One of the common ways for solving indoor navigation is known as Pedestrian Dead Reckoning (PDR), which employs inertial and magnetic sensors typically embedded in a smartphone carried by a user. Estimation of the pedestrian’s heading is a crucial step in PDR algorithms, since it is a dominant factor in the positioning accuracy. In this paper, rather than assuming the device to be fixed in a certain orientation on the pedestrian, we focus on estimating the vertical direction in the sensor frame of an unconstrained smartphone. To that end, we establish a framework for gravity direction estimation and highlight the important role it has for solving the heading in the horizontal plane. Furthermore, we provide detailed derivation of several approaches for calculating the heading angle, based on either the gyroscope or the magnetic sensor, all of which employ the estimated vertical direction. These various methods—both for gravity direction and for heading estimation—are demonstrated, analyzed and compared using data recorded from field experiments with commercial smartphones. Indoor navigation for pedestrians is essential for various location-based applications, such as commercial and emergency services. In the absence of Global Navigation Satellite System (GNSS) solution indoors, a common approach to handle this task is to use the inertial and magnetic sensors embedded in smartphones or other wearable devices, in a framework known as pedestrian dead reckoning (PDR) . TypicalCalculation of the horizontal turning rate, which is equivalent to the vertical projection of the three-axis gyroscope measurements, can be used for computing the heading angle.Extracting the horizontal acceleration signal out of the three-axis accelerometer measurements can be employed for identifying the dominant horizontal motion direction .Several step detection and step length estimation methods require only the vertical acceleration signal.In order to use magnetic sensors for solving the heading angle, information on the horizontal plane as embedded in the gravity direction needs to be incorporated as well.In PDR algorithms, gravity direction estimation has a central role, although this is usually not mentioned explicitly in the literature. The direction of gravity can be regarded as the unit vector perpendicular to the local horizontal  plane, pointing vertically downwards. When expressed in the sensor frame of reference , this vertical direction vector is generally time-varying due to rotational motion of the pedestrian. Once the gravity direction is estimated in this frame, it can be utilized to decompose any vector measurement into its vertical and horizontal components. This decomposition is essential for various PDR tasks, for example:That is, accurate estimation of the gravity direction is critical for these tasks.This paper is an extension of our initial work , in whicA comprehensive survey on indoor navigation was given in  with an Attitude of the sensor frame is closely related to the gravity direction, as the latter holds partial information on the former. Attitude estimation is investigated in a recent paper, , where vIn this work, we introduce and derive two approaches for gravity direction estimation, and utilize them as a baseline for heading determination. The first approach is based directly on the accelerometer measurements, while the second employs the gyroscope along with information on stationary intervals for relative attitude estimation. Additionally, by incorporating the estimated gravity direction, we derive and compare four different methods for computing the heading angle. The first one employs only gyroscopes and is based on the vertical component of the angular rate, as was proposed in  for examThe focus in this work is on the analytical derivation of these computational methods, as well as on qualitative analysis, comparisons and demonstrations using real-world experiments. It is noteworthy that the various approaches for heading estimation, as presented here, should be regarded as fundamental building blocks for PDR algorithms, rather than system-level solutions to the entire indoor navigation problem. In particular, we do not address absolute heading angle but merely relative ones, since the former requires additional information such as pedestrian forward direction in the sensor frame  constituting a framework for gravity direction estimation while emphasizing the central role it has for various tasks in PDR; (2) providing detailed derivation of several fundamental models for heading computation, all of which rely on the estimated direction of gravity; (3) demonstrating and analyzing all of the various methods, based on field experiments with newly established methodology for obtaining the ground truth. In addition, at the end of this paper, we also present an empirical result regarding different forms of implementing sequential processing. This analysis is demonstrated on gyroscope data, but in fact is relevant to any tri-axis sensor, particularly those used in PDR.The rest of this paper is organized as follows: 2. Although some highly accurate gravity models may have different expressions for The gravity vector, denoted by Thus, the problem of gravity direction determination can be formalized as the problem of computing or estimating the unit vector The accelerometer is a three-axis sensor that measures the non-gravitational acceleration vector , denoted by t is the instantaneous time and This can be expressed in the sensor frame, using Label , as(4)fApplying low-pass filtering to each axis of Denoting se Label  to writeOne of the trade-offs to be considered in the selection of the cutoff frequency of the filter is as follows: lower (higher) cutoff frequency will remove more (less) of the motion accelerations that “corrupt” the gravity component but will also result in slower (faster) response to changes in the true gravity direction, such as the periodic variations during the stride cycle or the change of sensor’s orientation when the pedestrian takes the device out of the pocket.T is the sampling interval of the sensor.In this work, for the purpose of low-pass filtering, we decided to use an infinite impulse response (IIR) filter. This is in contrast to using simple moving average, which is basically a rectangular window finite impulse response (FIR) filter. The advantages of an IIR filter are well-known  in signThere exist several possible sources of error in the accelerometer-based method for gravity direction estimation. One obvious source is the inherent measurement errors of the sensor, which include random noise, bias and possibly also scaling and misalignment of the axes. Regarding the random noise, most of its power would be removed during the low-pass filtering since this noise naturally has very wide bandwidth. The other factors—particularly the bias term—might be dominant sources of error when dealing with low-cost sensors (which is the common case for PDR applications). Another source of error is the residual accelerations due to pedestrian motion that were not entirely removed by the low-pass filtering. These residuals might include temporally short components such as the forward accelerations at the beginning of walking or the lateral accelerations during turns. Lastly, the filtering process—aimed at removing the motion accelerations—inherently removes also part of the variations in the true gravity direction as observed in the sensor frame, particularly the periodic changes during the walking cycle, in which the pedestrian-carried device exhibits rotational motion. In other words, these true variations might be “averaged-out” by the filter, depending on how low the cutoff frequency was selected. Dealing with this source of error is the motivation behind the other method for gravity direction estimation, which is presented next.A gyroscope measures the angular rate around its three sensitive axes. The sensor may exhibit angular rate resulting both from the pedestrian’s motion and from the rotational motion of the Earth. The latter can be safely neglected for our purpose, as it is much smaller than the accuracy of typical gyroscopes used in PDR applications.In the context of estimating gravity direction, the gyroscope measurements provide information that is useful in tracking the rotational dynamics of the sensor frame during the pedestrian’s walking cycle. This information may compensate for some of the limitations of the accelerometer-based method as was described in The average angular rate in each of the gyro’s axes can be treated as the residual bias error because one can expect these measurements to be zero when the sensor is at rest. This estimated bias can be removed from any future gyro’s measurements, thus reducing possible drifts when integrating them.Determining the gravity direction based on the accelerometer alone is more reliable when the sensor is stationary, since no motion accelerations are present (though bias and other sensor errors still degrade the accuracy). In this case, we suggest averaging the measurements within the stationary interval instead of applying the low-pass filter, thus obtaining a single gravity direction estimation, First, in order to initialize the filter, the method assumes the existence of a short stationary time period of the sensors prior to the pedestrian’s walking phase. Such an interval can naturally occur, for example when the pedestrian is standing still or when the device is initially placed on a table. The problem of automatic detection of stationary conditions is beyond the scope of this paper, but is relatively simple to solve, for instance based on thresholding the empirical variance of the accelerometer and gyroscope measurements in a sliding window . The int, we can express the previously-computed unit-vector t is computed asStarting from the end of the stationary interval at time q. The estimation error quaternion, denoted The attitude state of the sensor frame, relative to some reference frame, is represented by a quaternion, The process model of the filter is driven by the angular rate measurements of the gyro, as in . The liny, with additive measurement error vector, v, the model isy in the sensor frame and in the reference frame, respectively. The error vector is assumed to have covariance matrix The measurement model of the filter uses direction vectors that are measured continuously in the sensor frame, with prior information about their static representation in the reference frame. For a single direction vector, Since the gyroscope measurements inherently contain errors, such as random noise and bias, the simple integration described above will cause the computed attitude state to drift with time. For this reason, we propose to employ the gravity information obtained from the accelerometer in , by usink to denote any variable at the Initialization: Given an initial stationary time interval, P of n, using  with favPropagation step: The discretization of , which is a direct outcome of the angular motions experienced by the device and measured by these sensors. This heading angle does not take into account the relative orientation of the sensor frame with respect to the pedestrian’s body, while in fact this information is necessary for calculating the trajectory, as part of the dead reckoning process. Thus, the second type of heading angle is the one associated with the trajectory rather than the device. Moreover, even if the sensor frame is fixed on the pedestrian, it is still possible that the trajectory heading would change during walking, as a result of dynamic variations in the direction of motion relative to the forward axis of the pedestrian’s body .Within the scope of this work, we assume for simplicity that the trajectory heading differs from the device heading by a constant offset. Following this assumption, the methods for heading computation presented in this section attempt to achieve estimation of the relative heading of the device.t relative to its initial value .The proposed gyroscope-based methods for heading computation employ the vertical component of the angular rates vector, which can be extracted using the estimated direction of gravity. Denoting this vertical projection of the gyroscope by magnitude of the angular rate vector,The magnitude of vectors, when ignoring measurement errors, is independent of the frame in which they are measured, hence Although the turning rate of interest is disturbed by