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We show that the classification performance of graph convolutional networks (GCNs) is related to the alignment between features, graph, and ground truth, which we quantify using a subspace alignment measure (SAM) corresponding to the Frobenius norm of the matrix of pairwise chordal distances between three subspaces associated with features, graph, and ground truth. The proposed measure is based on the principal angles between subspaces and has both spectral and geometrical interpretations. We showcase the relationship between the SAM and the classification performance through the study of limiting cases of GCNs and systematic randomizations of both features and graph structure applied to a constructive example and several examples of citation networks of different origins. The analysis also reveals the relative importance of the graph and features for classification purposes., Comment: Published in IEEE Transactions on Neural Networks and Learning Systems; Date of Publication: 11 January 2021

Biodiversity offsetting is a policy approach that compensates for the ecological losses from development projects affecting biodiversity with equivalent gains through offsets, aiming at “No Net Loss” (NNL). Although offsets seem appealing in theory, several concerns have been raised about the difficulties reaching NNL in practice. While most of the discussion about offsets improvement is based on principles and strategies, we evaluated empirical evidence of offsets implemented, both from the procedure files (protected species and wetlands) and field surveys. Our objective was to evaluate whether offsets achieve NNL based on 91 procedure files in the Isere department in France. We identified that necessary data for assessing offsets gains, such as the location and offset sites' initial state, were not available in part (location) or all (initial state) procedure files investigated. We evaluated 59 offsets implemented for 22 development projects and where minimum data for monitoring offsets were available; we surveyed the presence or absence of target species and habitat from the offset site. The type of offsets (restoration, creation or maintenance of target habitat) was one of the characteristics that helped to explain both species and habitat absence, implying offset failure. Based on our analysis, we suggest three principal angles for progressing in NNL achievement: (i) collecting and publishing a set of essential information on offsets, (ii) requiring a management plan for each offset, and (iii) accumulating empirical evidence of offsets failure and success.

A heterodyne polariscope composed of a Zeeman laser, a half-wave plate, two quarter-wave plates, an analyzer, a photo-detector, and an x–y stage is proposed for determining the full-field principal angle and phase retardation of stressed plastic substrates. This paper introduces the configuration, measurement theory, and experimental setup of the polariscope. Additionally, this paper presents experiments as well as the results of these experiments. The experimental results show that the errors of the measured principal angle and phase retardation do not extend beyond 1.07° and 2.86°, respectively, and the results verify the validity and applicability of the proposed polariscope.

A simple technique for measuring the phase change on reflection with application to gauge block calibration is presented. The proposed technique measures the optical constants of the steel gauge block and the auxiliary platen which is made of fused silica using polarized light. The presented method considers the principal angle of incidence method and avoids the deviation of the quarterwave plate from exact λ/4 value. The phase change angle could be determined with high accuracy (±2o) and the uncertainty contribution of the phase change correction in the gauge block calibration could be reduced to (± 2 nm). The resulted phase change correction value was compared to the value obtained by interferometry using the stack method. The presented method is proposed to be convenient with the contactless measurement of gauge block length using the double ended interferometer

Principal angles of incidence and associated principal azimuths, at which incident linearly polarized light becomes circularly polarized upon reflection, are determined for an IR-transparent high-index Si film (of the proper thickness) which is deposited on a low-index SiO2 (glass) substrate at the near-IR wavelength λ = 1.55 μm. The results are substantially different from those reported by the author [Appl. Opt. 57, 9529–9532 (2018)] for the SiO2–Si (film-substrate) system at the same wavelength. The intensity reflectances for incident p- and s-polarized light, and the linear-to-circular polarization conversion efficiency are also calculated as functions of the principal angle.

Image-set classification has recently generated great popularity due to widespread application to challenging tasks in computer vision. The great challenges arise from measuring the similarity between image sets which usually exhibit huge inter-class ambiguity and intra-class variation. In this paper, based on the assumption that each image set as a linear subspace can be treated as a point on a Grassmann manifold, we propose discriminant Grassmann kernels (DGK) of principal angles between subspaces. To tackle the ambiguity and variation, we propose learning the DGK via kernel target alignment, which achieves kernels of great discrimination by maximizing correlations with class labels. The proposed DGK has been evaluated on two challenging datasets including the ETH-80 and UCSD datasets for object recognition and video-based traffic congestion recognition, respectively. Extensive experiments have shown that the proposed DGKs achieves state-of-the-art performance and surpasses most of previous methods, which demonstrates the great effectiveness of the DGKs for image-set classification.

