Two-dimensional Gaussian Function Research Articles

Evaluating wind-farm power generation using a new direct integration of axisymmetric turbine wake

Engineering wake models enable fast calculation of wind-farm power generation including wake-turbine interactions, which is critical for annual energy yield calculations and for farm optimisation either at the design phase or during operation. The operating point of a turbine is determined based on its rotor-averaged wind speed, which is impacted by upstream wakes. The rotor-averaged deficit is typically calculated numerically by using a set of averaging points across the rotor disk, for which a Gaussian shape function is commonly evaluated. This numerical approach can have uncertainty based on the distribution of the averaging points. Also, a large number of averaging points can increase computational cost, which is detrimental for studies that rely on many individual calculations of farm power generation, e.g. layout optimisation. To avoid these drawbacks, we present a novel analytical solution to the circular-disk integration of a two-dimensional axisymmetric Gaussian function depicting upstream wakes for an arbitrary lateral offset between the rotor and the wake source. The analytical expression is compared against numerical averaging of an upstream wake, and is applied to the Horns Rev 1 wind farm showing excellent accuracy. Wind-farm layout optimisation models, especially methods based on gradient-descent optimisation, can benefit from the presented expression significantly, as this provides an analytically differentiable formulation for the rotor-averaged wind speed.

Ecosystem Service Flow Perspective of Urban Green Land: Spatial Simulation and Driving Factors of Cooling Service Flow

The linking of ecosystem service flows (ESFs) with urban land management is still in its nascent stage. The spatial process modeling of ESFs plays a crucial role in establishing connections between urban land sustainability and human benefits. However, the spatial processes and driving mechanisms associated with urban cooling services (UCS) remain ambiguous. In this study, we selected the area within the 6th Ring Road of Beijing as the study area, where the population is highly concentrated and the urban greenery is relatively developed. We modeled the spatial processes of cooling service flow (UCSF) in this area and elucidated the contribution of landscape patterns to UCSF. Firstly, the cooling capacity, referred to as UCS, of the urban blue–green landscape, was estimated using the InVEST tool. Subsequently, the UCSF spatial process was simulated by employing a two-dimensional Gaussian function at the pixel level. In order to characterize the landscape features in the study area, eight landscape indices were selected, and Fragstats v4.2 was employed for their calculation. Finally, GeoDetector was utilized to explore the driving mechanisms of landscape patterns on UCSF. The predominant area for both UCS and UCSF lies between the 5th and 6th Ring Road in Beijing, exhibiting a declining trend from the 6th Ring Road toward the city center. The UCSF coverage area, which represents the beneficiary area, accounted for approximately 87.78% of the study area, with the largest increase occurring within the 2nd Ring Road. The Landscape Shape Index demonstrated the strongest individual contribution to UCSF, while its combined bivariate contribution was significant. Geometry exerted a greater influence on UCSF compared to landscape scale and spatial configuration. This study presents novel insights for assessing the omnidirectional flow of ESFs through the modeling of flow functions. The findings of this study can serve as a valuable reference for sustainable urban landscape management and planning.

A Fourier Galerkin method for ship waves

An efficient approximate solution to the problem of gravity–capillary ship-induced waves is derived using a Fourier Galerkin spectral approach. The proposed method solves linearised equations of finite-depth wave motion generated by a moving pressure distribution in a spectral framework. The method admits ship hulls of various shapes defined by their pressure signature and its Fourier transform in physical and spectral domains, respectively. For the present analysis, the ship hull of a given geometry is represented in a simplified form as a sum of two-dimensional Gaussian functions allowing analysis of interaction of wakes generated by bow-stern hull components or transverse multi-hull systems. An arbitrary movement of the pressure is realised through appropriate mathematical operations of translation and rotation of rigidly coupled component Gaussian forms directly in a spectral domain to preserve high model performance. Thus, the method is capable of giving fast predictions of wakes generated by an arbitrarily moving pressure distribution for different shapes of a simulated vessel.

Visual Axis and Stiles-Crawford Effect Peak Show a Positional Correlation in Normal Eyes: A Cohort Study.

