The evaporation of droplets is an important process not only in industrial and scientific applications, but also in the airborne transmission of viruses and other infectious agents. We derive analytical and semi-analytical solutions of the coupled heat and mass diffusion equations within a spherical droplet and in the ambient vapor phase that describe the evaporation process of aqueous free droplets containing nonvolatile solutes. Our results demonstrate that the solute-induced water vapor-pressure reduction considerably slows down the evaporation process and dominates the solute-concentration dependence of the droplet evaporation time. The evaporation-induced enhanced solute concentration near the droplet surface, which is accounted for using a two-stage evaporation description, is found to further slow-down the drying process. On the other hand, the presence of solutes is found to produce a lower limit for the droplet size that can be reached by evaporation and, also, to reduce evaporation cooling of the droplet, which tend to decrease the evaporation time. Overall, the first two effects are dominant, meaning that the droplet evaporation time increases in the presence of solutes. Local variation of the water diffusivity inside the droplet near its surface, which is a consequence of the solute-concentration dependence of the diffusion coefficient, does not significantly change the evaporation time. Crust formation on the droplet surface increases the final equilibrium size of the droplet by producing a hollow spherical particle, the outer radius of which is determined as well. Saturated water vapor concentrationSolute concentrationWater vapor concentrationLiquid water concentrationAmbient water vapor concentrationSolute concentration at the equilibrium stateInitial solute concentrationSolute concentration at the onset of crust formationInitial water concentration inside the dropletWater diffusion constant in airSolute diffusion constant in liquid waterWater diffusion constant in pure liquid waterWater diffusion constant in water solutionInitial water diffusion constant inside the dropletMolecular evaporation enthalpy of waterTotal evaporation fluxEvaporation flux densityTotal heat flux into the dropletCondensation reaction rate constantEvaporation reaction rate constantAvogadro constantDroplet radiusRadial distance from the droplet centerEquilibrium radius of the dropletRadius of the inner coreInitial radius of the dropletDroplet radius at the end of the first drying stageRadius of the inner core at the end of the first drying stageRelative humidityTimeDroplet surface temperatureAmbient temperatureLiquid water molecular volumeActivity coefficient of waterEvaporation induced temperature reduction at the droplet surfaceHeat conductivity of airEvaporation time of a pure water dropletEvaporation time of a solute-containing droplet in the presence of internal concentration gradientsThe time of the first drying stageThe time of the second drying stageMomentary volume fraction of solutesFinal equilibrium volume fraction of solutesVolume fraction of solutes in the inner coreInitial volume fraction of solutesInitial solute volume fraction at which the evaporation time is maximalEvaporation-cooling factorEvaporation-cooling factor in the absence of solutes °C Ref.  °C Ref. I.Evaporation from solute-containing droplets is an important process in many industrial and scientific applications ranging from pharmacology, agriculture, food, and cosmetics production to medical, biochemical, material, and soil sciences.In addition, the droplet-evaporation process plays an important role in the transmission of infectious diseases and respiratory viruses via the airborne route,Although there are many studieset al.et al.et al.et al.et al.et al.et al.et al.,R2-law, which assumes that the droplet temperature and thereby also the droplet evaporation rate remain constant during the evaporation process and, thus, the droplet surface decreases linearly in time.R2-law were accounted for in a model developed by Eslamian et al.,R2-law model.There are a few studies that address concentration gradients inside a solute-containing drying dropletThe evaporation-induced concentration gradients inside drying droplets are found to strongly affect the drying process not only by decreasing the evaporation rate,Depending on the type of solutes, the crust that forms might be either a dry crust μm.In this paper, a droplet evaporation model is presented by solving the heat and mass diffusion equations analytically or semi-analytically to describe the various effects nonvolatile solutes have on the drying process. Starting from the simple model scenario that only accounts for the solute-induced minimal water concentration inside the droplet, we gradually add complexities to our model to address different factors associated with the presence of solutes, i.e., the solute-induced water vapor-pressure reduction, effects of solutes on evaporation cooling, evaporation-induced water concentration gradients inside the droplet, and the dependence of the internal water diffusivity on the solute concentration. We also include the possibility of crust formation in our calculations to evaluate its effect on the droplet equilibrium size. All our calculations are done in the one-phase regime, meaning from the beginning of the evaporation process up to the point where crust formation sets in at the surface, and for a single droplet at rest, which defines the so-called stagnant-flow approximation and is valid for droplet radii up to 60 II.supplementary material Sec. B.1, the coupled diffusion equations for solvent and solutes within an incompressible solution are simultaneously solvable only when the diffusivities of the two components are the same. In the presence of slowly diffusing solutes, such as polymers or proteins, the water dynamics becomes slaved to the solute dynamics and thus one can approximately describe the diffusion dynamics by a single diffusion equation with a reduced effective diffusion coefficient, which is the approach we are taking. The exact problem is much more complicated and requires the solution of two coupled diffusion equations while accounting for the finite solution compressibility.The main objective of the present study is to model water evaporation from a solute-containing aqueous droplet that is sedimenting in air. For this, the time-dependent water–vapor concentration profile outside the droplet and the liquid–water and solute concentration profiles inside the droplet are determined from the diffusion equation, which for spherical symmetry depends on the radial coordinate and readsAn important factor that affects the droplet evaporation rate is the evaporation-induced cooling of the droplet, which decreases the temperature and thereby also the water vapor concentration at the droplet surface. To account for this, one needs to calculate the temperature distribution around the evaporating droplet from the heat-conduction equation in radial coordinates,supplementary material.In the following, the results obtained from solutions of the above equations at various approximation levels are discussed. Derivations of all equations are provided in the A.supplementary material Sec. A, the droplet radius R is determined by the differential equationsupplementary material Sec. A.4)supplementary material Sec. A.3)For later comparison with our main results, we first assume that the particle diffusion inside the droplet is sufficiently rapid, so that the water concentration at the droplet surface equals the mean water concentration in the droplet. If one furthermore assumes that the water diffusion in the vapor phase is fast so that at each instant of time a stationary water–vapor concentration profile is obtained, the time-dependent variation of the droplet radius due to the water evaporation results from the stationary solution of the coupled water–vapor-diffusion and heat-conduction equations Eqs.  outside supplementary material Sec. A.4 for more details), the radius-dependent evaporation time To make the differential equation R2-law that is defined by The