other angular rates, due to pedestrian’s motion, appropriate pre-filtering of Another approach for this task is to use the ested in , for exax, y or z) of the accelerometer is most dominant, and using the sign of the gyroscope in that axis to determine the direction of turning. Such a method might not be robust to situations in which two different axes are similarly dominant . We suggest a different approach to distinguish left turns from right turns, by directly observing the sign of signed-magnitude rate signalDiscriminating between left and right turns requires the sign of time. In , this waEquation , but witInitialize (a)Update the bias estimation by(b)Update the state indicator by thresholding with At time step The corrected turning rate at time The main problem in gyroscope-based heading is the potential accumulation of errors in the integral calculation of Equation , i.e., aThe relatively long time-constant above, The magnetometer is a three-axis sensor that measures the ambient magnetic field, which is a composition of the Earth’s geomagnetic field and local magnetic disturbances . In this section, we treat the computational problem of combining geomagnetic field vector with gravity direction vector, for the purpose of device heading angle calculation. Though the treatment of local disturbances is important in many real-world scenarios, we do not consider it here.We use By denoting the unit vectors of the geomagnetic field as ive from :(43)[Θ]TIn the last equation, the left-hand side is related to the gravity direction and the magnetic north direction in the sensor frame; while the right-hand side represents the unknown heading angle, el (WMM)  in whichFrom what we have observed in the literature, e.g., , the heaLooking back at Equation , we obtaWriting the right-hand side of  explicitThe tilt rotation matrix  side of  can be wThen, using –(47), itSubstituting  and 50)50) into Therefore, applying the four-quadrant arctangent function on these last two equations, the common term To summarize this subsection, once the gravity direction vector Equation , with μsThe various methods that have been developed in this paper—both for gravity direction estimation and for heading computation—are summarized in Conducting the experiments in natural outdoor environment with open sky-view, which allows GNSS measurements with satisfactory accuracy.Designing the walking path as a sequence of straight segments connected by short curves; this imitates a typical indoor environment made of straight corridors.In addition, based on such walking paths, we can obtain relatively accurate ground truth, by averaging the GNSS-based heading data during each straight segment.Obtaining ground truth for the heading angle in indoor environments is not trivial and usually requires dedicated infrastructure during experimental setup. In our previous work , we assuIt is important to emphasize that these outdoor experiments share the same pedestrian dynamics as typical indoor walking experiments, and that the GNSS data is only used here as a reference for evaluating the different methods.An error analysis of GNSS heading measurements is provided in . It showN measurements , the error STD drops by a factor of Assuming a typical STD of a single GNSS velocity measurement error to be around For gravity direction estimation, low-pass filtering of the accelerometer was done using a 2nd-order IIR filter as described in z-axis at the beginning. Then, the device was placed in the trousers’ pocket, after which the pedestrian started a normal walking. At the end of the experiment, it was taken out of the pocket and held in hand again.y-axis; but compared to the steady direction vector when using only accelerometer, fusion with the gyroscope shows large fluctuations in that signal. These reflect the angular motion of the leg during walking cycle, which causes the gravity unit-vector to change periodically in the sensor frame. The absence of these periodic changes in the accelerometer-based method confirms our notion from During walking, the gravity direction as estimated by both methods roughly aligns with the negative There is no obvious way to extract the ground truth of the gravity direction vector, at least in any real world setting involving unconstrained smartphones, primarily since this unit vector changes dynamically within the walking cycle. Therefore, in order to compare the accuracy of different methods for gravity direction estimation, we shall analyze them in terms of their influence on the relative heading angle, for which an approximate ground truth is more easily obtained.For deeper understanding of the differences among the various gyroscope-based heading methods, in Another analysis of interest, which is presented next, deals with a more general question of how to apply a sequence of processing steps on vector measurements—such as those obtained from tri-axis inertial sensors—when part of the processing is nonlinear. An important example for this is the computation of measurements magnitude when low-pass filtering (LPF) is required: there are two forms in which this can be implemented, as illustrated in the calculation of gravity direction based on the accelerometer, as was described in applying the magnitude calculation and LPF processing for the accelerometer  or for the magnetometer ;projecting the gyroscope measurements onto the vertical direction, as in Equation , which rAdditional examples for which the processing sequence is important, in the context of pedestrian navigation, are:The framework derived in this paper considers the direction of gravity as a central element in PDR algorithms, particularly for pedestrian heading determination. A detailed derivation of two approaches for estimating the gravity direction was presented, along with four methods to compute relative heading angle. While the accelerometer-based gravity direction gives an approximate but robust solution, the fusion-based one is designed to achieve higher accuracy by tracking the rotational motion within the walking cycles, but requires a more complex tuning of its parameters. The experimental results shown here—as well as those from our initial research, —indicateWhen implementing gyroscope-based approaches, one must incorporate additional information or assumptions in order to mitigate its natural drifting. The method we proposed in this context makes use of straight path segments in order to dynamically adapt the estimated bias of the horizontal turning rate. This relatively simple model appears to produce reasonable heading accuracy throughout the entire experiment.Using magnetic sensor for heading computation can be implemented by directly employing the gravity unit vector, as suggested in Considering the vertical projection of the gyroscope versus the magnitude of the angular rate vector, it seems that the former is more accurate as a means for computing the heading angle. Moreover, it naturally allows for distinguishing between left and right turns.Regarding the various methods for heading computation that were derived and compared in this work, we conclude the following:Finally, we have also demonstrated how the order of sequential processing—particularly magnitude calculation and low-pass filtering of gyroscope data—can be crucial for achieving the desired output signal. We conclude that such subtle design choices must be treated carefully when dealing with PDR algorithms in general.


The local size of computational grids used in partial differential equation (PDE)-based probabilistic inverse problems can have a tremendous impact on the numerical results. As a consequence, numerical model identification procedures used in structural or material engineering may yield erroneous, mesh-dependent result. In this work, we attempt to connect the field of adaptive methods for deterministic and forward probabilistic finite-element (FE) simulations and the field of FE-based Bayesian inference. In particular, our target setting is that of exact inference, whereby complex posterior distributions are to be sampled using advanced Markov Chain Monte Carlo (MCMC) algorithms. Our proposal is for the mesh refinement to be performed in a goal-oriented manner. We assume that we are interested in a finite subset of quantities of interest (QoI) such as a combination of latent uncertain parameters and/or quantities to be drawn from the posterior predictive distribution. Next, we evaluate the quality of an approximate inversion with respect to these quantities. This is done by running two chains in parallel: (i) the approximate chain and (ii) an enhanced chain whereby the approximate likelihood function is corrected using an efficient deterministic error estimate of the error introduced by the spatial discretisation of the PDE of interest. One particularly interesting feature of the proposed approach is that no user-defined tolerance is required for the quality of the QoIs, as opposed to the deterministic error estimation setting. This is because our trust in the model, and therefore a good measure for our requirement in terms of accuracy, is fully encoded in the prior. We merely need to ensure that the finite element approximation does not impact the posterior distributions of QoIs by a prohibitively large amount. We will also propose a technique to control the error introduced by the MCMC sampler, and demonstrate the validity of the combined mesh and algorithmic quality control strategy. The Bayesian statistical framework has been used extensively in the problem of system identification  or modelh-refinement) based on a posteriori error estimation techniques [At the root or our methodology is the estimation of errors due to the finite element approximation of the partial differential equation (PDE) of interest. Classically, spatial resolution of finite element models can be adaptively refined . The novel algorithmic approach proposed in this paper consists in running two Markov Chain Monte Carlo algorithms simultaneously to sample the posterior densities of the quantities of interest (component-wise MCMC). The first chain utilises the current finite element model, whilst the second chain runs with a corrected likelihood function that takes into account the discretisation error. The latter quantity may be constructed by making use of a posteriori finite element error estimates available in the literature. At any time during the sampling process, two empirical densities are available and may be compared to evaluate the effect of the discretisation error onto posterior densities. The second building block of our algorithmic methodology allows us to determine whether enough samples have been drawn by the component-wise MCMC. Using multiple parallel chains combined with bootstrap-based estimates of sampling errors, we automatically stop the MCMC algorithms when either (i) the required level of accuracy is achieved or (ii) we have generated enough statistical confidence in the fact that the discretisation error is too large for our purpose, and therefore that mesh refinement is necessary. While any available error estimate of the discretisation error may be used within the general strategy outlined above The paper is organised as follows. Although the methodology proposed in this paper is general, we will apply it to the verification of popular inverse finite element procedures used to monitor the integrity of structures during service. We will assume that the structure can be modelled as an elastic body, and that potential structural damage can be modelled by the evolution of a field of elastic constants characterised by a finite number of uncertain parameters. Assuming that the resonance frequencies of the structure have been measured physically, we attempt to identify this set of mathematical parameters, through the solution of an inverse problem. Significant deviations from the parameter values corresponding to an undamaged structural state may reveal structural failure. This is a typical condition monitoring method used for non-destructive structural testing.In order to clarify the proposed study, we will assume that we have measured the We assume that the quantities Seen from a different point of view, the probabilistic inversion setting amounts to the definition of a joint probability distribution for The formal expression of likelihood function sumption , namelyIt is now possible to formally proceed to the inversion itself by conditioning the joint distribution