Effect of Sensor Noise on the Resolution and Spatial Resolution of Displacement and Strain Maps Estimated with the Grid Method

International audienceThis paper deals with noise propagation from camera sensor to displacement and strain maps when the grid method is employed to estimate these quantities. It is shown that closed-form equations can be employed to predict the link between metrological characteristics such as resolution and spatial resolution in displacement and strain maps on the one hand and various quantities characterising grid images such as brightness, contrast and standard deviation of noise on the other hand. Various numerical simulations confirm first the relevance of this approach in the case of an idealised camera sensor impaired by a homoscedastic Gaussian white noise. Actual CCD or CMOS sensors exhibit, however, a heteroscedastic noise. A pre-processing step is therefore proposed to first stabilise noise variance prior to employing the predictive equations, which provide the resolution in strain and displacement maps due to sensor noise. This step is based on both a modelling of sensor noise and the use of the generalised Anscombe transform to stabilise noise variance. Applying this procedure in the case of a translation test confirms that it is possible to model correctly noise propagation from sensor to displacement and strain maps, and thus also to predict the actual link between resolution, spatial resolution and standard deviation of noise in grid images

Sensor noise measurement in the presence of a flickering illumination

International audienceRaw data from a digital imaging sensor are impaired by a heteroscedastic noise, the variance of pixel intensity linearly depending on the expected value. The most natural way of estimating the variance and the expected value at a given pixel is certainly empirical estimation from the variations along a stack of images of any static scene acquired at different times under the same camera setting. However, the relation found between the sample variance and the sample expectation is actually not linear, especially in the presence of a flickering illumination. The contribution of this paper is twofold. First, a theoretical model of this phenomenon shows that the linear relation changes into a quadratic one. Second, an algorithm is designed, which not only gives the parameters of the expected linear relation, but also the whole set of parameters governing an image formation, namely the gain, the offset and the readout noise. The rolling shutter effect is also considered

Towards deconvolution to enhance the grid method for in-plane strain measurement

The grid method is one of the available techniques to measure in-plane displacement and strain components on a deformed material. A periodic grid is first transferred on the specimen surface, and then images of the grid are compared before and after deformation. Windowed Fourier analysis-based techniques then permits to estimate the in-plane displacement maps and the strain components. In this report, we give a precise analysis of this estimation process. We show that the retrieved displacement maps and strain components are actually a tight approximation of the convolution of the actual displacements and strains with the analysis window. We also characterize the effect of digital image noise on the retrieved quantities and we prove that the resulting noise can be approximated by a stationary spatially correlated noise. These results are of utmost importance to enhance the metrological performance of the grid method, as shown in a separate report.La méthode de la grille est une des techniques de champs permettant de mesurer les déplacements ou les déformations à la surface d'un matériau subissant une sollicitation. Une grille périodique est transférée sur la surface de l'éprouvette considérée, et des images de la grille avant et après déformation sont comparées. Des techniques basées sur l'analyse de Fourier à fenêtre permettent alors d'estimer les cartes des composantes planes des déplacements et des déformations. Nous analysons dans ce rapport ce processus d'estimation. Nous montrons que les cartes estimées des déplacements et des déformations sont en fait bien approchées par la convolution des cartes réelles des déplacements et des déformations avec la fenêtre d'analyse. D'autre part, nous caractérisons la manière dont le bruit présent dans l'image de la grille se transfère sur les quantités estimées, et nous prouvons que le bruit résultant peut être approché par un bruit stationnaire spatialement corrélé. Ces résultats sont importants pour améliorer les performances métrologiques de la méthode de la grille, comme expliqué dans un rapport dédié

On noise reduction in strain maps obtained with the grid method by averaging images affected by vibrations

International audienceAny image-based contactless measurement system has a limited resolution because of sensor noise. If the sensor is rigorously static with respect to the imaged object, a possibility is to reduce noise by averaging images acquired at different times. This paper discusses images of a pseudo-periodic grid used in experimental solid mechanics to give estimations of in-plane displacement and strain components of a deformed flat specimen. Because of the magnification factor which is employed, the grid images are often affected by residual vibrations, thereby invalidating the assumption that the sensor is static. The averaged grid image is thus a biased estimator of the unknown noise-free image. In spite of this, we prove that the retrieved displacement and strain components still benefit from noise reduction by time-averaging. A theoretical model is discussed, and experiments on real and synthetic data sets are provided

Influence of the analysis window on the metrological performance of the grid method

International audienceThis paper deals with the grid method in experimental mechanics. It is one of the full-field methods available for estimating in-plane displacement and strain components of a specimen submitted to a load producing slight local deformation. This method consists in, first, depositing a regular grid on the surface of a specimen, and, second, comparing images of the grid before and after deformation. A possibility is to perform windowed Fourier analysis to measure these deformations as changes of the local grid aspect. The aim of the present study is to investigate the choice of the analysis window and its influence on the metrological performances of the grid method. Two aspects are taken into account, namely the reduction of the harmonics of the grid line profile, which are not pure sine because of manufacturing constraints, and the transfer of the digital noise from the imaged grid to the mechanical measurements. A theoretical study and a numerical assessment are presented. In addition, the interested reader can find in this paper a calculation of the Wigner–Ville transform of a triangular function which, to the best of the present authors’ knowledge, is not available in the existing literature

