Bayesian uncertainty evaluation applied to the tilted-wave interferometer.

The tilted-wave interferometer is a promising technique for the development of a reference measurement system for the highly accurate form measurement of aspheres and freeform surfaces. The technique combines interferometric measurements, acquired with a special setup, and sophisticated mathematical evaluation procedures. To determine the form of the surface under test, a computational model is required that closely mimics the measurement process of the physical measurement instruments. The parameters of the computational model, comprising the surface under test sought, are then tuned by solving an inverse problem. Due to this embedded structure of the real experiment and computational model and the overall complexity, a thorough uncertainty evaluation is challenging. In this work, a Bayesian approach is proposed to tackle the inverse problem, based on a statistical model derived from the computational model of the tilted-wave interferometer. Such a procedure naturally allows for uncertainty quantification to be made. We present an approximate inference scheme to efficiently sample quantities of the posterior using Monte Carlo sampling involving the statistical model. In particular, the methodology derived is applied to the tilted-wave interferometer to obtain an estimate and corresponding uncertainty of the pixel-by-pixel form of the surface under test for two typical surfaces taking into account a number of key influencing factors. A statistical analysis using experimental design is employed to identify main influencing factors and a subsequent analysis confirms the efficacy of the method derived.

Compressed FTIR spectroscopy using low-rank matrix reconstruction.

Fourier transform infrared (FTIR) spectroscopy is a powerful technique in analytical chemistry. Typically, spatially distributed spectra of the substance of interest are conducted simultaneously using FTIR spectrometers equipped with array detectors. Scanning-based methods such as near-field FTIR spectroscopy, on the other hand, are a promising alternative providing higher spatial resolution. However, serial recording severely limits their application due to the long acquisition times involved and the resulting stability issues. We demonstrate that it is possible to significantly reduce the measurement time of scanning methods by applying the mathematical technique of low-rank matrix reconstruction. Data from a previous pilot study of Leishmania strains are analyzed by randomly selecting 5% of the interferometer samples. The results obtained for bioanalytical fingerprinting using the proposed approach are shown to be essentially the same as those obtained from the full set of data. This finding can significantly foster the practical applicability of high-resolution serial scanning techniques in analytical chemistry and is also expected to improve other applications of FTIR spectroscopy and spectromicroscopy.

Compressed sensing FTIR nano-spectroscopy and nano-imaging.

Infrared scattering scanning near-field optical microscopy (IR s-SNOM) provides for spectroscopic imaging with nanometer spatial resolution, yet full spatio-spectral imaging is constrained by long measurement times. Here, we demonstrate the application of compressed sensing algorithms to achieve hyperspectral FTIR-based nano-imaging at an order of magnitude faster imaging speed to achieve the same spectral content compared to conventional approaches. At the example of the spectroscopy of a single vibrational resonance, we discuss the relationship of prior knowledge of sparseness of the employed Fourier base functions and sub-sampling. Compressed sensing nano-FTIR spectroscopy promises both rapid and sensitive chemical nano-imaging which is highly relevant in academic and industrial settings for fundamental and applied nano- and bio-materials research.

Bayesian estimation in random effects meta-analysis using a non-informative prior.

Pooling information from multiple, independent studies (meta-analysis) adds great value to medical research. Random effects models are widely used for this purpose. However, there are many different ways of estimating model parameters, and the choice of estimation procedure may be influential upon the conclusions of the meta-analysis. In this paper, we describe a recently proposed Bayesian estimation procedure and compare it with a profile likelihood method and with the DerSimonian-Laird and Mandel-Paule estimators including the Knapp-Hartung correction. The Bayesian procedure uses a non-informative prior for the overall mean and the between-study standard deviation that is determined by the Berger and Bernardo reference prior principle. The comparison of these procedures focuses on the frequentist properties of interval estimates for the overall mean. The results of our simulation study reveal that the Bayesian approach is a promising alternative producing more accurate interval estimates than those three conventional procedures for meta-analysis. The Bayesian procedure is also illustrated using three examples of meta-analysis involving real data. Copyright © 2016 John Wiley & Sons, Ltd.

Informative prior distributions for ELISA analyses.

