Recurrent inference machines as inverse problem solvers for MR relaxometry.

In this paper, we propose the use of Recurrent Inference Machines (RIMs) to perform T1 and T2 mapping. The RIM is a neural network framework that learns an iterative inference process based on the signal model, similar to conventional statistical methods for quantitative MRI (QMRI), such as the Maximum Likelihood Estimator (MLE). This framework combines the advantages of both data-driven and model-based methods, and, we hypothesize, is a promising tool for QMRI. Previously, RIMs were used to solve linear inverse reconstruction problems. Here, we show that they can also be used to optimize non-linear problems and estimate relaxometry maps with high precision and accuracy. The developed RIM framework is evaluated in terms of accuracy and precision and compared to an MLE method and an implementation of the Residual Neural Network (ResNet). The results show that the RIM improves the quality of estimates compared to the other techniques in Monte Carlo experiments with simulated data, test-retest analysis of a system phantom, and in-vivo scans. Additionally, inference with the RIM is 150 times faster than the MLE, and robustness to (slight) variations of scanning parameters is demonstrated. Hence, the RIM is a promising and flexible method for QMRI. Coupled with an open-source training data generation tool, it presents a compelling alternative to previous methods.

Atom column detection from simultaneously acquired ABF and ADF STEM images.

In electron microscopy, the maximum a posteriori (MAP) probability rule has been introduced as a tool to determine the most probable atomic structure from high-resolution annular dark-field (ADF) scanning transmission electron microscopy (STEM) images exhibiting low contrast-to-noise ratio (CNR). Besides ADF imaging, STEM can also be applied in the annular bright-field (ABF) regime. The ABF STEM mode allows to directly visualize light-element atomic columns in the presence of heavy columns. Typically, light-element nanomaterials are sensitive to the electron beam, limiting the incoming electron dose in order to avoid beam damage and leading to images exhibiting low CNR. Therefore, it is of interest to apply the MAP probability rule not only to ADF STEM images, but to ABF STEM images as well. In this work, the methodology of the MAP rule, which combines statistical parameter estimation theory and model-order selection, is extended to be applied to simultaneously acquired ABF and ADF STEM images. For this, an extension of the commonly used parametric models in STEM is proposed. Hereby, the effect of specimen tilt has been taken into account, since small tilts from the crystal zone axis affect, especially, ABF STEM intensities. Using simulations as well as experimental data, it is shown that the proposed methodology can be successfully used to detect light elements in the presence of heavy elements.

The maximum a posteriori probability rule for atom column detection from HAADF STEM images.

Recently, the maximum a posteriori (MAP) probability rule has been proposed as an objective and quantitative method to detect atom columns and even single atoms from high-resolution high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) images. The method combines statistical parameter estimation and model-order selection using a Bayesian framework and has been shown to be especially useful for the analysis of the structure of beam-sensitive nanomaterials. In order to avoid beam damage, images of such materials are usually acquired using a limited incoming electron dose resulting in a low contrast-to-noise ratio (CNR) which makes visual inspection unreliable. This creates a need for an objective and quantitative approach. The present paper describes the methodology of the MAP probability rule, gives its step-by-step derivation and discusses its algorithmic implementation for atom column detection. In addition, simulation results are presented showing that the performance of the MAP probability rule to detect the correct number of atomic columns from HAADF STEM images is superior to that of other model-order selection criteria, including the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). Moreover, the MAP probability rule is used as a tool to evaluate the relation between STEM image quality measures and atom detectability resulting in the introduction of the so-called integrated CNR (ICNR) as a new image quality measure that better correlates with atom detectability than conventional measures such as signal-to-noise ratio (SNR) and CNR.

Single Atom Detection from Low Contrast-to-Noise Ratio Electron Microscopy Images.

Single atom detection is of key importance to solving a wide range of scientific and technological problems. The strong interaction of electrons with matter makes transmission electron microscopy one of the most promising techniques. In particular, aberration correction using scanning transmission electron microscopy has made a significant step forward toward detecting single atoms. However, to overcome radiation damage, related to the use of high-energy electrons, the incoming electron dose should be kept low enough. This results in images exhibiting a low signal-to-noise ratio and extremely weak contrast, especially for light-element nanomaterials. To overcome this problem, a combination of physics-based model fitting and the use of a model-order selection method is proposed, enabling one to detect single atoms with high reliability.