The purpose of this study was to locate the visual axis and evaluate its correlation with the Stiles-Crawford effect (SCE) peak. Ten young, healthy individuals (20 eyes) were enrolled. An optical system was developed to locate the visual axis and measure SCE. To locate the visual axis, 2 small laser spots at 450 nm and 680 nm were co-aligned and delivered to the retina. The participants were asked to move a translatable pinhole until these spots were perceived to overlap each other. The same system assessed SCE at 680 nm using a bipartite, 2-channel (reference and test) Maxwellian-view optical system. The peak positions were estimated using a two-dimensional Gaussian fitting function and correlated with the visual axis positions. Both the visual axis (x = 0.24 ± 0.35 mm, y = -0.16 ± 0.34 mm) and the SCE peak (x = 0.27 ± 0.35 mm, y = -0.15 ± 0.31 mm) showed intersubject variability among the cohort. The SCE peak positions were highly correlated in both the horizontal and vertical meridians to the visual axes (R2 = 0.98 and 0.96 for the x and y coordinates, respectively). Nine of the 10 participants demonstrated mirror symmetry for the coordinates of the visual axis and the SCE peak between the eyes (R2 = 0.71 for the visual axis and 0.76 for the SCE peak). The visual axis and SCE peak locations varied among the participants; however, they were highly correlated with each other for each individual. These findings suggest a potential mechanism underlying the foveal cone photoreceptor alignment.

Atomic-Scale Strucural Analysis of Pt-Based Nanoparticles Using Scanning Transmission Electron Microscopy

Polymer electrolyte fuel cells (PEFCs) is proposed as an alternative to the power generation system in fuel cell vehicles. The performance of the PEFCs is significantly affected by the catalytic activity of the cathode, which contains platinum in general. Recent theoretical studies show that the Pt-Pt interatomic distance at the subsurface of the Pt-based catalysts might considerably influence the catalytic activity. However, experimental evidence remains a challenge due to the requirement of measurement accuracy in picometers at the subsurface. The present work attempts to achieve precise measurements on the Pt-Pt interatomic distance of Pt particles by high-angle annular dark-field (HAADF) imaging in scanning transmission electron microscopy (STEM). Further, we also aim to understand the effects of the change in Pt-Pt interatomic distance on the catalytic activity.A catalyst of 10 wt.% Pt-loaded on graphitized carbon (Sigma-Aldrich) was used in this work. TEM specimen was prepared by dispersing the catalyst on a carbon film supported copper grid. Structural characterizations were examined by a spherical aberration corrected scanning transmission electron microscope (JEM-2100F, JEOL) operated at 200 kV.In order to determine the Pt atomic coordinate accurately, images were first taken under the optimized conditions. Further, a series of images were taken and integrated to form one image with high signal-to-noise ratio and minimized drifting. Finally, the Pt atomic coordinates were extracted precisely on a picometer scale by fitting of the individual atomic columns to two-dimensional Gaussian functions using a custom code developed by MatLab software. To gain insight into the atomic changes in Pt particle, a map of Pt-Pt interatomic displacement vector was constructed and superimposed on the HAADF-STEM image. Intriguingly, a significant Pt-Pt interatomic displacement occurs at the surface of the Pt particle, corresponding to changes in surface morphology and particle size.The state-of-the-art methodology demonstrated here could greatly advance our understanding of surface chemistry of Pt-based catalysts such as Pt3Co alloys. Further understanding of the correlation between the interatomic distances on the catalyst surface and the catalytic activity may give considerable interests and guidelines for future developments of PEFCs.

