Fast Fourier Research Articles

Обробка даних на основі швидкого перетворення Фур’є в кореляційному течешукачі

Fast Fourier transform (FFT) algorithms are widely used in correlation leak detectors (CD) in the implementation of digital data processing. The combination of the computa-tional efficiency of FFT with modern high-performance mobile computing means pro-vides acceptable data processing time during leak detection. Due to the increase in the general wear and tear of pipeline networks in many countries, the requirements for reliabil-ity and efficiency in detecting and repairing damage are increasing. Meeting these re-quirements means complicating data processing algorithms to more fully take into ac-count the diversity of the acoustic environment. This is the reason for the special interest in FFT as a basic, universal algorithm, on the basis of which it is promising to further develop data processing in CD. The article describes the experience of the authors in the application of FFT in parametric CD, in particular, the use of FFT in spectral evaluation and digital filtering of signals, calculations of digital filters, calculation and further pro-cessing of correlation functions is considered.

Self-organized composite morphologies and interface instability of immiscible alloy thin films during physical vapor deposition: Insights from phase-field simulations

In this paper, we carried out phase-field simulations for investigating self-organized composite morphologies formation and interface instability of two phase immiscible binary alloy thin-film growth during physical vapor deposition. Predicted results show that for lower the deposition rate the solid-gas interface keeps planar and the lateral composition modulation (LCM) configuration occurs. As the deposition rate increases, the self-organized composite morphology gradually transfers to vertical composition modulations (VCM) configuration and the random composition modulations (RCM) configuration with the decrease of the boundary layer of the incident vapor and the unsteady solid-gas interface. Results show that the nanoprecipitate concentration modulation (NCPM) depends strongly on the nominally phase fraction. Moreover, we found that the historical dependence of self-organized composite morphologies and solid-gas interface stability, i.e., the initial condition can strongly influence the microstructure of the thin film during physical vapor deposition. Finally, we provide an analysis method for interface instability based on fast Fourier transform (FFT), and the frequency characteristics of cellular and bump structures are identified.

Extraction and evaluation of cable forces of a cable-stayed bridge based on amplitude and phase estimation method

In order to identify the time-varying frequency and amplitude of structural vibration based on the bridge structural health monitoring data and obtain the cable force of cable-stayed bridges in real time, a spectrum analysis method based on amplitude and phase estimation (APES) was proposed in this study. The amplitude spectrum of the acceleration data is first calculated by the APES method, the real-time spectrogram of the cable is obtained by the sliding window method. Then the modal frequency and amplitude are automatically extracted from the real-time spectrum by using a frequency extrusion post-processing technique, which can be regarded as the average value of the instantaneous frequency and amplitude respectively. Next, the fundamental frequency of the cable is extracted by using an automatic identification method, and the performance of the proposed method is verified. Finally, real-time scoring of cable forces and structural condition assessment is achieved with consideration of the moderation index model as well as the material strength. The results show that the APES method can use shorter calculation samples than the traditional Fast Fourier Transform (FFT) to obtain higher resolution and more accurate amplitude, which provides a theoretical basis for the real-time identification of fundamental frequency based on short-term monitoring data. The frequency extrusion post-processing-algorithm can reduce the spectrum recognition delay and improve timeliness of the cable force evaluation. The time-varying cable force with an interval of 10 s can be used to reflect the health status of the cable in time. The research results can provide technical support for the real-time extraction of cable force of long-span cable-stayed bridges, and can also provide an effective basis for component condition evaluation and bridge maintenance decision-making.

Effect of micro-clearance structure on the collapse of individual liquid hydrogen bubbles

Cavitation occurs in the micro-clearance of liquid-hydrogen-lubricated bearings owing to the pressure drop caused by high-speed shearing. The pressure undulation caused by cavitation collapse results in bearing surface erosion and significantly affects the bearing performance. In this study, a modified Z-G-B cavitation model was used to study the crushing process of a single liquid hydrogen bubble in a shear micro-clearance. Fast Fourier transform (FFT) and wavelet transform (WT) were applied to study the frequency characteristics of the pressure, mass transfer rate, and vapor mass fraction during bubble rupture in shearing micro-clearance. To obtain a deeper insight into the details of the effect of the shear micro-clearance structure on bubble collapse, the relationship between the flow field energy, attenuation rate, and frequency was investigated. The proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) methods were used to analyze the energy of each order mode of the flow field. The analysis results of the bubble vibration intensity with respect to time and frequency provide a theoretical basis for the optimization of the bearing structure.