Immunoassays are capable of measuring very small concentrations of substances in solutions and have an immense range of application. Enzyme-linked immunosorbent assay (ELISA) tests in particular can detect the presence of an infection, of drugs, or hormones (as in the home pregnancy test). Inference of an unknown concentration via ELISA usually involves a non-linear heteroscedastic regression and subsequent prediction, which can be carried out in a Bayesian framework. For such a Bayesian inference, we are developing informative prior distributions based on extensive historical ELISA tests as well as theoretical considerations. One consideration regards the quality of the immunoassay leading to two practical requirements for the applicability of the priors. Simulations show that the additional prior information can lead to inferences which are robust to reasonable perturbations of the model and changes in the design of the data. On real data, the applicability is demonstrated across different laboratories, for different analytes and laboratory equipment as well as for previous and current ELISAs with sigmoid regression function. Consistency checks on real data (similar to cross-validation) underpin the adequacy of the suggested priors. Altogether, the new priors may improve concentration estimation for ELISAs that fulfill certain design conditions, by extending the range of the analyses, decreasing the uncertainty, or giving more robust estimates. Future use of these priors is straightforward because explicit, closed-form expressions are provided. This work encourages development and application of informative, yet general, prior distributions for other types of immunoassays.

Evaluation of absolute form measurements using a tilted-wave interferometer.

Tilted-wave interferometry is a promising measurement technique for the highly accurate measurement of aspheres and freeform surfaces. However, the interferometric fringe evaluation of the sub-apertures causes unknown patch offsets, which currently prevent this measurement technique from providing absolute measurements. Simple strategies, such as constructing differences of optical path length differences (OPDs) or ignoring the piston parameter, can diminish the accuracy resulting from the absolute form measurement. Additional information is needed instead; in this paper, the required accuracy of such information is explored in virtual experiments. Our simulation study reveals that, when one absolute OPD is known within a range of 500 nm, the accuracy of the final measurement result is significantly enhanced.

Improved estimation of reflectance spectra by utilizing prior knowledge.

Estimating spectral reflectance has attracted extensive research efforts in color science and machine learning, motivated through a wide range of applications. In many practical situations, prior knowledge is available that ought to be used. Here, we have developed a general Bayesian method that allows the incorporation of prior knowledge from previous monochromator and spectrophotometer measurements. The approach yields analytical expressions for fast and efficient estimation of spectral reflectance. In addition to point estimates, probability distributions are also obtained, which completely characterize the uncertainty associated with the reconstructed spectrum. We demonstrate that, through the incorporation of prior knowledge, our approach yields improved reconstruction results compared with methods that resort to training data only. Our method is particularly useful when the spectral reflectance to be recovered resides beyond the scope of the training data.

Transfer Standard Uncertainty Can Cause Inconclusive Inter-Laboratory Comparisons.

Inter-laboratory comparisons use the best available transfer standards to check the participants' uncertainty analyses, identify underestimated uncertainty claims or unknown measurement biases, and improve the global measurement system. For some measurands, instability of the transfer standard can lead to an inconclusive comparison result. If the transfer standard uncertainty is large relative to a participating laboratory's uncertainty, the commonly used standardized degree of equivalence ≤ 1 criterion does not always correctly assess whether a participant is working within their uncertainty claims. We show comparison results that demonstrate this issue and propose several criteria for assessing a comparison result as passing, failing, or inconclusive. We investigate the behavior of the standardized degree of equivalence and alternative comparison measures for a range of values of the transfer standard uncertainty relative to the individual laboratory uncertainty values. The proposed alternative criteria successfully discerned between passing, failing, and inconclusive comparison results for the cases we examined.

Analytical Jacobian and its application to tilted-wave interferometry.

Tilted-wave interferometry (TWI) is a novel optical measurement principle for the measurement of aspherical surfaces. For the reconstruction of the wavefront and the surface under test, respectively, perturbation methods are applied, which require the calculation of the Jacobian matrix. For the practical use of the instrument, a fast and exact calculation of the Jacobian matrices is crucial, since this strongly influences the calculation times of the TWI. By applying appropriate approaches in optical perturbation methods we are able to calculate the required Jacobian matrices analytically when the nominal optical path through the system is given. As a result, calculation times for the TWI can be considerably reduced. We finally illustrate the improved TWI procedure and apply methods of optimal design to determine optimal positions of the surface under test. For such applications the fast calculation of the Jacobian matrices is essential.

Virtual experiment for near-field goniophotometric measurements.

Near-field goniometric measurements are employed to determine the photometric characteristics of light sources, i.e., the spatial and angular distribution of the emitted light. To this end, a complex measurement system consisting of a goniometer and a CCD-based imaging photometer is employed. In order to gain insight into the measurement system and to enable characterization of the whole measurement setup, we propose to apply a computer model to conduct virtual experiments. Within the computer model, the current state of all parts of the virtual experiment can be easily controlled. The reliability of the computer model is demonstrated by a comparison to actual measurement results. As an example for the application of the virtual experiment, we present an analysis of the impact of axial malpositions of the goniometer and camera.

A maximum likelihood approach to the inverse problem of scatterometry.