logarithmic term in Eq. As discussed above, Eq. 1.R2-law,To account for the solute-induced water–vapor pressure reduction, the logarithmic term in Eq.  see Eq.  decreaseR2-law is almost valid and thus Eq. 2.supplementary material Sec. A.3) asThe evaporation-induced cooling of the droplet decreases the temperature at the droplet surface compared to the ambient air by So far, we used Eq. To better understand how the solute-concentration dependence of evaporation cooling affects the droplet evaporation time, the evaporation time is now calculated from the numerical solution of Eq.  see Eq. . The appB.supplementary material Sec. B.1, the diffusion equations for solvent and solutes in an incompressible fluid mixture are simultaneously solvable only if the diffusion coefficients of all components in the liquid mixture are equal. Here, we use this assumption, which approximates the behavior of fast-diffusing solutes rather accurately. Accordingly, the liquid water diffusion coefficient In the previous section, the particle diffusivity inside the droplet was assumed to be sufficiently rapid, so that the solute remains evenly distributed. In the fast evaporation scenario see , howeverThe general problem one encounters when considering diffusion in radial coordinates is that the diffusion equation Eq.  has a siAdditionally, we consider a two-stage evaporation model. In the first stage , the drond stage . In thissupplementary material Sec. B)We first assume that the water diffusion coefficient in the liquid phase supplementary material Sec. B.4, mass conservation of water and solutes gives rise to the following differential equations that describe the time-dependent variation of the droplet radius in the two drying stages,supplementary material Sec. B.4 to yield Neglecting the probability of crust formation due to the increased solute concentration at the droplet surface (which is discussed later), one can calculate the droplet evaporation time as Ri° see . Figure supplementary material Sec. B.1 for more details). Using this assumption and neglecting the concentration dependence of the diffusion coefficient, which will be discussed next, the value reported in the Nomenclature for the water diffusion coefficient in pure water at 25 °C, As stated before, our diffusion model is valid only when we assume that the diffusion coefficients of water and solutes in the liquid phase are equal,  see Eq. . The relAs mentioned before, the differential equations Finally, the effect of an internal concentration gradient on the time-dependent variation of the droplet radius is investigated. The point to note here is that in the case of a droplet with a finite internal water diffusivity, the first drying stage is dominant. Indeed, when 1.The diffusion coefficient of water is knownAccording to Eq. supplementary material Sec. C to solve this equation along with the heat and mass conservation equations, the time-dependent radius of the evaporating droplet is determined. Panels (a)–(c) of By incorporating The presence of solutes inside a drying droplet has two effects on the internal water diffusivity: first, it changes the water diffusion coefficient in the droplet at the onset of evaporation, when the water concentration is homogeneous, and second, it induces a time-dependent diffusivity profile inside the droplet. Accordingly, Eq. alue see , csi is hell see . We firssupplementary material Sec. B) by As follows from Eq. So far, we examined various effects of nonvolatile solutes on the droplet evaporation process, without considering the possibility of crust formation (which is discussed next). Numerical examples for evaporation times obtained from the relevant equations are given in 2.The results presented so far are obtained without accounting for the possibility of crust formation. Depending on the evaporation rate, temperature, and type of the solutes, a solid solute crust can form, which dramatically affects the drying process and the time it takes for the droplet to reach the evaporation equilibrium state. It is, therefore, of importance to determine when a crust forms on the droplet surface and how it affects the droplet evaporation kinetics. It is also of interest to examine whether the concentration-dependent variation of the internal water diffusivity delays or accelerates the crust formation.A solid crust is known to form as a consequence of solute crystallization which, depending on the nucleation mechanism,Another assumption made here is that the outer radius of the droplet remains constant once the crust has formed, meaning that the possibility of crust collapse [file Eq. , the rat see Eq. .According to  see Eq. . Figure supplementary material Sec. C.2 for both To examine how the solute-concentration dependence of the internal water diffusivity affects the results, the equilibrium radius is calculated using the numerical method described in the  see Eq. . The resThe next step would be to model the evaporation process in the presence of a crust. Due to the diversity of possible crust structures and lack of experimental data on the crust properties, this step is difficult. As discussed in Sec. 1, depending on the type of solutes, the crust formed during the drying process might be a dry crust, which forms due to crystallization of salts in a salt solution, or a gel-like wet skin, which is expected to form on the surface of polymer-containing droplets. In the case of dry salt crusts, the evaporation process would continue through capillary action.III.R2-law, which assumes that the droplet temperature and thereby also the evaporation rate remains constant during the drying process.This paper reports analytical and semi-analytical solutions of the coupled heat and mass diffusion equations and investigates the effects of nonvolatile solutes on water evaporation from solute-containing aqueous droplets. The model accounts for evaporation cooling, internal water concentration gradients, solute-induced water vapor-pressure reduction, and the solute-concentration dependence of the water diffusivity inside the droplet. For this, the evaporating droplet is assumed to contain two regions: an outer shell, where the liquid phase exhibits an inhomogeneous water concentration, and an inner core, where the water concentration profile is uniform. Accordingly, the water evaporation is considered as a two-stage process. In the first stage, the outer shell grows from the surface toward the core of the droplet. When the shell covers almost the entire droplet, evaporation turns into the second stage, where the water concentration in the inner core gradually decreases to its equilibrium value. Our model accounts for solute-induced deviations from the classical Our investigations indicate that the presence of nonvolatile solutes within a drying droplet leads to opposing effects on the evaporation speed that partly compensate each other. The dominant effect is the water vapor-pressure reduction due to the reduced water concentration in the presence of solutes, which considerably slows down the evaporation process. Finite diffusivity of water and solute molecules within the droplet, which gives rise to a water concentration gradient in the droplet, further intensifies the solute-induced slowing down of the droplet evaporation. On the other hand, the presence of solutes produces a lower limit for the droplet radius that can be reached by evaporation, which tends to reduce the droplet evaporation time. Furthermore, the evaporation-induced increase in the solute concentration inside the droplet reduces the evaporation cooling, which slightly accelerates the droplet evaporation process. According to our results, the latter two effects are subdominant compared to the first two, but still considerable, meaning that the droplet evaporation time effectively increases in the presence of nonvolatile solutes.The solute-induced slowing down of the droplet evaporation process can be intensified by any factor that decreases the water diffusivity inside the droplet, such as the presence of strongly hydrated solutes. The inhomogeneous diffusivity profile inside the droplet arising