Atom-counting in High Resolution Electron Microscopy:TEM or STEM - That's the question.

In this work, a recently developed quantitative approach based on the principles of detection theory is used in order to determine the possibilities and limitations of High Resolution Scanning Transmission Electron Microscopy (HR STEM) and HR TEM for atom-counting. So far, HR STEM has been shown to be an appropriate imaging mode to count the number of atoms in a projected atomic column. Recently, it has been demonstrated that HR TEM, when using negative spherical aberration imaging, is suitable for atom-counting as well. The capabilities of both imaging techniques are investigated and compared using the probability of error as a criterion. It is shown that for the same incoming electron dose, HR STEM outperforms HR TEM under common practice standards, i.e. when the decision is based on the probability function of the peak intensities in HR TEM and of the scattering cross-sections in HR STEM. If the atom-counting decision is based on the joint probability function of the image pixel values, the dependence of all image pixel intensities as a function of thickness should be known accurately. Under this assumption, the probability of error may decrease significantly for atom-counting in HR TEM and may, in theory, become lower as compared to HR STEM under the predicted optimal experimental settings. However, the commonly used standard for atom-counting in HR STEM leads to a high performance and has been shown to work in practice.

Detecting and locating light atoms from high-resolution STEM images: The quest for a single optimal design.

In the present paper, the optimal detector design is investigated for both detecting and locating light atoms from high resolution scanning transmission electron microscopy (HR STEM) images. The principles of detection theory are used to quantify the probability of error for the detection of light atoms from HR STEM images. To determine the optimal experiment design for locating light atoms, use is made of the so-called Cramér-Rao Lower Bound (CRLB). It is investigated if a single optimal design can be found for both the detection and location problem of light atoms. Furthermore, the incoming electron dose is optimised for both research goals and it is shown that picometre range precision is feasible for the estimation of the atom positions when using an appropriate incoming electron dose under the optimal detector settings to detect light atoms.

Data distributions in magnetic resonance images: a review.

Many image processing methods applied to magnetic resonance (MR) images directly or indirectly rely on prior knowledge of the statistical data distribution that characterizes the MR data. Also, data distributions are key in many parameter estimation problems and strongly relate to the accuracy and precision with which parameters can be estimated. This review paper provides an overview of the various distributions that occur when dealing with MR data, considering both single-coil and multiple-coil acquisition systems. The paper also summarizes how knowledge of the MR data distributions can be used to construct optimal parameter estimators and answers the question as to what precision may be achieved ultimately from a particular MR image.

Iterative linear focal-plane wavefront correction.

We propose an efficient approximation to the nonlinear phase diversity (PD) method for wavefront reconstruction and correction from intensity measurements with potential of being used in real-time applications. The new iterative linear phase diversity (ILPD) method assumes that the residual phase aberration is small and makes use of a first-order Taylor expansion of the point spread function (PSF), which allows for arbitrary (large) diversities in order to optimize the phase retrieval. For static disturbances, at each step, the residual phase aberration is estimated based on one defocused image by solving a linear least squares problem, and compensated for with a deformable mirror. Due to the fact that the linear approximation does not have to be updated with each correction step, the computational complexity of the method is reduced to that of a matrix-vector multiplication. The convergence of the ILPD correction steps has been investigated and numerically verified. The comparative study that we make demonstrates the improved performance in computational time with no decrease in accuracy with respect to existing methods that also linearize the PSF.

Estimation of unknown structure parameters from high-resolution (S)TEM images: what are the limits?

Statistical parameter estimation theory is proposed as a quantitative method to measure unknown structure parameters from electron microscopy images. Images are then purely considered as data planes from which structure parameters have to be determined as accurately and precisely as possible using a parametric statistical model of the observations. For this purpose, an efficient algorithm is proposed for the estimation of atomic column positions and intensities from high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) images. Furthermore, the so-called Cramér-Rao lower bound (CRLB) is reviewed to determine the limits to the precision with which continuous parameters such as atomic column positions and intensities can be estimated. Since this lower bound can only be derived for continuous parameters, alternative measures using the principles of detection theory are introduced for problems concerning the estimation of discrete parameters such as atomic numbers. An experimental case study is presented to show the practical use of these measures for the optimization of the experiment design if the purpose is to decide between the presence of specific atom types using STEM images.