Imposing Dirichlet boundary conditions directly for FFT-based computational micromechanics

We discuss how Dirichlet boundary conditions can be directly imposed for the Moulinec–Suquet discretization on the boundary of rectangular domains in iterative schemes based on the fast Fourier transform (FFT) and computational homogenization problems in mechanics. Classically, computational homogenization methods based on the fast Fourier transform work with periodic boundary conditions. There are applications, however, when Dirichlet (or Neumann) boundary conditions are required. For thermal homogenization problems, it is straightforward to impose such boundary conditions by using discrete sine (and cosine) transforms instead of the FFT. This approach, however, is not readily extended to mechanical problems due to the appearance of mixed derivatives in the Lamé operator of elasticity. Thus, Dirichlet boundary conditions are typically imposed either by using Lagrange multipliers or a “buffer zone” with a high stiffness. Both strategies lead to formulations which do not share the computational advantages of the original FFT-based schemes. The work at hand introduces a technique for imposing Dirichlet boundary conditions directly without the need for indefinite systems. We use a formulation on the deformation gradient—also at small strains—and employ the Green’s operator associated to the vector Laplacian. Then, we develop the Moulinec–Suquet discretization for Dirichlet boundary conditions—requiring carefully selected weights at boundary points—and discuss the seamless integration into existing FFT-based computational homogenization codes based on dedicated discrete sine/cosine transforms. The article culminates with a series of well-chosen numerical examples demonstrating the capabilities of the introduced technology.

Machine learning-based weld porosity detection using frequency analysis of arc sound in the pulsed gas tungsten arc welding process

Automatic welding equipment has replaced human welders in the nuclear industry for safety issues and uniform and high welding quality. However, automatic welding equipment cannot predict porosity defects. So, the weldment must be inspected by non-destructive testing. This inspection was a costly and time-consuming process, and it applies to each weldment even if it welded same material. To improve the welding efficiency, a weld porosity detection system of the same weld material with different material thicknesses was needed. This paper proposed a machine-learned porosity detection system for 3.0 mm plates with welding arc sound data from the pulsed gas tungsten arc welding (P-GTAW) process of 1.6 mm plates. Ensemble-Empirical Mode Decomposition (EEMD) was used to divide the arc sound signal according to the pulse period of P-GTAW. Fast Fourier transform (FFT) was used to convert the arc sound into frequencies for features extraction according to porosity. The validity of these weld frequency features was confirmed through k-fold cross-validation across various machine learning techniques, with evaluation of F-1 scores against experimental weld sounds.

Lightweight Ghost Enhanced Feature Attention Network: An Efficient Intelligent Fault Diagnosis Method under Various Working Conditions.

Recent advancements in applications of deep neural network for bearing fault diagnosis under variable operating conditions have shown promising outcomes. However, these approaches are limited in practical applications due to the complexity of neural networks, which require substantial computational resources, thereby hindering the advancement of automated diagnostic tools. To overcome these limitations, this study introduces a new fault diagnosis framework that incorporates a tri-channel preprocessing module for multidimensional feature extraction, coupled with an innovative diagnostic architecture known as the Lightweight Ghost Enhanced Feature Attention Network (GEFA-Net). This system is adept at identifying rolling bearing faults across diverse operational conditions. The FFE module utilizes advanced techniques such as Fast Fourier Transform (FFT), Frequency Weighted Energy Operator (FWEO), and Signal Envelope Analysis to refine signal processing in complex environments. Concurrently, GEFA-Net employs the Ghost Module and the Efficient Pyramid Squared Attention (EPSA) mechanism, which enhances feature representation and generates additional feature maps through linear operations, thereby reducing computational demands. This methodology not only significantly lowers the parameter count of the model, promoting a more streamlined architectural framework, but also improves diagnostic speed. Additionally, the model exhibits enhanced diagnostic accuracy in challenging conditions through the effective synthesis of local and global data contexts. Experimental validation using datasets from the University of Ottawa and our dataset confirms that the framework not only achieves superior diagnostic accuracy but also reduces computational complexity and accelerates detection processes. These findings highlight the robustness of the framework for bearing fault diagnosis under varying operational conditions, showcasing its broad applicational potential in industrial settings. The parameter count was decreased by 63.74% compared to MobileVit, and the recorded diagnostic accuracies were 98.53% and 99.98% for the respective datasets.