from the solute-concentration dependence of the water diffusion coefficient influences the droplet evaporation process. Our results, however, reveal that this factor plays only a minor role in determining the droplet evaporation rate and can be safely neglected. In fact, the presence of strongly hydrated solutes on the one hand decreases the water-diffusion coefficient at the droplet surface due to the increased solute concentration at the surface. On the other hand, the presence of such solutes decreases the growth rate of the outer shell, where the water concentration is inhomogeneous, which leads to an increase in the water concentration gradient at the droplet surface. As a result, the water evaporation flux, which is the product of the water-diffusion coefficient and the water concentration gradient at the droplet surface, remains almost independent of the solute-induced variations of the internal water diffusivity.The physical–chemical effects that control the droplet evaporation process are dominated by various parameters, among which the initial solute volume fraction and the ambient relative humidity play key roles in determining the droplet evaporation time and the final equilibrium size of the evaporating droplet. The droplet evaporation time increases in humid environments due to the reduced evaporation rate. The effect of the initial solute volume fraction When the solute concentration at the droplet surface exceeds a critical supersaturation concentration, a solid crust forms on the droplet surface, which decisively influences the evaporation kinetics. The crust formation also affects the morphology of the dry residue that remains after evaporation by producing a hollow particle with a size larger than expected in the absence of crust formation. Assuming that the droplet retains its spherical shape during the evaporation process and neglecting the possibility of crust collapse or wrinkling, the outer radius of the hollow particle formed after crust formation is predicted analytically. Our results reveal three factors that can delay the crust formation and thus reduce the final size of the hollow particle: a decrease in the initial solute volume fraction, an increase in the relative humidity, or a decrease in the water activity coefficient. The latter factor represents a reduced volatility of water molecules due to non-ideal effects caused by water–solute and solute–solute interactions. Our results also indicate that the crust formation is slightly accelerated in the presence of solutes that decrease the internal water diffusivity, e.g., strongly hydrated solutes, and slightly delayed in the presence of solutes that increase the internal water diffusivity, e.g., weakly hydrated solutes. This effect is, however, insignificant and can be safely neglected.Although the present study provides insight into the effects of nonvolatile solutes on the evaporation process of solute-containing droplets, further work is required to answer a number of important open questions: (I) How does the water activity coefficient vary during the lifetime of the droplet and how do non-ideal effects due to solute–water interactions affect the drying process? (II) How does the evaporation process continue after crust formation? What is the exact mechanism of water evaporation in the presence of a dry crust? Does it involve capillary action? If so, more information about the crust porosity, its characteristic pore size, and the amount of water located inside the pores is required. (III) What happens after the formation of a gel-like skin that includes saturated polymers?supplementary material for the calculation of the droplet evaporation time with and without considering the evaporation cooling effect, the evaporation-induced concentration gradient inside the droplet, and the solute-concentration dependence of the internal water diffusivity.See the


Cardiac MRI left ventricular (LV) detection is frequently employed to assist cardiac registration or segmentation in computer-aided diagnosis of heart diseases. Focusing on the challenging problems in LV detection, such as the large span and varying size of LV areas in MRI, as well as the heterogeneous myocardial and blood pool parts in LV areas, a convolutional neural network (CNN) detection method combining discriminative dictionary learning and sequence tracking is proposed in this paper. To efficiently represent the different sub-objects in LV area, the method deploys discriminant dictionary to classify the superpixel oversegmented regions, then the target LV region is constructed by label merging and multi-scale adaptive anchors are generated in the target region for handling the varying sizes. Combining with non-differential anchors in regional proposal network, the left ventricle object is localized by the CNN based regression and classification strategy. In order to solve the problem of slow classification speed of discriminative dictionary, a fast generation module of left ventricular scale adaptive anchors based on sequence tracking is also proposed on the same individual. The method and its variants were tested on the heart atlas data set. Experimental results verified the effectiveness of the proposed method and according to some evaluation indicators, it obtained 92.95% in AP50 metric and it was the most competitive result compared to typical related methods. The combination of discriminative dictionary learning and scale adaptive anchor improves adaptability of the proposed algorithm to the varying left ventricular areas. This study would be beneficial in some cardiac image processing such as region-of-interest cropping and left ventricle volume measurement. Anatomical or tissue object detection has receiving increasing attention in medical image analysis field since it facilitates to isolate the area of interest from the background and determine the relevant position and category in medical images ,2,3. In So far, a limited number of automatic cardiac object detection methods have been proposed ,5 and soIn this paper, we propose a convolutional neural network (CNN) method specifically designed to detect the left ventricle object in cardiac MRI slices. Our approach takes the detection into coarse oversegmentation location, discriminative dictinary learning-based location adjustment, and fine CNN-based detection parts with interpretability. The unsupervised SLIC (simple linear iterative cluster) algorithm is introduced to obtain initial oversegmentation because it has compact structure, good edge fitness, low complexity, and flexible adjustment. In addition, it does not require a preliminary training process in comparison to the CNN-based region proposal subnetwork. The discriminative dictionary learning can efficiently represent the different sub-objects in left ventricle area, and the sequence tracking fast generates left ventricular scale adaptive anchors on the same individual to solve the problem of slow classification speed of discriminative dictionary. Once multi-scale adaptive anchors are generated in the target region for handling the varying sizes, they are trained in regional proposal network, and the left ventricle object is localized by the CNN based regression and classification strategy. In the following sections we will present related works, our motivation, the details of our method, and experimental validation.Traditionally, the object detection methods in the computer vision field can be directly applied for left ventricular detection. The representative method was originally developed for face detection, which introduces a sliding window to traverse the entire image, designs single or multiple handcrafted features  based detection methods have been investigated and they achieved more satisfactory performance in nature images, these methods can be mainly divided into one-phase based, two-phase based, and anchor-free based methods.One-stage object detection methods: This category aims to directly classify and regress the sliding windows, such as Yolo (you only look once) series ,14,15, STwo-stage object detection methods: This category intends to generate