to actually observed quantities. Applying Bayes’ formula, the posterior probability density of the model parameters isFind The numerical modelIn the previous variational statement, Solving the continuum mechanics model is equivalent to evaluating the mappingThere is, in general, no analytical solution to the continuous vibration problem, and a standard way to obtain approximate solutions is to substitute a finite element space Ω)⊃Uh(Ω) . Too coaWhilst using the continuum mechanics model exactly would deliver the posterior density Equation . SimilarThe finite element error is the mismatch between This section introduces the numerical examples that will be investigated in this paper, and aims to provide a first qualitative understanding of the effect of mesh refinement onto the quality of posterior probability densities.The stochastic field of Young’s modulus that will be used to exemplify the error control approach proposed in this paper is defined via a decomposition of domain Hence, the scalar parameters contained in vector The prior density is a multivariate Gaussian, and is given by equationThe prior mean Finally, we model the error The evaluation of the likelihood function appearing in solution  of the BTest 1(weakly informative data): only the first eigenvalue is measured, i.e., Test 2 (strongly informative data): the first three eigenvalues of the structure are measured. This can be interpreted as an inverse problem, where rich information is used to identify all the unknown of the model, and the probabilistic setting acts as a regulariser.Two tests will now be investigated:The two corresponding marginal posterior density It is interesting to notice that the posterior densities observed in Test 2 are much sharper than that of Test 1, owing to the quality of the data. The symmetry in the results are a consequence of structural and probabilistic symmetries  Markov Chain Monte Carlo (MCMC) algorithm see . The algThis is done by first drawing a random move from This particular transition is designed such that the following ergodic property holds:As a result, under some assumptions, the Markov chain is guaranteed to have ties see . Once foNotice that posed in ,41. Alteposed in ). In thiposed in ,44  by the MH algorithm, we discard the first 25% (burn-in) of these samples, resulting in Once The cumulative empirical predictive posterior density may be expressed asIt is now clear that both the finite element discretisation and the Monte-Carlo approximate sampling affect the quality of the resulting posterior densities. The total error, for the marginal distribution of a single element In the last expression, the first term is the pure finite element error, which would occur if we could run the Markov process for an infinite number of iterations, while the second term is a pure statistical error.We also define the error of an element The finite element method introduces an error in computational mapping For now, we assume that both these quantities can be estimated, at affordable numerical cost and in a reliable manner. Therefore, for any value of parameter It is now posible to sample the corrected posterior distributional model  into theal model . Notice Draw draw Accept Accept We formally define a component-wise MCMC were the uncorrected and corrected computational models are sampled at the same time. This will yield an estimate of the effect of the discretisation error on posteriori densities at any stage of the Markov process, which, in turn, will allow us to develop an early-stopping methodology. The Component-wise MCMC iteration proceeds as follows, given a current sample The MCMC algorithm is initialised by state In the field of a posteriori finite element error estimation, the error estimate is usually a post-processing operation of the coarse finite element solution. This adapts seamlessly to non-Markovian Monte-Carlo samplers, by post-processing each of the independent samples. Here, unfortunately, the finite element model has to be called twice at every iteration of the Markov process. Whether this can be avoided or not, for instance by making use of elements of sequential Monte-Carlo samplers, is unclear to us at this stage.At this stage, any numerical error estimator can be used, provided that it is goal-oriented. A method of choice could be a residual-based ,6 or smoIn this contribution, we develop and use a feature-based method, that finds its roots in data-science methodologies and is much more “black-box” than the previously mentioned strategies. The proposed technique is inspired by the work of ,36,37.slighlty corrected computational models, here generated by refining the mesh by a moderate factor. In our examples, the typical mesh size is divided by 1.5 . However, this very refined numerical model cannot be used at every iteration of MCMC as its evaluation is very costly. We propose to train a model that will map parameter Now, we train Each of the regression We sample artificial “data” by running the overkill computational model An example of fitted regression model is represented in n of the MCMC sampler, a Monte-Carlo estimate of the discretisation error for the posterior density of the ith component of At any iteration Crucially, n of the multiple-chain MCMC (MCi (at ambient temperature) isFollowing standard diagnostic approaches for MCMC samplers , we Equation  by(41)Πk varies between 1 and a large number The independence of the Xth and Xth percentile reads asXth and Yth percentile of the set passed as argument.Finally, the bootstrap confidence intervals are calculated by calculating the n, and not for the asymptotic limit. In fact, estimate n, each individual chain will be strongly affected by the initialisation of the chains. For small asymptotic errors, this can be expected to have a strong upward bias effect providing that the initialisation is disperse, which is often the case in practice . This can be visualised in the existence of the burn-in phase. For small the discrete evaluation of the KS distance itself, which generates an additional (upward) bias.It is important to understand that the derived bootstrap confidence interval stands for a chain of finite length n of the run.The variance of n of the Monte-Carlo solver, do:If Evaluate mesh convergence criterion -If -m should be an exponentially increasing function).otherwise perform If We now make use of the CI derived for Convergence of the finite element-based Bayesian inverse problem is achieved when Criterion n count even when the mesh becomes fine enough for n, the corrected and uncorrected posterior distributions can be statistically distinguished.The second criterion will help us determine whether mesh refinement is actually needed, or whether the statistical error is too large for use to take a robust decision regarding mesh refinement. The ideal criterion is the corrected and uncorrected posterior densities are identical.We postulate the following null hypothesis: Hypothesis 0 (H0): The rejection of the null hypothesis will indicate that the two densities are significantly different. Now, the criterion for the need for mesh refinement becomes the following:n and corresponding to the uncorrected finite element model only . Notice, to clarify the idea, that n tends to infinity: this is a measure of the statistical error only. The desired density can be estimated by resampling p-value The probability density of Z yields a less conservative indicator for the need for mesh refinement, which can be compensated by decreasing p-value Notice that the use of a larger percentile We now come back to the numerical examples introduced in n of the tempered MCThe results reported in In grey colours, we report evolution of The results are as follows. For the coarsest mesh, we detect a separation of the discretisation and statistical errors at iteration 60 . We can see here that the proposed algorithm allows us to stop the MCMC sampler early, moving directly to the level of mesh refinement that will yield the desired quality of the posterior densities.The second set of results correspond to Test 1, still. However, we now monitor the convergence of the posterior predictive density of the fourth eigenvalue. This is reported in the top two graphs of Finally, the last set of results concerns Test 2. Here, remember that the three first eigenvalues are used as measurements. We control the convergence of the fourth eigenvalue, which was controlled in the context of Test 1 in the previous subsection. We see here that for mesh We have presented a methodology to control the various sources of errors arising in finite-element based Bayesian inverse problems. We have focussed on a simple numerical approximation chain consisting of (i) a finite element discretisation of the continuum mechanics problem and (ii) a MCMC solver to draw samples from posterior density distributions. So far we did not consider further error sources such as that engendered by meta-modelling. We have showed that it was possible to drive the mesh refinement process in a goal-oriented manner, by quantifying the impact of the associated error onto posterior density distributions. In order to do so, we run two independent MCMC simultaneously, one of them using a corrected likelihood function that takes the discretisation error. Any a posteriori error estimate available for the PDE under consideration may be used to obtain this correction. Of course, the accuracy (effectivity) of the chosen error estimate will impact the accuracy of the methodology developed in this paper. The study and control of this effect will require further research to be carried out,We have shown that by using multiple replicates of the component-wise MCMC, it was also possible to (i) derive bootstrap-based confidence intervals for the error in posterior densities engendered by the spatial discretisation of the underlying PDE. Importantly, this approach allows us to stop the MCMC iteration as soon as we can be sufficiently confident in the fact that the posterior densities obtained with the current mesh are not accurate enough for our purpose. The approach is goal-oriented: the adaptivity is performed in the sense of the posterior distribution of either some of the latent parameters, or in the sense of the predictive posterior density of engineering QoIs. Finally, we have shown that the approach is uncertainty driven: it will only refine the mesh and/or request additional samples to be drawn by the MCMC algorithm if the effect on the QoIs can be felt when measured using a statistical distance between their posterior densities. In particular, model updating problems with wider priors tend to require less computational effort than parameter estimation problems with very rich observed data and/or narrow prior densities. The proposed methodology establishes a bridge between the field of reliability estimation for the sampling-based algorithms used to solve probabilistic inverse problems, and the field of deterministic, goal-oriented finite element error estimation.Although the results presented in this paper are encouraging, the proposed approach has shortcomings that need to be addressed in future research work. The error estimation procedure is relatively wasteful for two reasons. Firstly, multiple MCMC algorithms need to run independently for bootstrapping to be possible. These chains all exhibit their own burn-in phase, which increases the amount of discarded samples. Secondly, the finite element error estimation procedure is not merely a post-processing operation any longer; one cannot post-process the finite element results corresponding to the uncorrected chain in order to compute the corrected likelihood. This is because both chains are independent and require running the coarse finite element simulations at different points of the parameter domain. Finally, let us acknowledge that Bayesian finite element inverse problems should not be solved by a MCMC solver without constructing a surrogate model first. The number of finite element computations involved, even if the proposal distribution is well designed, is in the tens of thousands. We are currently investigating the use of Polynomial Chaos surrogates, which adds another layer of numerical approximation that needs to be controlled in a robust and efficient manner. In this context, an elegant approach to separate the sources of errors  may be found in , which m