LPI Sequences Optimization Method against Summation Detector Based on FFT Filter Bank

Waveform design is a crucial factor in electronic surveillance (ES) systems. In this paper, we introduce an algorithm that designs a low probability of intercept (LPI) radar waveform. Our approach directly minimizes the detection probability of summation detectors based on FFT filter banks. The algorithm is derived from the general quadratic optimization framework, which inherits the monotonic properties of such methods. To expedite overall convergence, we have integrated acceleration schemes based on the squared iterative method (SQUAREM). Additionally, the proposed algorithm can be executed through fast Fourier transform (FFT) operations, enhancing computational efficiency. With some modifications, the algorithm can be adjusted to incorporate spectral constraints, increasing its flexibility. Numerical experiments indicate that our proposed algorithm outperforms existing ones in terms of both intercept properties and computational complexity.

Machine learning assisted discovery of effective viscous material laws for shear-thinning fiber suspensions

AbstractIn this article, we combine a Fast Fourier Transform based computational approach and a supervised machine learning strategy to discover models for the anisotropic effective viscosity of shear-thinning fiber suspensions. Using the Fast Fourier Transform based computational approach, we study the effects of the fiber orientation state and the imposed macroscopic shear rate tensor on the effective viscosity for a broad range of shear rates of engineering process interest. We visualize the effective viscosity in three dimensions and find that the anisotropy of the effective viscosity and its shear rate dependence vary strongly with the fiber orientation state. Combining the results of this work with insights from literature, we formulate four requirements a model of the effective viscosity should satisfy for shear-thinning fiber suspensions with a Cross-type matrix fluid. Furthermore, we introduce four model candidates with differing numbers of parameters and different theoretical motivations, and use supervised machine learning techniques for non-convex optimization to identify parameter sets for the model candidates. By doing so, we leverage the flexibility of automatic differentiation and the robustness of gradient based, supervised machine learning. Finally, we identify the most suitable model by comparing the prediction accuracy of the model candidates on the fiber orientation triangle, and find that multiple models predict the anisotropic shear-thinning behavior to engineering accuracy over a broad range of shear rates.

An improved method planning path of an autonomous ground robot with using the MBD-RRT*FFT algorithm

The study analyzed the problems associated with planning the path of robots and increasing the accuracy and reliability of their guidance in real time. To solve this problem, a practical study was conducted, based on the obtained analysis, a modification of the asymptotically optimal BD-RRT*FT algorithm and was proposed and improved the MBD-RRT*FFT algorithm. was developed. Which, when applied in dynamic environments due to the use of multi-threaded computations, has better dynamic scheduling capabilities, and the use of Fitch's optimal search result selection algorithm provides a general tendency to choose the optimal straight-line path. To verify the effectiveness of the proposed algorithm improvement method, simulations were carried out using our own simulation program. A comparison of the MBD-RRT*FFT algorithm with other algorithms was carried out on three maps with an evaluation of performance indicators. In order to evaluate the behavior of the program, as well as to identify bottlenecks in the execution of the algorithm, the memory load of the decision-making system was monitored at all stages of the verification. This made it possible to see an increase in the consumption of memory resources by 12-15%, but taking into account the obtained results regarding the optimality and speed of calculations, conclusions were made regarding the minimal impact. Additional understanding of the need for careful selection of the hardware component of decision-making systems is provided. The scientific novelty of the method consists in the application of a scientific and methodological apparatus for improving the BD-RRT*FT algorithm, which ensured further development in terms of expanding and supplementing known data on algorithms with robot path planning and increasing accuracy and reliability in real time. The method includes methods of research, systematization and adjustment of new and previously acquired knowledge.