In this paper, we study the robust stabilization of a networked control system (NCS) with the communication channels described by cascade two-port networks. Simultaneous uncertainties are assumed to be in the plant, controller and two-port communication channels. The cascade two-port connections apply to the scenario where signals in the NCS are required to pass through bidirectional communication channels separated by several relays. Distortions and interferences are taken into account at each stage during the communication. In terms of robustness, we consider ℋ ∞ norm bounded uncertainties in the transmission matrices of the communication channels as well as the gap metric uncertainties in the plant and controller. A necessary and sufficient condition for the robust stability of the NCS is given in the form of an “arcsin” inequality, taking advantages of the properties of the canonical angles between subspaces defined on system graphs. With the analysis result, the synthesis problem can be solved through a tractable ℋ ∞ optimization.

Several Compression Field Theories have been developed for the design of structural concrete members to resist shear. These theories adopt the simplifying assumption that the direction of principal tensile stress and the direction of principal tensile strain coincide. This assumption is known as the Equal Principal Angles assumption (EPAs), or Wagner’s hypothesis. Experimental evidence indicates that this is a reasonable simplification. This paper investigates potential differences in the directions (or angles) of principal stress and principal strain and evaluates the relative influence of design parameters on the divergence of these angles. Within practical ranges of these design parameters, the transverse reinforcement content is found to have the greatest influence on the applicability of the EPA assumption.

A human femur model, submitted to static loads, is analyzed through the utilization of three material constitutive relationships, namely: isotropic, transversally isotropic and orthotropic. The influence of bone anisotropy with respect to principal stress/strain distribution on human femur external surface was accessed through the use of analytical and finite element approaches. The models results show that the principal angles at a medial path bone surface have a good correlation with human femur bone lamellae angles.

Many industrial processes possess multiple operating modes in virtue of different manufacturing strategies or varying feedstock. Direct application of many of the current multivariate statistical process monitoring (MSPM) techniques such as PCA (principal component analysis) and PLS (projection to latent structures) to such a process tends to produce inferior performance. This can most be attributed to the adopted assumption by most MSPM methodologies of only one nominal operating region for the underlying process. It is therefore reasonable to develop separate models for different operating modes. In this paper, based on metrics in the form of principal angles to measure the similarities of any two models, a multiple PLS model based process monitoring scheme is proposed. Popular multivariate statistics such as SPE (squared prediction error) and T2 can be incorporated in this framework straightforwardly. The proposed technique is assessed through application to the monitoring of an industrial pyrolysis furnace.

We recently developed a software package to autonomously detect coronal mass ejections (CMEs) in image sequences from large angle spectrometric coronagraph (LASCO). The detection is not done in the original images, instead we detect bright ridges in (time, height) plots using a modified version of the Hough transform. Experimental results on real-time data have shown that the developed technique can achieve excellent results in measuring starting time and principal angle and good results for the angular width and velocity measurement compared to the CMEs listed in the manually assembled catalog. The real-time output of the software can be found online at www.sidc.oma.be/cactus . With the present paper, we report on the first large scale application of the software to the LASCO archive. In an ongoing work, we have applied the software on a first sample of 29 months selected from archive data in the period July 1997 to December 2002. The results in this paper show that relevant characteristics of CMEs over the solar cycle are successfully recovered with the automated procedure. This proves that a completely automatically produced CME catalog is within reach.

We propose an alternative approach to the widely-used Bag-of-Features (BoF) for representing objects in terms of a collection of local features extracted from their images. In this new framework, called Subspaces-of-Features (SoF), first the sets of local features extracted from the images of the objects are represented as low-dimensional linear subspaces. Then the subspaces corresponding to different categories are orthogonalized, and the similarity between subspaces corresponding to different categories is calculated using the Grassmannian distances defined through the principal angles between the subspaces. The performance of SoF is illustrated on a standard generic object recognition benchmark.

In this paper we show how Grassmann distances and Grassmann kernels can be efficiently used to learn and classify face sequence videos. We propose two new methods, the Grassmann Distance Mutual Subspace Method (GD-MSM) which uses Grassmann distances to define the similarity between subspaces of images, and the Grassmann Kernel Support Vector Machine (GK-SVM), which applies two Grassmann kernels - the projection kernel and the Binet-Cauchy kernel - in a convex optimization scheme, using the Support Vector Machine (SVM) framework. GD-MSM and GK-SVM are compared in a face recognition task with several related methods using a large database of face image sequences from 100 subjects, containing expression changes related to a natural conversation setting. Additionally, we study the effect of combining all available training image sequences into a single subspace per category, in comparison with using multiple smaller subspaces, i.e. representing each category by several different subspaces, where each subspace is formed from image sequences taken under different conditions.

Electroencephalograph (EEG) is the most important method in diagnosis of brain diseases. Unfortunately, the doctor interprets EEG according to personal experiences, which results in over interpretation or incorrect judgment. In this paper, we derive a sparse kernel principal angle (SKPA). The SKPA is applied to disease diagnosis in examples: we select Electroencephalograph data of different diseases, calculate the principal angles between the datasets. Reading the principal angles, one can find diseases or even diagnose disease type. The electroencephalograph automatic diagnosis system based on kernel principal angle will be helpful to doctors.