This paper presents a despeckling method for multitemporal images acquired by synthetic aperture radar (SAR) sensors. The proposed method uses a scattering covariance matrix of each image patch as the basic processing unit, which can exploit both the amplitude information of each pixel and the phase difference between any two pixels in a patch. The proposed filtering framework consists of four main steps: (1) a prefiltering result of each image is obtained by a nonlocal weighted average using only the information of the corresponding time phase; (2) an adaptively temporal linear filter is employed to further suppress the speckle; (3) the final output of each patch is obtained by a guided filter using both the original speckled data and the filtering result of step 3; and (4) an aggregation step is used to tackle the multiple estimations problem for each pixel. The despeckling experiments conducted on both simulated and real multitemporal SAR datasets reveal the pleasing performance of the proposed method in both suppressing speckle and retaining details, when compared with both advanced single-temporal and multitemporal SAR despeckling techniques. The synthetic aperture radar (SAR) remote sensing technique has attracted wide attention since the launch of the advanced spaceborne sensors. These SAR satellites or sensors can provide images of the observed areas in different time phases in all day time and all weather conditions, and work in different imaging modes. The above advantages make the SAR a good tool to observe the Earth. However, speckle noise is inherent in SAR systems, as with most of the active imaging systems, which degrades the quality of the SAR data and hinders most applications using SAR data. In the imaging step of certain SAR systems, a multilook procedure is deployed to suppress the amount of speckle noise, which can, however, result in severe spatial resolution loss. Therefore, the application of a speckle reduction technique is often essential before using SAR data, especially for most satellite radar systems which provide single-look data. This paper focuses on the despeckling of single-polarization SAR data; fully polarimetric SAR or interferometric SAR despeckling is outside the scope of this paper.During the past four decades, a number of single-temporal SAR despeckling methods were proposed. In the early years, many SAR filters were developed based on minimum mean-square error theory , such asIn the last ten years, the nonlocal means (NLM) scheme  has attrThanks to the newly launched satellite radar systems, multitemporal SAR images of the same area are now available in many cases. Theoretically, when dealing with the despeckling issue of multitemporal images, exploiting both the temporal and spatial information can provide better results than the single-temporal filters. In recent years, some multitemporal SAR filters have been investigated. In 2014, Su et al.  proposedFor most single-polarization SAR filters, to estimate the true values of the intensity or amplitude data, only the statistical information of the intensity or amplitude is considered. However, for single-look complex (SLC) SAR data, the phase information is available and can also help to characterize the scattering traits of the targets. Consider the phase information when despeckling the amplitude data can provide better processing results. Inspired by this idea, this paper presents a multitemporal SAR filter for SLC data, which is developed under the NLM framework. One of the main innovations of this work is that the filtering framework is based on a scattering covariance matrix (SCM) constructed by the complex data of the pixels in each image patch. By doing so, both the statistical information of the amplitude of each pixel and the phase difference between any two pixels in a patch are utilized to guide the filtering process. In addition, differing from most multitemporal SAR filters that use a change detection approach to exploit the temporal redundancy, the proposed method utilizes an adaptive temporal filter based on the stationarity of the patches, which can avoid the issue of determining the change detection threshold.In the first step of the proposed method, the basic estimations of the patches of each image in the different time phases are independently estimated by a nonlocal weighted average. Then, by searching and grouping the similar patches in time and obtaining the stationarity information of the patch groups, a simple but effective temporal linear filter is used to further suppress the speckle level. In the final step, another weighted average is employed to obtain the final estimation of the patches, and an aggregation process is used to tackle the multiple estimations issue for each pixel.The rest of this paper is organized as follows. The idea of the scattering covariance matrix of patch is introduced in It needs to be noted that, although this paper focuses on the despeckling issue of single-polarization SAR data, it is still necessary to recall the statistical trait of fully polarimetric SAR (PolSAR) data, since the idea of SCM employed in the proposed method stems from it.For single-look PolSAR data, the polarimetric information of a pixel can be characterized by a polarimetric covariance matrix (PCM), which is constructed by the outer product of the scattering vector spose αK :(1)C=ααKTo compare the structure similarity between two image patches, only intensity or amplitude information is considered in most NLM-based SAR filters. A direct problem is that, as mentioned in the introduction, the phase information in SLC SAR data can also help to characterize the scattering traits of the targets and considering such information when comparing patches can provide better results. As revealed in , considex are statistically independent, we introduce a scattering vector for this patch by jointing the back-scattering signals of all the pixels:nth pixel. In this paper, we fix the size of all the patches as Based on the above analyses and inspired by the PCM of PolSAR data, we propose to use the SCM to represent the complex scattering information of all the pixels in an image patch. Assuming that all the pixels in a patch centered at pixel L is the number of looks, and It is well known that the PCM of PolSAR data follows a complex Wishart distribution . As we cx and y have the same true SCM), then the test statistic for the single-look data can be derived as:Under the NLM framework, the proposed method is developed using the SCM of an image patch as the basic processing unit. Hence, the first question is how to measure the similarity between two patches based on their SCMs. To handle this issue, we employ a likelihood-ratio test method designed for complex Wishart-distributed data . If we aTaking the logarithm of Equation (7), we have:t variables can be derived as [The idea of the likelihood-ratio test for two patches can be extended to measure the stationarity of a patch group . With the hypothesis that all the patches of a group are equal, the test statistic for rived as ,26:(9)F filter is shown in x; The SCM-MSAR filter includes two spatial filtering steps . By doing so, the most similar pixel with respect to x in all the time sequences is obtained. However, we found that the patch grouping process is not good enough due to the influence of speckle, and considering the information of the neighbors of x can refine the estimation process , and find the patch in each time which is considered as the similar patch most frequently.For a given patch Taking into account the redundant information in the temporal domain when designing, multitemporal SAR filters can achieve better results than single-temporal filters. To achieve this goal, most of the current multitemporal SAR filters use a change detection step, often followed by a simple temporal averaging using the unchanged pixels, to further suppress the speckle or to compare the similarity between pixels more precisely. As we stated before, the phase information is not considered in these methods. Furthermore, determination of a proper threshold for change detection is difficult in many cases.In this paper, we propose a simple but effective temporal filter based on the stationarity of a similar patch group in time .In this section, to illustrate the effectiveness of the proposed SCM-MSAR filter, we describe the experiments carried out on simulated images and two real multitemporal SAR datasets, and make comparisons between the proposed filter and three state-of-the-art filters: the single-temporal SAR-BM3D filter , the mul(1)speckle is reduced to a large degree in homogeneous areas;(2)image features and radiometric information are well preserved; and(3)the result should be without artifacts.Generally speaking, an outstanding SAR filter should have the following characteristics ,33:(1)sp(1)Equivalent number of looks (ENL): the ENL indicator is widely used in the literature to evaluate the level of speckle, and the higher the ENL value, the better the filter reduces the speckle.(2)Figure of merit (FOM): the FOM index  is used (3)Edge-preservation degree based on the ratio of average (EPD-ROA): for the real SAR datasets, EPD-ROA  is one o(4)Mean of ratio (MOR): due to the statistical trait of speckle, a MOR value between the speckled and the despeckled data should be very close to one if the filter can effectively preserve the radiometric information.To quantify the performances of the filters in the above aspects, we use the following four indicators:In this part, we use a framework proposed by Di Martino et al.  to simulFirst, we simulated three datasets which all have four time sequences. To inspect the influence of the changes between the images in the target time and the other times  on the despeckling results, we randomly chose some pixels with a certain ratio and changed their values by a mean filter before exerting speckle for each complementary time. The change ratios were set as 0.4, 0.2, and 0.1 for the three simulated datasets, respectively. The simulated images and the filtered results of the dataset with The above analyses are backed up by the quantitative assessment results listed in We also simulated a dataset having eight time sequences with Experiments were also carried out on two real multitemporal datasets, to verify the effectiveness of the proposed method. The first dataset has two time sequences which were acquired by the TerraSAR-X system in Ruhr, Germany, on 20 February 2008 and 4 March 2008, respectively. The imaging mode is strip mode with a spatial resolution of 3 m × 3 m working on X band, and the polarization type is HH. From The second real SAR dataset shown in Preservation of signals from strong scatterers is essential for target and man-made structure detection. To show the capability of the filters in retaining the information of strong scatterers, two lines were taken across two high-return targets, respectively (marked by the red rectangle in The proposed SCM-MSAR filter consists of three filtering steps and a multiple estimations aggregation step. To reveal the influence of the different steps on the filtering result, we display the filtered image of each step in In this paper, a multitemporal SAR filter was proposed for the despeckling of SLC SAR data, using a SCM of each patch as the basic processing unit, which can better take advantage of the phase difference between pixels. The proposed method includes two spatial filtering approaches. One is a nonlocal filtering scheme, and the other is a guided filtering scheme. In addition, an adaptive temporal filter is employed in the proposed method based on the temporal stationarity of patches, to better exploit the redundant temporal information. We compared the proposed method with some state-of-the-art single-temporal and multitemporal SAR filters, in experiments conducted on both simulated and real datasets. The experimental results revealed that the proposed method can not only obtain a good balance between reducing speckle and retaining image structure, but also show an outstanding performance in preserving radiometric information.It must be pointed out that, since the output of each spatial and temporal filtering step of the proposed method is the matrix image, the phase of each pixel in a patch cannot be directly retrieved using the filtered SCM; that is to say, the proposed filter only tackles with the despeckling issue of amplitude or intensity. Our future work will focus on the despeckling problem of both amplitude and phase for complex SAR data.