Estimation of dynamic neural activity using a Kalman filter approach based on physiological models.

This paper presents a new method to estimate dynamic neural activity from EEG signals. The method is based on a Kalman filter approach, using physiological models that take both spatial and temporal dynamics into account. The filter's performance (in terms of estimation error) is analyzed for the cases of linear and nonlinear models having either time invariant or time varying parameters. The best performance is achieved with a nonlinear model with time-varying parameters.

Likelihood-based hypothesis tests for brain activation detection from MRI data disturbed by colored noise: a simulation study.

Functional magnetic resonance imaging (fMRI) data that are corrupted by temporally colored noise are generally preprocessed (i.e., prewhitened or precolored) prior to functional activation detection. In this paper, we propose likelihood-based hypothesis tests that account for colored noise directly within the framework of functional activation detection. Three likelihood-based tests are proposed: the generalized likelihood ratio (GLR) test, the Wald test, and the Rao test. The fMRI time series is modeled as a linear regression model, where one regressor describes the task-related hemodynamic response, one regressor accounts for a constant baseline and one regressor describes potential drift. The temporal correlation structure of the noise is modeled as an autoregressive (AR) model. The order of the AR model is determined from practical null data sets using Akaike's information criterion (with penalty factor 3) as order selection criterion. The tests proposed are based on exact expressions for the likelihood function of the data. Using Monte Carlo simulation experiments, the performance of the proposed tests is evaluated in terms of detection rate and false alarm rate properties and compared to the current general linear model (GLM) test, which estimates the coloring of the noise in a separate step. Results show that theoretical asymptotic distributions of the GLM, GLR, and Wald test statistics cannot be reliably used for computing thresholds for activation detection from finite length time series. Furthermore, it is shown that, for a fixed false alarm rate, the detection rate of the proposed GLR test statistic is slightly, but (statistically) significantly improved compared to that of the common GLM-based tests. Finally, simulations results reveal that all tests considered show seriously inferior performance if the order of the AR model is not chosen sufficiently high to give an adequate description of the correlation structure of the noise, whereas the effects of (slightly) overmodeling are observed to be less harmful.

Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: a theoretical framework.

This paper is the first part of a two-part paper on maximum likelihood (ML) estimation of structure parameters from electron microscopy images. In principle, electron microscopy allows structure determination with a precision that is orders of magnitude better than the resolution of the microscope. This requires, however, a quantitative, model-based method. In our opinion, the ML method is the most appropriate one since it has optimal statistical properties. This paper aims to provide microscopists with the necessary tools to apply this method so as to determine structure parameters as precisely as possible. It reviews the theoretical framework, including model assessment, the derivation of the ML estimator of the parameters, the limits to precision and the construction of confidence regions and intervals for ML parameter estimates. In a companion paper [Van Aert et al., Ultramicroscopy, this issue, 2005], a practical example will be worked out.

Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part II: a practical example.

This paper is the second part of a two-part paper on maximum likelihood (ML) estimation of structure parameters from electron microscopy images. In order to show the practical applicability of the theoretical methods described in the first part of this two-part paper, an experimental study of an aluminium crystal is presented. In this study, structure parameters, atom column distances in particular, are estimated from high-resolution transmission electron microscopy (HRTEM) images using the ML method. The necessary steps to be made in the application of this method will be worked out one by one, including model assessment, the computation of the ML parameter estimates, and the construction of confidence intervals for these parameter estimates.

Generalized likelihood ratio tests for complex fMRI data: a simulation study.

Statistical tests developed for the analysis of (intrinsically complex valued) functional magnetic resonance time series, are generally applied to the data's magnitude components. However, during the past five years, new tests were developed that incorporate the complex nature of fMRI data. In particular, a generalized likelihood ratio test (GLRT) was proposed based on a constant phase model. In this work, we evaluate the sensitivity of GLRTs for complex data to small misspecifications of the phase model by means of simulation experiments. It is argued that, in practical situations, GLRTs based on magnitude data are likely to perform better compared to GLRTs based on complex data in terms of detection rate and constant false alarm rate properties.