proposals separately and then perform regression and classification. Representative algorithms include RCNN  , Fast RCAnchor-free methods: This category is to directly predict the position of objects by inferring key point locations, such as Cornernet  and CentThere are less reports of traditional methods on the detection of left ventricle in cardiac MRI. Focusing more on the location of left ventricle , automatThe design of many general frameworks in target detection does not depend on image categories. Although they can be transferred to medical MRI after modification, it is worth noting that medical MRI can only be obtained through professional medical equipment, unlike natural images, which can easily obtain a large number of data. In many cases, the natural images are sent to deep CNN such as Resnet152, Resnet101, or Resnet50  for achiBased on the above considerations, this paper focuses on the two-stage convolutional network algorithms for left ventricle detection in cardiac MRI, where vgg-16  is takenAmong the classical two-stage detection algorithms, Faster RCNN shows state-of-the-art performance. However, it has certain limitations when it is directly applied to cardiac MRI left ventricle detection. (1) The shape and size of the left ventricle vary greatly in cardiac MRI slices. As shown in (2) The capture of perception area of RPN in Faster RCNN is weak. As shown in (3) The distribution of organs and tissues in cardiac MRI is complicated, and some organs and left ventricle have little difference. RPN generates non-differential anchors in the original image, which leads to misdetection of similar regions of the left ventricle. At the same time, due to the lack of adaptability of the generated anchors to small-sized ventricular regions, missed detection is prone to occur.In view of the above problems, this paper proposes a CNN left ventricular detection algorithm combining discriminant dictionary learning and sequence tracking. The algorithm applies a discriminant dictionary learning to classify the superpixel oversegmented regions, then merge the target region of left ventricle, and generate multiscale adaptive anchors in the target region. Combining with non-differential anchors in RPN, the left ventricle is detected by CNN regression and classification strategy. In order to solve the problem of slow classification speed of discriminative dictionary, a fast generation module of left ventricular scale adaptive anchors based on sequence tracking is proposed on the same individual. Specifically, this paper provides three contributions as follows.(1) During the training phase, this paper classifies the segmented superpixel regions by discriminative dictionary, fuses the same labels to construct the left ventricular candidate region, generates scale adaptive anchors in the candidate region, and combines anchors in RPN training to increase the number of positive samples and alleviate the imbalance problem of positive and negative samples, randomly perturb the scale adaptive anchors, construct the diversity of positive samples, obtain some high-quality negative samples, add them into the proposal layer to enhance the network’s discrimination of hard negative samples.(2) During the detection phase, the adaptive anchors of the left ventricular region scale are mapped to RPN and proposal layers, respectively, to effectively prevent false detection and missed detection and enhance the robustness of the network to small-sized ventricular regions.(3) Considering the slow classification speed of discriminative dictionary, this paper proposes a rapid generation module of scale adaptive anchors applicable to the left ventricle region in the same individual according to the idea in discriminative scale space tracking . It can The rest of this article is organized as follows. The second section presents the details of the proposed method, and the third section reports the experimental results and makes discussion. Finally, the fourth section concludes the study.The proposed overall framework of left ventricular detection in cardiac MRI is shown in (1) Left ventricle candidate region building module. This module adopts the unsupervised SLIC algorithm  to segme(2) Detection network module. This module follows the network structure of Faster RCNN, and uses vgg-16 to extract features. In the training phase, the left ventricular candidate region scale adaptive anchors are added to the RPN, and the diversity of positive samples is constructed by randomly perturbing the scale adaptive anchors, while generating some high-quality negative samples to be added to the proposal layer. During the test phase, anchors are mapped to the RPN and the proposal layer.(3) Fast generation module of left ventricular scale adaptive anchors. According to the detection results of the current frame, all MRI slices of the same individual are stacked along the short axis. The module can quickly generate left ventricular scale adaptive anchors on each cardiac MRI along the frame, which significantly saves time cost and achieves high accuracy.The details of the modules will be presented as follows.This module aims to automatically build the region candidates containing left ventricles based on oversegmentation and region merging techniques. As shown in The initial oversegmentation is to generate a pile of superpixel candidates fast in an explainable way by borrowing the merits of the structural tightness and edge fitness of the cardiac images.Due to the imperfection of magnetic resonance equipment, cardiac MRI is prone to artifacts and noise. In the preprocessing stage, the SLIC superpixel segmentation algorithm utilizes the redundant information between pixels and the effects of artifacts and noise can be eliminated through feature similarity. We use unsupervised SLIC oversegmentation to obtain initial superpixels because it has compact structure, good edge fitness, and flexible adjustment. In addition, the algorithm has low complexity and does not require a preliminary training process. Nevertheless, it is necessary to fuse them since the structural organization is significantly different and the superpixels do not cover the true left ventricle region exactly. Furthermore, according to the results of K-means clustering, the superpixel regions can be coarsely divided into six types related to left ventricle regions and one to background region. In this way, these seven types of regions will be further classified to their desired labels by learning strategies.The oversegmented regions usually differ from the true regions due to the large gray discrepancy between the myocardium and the blood pool as well as the intensity inhomogeneity in the surrounding tissues. To handle this, we propose a discriminative dictionary learning method to fuse the myocardium and blood pool into the same region and then improve the accuracy of scale adaptive anchors generation which mainly depends on the results of these oversegmented regions. Traditional dictionary learning method is not suitable for this problem since it lacks discriminative power for each type of data. If the dictionary specific to each kind of data is learned, the sub-dictionaries are able to represent the data with same properties , while their representation ability for other kinds is poor. However, the nearby superpixels in the sub-dictionaries are usually overlapped to represent the partial object and when the left ventricle label is incorporated to enhance the class information in advance, the myocardium and blood pool will be taken into a same type for training a certain type of sub-dictionary in the dictionary, so it becomes possible to put both into one category in testing stage.i-th sub-dictionary of D, which corresponds to a certain type of superpixel region, denoting the superpixel region, D, and i-th sub-dictionary of F-norm of the remaining sub-dictionaries as small as possible.The discriminative dictionary learning can explicitly constrain the type of atoms in the dictionary to build a discriminative sub-dictionary , so we uf is defined asm and i-th class, respectively. i-th class. By making the intra-class error as small as possible and the inter-class error as large as possible in the sparse coding, The discriminant sparse coding term So Equation  can be tD is fixed, X can be updated by using the IMP algorithm [The solution of the model is divided into two parts. lgorithm , that ism and k-th class of sparse coding, respectively. When X is fixed, then Y on It should be noted the remaining D. In each type of sub-dictionary n atoms as D, and there are totally (1) Initialize dictionary D, update the sparse coding (2) Fix Equation .X, update dictionary D, and calculate it by using Equation ((3) Fix Equation .D and X in two adjacent iterations are close enough, or until the maximum number of iterations, then output X and D.