Albulina metallica butterfly is remarkable as both the dorsal (blue) and ventral (gold-green) cover scales are colored by the same type (pepper-pot) of photonic nanoarchitecture, exhibiting only a short-range order. The vapors of ten different volatiles were tested for sensing on whole wing pieces and some of the volatiles were tested on single scales as well, both in reflected and transmitted light. Chemically-selective responses were obtained showing that selectivity can be increased by using arrays of sensors. The sensing behavior is similar in single scales and on whole wing pieces, and is similar in reflected and transmitted light. By immersing single scales in an index-matching fluid for chitin, both the light scattering and the photonic nanoarchitecture were switched off, and the differences in pigment content were revealed. By artificially stacking several layers of blue scales on top of each other, both the intensity of the characteristic photonic signal in air and the magnitude of the vapor sensing response for 50% ethanol vapor in artificial air were increased.Fast, chemically-selective sensing of vapors using an optical readout can be achieved with the photonic nanoarchitectures occurring in the wing scales of butterflies possessing structural color. These nanoarchitectures are built of chitin and air. The  Natural photonic nanoarchitectures are valuable sources of information since they were optimized during many millennia of biological evolution ,2. TheseThe photonic nanoarchitectures occurring in insect cuticles or bird feathers show an optical response when the vapor composition of the surrounding atmosphere changes ,19,20. TRecently, we investigated the vapor sensing properties of three-dimensional nanoporous “pepper-pot”-type nanoarchitectures occurring in the wing scales of polyommatinae blue butterflies . It was Albulina metallica butterfly species is special, as the males have similarly colored dorsal wing surfaces as compared to the polyommatinae blues , while the ventral side has green coloration, also of structural origin [The l origin . In our l origin . We showl origin . Direct l origin .Most often the wings of butterflies are covered by several layers of scales, so that they are built up as follows, dorsal cover scales/dorsal ground scales/wing membrane/ventral ground scales/ventral cover scales ,32. LighAlbulina metallica are reported. It was found that the dorsal (blue) and ventral (green) sides show characteristic optical responses to the test volatiles. Using principal component analysis (PCA), the vapor sensing spectra of the whole wing measurements were evaluated. We found that the structural colors of the two sides, and therefore the photonic nanoarchitectures generating them, have different vapor sensing properties. We show for the first time that separated single wing scales containing photonic nanoarchitectures can be used for optical vapor sensing. Using an optical microscope setup, the vapor sensing properties of the single blue and green scales were investigated. It was found that the optical responses are also vapor specific, similar to the measurements on the whole wings, and sensitivity to the ethanol test vapor of the single scales was enhanced by stacking several scales onto each other. These findings show that naturally tuned photonic nanoarchitectures can be used to construct sensor arrays consisting of whole wings, or consisting of one or a few scale stacks with different chemical selectivity, enabling the production of small (approximately 100 µm size) and material specific vapor sensors with optical readout.In this work, the vapor sensing properties of single scales and of whole wings for the two differently colored wing sides of the male Albulina metallica is not subjected to any restriction. The samples used in this work belong to the curated collection of the Hungarian Natural History Museum. For the whole wing optical spectroscopy measurements, we used the Avantes  modular fiberoptic system . All chReflected and transmitted light optical microscopy were carried out using a Zeiss AxioImage A1 . The index-matching oil was Zeiss Immersol 518 F. For microspectrometry, the light was collected through an adapter tube attached to the upper port of the microscope and conducted with an optical fiber to the Avantes (HS1024 × 122TEC) spectrometer.Scanning electron microscopy images were taken using an LEO 1540 XB  microscope on wing pieces attached with conductive tape without any preparation.We carried out all data processing (including the PCA) and representation using Origin 2018 software .Albulina metallica males can be revealed by DSA image processing of the electron micrographs [To compare the two sides of the butterfly wings, we investigated wing pieces under a scanning electron microscope. In the wing scales, similar pepper-pot-type photonic nanoarchitectures were observed . The colrographs  based onThe reflectance spectrum is determined by the periodicity and the refractive index contrast of the materials building up the nanoarchitecture. A relatively high spectral shift occurs when the air voids in the chitinous matrix are filled with a liquid with different refractive index than the initial (air). To present this effect, single separated wing scales from the dorsal and ventral sides of the male specimen were placed on a microscope slide. During whole wing vapor exposure measurements, the optical response for ten vapors were measured at increasing concentrations from 0% to 50%, in 5% steps. A total of eight measurements were conducted: the two operators measured the vapor sensing properties of the two sides of the butterfly wing independently, each measurement was repeated twice. To facilitate the presentation and the interpretation of the measured data, the relative reflectance spectra were introduced  using thOn both wing surfaces, we carried out a total number of four measurements that were averaged to follow the characteristic evolution of the response signal. As an example, we show the averaged response signal from 0% to 50% in the case of the vapors of chemically different compounds: water, chloroform, isopropyl alcohol, and ethanol . To compTo obtain insight in the processes taking place on single scales, we separated and placed them on glass microscope slides to carry out reflectance and transmittance measurements. For reference, we used the light reflected from and transmitted through the glass slide. In After the whole wing vapor sensing measurements, single scale measurements were carried out. The butterfly wing in the gas cell was investigated using an optical microscope. The 20× objective had enough working distance to be able to focus on single scales inside the cell. We performed vapor sensing measurements on ten blue and ten gold-green wing scales while applying a 50% concentration of chloroform and water vapors. The average of the ten measurements on each side and for each vapor is plotted in The effect of stacked wing scales on the vapor-sensing properties was also investigated. Individual blue scales were placed on top of each other on a microscope cover slide  using a sharp needle. In this setup, we measured transmittance of light across one, two, and three stacked scales. A. metallica, the morphology of the wing scales on the dorsal and ventral sides is very similar; the internal nanoarchitectures are hard to distinguish by only qualitative analysis of the scanning electron microscope (SEM)   and tranPhotonic crystals, understood in a strict sense, suppose a well-ordered structure; however, as was demonstrated by theoretical modeling, disordering the structure while preserving its periodicity does not destroy the photonic band gap . In our The color of the scales is given by the particular nanostructure and refractive index contrast of the building materials. We obtained a considerable color change on single scales from the dorsal and ventral sides of the wings by immersing them in liquid ethanol under an optical microscope . The colIn Based on the first report on chemical sensing of photonic nanoarchitectures occurring in butterfly scales , chemican = 1.518, close to the refractive index of the chitin, one can observe in transmitted light the contribution of the absorption  enable transmittance measurement. Such an arrangement may be more advantageous for the construction of miniaturized sensors. As we presented before, the transmitted spectra are nearly complementary to the reflected ones, therefore one expects a similar sensing behavior in transmittance, too. Furthermore, such an arrangement allows for the increase of the signal by stacking several scales on top of each other. Here, we placed single dorsal blue scales on top of each other  and ventral (gold-green) cover scales in the wings of the male As the complete wings are optically complex objects, we also investigated the behavior of single scales, both by using a microscope and collecting the signals not from the entire wing, but from one single scale only, by removing single scales from the wing and placing them on microscope slides for individual characterization. We found that the reflected and transmitted light were complementary to a good extent, but towards the short wavelength side an increasing amount of light was missing. It was found that, in the case of the gold-green scales, more light was missing as compared to the blue scales. This is attributed to the higher concentration of pigment present in the gold-green scales. In both types of scales, light scattering also contributes to the loss in intensity of the reflected and transmitted beams. The single scales exhibit similar chemically-selective sensing both in reflected light and transmitted light as was found on the whole wings. By artificially stacking several layers of scales possessing photonic nanoarchitectures, we demonstrated that the magnitude of the sensing signal can be increased.