Structure-Based Analysis of Protein Cluster Size for Super-Resolution Microscopy in the Nervous System

To overcome the diffraction limit and resolve target structures in greater detail, far-field super-resolution techniques such as stochastic optical reconstruction microscopy (STORM) have been developed, and different STORM algorithms have been developed to deal with the various problems that arise. In particular, the effect of the local structure is an important issue. For objects with closely correlated distributions, simple Gaussian-based localization algorithms often used in STORM imaging misinterpret overlapping point spread functions (PSFs) as one, which limits the ability of super-resolution imaging to resolve nanoscale local structures and leads to inaccurate length measurements. The STORM super-resolution images of biological specimens from the cluster-forming proteins in the nervous system were reconstructed for localization-based analysis. Generally, the localization of each fluorophore was determined by two-dimensional Gaussian function fitting. Further, the physical shape of the cluster structure information was incorporated into the size parameter of the localization structure analysis in order to generate structure-based fitting algorithms. In the present study, we proposed a novel, structure-based, super-resolution image analysis method: structure-based analysis (SBA), which combines a structural function and a super-resolution localization algorithm. Using SBA, we estimated the size of fluorescent beads, inclusion proteins, and subtle synaptic structures in both wide-field and STORM images. The results show that SBA has a comparable and often superior performance to the commonly used full width at half maximum (FWHM) parameter. We demonstrated that SBA is able to estimate molecular cluster sizes in far-field super-resolution STORM images, and that SBA was comparable and often superior to FWHM. We also certified that SBA provides size estimations that corroborate previously published electron microscopy data.

A stellar energy extraction method based on exponential intra-pixel sensitivity model

The stability and accuracy of stellar energy extraction are important factors affecting the accuracy of in-orbit radiation calibration based on stars. However, the point spread function (PSF) and the intra-pixel sensitivity (IPS) of the infrared detector will have a great influence on the stability and accuracy of the point target energy extraction. To solve the above problems, this paper proposes a stellar energy extraction method based on an exponential intra-pixel sensitivity model. First, reconstruct the background image based on the multi-frame images to remove the influence of the instrument background. Then, each pixel was divided into 10 × 10 sub-pixels, and the value of each sub-pixel was obtained by fitting the target response value with a two-dimensional Gaussian function. Then, the exponential intra-pixel sensitivity model was established to obtain the IPS value of each sub-pixel, and the target response was deconvolved with the established IPS model. Finally, the star energy is obtained by the aperture photometry method. According to the simulation results, the accuracy of the proposed method is 0.4%, which is 15.6% higher than that without the IPS model; the stability of the proposed method is 0.5%, which is 5.6% higher than that without the IPS model. According to the experimental results using infrared star images observed in orbit, the accuracy of the proposed method is better than 3%, which is 13% higher than that without the IPS model; the stability of the proposed method is better than 3%, which is 5% higher than that without the IPS model. This method can provide a useful reference for the high-accuracy on-orbit star calibration of a large-aperture infrared remote sensing camera in space.

Displacement separation analysis from atomic-resolution images

Structural distortions frequently occur in materials, either periodically (ferroelectric or antiferroelectric) or in local areas (domain boundaries, surfaces/interfaces, dislocations). Measuring atomic displacements from an average lattice is of crucial importance for analyzing structural distortions and their connections to physical properties. Conventionally, the displacements are measured atom-by-atom by fitting atomic-resolution images with two-dimensional gaussian functions. Here, we exhibit an efficient method, named Displacement Separation Analysis, DSA in short, to directly separate atomic displacements from an average lattice based on Fourier space filtering. Using antiferroelectric AgNbO3 as a model system, we demonstrate the consistence between DSA and gaussian fitting. The suppression of polarization at interfacial region of h-LuFeO3/α-Al2O3 heterostructure and the emergence of modulation structure in LuFe2O4+x is then revealed using DSA, attesting the implication of DSA in unveiling structural distortions either locally or periodically. Inspired by the simple principle of DSA, such method can be used for any atomic-resolution images, including TEM, STM, and AFM images to exhibit the atomic displacement intuitively.

GmFace: An explicit function for face image representation

Establishing mathematical models is a ubiquitous and effective method to understand the objective world. Due to the complex physiological structures and dynamic behaviors, the mathematical representation of the human face is an especially challenging task. In this paper, an explicit function called GmFace is proposed for face image representation in the form of a multi-Gaussian function. The model utilizes the advantages of two-dimensional Gaussian function which provides a symmetric bell surface with a controllable shape. The GmNet is then designed using Gaussian functions as neurons, with parameters that correspond to each of the parameters of GmFace in order to transform the problem of GmFace parameter solving into a network optimization problem of GmNet. Furthermore, using GmFace, several face image transformation operations can be realized mathematically through simple parameter computation. Experimental results demonstrate that GmFace has a superior representation ability for face images compared to convolutional autoencoder (CAE), principal component analysis (PCA) and discrete cosine transform (DCT) method.