Towards deconvolution to enhance the grid method for in-plane strain measurement

International audienceThe grid method is one of the techniques available to measure in-plane displacement and strain components on a deformed material. A periodic grid is first transferred on the specimen surface, and images of the grid are compared before and after deformation. Windowed Fourier analysis-based techniques permit to estimate the in-plane displacement and strain maps. The aim of this article is to give a precise analysis of this estimation process. It is shown that the retrieved displacement and strain maps are actually a tight approximation of the convolution of the actual displacements and strains with the analysis window. The effect of digital image noise on the retrieved quantities is also characterized and it is proved that the resulting noise can be approximated by a stationary spatially correlated noise. These results are of utmost importance to enhance the metrological performance of the grid method, as shown in a separate article

On noise prediction in maps obtained with global DIC

International audienceA predictive formula giving the measurement resolution in displacement maps obtained using Digital Image Correlation was proposed some years ago in the literature. The objective of this paper is to revisit this formula and to propose a more general one which takes into account the influence of subpixel interpolation for the displacement. Moreover, a noiseless DIC tangent operator is defined to also minimize noise propagation from images to displacement maps. Simulated data enable us to assess the improvement brought about by this approach. The experimental validation is then carried out by assessing the noise in displacement maps deduced from a stack of images corrupted by noise. It is shown that specific image pre-processing tools are required to correctly predict the displacement resolution. This image pre-processing step is necessary to correctly account for the fact that noise in images is signal-dependent, and to get rid of parasitic micro-movements between camera and specimen that were experimentally observed and which corrupt noise estimation. Obtained results are analyzed and discussed

On the propagation of camera sensor noise to displacement maps obtained by DIC - an experimental study

International audienceThis paper focuses on one of the metrological properties of DIC, namely displacement resolution. More specifically, the study aims to validate, in the environment of an experimental mechanics laboratory, a recent generalized theoretical prediction of displacement resolution. Indeed, usual predictive formulas available in the literature neither take into account sub-pixel displacement, nor have been validated in an experimental mechanics laboratory environment, nor are applicable to all types of DIC (Global as well as Local). Here, the formula used to account for sub-pixel displacements is first recalled, and an accurate model of the sensor noise is introduced. The hypotheses required for the elaboration of this prediction are clearly stated. The formula is then validated using experimental data. Since rigid body motion between the specimen and the camera impairs the experimental data, and since sensor noise is signal-dependent, particular tools need to be introduced in order to ensure the consistency between the observed image noise and the model on which prediction hypotheses are based. Pre-processing tools introduced for another full-field measurement approach, namely the Grid Method, are employed to address these issues

The grid method for in-plane displacement and strain measurement: a review and analysis

International audienceThe grid method is a technique suitable for the measurement of in-plane displacement and strain components on specimens undergoing a small deformation. It relies on a regular marking of the surfaces under investigation. Various techniques are proposed in the literature to retrieve these sought quantities from images of regular markings, but recent advances show that techniques developed initially to process fringe patterns lead to the best results. The grid method features a good compromise between measurement resolution and spatial resolution, thus making it an efficient tool to characterise strain gradients. Another advantage of this technique is the ability to establish closed-form expressions between its main metrological characteristics, thus enabling to predict them within certain limits. In this context, the objective of this paper is to give the state of the art in the grid method, the information being currently spread out in the literature. We propose first to recall various techniques that were used in the past to process grid images, to focus progressively on the one that is the most used in recent examples: the windowed Fourier transform. From a practical point of view, surfaces under investigation must be marked with grids, so the techniques available to mark specimens with grids are presented. Then we gather the information available in the recent literature to synthesise the connection between three important characteristics of full-field measurement techniques: the spatial resolution, the measurement resolution and the measurement bias. Some practical information is then offered to help the readers who discover this technique to start using it. In particular, programmes used here to process the grid images are offered to the readers on a dedicated website. We finally present some recent examples available in the literature to highlight the effectiveness of the grid method for in-plane displacement and strain measurement in real situations

Caractérisation expérimentale d'assemblages de plaques boulonnées par mesures de champs sans contact

International audienceLes assemblages de plaques boulonnées sont largement utilisés en construction métallique. Ils permettent le transfertde différents efforts internes tels que flexion, efforts axiaux et de cisaillement. Le comportement mécanique de ces assemblagesest complexe à cause du transfert d'efforts entre boulons et plaques. Cela est dû à la fois à la nature hyperstatique de la liaison,à l'évolution du contact entre les boulons et leurs trous et au comportement élastoplastique des plats métalliques constituantsl'assemblage. L'interaction entre les trous d'une même plaque est aussi très importante au regard des différentes configurationsdes liaisons et des positions des trous. Les différents modes de ruine sont la rupture de section nette, la déformation plastiquedes sections brutes, le cisaillement des boulons et la pression diamétrale sur le trou de la plaque. Parmi ces modes, la pressiondiamétrale est la plus difficile à prédire car elle combine compression locale des trous par les boulons et cisaillement de lazone chargée située entre le trou et le bord de la plaque. De nombreuses études analysent le comportement d'assemblages deplaques en combinant des calculs par éléments finis avec des expérimentations instrumentées par un nombre limité de jauges dedéformation pour estimer la distribution des déformations dans les composants des joints. Dans le présent travail, la méthodede la grille est utilisée pour analyser en détail la distribution des déformations des plaques pour différentes configurations depositions de trous chargés ou non par des boulons. Cette approche permet d'observer les champs de déformations dans unelarge zone autour des boulons dans les phases élastiques et plastiques. Les résultats sont utiles pour comprendre à la fois lelien entre les zones affectées par la ruine et le mode de ruine par pression diamétrale. Cette méthode peut également servir deréférence pour calibrer des modèles complexes d'éléments finis.</p