A short time series rolling bearing fault diagnosis method based on FMTF-CNN

The signal characteristics are extracted directly from the convolutional level when the Convolutional Neural Network (CNN) is used as a fault diagnosis method in most instances. Some scholars have proposed to use the Markov Transition Field (MTF) to extract the signal characteristics before convolution. However, for time-domain signals, the fault characteristics extraction is difficult, and the training process of the neural network parameters is slow. Therefore, in this paper, based on Fast Fourier Transform (FFT) and MTF algorithm, the Fourier Markov Transition Field (FMTF) characteristics extraction method in frequency domain is proposed, and a large number of experiments are carried on the public bearing data set of Jiangnan University. The final verification shows that the FMTF-CNN method has higher accuracy, faster training process and more accurate characteristics extraction than the traditional MTF-CNN diagnosis method.

Design of Fully Spectral CNNs for Efficient FPGA-Based Acceleration.

Computing convolutional layers in the frequency domain using fast Fourier transformation (FFT) has been demonstrated to be effective in reducing the computational complexity of convolutional neural networks (CNNs). Nevertheless, the main challenge of this approach lies in the frequent and repeated transformations between the spatial and frequency domains due to the absence of nonlinear functions in the spectral domain, as such it makes the benefit less attractive for low-latency inference, especially on embedded platforms. To overcome the drawbacks in the existing FFT-based convolution, we propose a fully spectral CNN using a novel spectral-domain adaptive rectified linear unit (ReLU) layer, which completely removes the compute-intensive transformations between the spatial and frequency domains within the network. The proposed fully spectral CNNs maintain the nonlinearity of the spatial CNNs while taking into account the hardware efficiency. We then propose a deeply customized and compute-efficient hardware architecture to accelerate the fully spectral CNN inference on field programmable gate array (FPGA). Different hardware optimizations, such as spectral-domain intralayer and interlayer pipeline techniques, are introduced to further improve the performance of throughput. To achieve a load-balanced pipeline, a design space exploration (DSE) framework is proposed to optimize the resource allocation between hardware modules according to the resource constraints. On an Intel's Arria 10 SX160 FPGA, our optimized accelerator achieves a throughput of 204 Gop/s with 80% of compute efficiency. Compared with the state-of-the-art spatial and FFT-based implementations on the same device, our accelerator is 4×∼ 6.6× and 3.0×∼ 4.4× faster while maintaining a similar level of accuracy across different benchmark datasets.

Effectiveness of Air Admission through Short Pipes in Reducing Pressure Fluctuation of Propeller Hydro-turbine Flow Channel

Abstract When the hydro-turbine operates under part-load conditions, a large-scale vortex flow will form in the draft tube. This vortex can cause low-frequency pressure pulsations that endanger the safe operation of the hydro-turbine. By injecting air into the draft tube, the structure of the vortex can be effectively disrupted, reducing the amplitude of the pressure pulsations. In this paper, a propeller hydro-turbine was studied, and the SST k-ω turbulence model based on the Reynolds averaged Navier-Stokes equations (RANS) was utilized to perform a full-channel unsteady numerical simulation on a part-load condition with the guide vane opening (GVO) of 35°. Then, the fast Fourier transform (FFT) was applied to analyze the pressure fluctuations at monitoring points on the draft tube wall, the head cover, and the nose of the spiral casing. The results indicated that injecting air with an appropriate flow rate into the draft tube through short pipes can reduce the pressure fluctuations on the draft tube wall by 51.4% to 72.5% and decrease the pressure fluctuations on the head cover and the nose of the spiral casing by approximately 40% to 42%. This study proposes a new air admission structure that can enhance the operating stability of the propeller hydro-turbine under part-load conditions.

Nano-architectonics of Pt single-atoms and differently-sized nanoparticles supported by manganese-oxide nanosheets and impact on catalytic and anti-biofilm activities