Generation of Augmented Capillary Network Optical Coherence Tomography Image Data of Human Skin for Deep Learning and Capillary Segmentation.

The segmentation of capillaries in human skin in full-field optical coherence tomography (FF-OCT) images plays a vital role in clinical applications. Recent advances in deep learning techniques have demonstrated a state-of-the-art level of accuracy for the task of automatic medical image segmentation. However, a gigantic amount of annotated data is required for the successful training of deep learning models, which demands a great deal of effort and is costly. To overcome this fundamental problem, an automatic simulation algorithm to generate OCT-like skin image data with augmented capillary networks (ACNs) in a three-dimensional volume (which we called the ACN data) is presented. This algorithm simultaneously acquires augmented FF-OCT and corresponding ground truth images of capillary structures, in which potential functions are introduced to conduct the capillary pathways, and the two-dimensional Gaussian function is utilized to mimic the brightness reflected by capillary blood flow seen in real OCT data. To assess the quality of the ACN data, a U-Net deep learning model was trained by the ACN data and then tested on real in vivo FF-OCT human skin images for capillary segmentation. With properly designed data binarization for predicted image frames, the testing result of real FF-OCT data with respect to the ground truth achieved high scores in performance metrics. This demonstrates that the proposed algorithm is capable of generating ACN data that can imitate real FF-OCT skin images of capillary networks for use in research and deep learning, and that the model for capillary segmentation could be of wide benefit in clinical and biomedical applications.

A new damage estimation method for carbon fiber reinforced polymer based on electrical impedance tomography.

Damage estimation is vital for monitoring the remaining life of carbon fiber reinforced plastic/polymer (CFRP). As a non-invasive, non-radiative, and low-cost method, electrical impedance tomography (EIT) is increasingly investigated for the purpose of structural health monitoring of CFRP. The commonly used EIT method is limited by the image accuracy since it estimates the damage just through a reconstructed image. In this paper, a damage estimation method (DEM) is proposed to quantify the damage location and area. First, each damage is fitted into a two-dimensional Gaussian function through edge fitting. Then, the parameters of the Gaussian function are optimized with the two-norm regularization method. Finally, the damage location and area are estimated with the parameters of the Gaussian function. The accuracy of the DEM is directly evaluated in terms of location error and area error. Both simulation and experimental results demonstrated the potential of the proposed method in providing damage estimation information.

Pipeline corrosion defect parameterisation with magnetic flux leakage inspection: a contextual representation approach

Corrosion is one of the significant reasons for oil and gas pipeline failures. In pipeline integrity management programmes, the magnetic flux leakage (MFL) technique is widely used to detect and quantify corrosion defects. The inspection results of MFL list the profiles of individual corrosion defects; however, the structural safety of a pipeline not only depends on the size of individual corrosion defects but also the pattern of closely spaced defects. To achieve a contextual defect representation, the concept of parameterisation, which considers the adjacent defects as additional information of the central defect, is proposed in this study. The process through which to realise this contextual representation is described in this paper. Three parameterisation models are proposed and a two-dimensional Gaussian function is employed to model the interaction strength between adjacent defects. The experimental results demonstrate that the shape context (SC) model associated with the interaction strength function (ISF) shares the highest similarity with human inspectors in comparison with the other two models. Thus, the proposed parameterisation approach can be used to retrieve similar corrosion defects and analyse defect population distribution along a pipeline.