Scatterometry is frequently used as a non-imaging indirect optical method to reconstruct the critical dimensions (CD) of periodic nanostructures. A particular promising direction is EUV scatterometry with wavelengths in the range of 13 - 14 nm. The conventional approach to determine CDs is the minimization of a least squares function (LSQ). In this paper, we introduce an alternative method based on the maximum likelihood estimation (MLE) that determines the statistical error model parameters directly from measurement data. By using simulation data, we show that the MLE method is able to correct the systematic errors present in LSQ results and improves the accuracy of scatterometry. In a second step, the MLE approach is applied to measurement data from both extreme ultraviolet (EUV) and deep ultraviolet (DUV) scatterometry. Using MLE removes the systematic disagreement of EUV with other methods such as scanning electron microscopy and gives consistent results for DUV.

Bayesian analysis of an international ELISA comparability study.

BACKGROUND: Immunoassays are biochemical tests applied to measure even very small amounts of substance using the highly specific binding between an antibody and its antigen. They have a wide range of applications. The measurement however, might be associated with substantial uncertainty; this can have significant consequences for any diagnosis, or clinical decision. An international comparability study was thus performed to assess the sources of uncertainty involved in the estimation of a protein cytokine concentration using a fluorescent ELISA. METHODS: In contrast to the original publication for this international comparability study, we reanalyse the data using Bayesian inference. This provides a statistically coherent approach to estimate ELISA concentrations and their associated uncertainties. RESULTS: The Bayesian uncertainties of individual ELISAs and laboratory estimates are considerably larger than previously reported uncertainties. The average concentrations estimated here differ from the ones estimated by each study participant. In general, this leads to different conclusions about the study. In particular, the inter- and intra-laboratory consistency is increased, and repeatability problems occur for fewer laboratories. CONCLUSIONS: Decisions which are based on plausible ranges of measurements (such as credible intervals), are generally superior to those solely based on point estimates (such as the mean). Reliable uncertainties are thus vital, and not only in metrology. In this paper, a general method is developed to derive concentration estimates and valid uncertainties for ELISAs. Guidance on applying this Bayesian method is provided and the importance of reliable uncertainties associated with ELISAs is underlined. The applicability and virtues of the presented method are demonstrated in the context of an international comparability study.

Comparison and assessment of procedures for calculating the R(12) line strength of the ν1 + 2 ν2 + ν3 band of CO2.

Recently, results for the CO(2) R(12) line strength parameter have been reported, which differ significantly and are inconsistent with respect to quoted uncertainties. We investigate to what extent this inconsistency might be caused by the chosen data analysis methods. To this end, we assess and compare a parametric fitting procedure and a non-parametric approach. We apply the methods to simulated and measured line spectra, and we specify the conditions required for the safe application of the two procedures. For our present data, the corresponding conditions are satisfied for both methods, and consistent results are obtained. However, the simulations reveal that the fitting procedure can show shortcomings when the uncertainty in the wavenumber is large.

Bayesian uncertainty quantification for magnetic resonance fingerprinting.

Magnetic Resonance Fingerprinting (MRF) is a promising technique for fast quantitative imaging of human tissue. In general, MRF is based on a sequence of highly undersampled MR images which are analyzed with a pre-computed dictionary. MRF provides valuable diagnostic parameters such as theT1andT2MR relaxation times. However, uncertainty characterization of dictionary-based MRF estimates forT1andT2has not been achieved so far, which makes it challenging to assess if observed differences in these estimates are significant and may indicate pathological changes of the underlying tissue. We propose a Bayesian approach for the uncertainty quantification of dictionary-based MRF which leads to probability distributions forT1andT2in every voxel. The distributions can be used to make probability statements about the relaxation times, and to assign uncertainties to their dictionary-based MRF estimates. All uncertainty calculations are based on the pre-computed dictionary and the observed sequence of undersampled MR images, and they can be calculated in short time. The approach is explored by analyzing MRF measurements of a phantom consisting of several tubes across which MR relaxation times are constant. The proposed uncertainty quantification is quantitatively consistent with the observed within-tube variability of estimated relaxation times. Furthermore, calculated uncertainties are shown to characterize well observed differences between the MRF estimates and the results obtained from high-accurate reference measurements. These findings indicate that a reliable uncertainty quantification is achieved. We also present results for simulated MRF data and an uncertainty quantification for anin vivoMRF measurement. MATLAB®source code implementing the proposed approach is made available.

Improving the lateral resolution of a multi-sensor profile measurement method by non-equidistant sensor spacing.

We present a method to enhance the achievable lateral resolution of a multi-sensor scanning profile measurement method. The relationship between the profile measurement method considered and established shearing techniques is illustrated. Simulation and measurement results show that non-equidistant sensor spacing can improve the lateral resolution significantly.