In this paper we address the problem of classifying vector sets. We motivate and introduce a novel method based on comparisons between corresponding vector subspaces. In particular, there are two main areas of novelty: (i) we extend the concept of principal angles between linear subspaces to manifolds with arbitrary nonlinearities; (ii) it is demonstrated how boosting can be used for application-optimal principal angle fusion. The strengths of the proposed method are empirically demonstrated on the task of automatic face recognition (AFR), in which it is shown to outperform state-of-the-art methods in the literature.

One of the fundamental challenges of recognizing actions is accounting for the variability that arises when arbitrary cameras capture humans performing actions. In this paper, we explicitly identify three important sources of variability: (1) viewpoint, (2) execution rate, and (3) anthropometry of actors, and propose a model of human actions that allows us to investigate all three. Our hypothesis is that the variability associated with the execution of an action can be closely approximated by a linear combination of action bases in joint spatio-temporal space. We demonstrate that such a model bounds the rank of a matrix of image measurements and that this bound can be used to achieve recognition of actions based only on imaged data. A test employing principal angles between subspaces that is robust to statistical fluctuations in measurement data is presented to find the membership of an instance of an action. The algorithm is applied to recognize several actions, and promising results have been obtained.

An analytical method has been developed for directly calculating the principal angle theta(p) at which the phase difference between the two reflection coefficients is equal to 90 degrees and at which the ratio of /r(p)/ to /r(s)/ is equal or close to a minimum. The equations given here can be used in many optical applications. For example, ellipsometric data measured at an incidence angle theta(p) will have higher precision than data measured at other incidence angles. theta(p) is the principal angle. Instead of three principal angles, there is only one principal angle, which can be found in the region of 0or = theta(p)or = 90 degrees for most metallic materials used in applications. Results show good agreement between the measured and the calculated spectra of delta(p) and rho(po).

The head and skull in patients with Mediterraneen anaemia show often a marked change of the external shape due mainly to the considerable enlargement of the cranial bones. From the abnormal external appearance the suspicion may arise that the form and space of the internal cranial cavity are likewise affected and even the basic structure of the skull is altered. However, a detailed investigation of the principal angles, distances and characteristic structures of the skull made on lateral roentgenograms of these patients, a comparison of the results between cases with different intensity of the osseous changes and a confrontation with the findings in normal individuals proved that there were no major discrepancies. The essential features of the human skull were always preserved. Only in some cases with severe alterations of the cranial bones a few small deviations from the normal values could be noticed and in a little number of them signs of a slight narrowing of the cranial cavity werde to be observed. This way was probably caused by a very early, rapid and extreme enlargement of the cranial bones resulting in a kind of "premature panzer-skull."

In this communication we wish to point out the non-applicability of two significant assumptions in traditional ellipsometric literature, in the context of infrared ellipsometry of metals. The first concerns the choice of the optimum angle of incidence to maximize experimental accuracy and the second is related to the definition of θp. the principal angle of incidence.

In principal angle-of-incidence ellipsometry, or ϕP - K ellipsometry, one makes measurements of the principal angle of incidence, ϕP, and the imaginary part of the ellipsometric function ϱ for this angle of incidence, K, in place of Δ and tan ψ. The parameter K has often been referred to as the ellipticity at principal angle incidence. Following a discussion of the sensitivity of ϕP-K ellipsometry, the results obtained in recent years by ϕP-K ellipsometry are reviewed. A return-path type ellipsometer for use in ϕP-K ellipsometry is described, which requires no compensator and can be a powerful tool for spectroscopic ellipsometry.

Pfund's half-shade analyzer, which originally was devised for the measurement of the Brewsterian angle, ϕ B , can be applied to the measurement of the principal angle of incidence, ϕ p . Its sensitivity, h , and possible sources of systematic errors are discussed. It is shown that, in the measurement of ϕ B , h can be made to approach ∞, but in the measurement of ϕ p , h is finite and sometimes very small. A superior half-shade device for the precise determination of ϕ p is proposed, which consists of a λ/4-plate, a half-shade plate of the type of Bravais bi-plate, and an analyzer (polarizing plate). Its sensitivity is fully discussed and the systematic errors are analyzed. Experimental test of the new device is briefly described.

An undergraduate experiment for the optics laboratory or for an independent study project is described. Thermally evaporated thin aluminum films are formed in a vacuum of about 10−5 Torr. Two methods of film thickness measurements are presented. Ellipsometric measurements of the principal angle of incidence φ and the principal azimuth ψ are made on aluminum films of various thicknesses.