The aim of structural identification is to provide accurate knowledge of the behaviour of existing structures. In most situations, finite-element models are updated using behaviour measurements and field observations. Error-domain model falsification (EDMF) is a multi-model approach that compares finite-element model predictions with sensor measurements while taking into account epistemic and stochastic uncertainties—including the systematic bias that is inherent in the assumptions behind structural models. Compared with alternative model-updating strategies such as residual minimization and traditional Bayesian methodologies, EDMF is easy-to-use for practising engineers and does not require precise knowledge of values for uncertainty correlations. However, wrong parameter identification and flawed extrapolation may result when undetected outliers occur in the dataset. Moreover, when datasets consist of a limited number of static measurements rather than continuous monitoring data, the existing signal-processing and statistics-based algorithms provide little support for outlier detection. This paper introduces a new model-population methodology for outlier detection that is based on the expected performance of the as-designed sensor network. Thus, suspicious measurements are identified even when few measurements, collected with a range of sensors, are available. The structural identification of a full-scale bridge in Exeter (UK) is used to demonstrate the applicability of the proposed methodology and to compare its performance with existing algorithms. The results show that outliers, capable of compromising EDMF accuracy, are detected. Moreover, a metric that separates the impact of powerful sensors from the effects of measurement outliers have been included in the framework. Finally, the impact of outlier occurrence on parameter identification and model extrapolation  is evaluated. Sensing in the built environment has shown the potential to improve asset management by revealing intrinsic resources that can be exploited to extend the service life of infrastructure . HoweverIn the field of structural identification, measurements are employed to update parameter values affecting structure behaviour. Residual minimization, also known as model-calibration, consists of adjusting model parameters to minimize the difference between predicted and measured values. This approach is the most common data-interpretation technique and has been a research topic for several decades ,3,4. HowBayesian model updating is the most common population-based structural-identification approach. This approach updates the initial knowledge in terms of probabilistic distributions of parameters by including the probabilistic distribution of measurement observations. The accurate identification of parameter values can be obtained using non-traditional implementations of Bayesian model updating . HoweverError-domain model falsification (EDMF)  is an enThe EDMF performance in identifying parameter values depends on factors such as the adopted sampling approach to generate the initial model population, the selection of relevant parameters, and the sensor configuration. An adaptive EDMF-compatible sampling approach that is able to outperform traditional sampling techniques has been recently proposed in Reference . Some alA basic hypothesis of all structural-identification methodologies is that measurement datasets do not include wrong data. Anomalous values in measurements, which are often called outliers, can occur due to faulty sensors or unexpected events during the monitoring process . From a In the field of damage detection, the presence of even a small number of wrong measurements or missing data reduced the performance of most algorithms ,20. MoreIn structural identification, the presence of outliers reduced the performance of current methods in terms of identification accuracy and prediction reliability. In Bayesian model updating, the classes of methods for outlier detection were proposed. The main two classes were based respectively on probabilistic measures such as posterior probability density function of errors  and L1 oPasquier and Smith in  proposedThis paper presents a new outlier-detection framework that is compatible with population approaches such as EDMF. The proposed strategy is based on a metric used to evaluate the expected performance of sensor configurations that are often employed to optimize sensor placement. Additionally, a context metric that separates the impact of powerful sensors from the effects of measurement outliers has been included in the framework. The new approach is, therefore, suitable to analyse data sets that consist of sparse non-time-dependent measurements and overcomes limitations that characterise existing outlier-detection methodologies.The remainder of the paper is organised as follows. Error-domain model falsification (EDMF)  is a recLet In Equation (2), the difference between a model prediction and a measured value at location Measurements errors Plausible behaviour models are selected by falsifying those for which residuals exceeds the thresholds boundaries that are defined in the uncertainty domain . Being a falsification approach, EDMF initially requires that a set of model instances is generated by assigning parameter values to the model class. Then, the threshold bounds are defined at each sensor location as the shortest interval Models for which residuals are within the threshold bounds  each sensor individually, and (ii) the entire sensor configuration. Sensors that are deemed to be suspicious are removed. Finally, the CMS is computed using only reliable measurements. The proposed model–class validation and outlier-detection methodology are described in detail in the next sections.Interpolation tasks  and, mostly, extrapolation tasks  represent the ultimate aims of structural identification. The extrapolation tasks are intrinsically more demanding since the fictitious parameter values do not compensate for model–class errors . The outModel–class accuracy is checked by comparing prediction ranges that are computed using the initial model population with measured values at each sensor location. A qualitative comparison between the two model classes, namely Each vertical axis represents a sensor and the prediction ranges are depicted using interval bounds. Measured values eference . However, the situation depicted in  dataset .Unlike continuous monitoring in which a large amount of data is collected over time, datasets obtained during static tests often consist of a few measurements that are related to specific static configurations. Even when the same test is performed multiple times—usually to assess the measurement repeatability under site conditions—the amount of values collected from each measurement is insufficient to carry out statistical analyses. Therefore, anomaly detection cannot be performed by uniquely analysing the dataset.Then, using EDMF for each set of simulated measurements and the corresponding number of candidate models in the CMS  is recorded. This number represents the expected dimension of the CMS population if a specific set of Given a sensor In step 2, two CDFs are computed: (i) one using the entire sensor network, and (ii) the second one using the network without sensor When sensor Finally, in step 3, the sensor that is deemed to be an outlier is removed from the sensor configuration and step 2 is performed iteratively until no suspicious data are found. Removing the suspicious sensors is an effective solution to avoid false-negative identification since the CMS obtained after excluding a sensor always includes the original CMS. Therefore, traditional outlier-correction strategies are not needed.The Exeter Bascule Bridge (UK) has a single span of 17.3 m and was designed in 1972 to be lifted in order to allow the transit of boats along the canal. The light-weight deck, which consists of a series of flanked aluminium omega-shaped profiles, is connected to 18 secondary beams (type UB 533.210.82) that are bolted to two longitudinal girders (type UB 914.305.289). The bridge has a total width of about 8.2 m and carries the carriageway and a footway. The North-bank supports are hinges, while, on the South bank, the structure is simply supported. Two hydraulic jacks, which are activated during lifting manoeuvres, are connected to the two longitudinal girders on the North-bank side.A static load test was performed to collect the mid-span vertical displacements and strain measurements at several locations. Three parameters that influence the structural behaviour are selected for model updating, namely: the equivalent Young’s modulus of the aluminium deck  could have the opposite effect, leading to a more flexible behaviour than the real one. The latter omission has a smaller influence on the bending behaviour, thus, the model uncertainty range is asymmetric. The bounds for this source of uncertainty have been defined using conservative engineering judgments, as recommended in Reference .Typical uncertainties that relate to the FE method such as the mesh refinement and additional uncertainties are estimated according to the technical literature. The mesh-refinement uncertainty has been quantified through a convergence analysis, by increasing the mesh density until the model response converged asymptotically and the prediction variations were lower than 1%. An analogous practice is described in Reference . AdditioThe adopted sensor configuration consists of six strain gauges that are glued to the main girders and a selected secondary beam. Additionally, a deflection target was installed on the East girder at mid-span and a precision camera was used to record the vertical displacements. The sensor configuration and the truck position are depicted in Uncertainties associated with the sensor configuration are reported in In order to perform the model–class validation, the two model classes depicted in The updated model class includes friction connections on the North-bank side of the bridge, the two hydraulic jacks that are used for lifting, and the presence of a 10-mm gap between the base plates of the main girders and the abutment at the South-West support. The presence of the gap was observed during the visual inspection of the structure, confirming the iterative nature of structural identification.The model–class validation described in When the initial model class is employed , the preThe detection of suspicious values in the measurement datasets is carried out according to the two-step methodology presented in The analysis of each sensor is performed individually in step 1. In Although removing In step 2, the expected performance of each sensor is assessed and compared with the actual values of falsification performance. In In Similarly, in The falsification performance of It is worth noting that a large variation of #CMs is not always connected to anomalous measurements. For example, #CMs variations for sensor Finally, in step 3, the sensor network is updated by removing sensor For comparison, Since sensor Simulated outliers are used in this section to test the proposed methodology. In all the scenarios, the proposed methodology is able to detect the simulated outliers. In scenario 2, a reduction of 20% of the true measurement leads to the complete falsification of the model class, while in scenario 5, 8 candidate models are found despite the fact that the When the variations of eference , severalIn scenario 1, sensor The presence of outliers in the measurement datasets can reduce the accuracy of the structural-identification methodologies such as EDMF. The reserve capacity of an existing structure can be defined, for a defined limit state, as the ratio between the design load—given by codes—and the as-built maximum loads—computed using models. In the model class, the test loads are replaced by design load configurations, in which all the relevant safety factors are applied. The serviceability limit state (SLS) of stress control is investigated for the Exeter Bascule bridge by checking that under characteristic design loads, the maximum Von Mises stress in each element is lower than the yield strength . However, if a false positive occurs and the sensor is removed, the resulting CMS becomes larger, thus, including the true model that would have been identified using sensors that were incorrectly removed.The framework presented in this paper compares the expected and current performance of the sensor configuration by mapping the effects that outliers have on the CMS. Therefore, data sets that consist of sparse static measurements can be validated. A context metric (The results presented in nd Smith  leads toThe following limitations of the proposed framework are recognised. The sampling technique adopted to generate the model population and the assessment of uncertainties influence identification results and the generation of simulated measurements. Additionally, alternative approaches may be employed to generate simulated measurements. Moreover, accurate parameter identification and successful outlier detection are possible only when the reasonable model classes are adopted. Model–class features and model uncertainties should always be verified through visual inspection and iterative model–class updating when new information becomes available.The environmental conditions under which the test is performed may affect the values of the identified parameters. In the EDMF methodology, the environmental variability can be accounted for by explicit modelling and the measurement of environmental effects, including additional sources of uncertainties and by repeating the test multiple times in various conditions. The environmental variability should not affect the outlier detection methodology since the analysis is based on the variation of the CMS population when a sensor is removed from the network. While different test conditions may provide varying CMS populations for points A and B, the relative variation of the CMS population between the two points when a sensor is removed should not be affected.Partial sensor redundancy is crucial to ensuring the robustness of the sensor configurations ,43. WhenPopulation methods for structural identification are not robust when there are outliers in the measurements. The proposed methodology, based on the expected performance of sensor identification, helps reveal the outliers that compromise the accuracy of data interpretation. Compared with previous algorithms, suspicious measurements are more efficiently checked using the information provided by the entire sensor configuration. A metric that separates the impact of powerful sensors from the effects of measurement outliers provides a useful tool for asset managers.