Fast and robust pavement crack distress segmentation utilizing steerable filtering and local order energy

Pavement crack plays an important role in estimating pavement conditions and implementing pavement maintenance management. There are three shortcomings summarized. (1) Low contrast between crack pixels with surrounding pixels; (2) intensity values along crack discontinuous; (3) complexity structure instead of unitary and unbranched cracks. The proposed algorithm, including three essential steps to address the above disadvantages. A fast nonlocal means denoising method (FNLM) is first proposed to eliminate Gaussian isolated noise while retains target pixels. Second, the basic filter named steerable filter (SF), which calculates the maximal response with different orientations at a certain position, is made up of the second partial derivatives of a two-dimensional Gaussian function. Then the crack saliency map is obviously generated. Last, the local order energy (LOF) is applied to extract crack features from the saliency map to generation binary image, and the mathematical morphology operation method is utilized to locate the location and orientation of crack. Different types of pavement images are applied to estimate the performance of the proposed algorithm, and the experimental results represent the proposed method can achieve fast and robust performance.

An empirical, Bayesian approach to modelling crop yield: Maize in USA

We apply an empirical, data-driven approach for describing crop yield as a function of monthly temperature and precipitation by employing generative probabilistic models with parameters determined through Bayesian inference. Our approach is applied to state-scale maize yield and meteorological data for the US Corn Belt from 1981 to 2014 as an exemplar, but would be readily transferable to other crops, locations and spatial scales. Experimentation with a number of models shows that maize growth rates can be characterised by a two-dimensional Gaussian function of temperature and precipitation with monthly contributions accumulated over the growing period. This approach accounts for non-linear growth responses to the individual meteorological variables, and allows for interactions between them. Our models correctly identify that temperature and precipitation have the largest impact on yield in the six months prior to the harvest, in agreement with the typical growing season for US maize (April to September). Maximal growth rates occur for monthly mean temperature 18 °C–19 °C, corresponding to a daily maximum temperature of 24 °C–25 °C (in broad agreement with previous work) and monthly total precipitation 115 mm. Our approach also provides a self-consistent way of investigating climate change impacts on current US maize varieties in the absence of adaptation measures. Keeping precipitation and growing area fixed, a temperature increase of 2 °C, relative to 1981–2014, results in the mean yield decreasing by 8%, while the yield variance increases by a factor of around 3. We thus provide a flexible, data-driven framework for exploring the impacts of natural climate variability and climate change on globally significant crops based on their observed behaviour. In concert with other approaches, this can help inform the development of adaptation strategies that will ensure food security under a changing climate.

Mpfit: a robust method for fitting atomic resolution images with multiple Gaussian peaks

The standard technique for sub-pixel estimation of atom positions from atomic resolution scanning transmission electron microscopy images relies on fitting intensity maxima or minima with a two-dimensional Gaussian function. While this is a widespread method of measurement, it can be error prone in images with non-zero aberrations, strong intensity differences between adjacent atoms or in situations where the neighboring atom positions approach the resolution limit of the microscope. Here we demonstrate mpfit, an atom finding algorithm that iteratively calculates a series of overlapping two-dimensional Gaussian functions to fit the experimental dataset and then subsequently uses a subset of the calculated Gaussian functions to perform sub-pixel refinement of atom positions. Based on both simulated and experimental datasets presented in this work, this approach gives lower errors when compared to the commonly used single Gaussian peak fitting approach and demonstrates increased robustness over a wider range of experimental conditions.

Automatic detection of risk zones in diabetic foot soles by processing thermographic images taken in an uncontrolled environment

Diabetes mellitus has become a global healthcare issue with incidence levels growing exponentially each year. Diabetic foot is one of the complications related with the ailment, and if it is not attended in time, it can progressively deteriorate to a condition that necessitates foot amputation. This study aimed to establish a non-invasive monitoring system for diabetic foot. This proposed system detects and classifies temperature differences in foot sole zones as ulcerous if >2.2 °C and necrotic if <−2.2 °C, by processing thermographic images of foot soles. This is achieved without homogeneous background or room temperature control. There are reports of systems that are designed to work under controlled environments; however, their performance declines or falters altogether under uncontrolled, home environments.The system proposed in this paper combines step-by-step and end-to-end algorithms that compensate for the limitations in data and, at the same time, enhance performance. It uses deep-learning techniques to segment visible-spectrum images using a retrained Mask R-CNN model, which is adjusted with 141 images to segment foot soles. These results are used over the temperature matrix in order to isolate the foot sole temperatures. The visualisation and classification methods use step-by-step algorithms with comparisons of homologous foot sole regions, convolutions with a two-dimensional (2D) Gaussian function, filters that process and compare areas of 1.5 cm2 over each foot sole, and a thermal threshold to differentiate ulcerous from necrotic zones.The results illustrated a detection accuracy of 90% for ulcers and 88% for necrosis, while the labelled areas had an error of 7.05% and 10%, respectively. These results demonstrated that the system is capable of successfully detecting and visualising the specific temperature differences over samples under an uncontrolled environment.