Hybrid-nanozymes are promising in various applications, but comprehensive comparison of hybrid-nanozymes composed of single-atoms or nanoparticles on the same support has never been made. Here, manganese-oxide nanosheets were loaded with Pt-single-atoms or differently-sized nanoparticles and their oxidase- and-peroxidase activities compared. High-resolution Transmission-Electron-Microscopy and corresponding Fast Fourier Transform imaging showed that Pt-nanoparticles (1.5 nm diameter) had no clear (111) crystal-planes, while larger nanoparticles had clear (111) crystal-planes. X-ray Photo-electron Spectroscopy demonstrated that unloaded nanosheets were composed of MnO2 with a high number of oxygen vacancies (Vo/Mn 0.4). Loading with 7.0 nm Pt-nanoparticles induced a change to Mn2O3, while loading with 1.5 nm nanoparticles increased the number of vacancies (Vo/Mn 1.2). Nanosheets loaded with 3.0 nm Pt-nanoparticles possessed similarly high catalytic activities as Pt-single-atoms. However, loading with 1.5 nm or 7.0 nm Pt-nanoparticles yielded lower catalytic activities. A model is proposed explaining the low catalytic activity of under- and over-sized Pt-nanoparticles as compared with intermediately-sized (3.0 nm) Pt-nanoparticles and single-atoms. Herewith, catalytic activities of hybrid-nanozymes composed of single-atoms and intermediately-sized nanoparticles are put a par, as confirmed here with respect to bacterial biofilm eradication. This conclusion facilitates a balanced choice between using Pt-single-atoms or nanoparticles in further development and application of hybrid-nanozymes.

SegCFT: Context-aware Fourier Transform for efficient semantic segmentation

Semantic segmentation has been one of the most critical tasks in computer vision. Recent works mainly focus on improving segmentation performance by designing high-capacity transformer architectures. They try to solve the high data consumption and computing costs required for model training and deployment in the cloud, but the high computation overhead still makes it difficult to be directly applied to limited resource devices. In this paper, we propose a novel fast Fourier Transform (FFT) based Context-aware Feature Mixer under the transformer-like architecture for precise and efficient semantic segmentation, called SegCFT. Different from the self-attention-based transformer, SegCFT uses a Hierarchical Fourier Transform (HFT) to reduce computational cost via non-parametric calculation and promote segmentation performance by fusing the channel-wise and pixel-wise contexts. To integrate the features from the frequency domain of DFT into the spatial domain of the transformer-like architecture, an Adaptive Modulation Unit (AMU) is designed to modulate the frequency-domain features and ensure consistency between the frequency domain and the spatial domain. Experimental evaluation on two semantic segmentation benchmarks, ADE20k and Cityscapes, shows that SegCFT achieves competitive segmentation performance, while the training and inference costs are superior to the previous methods.

A coupled FEM-FFT concurrent multiscale method for the deformation simulation of CFRPs laminate

Carbon fiber reinforced polymer composites (CFRPs) laminate is increasingly used in aircraft structures and its assembly deformation has great influence on the aircraft aerodynamic profile and mechanical performance. Due to the inherent multiscale nature of CFRPs, the traditional macroscopic models are unable to incorporate their true microstructural details, resulting in them being unable to provide both the macro and micro deformations simultaneously. To address this issue, this work proposed a coupled FEM-FFT concurrent multiscale method for the deformation simulation of CFRPs laminate, where the mechanical response of each macro point of CFRPs laminate was from the corresponding unidirectional representative volume element (UD RVE). The FEM (Finite Element Method) was used for the analysis at the macroscopic scale, while the FFT (Fast Fourier Transform) based method was employed in the UD RVE to speed up the calculation. The elastoplastic behavior of resin matrix in the UD RVE was modelled with the parabolic yield criterion. The proposed concurrent multiscale method was validated respectively by the tension, compression and shearing experiments of CFRPs laminate. The results show that there is a good agreement both for the full strain field and load–displacement curve, demonstrating the effectiveness of the proposed concurrent multiscale method.

An efficient sub-aperture millimeter-wave imaging technique based on boundary-type MIMO array

Millimeter-wave (MMW) 3-D imaging based on multiple-input-multiple-output (MIMO) radar has been widely studied due to its unique advantages in human security screening. However, more research has been done on the use of scanning 1-D MIMO array and the fast imaging methods that match them. This limits the rate of data acquisition and imaging speed to some extent. It is necessary to study frame rate imaging system as well as efficient imaging methods. In this article, a boundary-type MIMO array is designed, based on which an efficient frequency-domain imaging algorithm with sub-aperture is proposed. The proposed algorithm solves the target reflectivity function by dividing the uniform array into two sub-apertures and solving the echo signals of each sub-aperture separately. The final target reflectivity function is obtained by superposing all the solutions. Compared with the advanced RMA, FSA and FωK algorithms, the proposed sub-aperture frequency-domain imaging algorithm (SFA) does not require a large amount of zero-padding to satisfy the requirements of fast Fourier transform (FFT) in all dimensions. It greatly improves the speed of imaging and saves the space required for computation. We derive the theoretical formulation of SFA applied to MIMO array and analyze its computational complexity and phase error. Simulation and experiment verify the efficiency of the proposed SFA.