Dynamic uncertainty for compensated second-order systems.

The compensation of LTI systems and the evaluation of the according uncertainty is of growing interest in metrology. Uncertainty evaluation in metrology ought to follow specific guidelines, and recently two corresponding uncertainty evaluation schemes have been proposed for FIR and IIR filtering. We employ these schemes to compare an FIR and an IIR approach for compensating a second-order LTI system which has relevance in metrology. Our results suggest that the FIR approach is superior in the sense that it yields significantly smaller uncertainties when real-time evaluation of uncertainties is desired.

Mammography Image Quality Assurance Using Deep Learning.

OBJECTIVE: According to the European Reference Organization for Quality Assured Breast Cancer Screening and Diagnostic Services (EUREF) image quality in mammography is assessed by recording and analyzing a set of images of the CDMAM phantom. The EUREF procedure applies an automated analysis combining image registration, signal detection and nonlinear fitting. We present a proof of concept for an end-to-end deep learning framework that assesses image quality on the basis of single images as an alternative. METHODS: Virtual mammography is used to generate a database with known ground truth for training a regression convolutional neural net (CNN). Training is carried out by continuously extending the training data and applying transfer learning. RESULTS: The trained net is shown to correctly predict the image quality of simulated and real images. Specifically, image quality predictions on the basis of single images are of similar quality as those obtained by applying the EUREF procedure with 16 images. Our results suggest that the trained CNN generalizes well. CONCLUSION: Mammography image quality assessment can benefit from the proposed deep learning approach. SIGNIFICANCE: Deep learning avoids cumbersome pre-processing and allows mammography image quality to be estimated reliably using single images.

Suppression of aliasing in multi-sensor scanning absolute profile measurement.

The task of anti-aliasing in absolute profile measurement by multi-sensor scanning techniques is considered. Simulation results are presented which demonstrate that aliasing can be highly reduced by a suitable choice of the scanning steps. The simulation results were confirmed by results obtained for interferometric measurements (Nyquist frequency 1/646 microm(-1)) on a specifically designed chirp specimen with sinusoidal waves of amplitude 100 nm and wavelengths from 2.5 mm down to 19 microm.

Assessment of CT Image Quality Using a Bayesian Framework.

In computed tomography, there is a tradeoff between the quality of the reconstructed image and the radiation dose received by the patient. In order to find an appropriate compromise between the image quality of the reconstructed images and the radiation dose, it is important to have reliable methods for evaluating the quality of the reconstructed images. A successful family of methods for the assessment of image quality is task-based image quality assessment, which often involves the use of model observers, and which assesses the quality of the image reconstruction by deriving a figure of merit. Here, we present a Bayesian framework that can be used in task-based image quality assessment. Our framework is applicable to binary classification problems with normally distributed observations, and we make the additional assumption that the covariance matrix is the same in both image classes. We choose a particular non-informative prior for the parameters of our model, which allows us to derive an expression for the Bayes factor for the binary classification problem which to the best of our knowledge is novel. We introduce a novel model observer based on this Bayes factor. Further, we have developed a methodology for estimating the posterior distribution of the figure of merit for this type of classification problem. Compared with classical statistical approaches, our Bayesian approach has the advantage that it provides a full characterization of the uncertainty of the figure of merit. Our choice of prior allows us to design a simple Monte Carlo algorithm to efficiently sample the posterior of the figure of merit of the ideal observer, in contrast to common Bayesian procedures which rely on computationally expensive Markov chain Monte Carlo sampling. We have shown that for training samples of sufficient size, our estimated credible intervals for the figure of merit have coverage probabilities close to their credibility, so that our approach can reasonably be used within a classical statistical framework as well.

Pixel-wise quantification of myocardial perfusion using spatial Tikhonov regularization.

Quantification of myocardial perfusion by contrast-enhanced cardiovascular magnetic resonance imaging (CMR) aims for an observer independent and reproducible risk assessment of cardiovascular disease. Currently, the data used for the pixel-wise analysis of cardiac perfusion are either filtered prior to a fitting procedure, which inherently reduces the spatial resolution of data; or all pixels are considered without any regularization or prior filtering, which yields an unstable fit in the presence of low signal-to-noise ratio. Here, we propose a new pixel-wise analysis based on spatial Tikhonov regularization which exploits the spatial smoothness of the data and ensures accurate quantification even for images with low signal-to-noise ratio. The regularization parameter is determined automatically by an L-curve criterion. We study the performance of our method on a numerical phantom and demonstrate that the method reduces significantly the root-mean square error in the perfusion estimate compared to a non-regularized fit. In patient data our method allows us to recover the myocardial perfusion and to distinguish between healthy and ischemic regions.