By the aid of a goniometer with the optic of a microscope it is possible to measure well reproducible the principal angle of incidence of a polished ore section. It may be used as a standard material constant for identifications. By the use of objective methods, with the aid of photomultiplier tubes and analysis by computation an accuracy of some few minutes of arc is within reach. Whereby certainly the quality of the sections and the polishing have to satisfy high requirements. The measurements are done without using a standard of comparison. They may be used as complement to the common measurements by vertical illumination. Details of the object are measurable down to a minimum diameter of about 15 μm.

We present a modification of O'Bryan's return-path ellipsometer (J. Opt. Soc. Am. 26 (1936) 122) that extends its applicability to arbitrary surface retardations (not to be restricted to ±90°) at angles of incidence other than the principal angles. This modified version uses a phase-retarding corner-reflector to return the light beam for a second reflection at the sample surface, with a single optical component acting as both polarizer and analyzer. The operation of the ellipsometer is visualized using the complex-plane representation of polarized light.

The apparatus is designed for ellipsometry based on the measurement of the principal angle of incidence. It is of the return-path type, somewhat like the Littrow-type spectrometer, requiring only one polarizer and no compensator.

Reflection of light at the surfaces of a λ/4 plate seriously affects the accuracy of ellipsometric measurements. The ellipsometer described here, which is to be used for the determination of the principal angle of incidence, P, and the reflectivity ratio of the p- and s-components at this angle of incidence, P, employs only one polarizer and no compensator, and is free from this nuisance. Its extinction sensitivity is roughly twice as large as that of a conventional ellipsometer. The requirements that have to be fulfilled in order that P and ρP can be determined to ±3'' and ±10'' respectively are given in detail.

A computer-aided method of calculation for obtaining n and d directly from the observed values of tan Ψ and Δ is described. Two equations giving tan Ψcos Δ and tan Ψsin Δ as respective functions of Fresnel coefficients and δ are solved for δ and n. Here δ=4πd√n2-sin 2/λ ( is the angle of incidence). As a means for solution a linear equation for cos δ is conveniently utilized, which is derived from the first of the above equations, and another equation giving tan 2Ψ as a function of Fresnel coefficients and δ. A similar method of calculation is also given for the case where n and d are to be determined from the principal angle of incidence, P, and the ellipticity of light reflected at this angle of incidence, ρP.

A high-precision ellipsometer is described capable of following the thickness and refractive index of condensing films and of revealing small departures from homogeneity. Applied to a growing film of magnesium fluoride, the instrument reveals variations in the refractive index of the order 0.005–0.01.

The reflection coefficients for perpendicular and parallel polarization of a germanium-mercury interface have been measured as a function of angle of incidence at a wavelength of three microns. There is in general good agreement with theory although at the principal angle, the parallel component of polarization is absorbed more strongly than indicated by theory.

The optical constants of opaque materials are frequently determined by measuring the principal angle of incidence, ϕ¯, and the principal azimuth ψ¯. The approximation furnished by Brewster’s law is examined. The slope at ϕ¯ of the graph of Rp/Rs against ϕ is 12cot3ϕ¯sin4ψ¯; previous investigations omitted the factor 12.The variation of ψ with ϕ in the neighborhood of ψ¯ and ϕ¯ is expanded in the form ψ−ψ¯=a(ϕ−ϕ¯)+b(ϕ−ϕ¯)2+c(ϕ−ϕ)2.The parabolic term is the most important in the infrared. Therefore, whether ϕ−ϕ¯ be positive or negative, ψ−ψ¯ will be positive. Such a systematic error in ψ¯ leads to a positive error in the measurement of refractive index. The magnitude of such an error depends on the signal to noise ratio. It is not significant in the visible region but is relevant in the infrared. Experimental values are examined and a graphical chart enables a ready estimate to be made in any practical case.

The optical constants of a surface from which plane polarized light is reflected may be determined by measuring the principal angle of incidence, ϕ¯, and the principal azimuth ψ¯. If the polarizer be rotated and the reflected light be examined by an analyzer, the intensity is modulated at twice the frequency of rotation. The resultant signal vanishes at ϕ¯,ψ¯, a tuning condition which is unique, is easily found and is not affected by polarizing influences subsequent to the analyzerThe effect of stray polarization is examined and it is shown that the quality of the polarizer affects only the sensitivity. The bearing of the signal to noise ratio on experimental errors is considered since this ratio is limited in the infrared. It is shown that the method is valuable if the extinction coefficient lies in the range 0.4≤κ≤5.5. Outside this range the accuracy rapidly deteriorates. This limitation is shared by all techniques based on measurement of the principal angles. Experimental details and results are presented.

The refractive index n(f) and thickness d(f) of the 'polish layer' on the surface of various kinds of glass polished under standard conditions were determined by ellipsometry. The method of ellipsometry employed was based on the measurement of the principal angle of incidence and the ellipticity of the light reflected at this angle. (Author)