A new scheme based on Kalman filtering to optimize the waveforms of an adaptive multi-antenna radar system for target impulse response (TIR) estimation is presented. This work aims to improve the performance of TIR estimation by making use of the temporal correlation between successive received signals, and minimize the mean square error (MSE) of TIR estimation. The waveform design approach is based upon constant learning from the target feature at the receiver. Under the multiple antennas scenario, a dynamic feedback loop control system is established to real-time monitor the change in the target features extracted form received signals. The transmitter adapts its transmitted waveform to suit the time-invariant environment. Finally, the simulation results show that, as compared with the waveform design method based on the MAP criterion, the proposed waveform design algorithm is able to improve the performance of TIR estimation for extended targets with multiple iterations, and has a relatively lower level of complexity. Cognitive radars have received a lot of attention in recent years. Similar to brain-empowered system architectures, cognitive radar employs adaptive feedback principle to facilitate adaptive detection of the time-invariant target scene. Subsequently, the target feature information in the backscatter signal is exploited to allocate the power or spectrum of the probing signal at the transmitter ,2. CogniThe transmitted waveforms of the cognitive radar are constantly adjusted in order to extract target feature information in a time variant environment. Many methods to design the cognitive waveform have been proposed, in several recent works. The literature  developeRecent researches have also indicated that multiple antennas radar can take full advantage of spatial diversity gains by detecting the extended target in different directions ,33,34,35(1)We extract the target feature information derived from successive received signals at the receiver. The TIR can be considered as a temporally-correlated function during pulse repetition interval (PRI).(2)By utilizing this temporal correlation of the TIR changes, the waveform design problem is modeled by minimizing the MSE of TIR estimation. The MSE of TIR estimation can be obtained during the Kalman Filtering-based iteration process.In this paper, we analyze the performance of an adaptive multi-antenna radar system that contains the idea of “Kalman Filtering” and “TIR estimation”. A novel method based on Kalman filtering optimizing the waveforms of an adaptive multi-antenna radar system for TIR estimation is presented. The receiver adopts the Kalman Filtering approach by updating the target parameters. The radar system updates the TIR estimation and utilizes this information to choose the optimal waveform for transmission. An adaptive feedback loop enables the delivery of the estimated value of TIR to the transmitter. The transmitter adapts its transmitted waveform to suit the time-invariant environment. The proposed waveform design algorithm is studied in order to improve the performance of TIR estimation, which can be summarized as follows:(1)We present an adaptive multi-antenna radar system model, based on the idea of the MSE of TIR minimization.(2)We present a Kalman filtering-based waveform design approach by making use of temporal correlation of the TIR derived from successive received signals.(3)We provide performance analysis of the adaptive multi-antenna system in terms of the TIR estimation by the proposed iteration steps. The proposed algorithm has relatively lower complexity, due to the product operation in the time domain.The main contributions of this paper are summarized as follows:The structure of this paper is as follows. In Throughout this paper, the following notations will be used. Vectors are denoted by boldface lowercase letters and matrices by boldface uppercase letters. We consider that an adaptive multi-antenna radar system is equipped with a transmit array and a receive array comprising In the intelligent transportation scenario, we analyze the performance of an adaptive multi-antenna radar system that contains the idea of “Kalman filtering” and “TIR estimation”. The system architecture of the adaptive multi-antenna radar is presented in q-th receiver, due to the probing signal sent from the n-th transmitter and reflected from the m-th scatterer (excluding noise), is denoted as:n-th transmitter to the q-th receiver via the m-th scatterer. n-th transmitter to the m-th scatterer, and from the m-th scatterer to the q-th receiver, respectively. It is worth noting that the target moves with a constant velocity. As the radar carrier, Doppler is not useful for waveform optimization design; this is ignored in this paper.In this work, we assume that the target was contained in a single range cell, which consists of n-th transmitter to the q-th receiver via the target. The vector version of the backscattering signal at q-th receiver can be expressed as:In order to simplify the discussion, we assume that the target has a center of gravity at n-th transmitter to the q-th receiver via the target. q-th receiver. We further assume the scenario that The term At time k-th pulse, the time dynamic characteristic of the TIR can be described as:The relative angle between the multi-antenna radar and the target is time-invariant during the PRI. Since we will consider the waveform design problem for target parameter estimation in the time variant radar scene, the dynamic model of the TIR should be established. From the literature , the TIRWe aim to explore the waveform optimization for an adaptive multi-antenna radar system. By employing the spatial diversity technology, the multi-antenna systems can improve the parameter estimation capability and target detection capability. From the literature , the TIRIn the first step, the received signals are used to extract the TIR via the Kalman filtering-based approach. The MSE matrix of TIR estimation can be obtained during the Kalman filtering-based iteration process.In the second step, TIR is considered as a temporally-correlated function during pulse repetition interval (PRI). By utilizing this temporal correlation of the TIR changes, the waveform design problem is modeled by minimizing the MSE of TIR estimation.The cognitive processing is summarized as follows: The receiver employs the Kalman filtering approach update the target parameters. The radar system updates TIR estimation and utilizes this information to choose the optimal waveform for transmission. An adaptive feedback loop enables the delivery of estimated value of TIR to the transmitter. The transmitter adapts its probing signals to suit the time-invariant environment. The process of waveform optimization for the target parameter estimation is depicted in k-th pulse sample, TIR estimation based on maximum a posteriori (MAP) criterion can be written as:In this section, we intend to estimate TIR in order to improve the performance of target detection in the adaptive multi-antenna radar system. During the The received waveform We haveAfter the simplification, TIR estimation based on MAP criterion in AWGN channel can be written as:We haveThe MSE matrix of TIR estimation based on MAP criterion can be expressed as:Algorithm 1. Kalman filtering for TIR estimation in cognitive radar systemStep 1: Initializing iteration index Step 2: The predicted matrix of TIR can be expressed as Step 3: The predicted MSE matrix of TIR can be obtained as Step 4: The Kalman gain matrix can be expressed as Step 5: The estimated matrix of TIR can be expressed as Step 6: The MSE matrix of TIR is updated as The temporal correlation characteristic of TIR cannot be utilized by using MAP criterion in the process of calculating estimation of parameters. In this work, we present a Kalman filtering-based method for TIR estimation under multiple antennas scenario. This method has two steps: firstly, we utilize the MAP criterion to initialize the estimated parameters, i.e., the observed state of TIR k-th pulse, the MSE matrix of TIR estimation via Kalman filtering-based approach is preliminary described as follows:As extensively discussed in the existing literature ,43, the Under transmitted power and detection probability constraints, the above waveform design problem (13) can be described as:terature , we can As we can see from the above waveform design problem (15), the above waveform optimization problem is a convex one under AWGN channel. However, the proposed scheme is a non-convex and non-linear optimization problem, considering the clutter and jamming. It is impossible to find the theoretical solution by using an algorithm with polynomial complexity. The goal of this paper is to obtain the optimal  solutions of the above waveform design problem (15).We propose a two-step approach to solve the above problem (15), which is based on the methods of semi-definite programming. Firstly, according to the literature , the aboterature , the radThe symbols “Q-function and Q-function describes a monotonic decreasing and continuous function, the expression As the number of Kalman filtering iterations increases, the estimated value of TIR is close to the real measured value. Since From the literature , an algoWe assume that the transmitted powers of all of k-th pulse sample, we assume that n-th transmit antenna and the q-th receive antenna. We use the normalized MSE to define the estimation performance:We compare the performance gain of TIR estimation for optimization waveforms designed by the proposed