Physical limits on atomic resolution.

It is shown that the ultimate resolution is not limited by the bandwidth of the microscope but by the bandwidth (i.e., the scattering power) of the object. In the case of a crystal oriented along a zone axis, the scattering is enhanced by the channeling of the electrons. However, if the object is aperiodic along the beam direction, the bandwidth is much more reduced. A particular challenge are the amorphous objects. For amorphous materials, the natural bandwidth is that of the single atom and of the order of 1 angstrom(-1), which can be reached with the present generation of medium voltage microscopes without aberration correctors. A clear distinction is made between resolving a structure and refining, that is, between resolution and precision. In the case of an amorphous structure, the natural bandwidth also puts a limit on the number of atom coordinates that can be refined quantitatively. As a consequence, amorphous structures cannot be determined from one projection, but only by using atomic resolution tomography. Finally a theory of experiment design is presented that can be used to predict the optimal experimental setting or the best instrumental improvement. Using this approach it is suggested that the study of amorphous objects should be done at low accelerating voltage with correction of both spherical and chromatic aberration.

How to optimize the experimental design of quantitative atomic resolution TEM experiments?

A quantitative measure is proposed to evaluate and optimize the design of quantitative atomic resolution TEM experiments. It aims at precise measurement of unknown structure parameters. Specifically, the proposed measure quantifies the statistical precision with which positions of atom columns can be estimated. The optimal design is then given by the combination of microscope settings for which this precision is highest. The proposed measure is also used to find out if new instrumental developments improve the precision as compared to existing methods.

Maximum likelihood estimation of signal amplitude and noise variance from MR data.

In MRI, the raw data, which are acquired in spatial frequency space, are intrinsically complex valued and corrupted by Gaussian-distributed noise. After applying an inverse Fourier transform, the data remain complex valued and Gaussian distributed. If the signal amplitude is to be estimated, one has two options. It can be estimated directly from the complex valued data set, or one can first perform a magnitude operation on this data set, which changes the distribution of the data from Gaussian to Rician, and estimate the signal amplitude from the obtained magnitude image. Similarly, the noise variance can be estimated from both the complex and magnitude data sets. This article addresses the question whether it is better to use complex valued data or magnitude data for the estimation of these parameters using the maximum likelihood method. As a performance criterion, the mean-squared error (MSE) is used.

Is atomic resolution transmission electron microscopy able to resolve and refine amorphous structures?

Atomic resolution transmission electron microscopy, even with an aberration free microscope, is only able to resolve and refine amorphous structures at the atomic level for very small foil thicknesses. Then, a precision of the order of 0.01 A is possible, but this may require long recording times, especially for light atoms. For larger thicknesses, amorphous structures can in principle only be resolved and refined using electron tomography.

High-resolution electron microscopy and electron tomography: resolution versus precision.

The performance of high-resolution electron microscopy and electron tomography is usually discussed in terms of two-point resolution, expressing the possibility of perceiving separately two image points of an object. However, the concept resolution obtains another meaning if one uses prior knowledge about the object and the imaging procedure in the form of a parametric model describing the expectations of the observations. The unknown parameters, such as the positions of the components in an object, can be measured quantitatively by fitting this model to the observations. Due to the statistical nature of the experiment, the resulting solutions for the positions of the components and therefore for the distance between the components will never be exact. An alternative to resolution is then the precision with which the distance can be measured. In the present paper, it is shown that the precision depends on the size of the components, the distance between the components, the resolution of the instrument, and the number of electron counts. For electron tomography, it also depends on the orientation of the object with respect to the rotation axis.

Optimal experimental design of STEM measurement of atom column positions.

A quantitative measure is proposed to evaluate and optimize the design of a high-resolution scanning transmission electron microscopy (STEM) experiment. The proposed measure is related to the measurement of atom column positions. Specifically, it is based on the statistical precision with which the positions of atom columns can be estimated. The optimal design, that is, the combination of tunable microscope parameters for which the precision is highest. is derived for different types of atom columns. The proposed measure is also used to find out if an annular detector is preferable to an axial one and if a C(s)-corrector pays off in quantitative STEM experiments. In addition, the optimal settings of the STEM are compared to the Scherzer conditions for incoherent imaging and their dependence on the type of object is investigated.