(4) Return to step 2 and iteratively update until the values of y and excludes the influence of the remaining sub-dictionaries. Let y on dictionary D and it is solved as followsIn the testing stage, a single class of sub-dictionary is used to reconstruct a certain superpixel region Furthermore, the label is estimated by combining reconstruction error and sparse coding error as follows,Since initial superpixel region of each image is able to be labeled according to the discriminative dictionary learning, the objective of fusion strategy is to merge the regions with the same label. In the ideal case, the left ventricular region can be completed distinguished by discriminating the superpixel region of the ventricle through a discriminative dictionary, but in practice, due to the complex distribution of various tissues and organs in cardiac MRI images, as well as the subtle differences between some tissues and organs and the left ventricle, some organs that scatter in different positions in the image are easily misjudged as left ventricle, so in the following the detection network is described for accurate classification and localization.The detection network in this paper adopts the regional proposal network (RPN) that is modified with scale adaptive anchors. In order to match the number of anchors slipped out of each point of the RPN, the scale adaptive anchors generated correspond to 9 anchors for each point. In the training phase, the Euclidean distances are computed between the center of the label and all the centers in the training images that are identified as left ventricular candidate regions, then the smallest distance is regarded as a significant left ventricular candidate region. In short-axis cardiac MRI images, the left ventricle region is close to a circle, so the precise bounding box should be close to a square. The minimum bounding box is generated in the salient left ventricle region in each training image, which is called the saliency anchor. If this saliency anchor is rectangular, a long and short side of the saliency anchor are used to generate two square enclosing boxes in the same center, which are three basic enclosing boxes. If the saliency anchor is square, the diagonal length and semi-diagonal length of the saliency anchor are used to generate two square enclosing boxes in the same center, which is the three basic bounding boxes. The length and width of the 3 basic bounding boxes are multiplied by the scale scaling factor In order to further alleviate the shortage of positive samples, the sample augmentation was taken, where nine scale adaptive anchors’ centers were generated by randomly disturbance in the significance anchor and replacing the candidate boxes corresponding to the points around the current RPN points. In this way, the positive samples with diversity are constructed, and the high-quality negative samples around the threshold boundary are also built to join the proposal layer, so as to enhance the network’s ability to identify the hard to distinguish negative samples. In the testing stage, the scale adaptive anchors are mapped to the RPN and proposal layer on all left ventricular candidate regions for detection.The cardiac MRI slices contain a series of images of multiple individuals at different times. To speed up the detection, this paper proposes a rapid generation module of adaptive anchors suitable for left ventricular regional scale for the same individual. The overall framework is shown in h by establishing a minimized cost function as followsg is Gaussian response and l represents a certain dimension of the feature. f spectrum. It is empirically set as 0.01. This filter can be solved by a transformed into the frequency domain as follows,G is the conjugate of the two-dimensional Gaussian response to the two-dimensional Fourier spectrum; k-th dimension of the current frame, k-th dimension feature of the current frame, H is split into numerator A and denominator B, and it is iteratively updated as followsDue to the complex tissue distribution and low discrimination of various organs in medical MRI images, it is easy to make the estimated center position significantly different from the true center if the dictionary learning algorithm is directly applied. The scale adaptive anchors in this paper are generated in the same center; the accurate results are hard to obtain if the center position is not accurate. To handle this, the detection boxes are divided into 9 small blocks, and the estimated center positions of these blocks are averaged to obtain the prediction center. The final central location is computed as the weighting location of the prediction center and the prior center respectively. At the new center position, the left ventricular saliency anchor is obtained through a multiscale filter template, and the scale anchors are set on the saliency anchor in the same manner. Subsequent detection does not need to generate significant anchors through discriminative dictionary classification and label fusion, which effectively improves the detection speed and achieves competitive results. The training of template constructs the optimal correlation filter k-th dimension feature of the new frame. Once the new center point is estimated, candidate patches of different scales are obtained at the new center, and the best matching scale is obtained through the one-dimensional scale filtering template. The method of obtaining is similar to the position estimation, the choice of scale is shown as w and h are the width and height of the previous frame, respectively, and In the above formula, The tracking of each individual is mainly divided into two steps: tracking of location and tracking of changes in scale. The flowchart of the main tracking algorithm is as follows:(i) Location estimation.(1) According to the detection position of the current frame, the detection region is averagely divided into 9 regions (2) At the next frame, using 9 (3) Calculating (ii) Scale estimation.(1) At the new center of the target, extracting samples (2) Calculating (3) Calculating (iii) Update.Using the current frame information, the position template and the scale template are updated by the above update equations, where The experimental data are from the cardiac atlas project (CAP) database. This data set was created by a collaborative database of multiple organizations , among wThe myocardium region in each image was manually labeled by two well-trained students. Since this labeling was a non-trivial and challenging problem due to the fuzzy organ boundaries and large anatomical variability, their results were carefully cross-checked and further checked by a radiologist to made a final result as gold standard for evaluation. For each subject, this processing took about 2 h.The detection performance of the model is comprehensively evaluated by the following measurements. Considering accuracy and recall measures usually received special attention in the medical image processing field, in our experiment the precision (P), recall (R), F1 score, AP50 and AP75 are commonly taken as evaluation indexes in our object detection tasks. In order to reflect the adaptability of the model to different individuals, 10 different individuals were randomly selected from the test set and evaluated using the AP50, AP75, and mAP indicators. In order to ensure the uniformity of the evaluation results, the