eigenfunctions are linearly dependent and that an eigenfunction expansion representation of a suitably smooth function f ( y , z ) converges point-wise to that function. Orthogonality results for the derivatives ψ n y ( a , z ) are derived which, together with a partial orthogonality relation for ψ n ( y , z ), enable the solution of a wide range of acoustic scattering problems. Two prototype problems, of the type typically encountered in two-part scattering problems, are solved, and numerical results showing the displacement of the elastic walls are presented.

Asymptotic properties of bridge estimators in sparse high-dimensional regression models

We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero. We show that under appropriate conditions, bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance. Thus, bridge estimators have an oracle property in the sense of Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] and Fan and Peng [Ann. Statist. 32 (2004) 928--961]. In general, the oracle property holds only if the number of covariates is smaller than the sample size. However, under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients, we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size.

Blind source separation of positive and partially correlated data

Blind source separation (BSS) consists of processing a set of observed mixed signals to separate them into a set of unobservable original components. Various approaches have been employed to solve BSS problems using the strong assumption focusing on mutually uncorrelated (or orthogonal) source signals. However, in many real-life problems, signal orthogonality is not guaranteed. This paper introduces a new approach to BSS that can be applied to nonorthogonal signals. The orthogonality requirement is replaced by a partial orthogonality and a nonnegativity constraint which are well-suited for many real-world signals. An algebraic property is then exploited to express BSS problems in terms of constrained optimization. An efficient algorithm implementing the approach is reported and applied to examples from nuclear magnetic resonance spectroscopy. Résumé La séparation aveugle de sources consiste à traiter un ensemble de signaux en mélange pour en extraire les composantes inobservables d’origine. Diverses approches utilisent l’hypothèse très restrictive de non-corrélation mutuelle des signaux sources, c’est-a-dire d’orthogonalité. Cependant, dans nombreuses situations réelles, la validité de l’hypothèse n’est pas garantie. Cet article propose une approche originale de la séparation de sources applicable à des sources non-orthogonales. La condition de non-corrélation est remplacée par une hypothèse de positivité, adaptée à de nombreux problèmes réels. Une propriété algébrique permet de transposer le problème en termes d’optimisation sous contrainte. Un algorithme efficace mettant en œuvre cette approche est appliqué à la spectroscopie de résonance magnétique nucléaire.

Electromagnetic Scattering from A Dielectric Coated Conducting Elliptic Cylinder Loading A Semi-Elliptic Channel In A Ground Plane

Analytic solution to the problem of scattering of a plane electromagnetic wave by a dielectric coated confocal conducting elliptic cylinder loading a semi-elliptic channel is investigated. The incident, scattered and transmitted fields in every region are expressed in terms of Mathieu functions. Applying the boundary conditions at various interfaces along with the partial orthogonality properties of angular Mathieu functions, the unknown scattered and transmitted field coefficients are obtained and written in matrix form. The presented numerical results show that the presence of a dielectric coated conducting elliptic cylinder changes significantly the echo patterns width of the elliptic channel.