The 3D Structure of CO Depletion in High-mass Prestellar Regions

Abstract Disentangling the different stages of the star formation process, in particular in the high-mass regime, is a challenge in astrophysics. Chemical clocks could help alleviate this problem, but their evolution strongly depends on many parameters, leading to degeneracy in the interpretation of the observational data. One of these uncertainties is the degree of CO depletion. We present here the first self-consistent magneto-hydrodynamic simulations of high-mass, star-forming regions at different scales, fully coupled with a nonequilibrium chemical network, which includes C–N–O bearing molecules. Depletion and desorption processes are treated time dependently. The results show that full CO depletion (i.e., all gas-phase CO frozen-out on the surface of dust grains) can be reached very quickly, in one-third or even smaller fractions of the freefall time, whether the collapse proceeds on slow or fast timescales. This leads to a high level of deuteration in a short time, both for typical tracers like N2H+, as well as for the main ion H 3 + , the latter being in general larger and more extended. N2 depletion is slightly less efficient, and no direct effects on N-bearing molecules and deuterium fractionation are observed. We show that CO depletion is not the only driver of deuteration, and that there is a strong impact on D frac when changing the grain size. We finally apply a two-dimensional Gaussian point-spread function to our results to mimic observations with single-dish and interferometers. Our findings suggest that the low-values observed in high-mass star-forming clumps are in reality masking a full-depletion stage in the inner 0.1 pc region.

EL: Local Image Descriptor Based on Extreme Responses to Partial Derivatives of 2D Gaussian Function

We propose a two-part local image descriptor EL (Edges and Lines), based on the strongest image responses to the first- and second-order partial derivatives of the two-dimensional Gaussian function. Using the steering theorems, the proposed method finds the filter orientations giving the strongest image responses. The orientations are quantized, and the magnitudes of the image responses are histogrammed. Iterative adaptive thresholding of histogram values is then applied to normalize the histogram, thereby making the descriptor robust to nonlinear illumination changes. The two-part descriptor is empirically evaluated on the HPatches benchmark for three different tasks, namely, patch verification, image matching, and patch retrieval. The proposed EL descriptor outperforms the traditional descriptors such as SIFT and RootSIFT on all three evaluation tasks and the deep-learning-based descriptors DeepCompare, DeepDesc, and TFeat on the tasks of image matching and patch retrieval.

Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

Ring-shaped acoustic black holes for broadband vibration isolation in plates

Acoustic black holes (ABHs) in plates have shown great potential in a variety of applications that range from passive noise and vibration reduction, to energy harvesting thanks to localization, or to unusual flexural wave manipulation like lensing, or negative refraction and bi-refraction. The ABH effect can be typically achieved in plates by embedding regular arrays of circular cuneate indentations, with power-law profile. In this paper, we suggest new ring-shaped ABH designs that may be used for vibration isolation. Many common situations in the vibroacoustics of built-up structures involve wave propagation in plates excited at a small source area. This could be the case, for instance, of a beam/plate connection. It is herein proposed to surround the plate excitation region by means of ring-shaped ABHs to prevent the transmission of vibrations outside them. Several configurations of ABHs are tested, from concentric annular ABHs in different number and sizes to traditional circular ABHs in a ring arrangement. The inclusion of stiffeners to prevent excessive structural plate weakness due to the ABH indentations is also addressed. The performance of the ABH designs are analyzed by means of a semi-analytical approach that uses two-dimensional Gaussian functions to approximate the plate flexural displacement field, in the framework of the Rayleigh-Ritz method. The annular ABHs are shown to exhibit remarkable good isolation for the whole frequency range. An explanation is provided for their behavior.