threshold score of 0.7 in Faster RCNN is used in the experiment. The prediction confidence score is greater than 0.7 as the correct detection, otherwise as the false detection.When calculating P and R, the indicators are defined as follows. The number of samples in which the left ventricular region is correctly predicted as the left ventricular region is The recall rate is the ratio of the number of positive samples that are correctly predicted in the left ventricle area to the number of all true positive samples. i.e.,P and R measures. It can be used to measure the similarity of the two sets, i.e.,The experiments were carried out on 2.0 GHz Intel CPU, 48GB RAM, NVIDIA RTX 2080Ti, Linux 64 bit PC. the programming environment was Anaconda 5.0.1 (Python 3.6), tensorflow1.4, keras2.0.8.M. If M is too large, the fitting degree of the edges of various tissues and organs will not be accurate, and the resulting IOU after the merging processing will be low and vice versa. In order to get optimal parameters, the relationship between the IOU after merging and the initial number of segmentation was experimentally investigated and the results are shown in M is 400. Therefore, the number of initial superpixel is finally determined as 400.For the dictionary training data set, 10 different individual legends are randomly selected from the total training set, and a complete slice sequence is randomly selected from the 10 different legends to build the discriminatice dictionary. In the process of experiment, the weight coefficient follows the original model. Before training of discriminative dictionary learning, PCA is used to reduce the dimensionality of various training data and retain core atoms. The initial segmentation number in the experiment is set to In the network parameter setting, the data augment approaches such as horizontal flip-up extended data set were used. The optimization approach was taken as the stochastic gradient descent method and the learning rate was set to 0.001 and the training batch was 256 anchors. According to the candidate box IOU as a positive and negative sample partitioning criterion, the RPN settled IOU > 0.7 as a positive sample, IOU < 0.3 as a negative sample, and the NMS  thresholThe training loss curves including regression part and classification part are shown in In the left ventricular scale adaptive anchors rapid generation module in the experiments, The scale scaling factor in our model is a key parameter in generating scale adaptive anchors; if it is not set properly, the anchors will be difficult to fit different scale ventricular regions. The relationship between the average IOU and the scaling factor was explored by randomly selecting 5000 images in the training data set and then calculating the average IOU of the scale adaptive anchors under different scale factors. The results are shown in To thoroughly investigate the performance of the proposed method, three variant methods were built as follows. (1) Discriminant dictionary+Faster method. This variant one focuses on the effect of discriminant dictionary module, by combining it to the Faster RCNN; (2) Correlation filter + Faster method. This variant one focuses on the correlation tracking module; (3) Correlation filter + Proposal method. This variant one combines the left ventricular scale adaptive anchors generation module and the proposal layer to improve the detection speed. Furthermore, to more comprehensively evaluate the proposed algorithm, the state-of-the-art two-stage methods such as Faster RCNN , one-staIn order to explore the adaptability of the detection algorithm to different individuals, the model was tested on 10 different individuals randomly selected in the test set. Using the ground-truth images as the standard, the test results were evaluated using AP50, AP75, and mAP. The comparison results are reported in As can be seen from the above table, the model has different adaptability to different individuals. The proposed method and its variant can achieve significant performance improvements based on Faster RCNN on most individuals. In the evaluation of the mAP50 index, the method Correlation filter + Faster in this paper achieved competitive results in all comparison methods, which is 4.1% higher than Faster RCNN. In mAP75, the method and its variants outperforms 2.2∼3.6% improvement in comparison to Faster RCNN. but less than RetinaNet.In order to compare the P-R curves of each method, the enlarged parts of the curves are plotted in the right column. It can be seen from the zoomed parts and the larger area under the PR curves that when IOU < 0.7, the proposed Correlation filter + Faster method achieves better performance than the compared methods.As can be seen from In the detection stage, this paper explores whether the number of scale adaptive anchors is better. At the scale scaling factor of 0.4 to 1.5, two sets of scale adaptive anchors are added each time, and RPN and the proper layer are gradually added. After experiments were performed on the testing set, AP50 and AP75 were computed for evaluation by using the label box as the standard. The results are shown in As can be seen from the figure, the AP50 and AP75 indicators have no significant relationship with the number of groups of scale scaling factors, and the indicators change within a small range as the number of groups changes.The proposed method and variants of this article have presented competitive performance in the experiments. As can be seen from the tables, they have been improved to varying degrees on the basis of Faster RCNN. In the comparison of the AP50 and AP75 indicators, the variant method Correlation filter + Faster achieved the most advanced results on the AP50 indicator, which increased by 2.1% on the basis of Faster RCNN. The variant method Correlation filter + Proposal of this paper has achieved comparable detection results with retinanet on the AP75 indicator and has increased by 1.3% on the basis of Faster RCNN. In terms of the recall rate index, the method and variants of this article are better than Faster RCNN. Under the condition of IOU = 0.5, the Correlation filter + Faster method has achieved a competitive result, which is 2% higher than Faster RCNN. Under the condition of 0.75, the variation method Correlation filter + Proposal in this paper achieved a detection performance comparable to Retinanet, which improved 1.2% on the basis of Faster RCNN. The experimental results show that the left ventricular scale adaptive anchors proposed in this paper can prevent the left ventricle area from being missed to some extent due to the adaptability to the left ventricular area of different scales.In terms of accuracy index, the method Discriminant dictionary + Faster and the variant method Correlation filter + Proposal both achieved certain improvement on Faster RCNN. The main reason is that after the left ventricular scale adaptive anchors are added to the non-differential anchors in the RPN, the candidate boxes around the left ventricle region are more significant, making this part of the candidate boxes more prominent in the network during training and testing and in MRI; although there are regions similar to the left ventricular region, the surrounding candidate anchors are not significant, so the occurrence of false detection is suppressed to a certain extent. At the same time, it should be noted that the Correlation filter + Faster is lower than the Faster RCNN in the accuracy index, and the performance at IOU = 0.75 is lower than IOU = 0.5. The possible reasons are conjectured as follows. In the training, the scale adaptive anchors generated by dictionary learning are used. While in the detection, the scale adaptive anchors generated by tracking are used. The scale changes of these two anchors during the regression process and the change of the moving distance are different, which leads to the low IOU of the detection boxes. Some of the detection boxes were not included in the correct number of detection due to the low IOU, so that certain accuracy was lost.The proposed method was compared with a series of classic detection