Image driven deep learning method with FFT solver for predicting the microscale full field stress of stochastic boundary composites

AbstractA novel image‐driven deep learning approach embedded with model‐data‐knowledge information can directly predict the full field stress distribution and mechanical properties of composites solely based on initial scanning geometry images. The fiber spatial distribution and morphological features are identified by Scanning Electron Microscopy (SEM) initially. An effective random fiber generation algorithm is further utilized to generate images database comprising equivalent geometric models of various microstructures. Subsequently, the geometry model images database is analyzed by fast Fourier transform (FFT) to produce the database including the elastic modulus parameters and full field stress distributions of composites with distinct microstructures. Finally, an innovative image‐learning comprehensive paradigm uniting convolutional neural networks and convolutional autoencoders is elaborated systematically to learn the inherent laws of geometry images and field images. The results show that the proposed method have capacity to effectively and accurately predict the elastic modulus and full field stress distributions combined with error analysis even for stochastic boundary composites. The proposed methodology is a symbiosis of cutting‐edge image learning and non‐destructive testing techniques for composites, which can directly use the local non‐destructive or scanning images of composite structural components to quickly obtain field information and further predict damage evolution online.Highlights Proposing a novel deep learning approach combined with FFT directly to predict stress distribution of stochastic boundary composites solely based on micro‐images. Adopting equivalent geometric models dispersed by higher pixels mesh density considering extreme thin interface. Incorporate adaptive algorithms for streamlined training optimization. An innovative integration of deep learning and non‐destructive testing technology for the stress distribution and properties prediction online.

Health Status Detection for Motor Drive Systems Based on Generalized-Layer-Added Principal Component Analysis

Health status detection for motor drive systems includes detecting the working status of the motor and diagnosing open-circuit (OC) faults in the inverter. This paper proposes a generalized-layer-added principle component analysis (GPCA) to determine the load-up/load-shedding status of a motor and diagnose faults in its inverter. Most current methods for detecting OC faults are constrained by changes in the current amplitude and frequency, potentially leading to misjudgments during load-up/load-shedding transient states. The proposed method addresses this issue. Initially, this paper employs a homogenization method to process current data, eliminating the impact of transient processes during motor load-up/load-shedding states on inverter fault diagnosis. Subsequently, the fast Fourier transform (FFT) is used to extract the frequency domain characteristics of the data. If the PCA method is trained with a singular matrix, this can lead to an unreliable result. This paper introduces a generalization layer based on the PCA method, leading to the GPCA method, which enables training with singular matrices. The GPCA method is then developed to compute data features. By presetting thresholds and utilizing the prediction error value and contribution rate index of the GPCA method, the relevant state of the motor drive system can be determined. Finally, through simulations and experiments, it has been demonstrated that the method, using data from the stable working state, can effectively detect the working status of a motor and diagnose OC faults in its inverter, with a diagnostic time of 0.05 current cycles.

Phase division and recognition of crystal HRTEM images based on machine learning and deep learning

The High Resolution Transmission Electron Microscope (HRTEM) images provide valuable insights into the atomic microstructure, dislocation patterns, defects, and phase characteristics of materials. However, the current analysis and research of HRTEM images of crystal materials heavily rely on manual expertise, which is labor-intensive and susceptible to subjective errors. This study proposes a combined machine learning and deep learning approach to automatically partition the same phase regions in crystal HRTEM images. The entire image is traversed by a sliding window to compute the amplitude spectrum of the Fast Fourier Transform (FFT) in each window. The generated data is transformed into a 4-dimensional (4D) format. Principal component analysis (PCA) on this 4D data estimates the number of feature regions. Non-negative matrix factorization (NMF) then decomposes the data into a coefficient matrix representing feature region distribution, and a feature matrix corresponding to the FFT magnitude spectra. Phase recognition based on deep learning enables identifying the phase of each feature region, thereby achieving automatic segmentation and recognition of phase regions in HRTEM images of crystals. Experiments on zirconium and oxide nanoparticle HRTEM images demonstrate the proposed method achieve the consistency of manual analysis. Code and supplementary material are available at https://github.com/rememberBr/HRTEM2.