scheme to the gain for random waveform, and compare this result with the optimized waveform based on MAP criterion. Firstly, we consider that the backscatter signals are subject to interference from white Gaussian noise. As the Kalman filtering approach utilizes the temporal correlation of the TIR during the pulse interval, the proposed radar system adapts its probing signal better than waveforms based on MAP criterion to the fluctuating target radar cross section (RCS). On the other hand, optimized waveforms based on MAP criterion are unable to match the time-varying TIR after multiple iterations. Hence, the TIR estimation performance is suboptimal in this case.Secondly, we assume that the backscattering signals are disturbed by the clutter. We compare the performance gain of TIR estimation for optimization waveforms designed by the proposed scheme to the gain for random waveform, and compare this result with the optimized waveform designed by the proposed scheme, without considering the clutter. Similarly, as shown in In this paper, we have developed a novel waveform design algorithm for adaptive multiple antennas radar system, which can improve the performance of TIR estimation. The receiver adopts the Kalman filtering approach by updating the target parameters. An adaptive feedback loop enables the delivery of the estimated value of TIR to the transmitter. The radar system updates the TIR estimation and utilizes this information to choose the optimal waveform for transmission. Finally, the simulation results show that the proposed scheme provides higher performance gains in terms of TIR estimation for multiple antenna scenarios under the transmitted power and detection probability constraints. As compared with the waveform design algorithm for TIR estimation based on MAP criterion, the proposed approach has relatively lower computational complexity. The next step will focus on the tradeoff between the complexity of computing and the performance improvement offered by the scheme.


Adverse Local Tissue Reactions (ALTRs) are one of the main causes of hip implant failures. Although the metal release from the implants is considered as a main etiology, the mechanisms, and the roles of the released products are topics of ongoing research. The alloys used in the hip implants are considered biocompatible and show negligible corrosion in the body environment under static conditions. However, modularity and its associated mechanically assisted corrosion have been shown to release metal species into the body fluids. ALTRs associated with metal release have been observed in hip implants with metal-on-metal articulation initially, and later with metal-on-polyethylene articulation, the most commonly used design in current hip replacement. The etiological factors in ALTRs have been the topics of many studies. One commonly accepted theory is that the interactions between the metal species and body proteins and cells generate a delayed type IV hypersensitivity reaction leading to ALTRs. However, lymphocyte reactions are not always observed in ALTRS, and the molecular mechanisms have not been clearly demonstrated. A more accepted mechanism is that cell damage generated by metal ions may trigger the secretion of cytokines leading to the inflammatory reactions observed in ALTRs. In this inflammatory environment, some patients would develop hypersensitivity that is associated with poor outcomes. Concerns over ALTRS have brought significant impact to both the clinical selection and development of hip implants. This review is focused on the mechanisms of ALTRs, specifically, the metal release process and the roles of the metal species released in the etiology and pathogenesis of the disease. Hopefully, our presentation and discussion of this biological process from a material perspective could improve our current understanding on the ALTRs and provide useful guidance in developing preventive solutions. Over one million people receive hip replacements every year to relieve pain and restore hip functions from osteoarthritis, a degenerative disease that affects most people in their seventh decade of life RIPO, . AlthougThe main symptoms of ALTRs are pain and swelling. They can generate extensive destruction to the soft tissues of the hip, challenging the prognosis of further clinical solutions  on the implant surfaces  is preferentially released as Co3+ at extremely low rate in static immersion tests compared with Co2+ and Mo  is the major elemental component in the passive layer formed on the CoCrMo surfaces. It is released from the implants as CrMolybdenum (Mo) is rarely studied in the research associated to ALTRs. However, the lack of studies in synovial fluid does not exclude the possible adverse effects of Mo on the periprosthetic tissues. Molybdenum species are stable in passivity at low pH, but a pH higher than 7.0 is thermodynamically favorable for Mo dissolution  than patients with MoM total hip replacement. One study revealed 1.73 and 2.15 of mean Cr/Co in serum from total 160 cases of two types of hip resurfacings  have been histologically described as the growth of cystic or solid fibrotic masses originating in the synovial membrane of patients with hip implants  released from the implants as a consequence of corrosion or wear, have been accepted as the main etiological factor of ALTRs  and generates structural changes in the molecule. This modified-HLA molecule is recognized by the T-cell receptor, triggering the immune reaction responsible for severe lung destruction due to chronic inflammation in 1–15% of people exposed to beryllium  has been recently described for beryllium (Be). The Bein vitro.” Studies carried out at the Rizzolli Orthopedic Institute had shown that high concentrations of Co2+ ions generated necrosis in human peripheral mononuclear cells, while lower concentrations induced apoptosis in the same cell population. On the other hand, Cr3+ ions at high concentrations induced apoptosis in the cells and no effects were observed in lower concentrations , cultured with different concentration of nanoparticles of Cr3+ in synovial tissues and fluids, which differs from the carcinogenic Cr6+ that is soluble and uses phosphate channels to penetrate the cell membranes ; it can penetrate cells, and interact with organelles and proteins , ATP synthesis has shown to be inhibited by cobalt in a dose-dependent manner. This happens not as the result of the inhibition of ATP synthase or the respiratory chain, but as consequence of the opening of the mitochondrial permeability transition pore (MPT), which allows the H+ to migrate from the intermembrane space to the mitochondria matrix, thus decreasing the electromotive force that activates the ATP synthase , and an increased expression of the HIF-1α regulated genes (Karovic et al., In one of the early studies in hip pathology associated with MoM hip implants, Campbell et al.  observedThe observations explained above suggest that there are two main events that explain two types of possibly interrelated reactions. The first is the a physiopathological event of metal-induced inflammation of the hip joints, which affects a larger number of patients, mostly with MoM implants. The second event is associated with hypersensitivity reaction, which is less prevalent in MoM ALTRs, but correspond to most of ALTRs in MoP ALTRs and low wear MoM.As described previously, cobalt ions can induce MPT, which is not a constitutional structure of the mitochondria, but is stabilized under pathologic conditions. The increased permeability of the mitochondrial internal membrane decreases the electrochemical gradient between the intermembrane space and the mitochondrial matrix. Subsequently the proton-motive force is reduced, and the generation of ATP is impaired . The redSome patients develop a hypersensitivity reaction, initiated by the activation of T-cells, driving to a predominant Th1 reaction (Catelas et al., It is accepted that CoCrMo over release from either tribocorrosion at the articular surfaces or fretting corrosion at the modular junctions of hip implants trigger adverse local tissue reactions. These mechanisms of degradation results in elevated metals ions in the periprosthetic tissues and fluids, which are the most likely species to induce cell damage. The generation of particles has an important role in metal ion release by re-passivating exposed surfaces, increasing the exposed surface, and inducing phagocytosis that expose the particles to an oxidative environment. Metal ions, mainly cobalt, induces mitochondrial stress, by opening transition pores in the mitochondrial internal membrane. The presence of pores in the membrane decrease the electrochemical potential reducing the ATP synthesis generating a hypoxia-like condition in the cells. Subsequently the transcription factors HIF-1α and NFκB trigger the synthesis and secretion of cytokines that elicit the inflammation in the periprosthetic tissues. The mechanisms of hypersensitivity are not clear, but activation of T-cell is highly specific, thus the presence of specific antigens that activates the lymphocyte response is needed. In summary, ALTRs are complex pathology, in which two major physiopathological events are involved: the mitochondrial stress induced by metal ions, and the development of hypersensitivity in a pre-existing chronic inflammation setting.All authors contributed to reviewing the relevant literature and drafting the manuscript. Figures were obtained and processed by QW and FE.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.