algorithms on the detection time index. During the experiment, the models were performed under the same experimental conditions as much as possible. The results are reported in As can be seen from the table above, in the one-stage algorithm, the detection speed of Yolov3 has achieved a leading advantage. The method of this paper, Discriminant dictionary + Faster and Correlation filter + Faster, has a higher time cost than Faster RCNN. In particular, in the Discriminant dictionary + Faster method, each time a discriminative dictionary classifies a superpixel region, iteration is needed, resulting in high time cost. After using the left ventricular region scale anchors to quickly generate the module, the time cost is significantly saved, and the variant Correlation filter + Proposal method, because the previous RPN and other convolutional layers are removed, directly cascades the left ventricular-scale adaptive anchors module with the proposal, effectively increasing the speed and gaining 7% on Faster RCNN.The detection performance of the proposed method is significantly improved in comparison to Faster RCNN, which can effectively prevent false detection and missed detection. The comparison with a series of detection methods validates the effectiveness of the improved method in this paper. However, due to the complex tissue and organ distribution in MRI, the shape characteristics of some left ventricle areas are not obvious, and some areas are highly similar to the left ventricle; it is inevitable that a small amount of missed or false detection will occur. As can be seen from the figures above, the occurrence of pseudo-detection is mainly in areas where the appearance is similar to that of the left ventricle. Missing detection mainly occurs in the area where the the left ventricle shape features are not obvious and the size is too small. In further research, the adaptive and discriminant dictionary learning will be investigated for this challenging case.It should be noted that this study treated 3D image volumes as 2D images for left ventricle detection in the training stage and testing stage. The 3D CNN was not applied to build the detect network mainly due to two facts. One was that the anisotropic property in the volumes makes 3D CNN hard to extract the feature along the short axis. Typically, the Z-interval is 6mm while the XOY spacing is 0.7mm in the data set. The other reason was based on the consideration of the training time of 3D CNN in comparison to 2D CNN.Nevertheless, the structure of the proposed method cannot be called independent 2D CNN since a fast-scale adaptive anchors generation module is designed using correlation filtering which intrinsically belongs to a tracker. In each individual, the tracking of location and changes in scale is based on the observation that the positions of left ventricles in the images of the same individual are constrained in a similar location. In our future work, the proposed method will be extended to handle the structure of 3D CNN for detecting left ventricle and the tracking module will focus on the image volumes at different time points.In this paper, in view of the large span of the left ventricle area in cardiac MRI, varying size of ventricular area, and the remarkable gray level and shape difference of the myocardial blood pool, a CNN detection method based on discriminative dictionary learning and sequence tracking is proposed for left ventricular localization. The left ventricle target region was fused by discriminating dictionary-classified superpixel region labels, and multi-scale adaptive anchors were generated in the target region. Combining with non-differential anchors in region proposal network, the left ventricle is detected by regression and classification strategy. To solve the problem of slow classification efficiency in discriminative dictionary learning, a sequence-tracking-based left ventricular scale adaptive anchors rapid generation module was proposed in the same individual. The proposed method and its variants were performed and compared to some classical algorithms in target detection on the public available heart atlas data set (CAP). The experimental results show the effectiveness of the proposed method in a series of evaluation indicators and that this method has the most competitive results on some evaluation indicators.Considering the left ventricle in cardiac MRI is a three-dimensional tissue, our future work will focus on extending the proposed method to the three-dimensional detection of cardiac tissue and further estimate the posture of the heart in three-dimensional space.


Continuous ultrasonic welding is a promising technique for joining thermoplastic composites structures together. The aim of this study was to gain further insight into what causes higher through-the-thickness heating in continuous ultrasonic welding of thermoplastic composites as compared to the static process. Thermocouples were used to measure temperature evolutions at the welding interface and within the adherends. To understand the mechanisms causing the observed temperature behaviours, the results were compared to temperature measurements from an equivalent static welding process and to the predictions from a simplified heat transfer model. Despite the significantly higher temperatures measured at the welding interface for the continuous process, viscoelastic bulk heat generation and not thermal conduction from the interface was identified as the main cause of higher through-the-thickness heating in the top adherend. Interestingly the top adherend seemed to absorb most of the vibrational energy in the continuous process as opposed to a more balanced energy share between the top and bottom adherend in the static process. Finally, the higher temperatures at the welding interface in continuous ultrasonic welding were attributed to pre-heating of the energy director due to the vibrations being transmitted downstream of the sonotrode, to reduced squeeze-flow of energy director due to the larger adherend size, and to heat flux originating downstream as the welding process continues. Because of their high specific mechanical properties, fibre-reinforced polymer composite materials are interesting for industries in which weight is of utmost importance, such as the aerospace industry. The polymers used as matrices in these composite materials are typically thermoset or thermoplastic. Thermoplastics, in contrast to thermosets, do not form a cross-linked network and they rely on the entanglement of linear polymer chains to form a solid structure. These thermoplastic polymer chains become mobile when heated sufficiently above the melting temperature for semi-crystalline polymers or above the glass transition temperature for amorphous polymers. This allows the use of cost-effective manufacturing and joining processes such as press forming and welding for thermoplastic composite parts and structures.Welding has significant advantages over the more traditional joining methods mechanical fastening and adhesive bonding. For mechanical fastening drilling of holes results in breakage of fibres, and adhesive bonding requires intensive surface treatments and long curing cycles . The mosUltrasonic welding of thermoplastic composites can be performed in either a static or a continuous fashion. Static ultrasonic welding, e.g., spot welding a is defiExcessive resin and fibre squeeze out are undesirable since they may compromise the structural integrity of the adherends and, hence, of the welded assembly. Consequently, the focus of the present study is gaining a better insight into what causes higher through-the-thickness heating in continuous ultrasonic welding of thermoplastic composites as compared to the static process. In addition, this leads to a better insight into the process and heating mechanisms and thus to high-quality welds. To this end, thermocouples were used to measure the temperature evolution within the adherends and at the weld interface for different welding configurations and combinations of welding parameters. The results were compared to temperature measurements from an equivalent static welding process and to the predictions from a simplified numerical heat transfer model. To further understand the through-the-thickness heating, the total energy input was also reduced by either increasing the welding speed or decreasing the vibration amplitude. Additionally, to understand heating ahead of the sonotrode a damping unit was introduced to dampen vibrations travelling downstream. Finally, the effect of the adherend size, i.e., large adherends for the continuous process versus small coupon sized adherends for the static process, on the temperature evolution was studied by means of static welds on different sized adherends.The thermoplastic composite laminates used for the welding experiments in this study were made out of carbon fibre fabric (five harness satin weave) impregnated with polyphenylene sulphide powder, CF/PPS semipreg . The laminates were stacked according to a nterface ,24.The custom-built ultrasonic welding machine shown in er block b. The coer block b, while er block b and weler block b. FigureThe same experimental setup as described in the previous subsection  was usedTemperatures were measured at the welding overlap and within the adherends using K-type thermocouples . The sleeved thermocouples had a total diameter of 0.70 mm, while the diameter of the thermocouple wires was 0.10 mm. An analogue thermocouple output amplifier (Adafruit AD8495) was used to simultaneously sample temperature readings at 1 kHz from a maximum of five thermocouples. A moving average filter (10 points for the static process and 25 or 60 points for the continuous process) was applied in MATLAB to filter out high-frequency fluctuations from the temperature data. For the temperature measurements at the weld interface, the thermocouples were placed in the middle of the overlap, sandwiched between the bottom adherend and the energy director. For the measurements within the adherends, the thermocouples were inserted into 0.7 mm diameter holes drilled up to a depth of approximately 7 mm. The placement and depth of each hole ensured that the tip of the thermocouple was located approximately in the middle of the overlap width (directly above or below the thermocouple at the weld interface) and approximately midway through the thickness of the adherend. Note that the diameter of the hole and of the sleeved thermocouple were the same to ensure a press fit. To measure the temperature at the weld interface and through the thickness in the top or bottom adherends, different thermocouple configurations shown in Whenever necessary, the continuously welded adherends were cut into six 25.4 mm wide single lap shear samples with a diamond saw of which five were used for mechanical testing. The remaining sample was used for cross-sectional microscopy. The 28.8 mm wide edges at the start and at the end of the continuous welds were discarded. The single-lap shear samples were mechanically tested with a Zwick/Roell 250 kN  universal testing machine with a cross-head speed of 1.3 mm/min. The grips were given the necessary offset to minimize secondary bending. The apparent lap shear strength (LSS) was calculated by dividing the maximum load by the overlap area. After mechanical testing, a non-contact roughness measurement/profiler system  was used for the analysis of fracture surfaces. To obtain cross-sectional views from the welded adherends, specimens were cut and embedded in epoxy resin. They were ground and polished with a Struers Tegramin-20 polisher . A 3D laser scanning confocal microscope  was used for inspecting the cross-sections.T is temperature, t is time, and k are the density, thermal capacity, and thermal conductivity of CF/PPS, respectively.A 2D transient heat transfer model representing the static welding process was created in COMSOL Multiphysics 5.5  to estimate the temperature increase in the adherends resulting from only heat conduction from the weld interface in both the static and the continuous process. Since the energy director is very thin compared to the adherends, it was not included in the model. The heat transfer model is based on the following heat transfer equation:The temperature evolution at the weld interface for the continuous process is shown in The aim of this study was gaining further insight into what causes higher through-the-thickness heating in continuous than in static ultrasonic welding. Through-the-thickness heating combined with the application of the welding/consolidation pressure may cause matrix and fibre squeeze flow as well as porosity in the adherends  and hencA plausible cause of higher through-the-thickness heating is the higher overall temperatures at the welding interface  resultinContrarily, temperature measurements in the bottom adherend during the continuous process b show trAdditionally, given the importance of the temperature evolution at the welding interface in the quality of the welded joints, it is important to understand what causes the temperature differences between the continuous and the static process  and theiFirstly, the pre-heating experienced by the energy director, which is consistent with previous observation of thermal effects in the energy director downstream of the sonotrode , is belionotrode . It doesSecondly, the temperature at the interface experiences a continuous increase during the time the sonotrode is moving above a certain location which results in higher maximum temperatures as compared to the static process . Based oThirdly, longer times needed to cool down the interface below TFinally, regarding the quality of the welds, it is interesting to note that in those cases in which the strength of continuously welded joints is comparable (or higher) to that of the static welds (i.e., reference and lower amplitude case, see The aim of this study was to gain further insight into what causes higher through-the-thickness heating in continuous ultrasonic welding of thermoplastic composites as compared to the static process. In the continuous process, temperatures at the welding interface were indeed found to be significantly higher than in the static process. This was attributed to a combination of pre-heating of the energy director due to the vibrations being transmitted downstream of the sonotrode, reduced squeeze-flow of the energy director due to the larger adherend size, and heat flux originating downstream as the welding process continues. Thermal conduction from the hotter interface was however found to not be the main cause of higher through-the-thickness heating in the top adherend, which was in turn attributed to viscoelastic bulk heat generation. The top adherend seemed to indeed absorb most of the vibrational energy in the continuous process as opposed to a more balanced energy share between the top and bottom adherend in the static process. On the contrary, the bottom adherend showed a temperature evolution similar to what could be expected from predominant thermal conduction from the welding interface. Consequently, reducing the total vibration energy introduced in the material in the continuous welding process proved to have a substantial effect on reducing through-the-thickness heating.


The stochastic model, together with the functional model, form the mathematical model of observation that enables the estimation of the unknown parameters. In Global Navigation Satellite Systems (GNSS), the stochastic model is an especially important element as it affects not only the accuracy of the positioning model solution, but also the reliability of the carrier-phase ambiguity resolution (AR). In this paper, we study in detail the stochastic modeling problem for Multi-GNSS positioning models, for which the standard approach used so far was to adopt stochastic parameters from the Global Positioning System (GPS). The aim of this work is to develop an individual, empirical stochastic model for each signal and each satellite block for GPS, GLONASS, Galileo and BeiDou systems. The realistic stochastic model is created in the form of a fully populated variance-covariance (VC) matrix that takes into account, in addition to the Carrier-to-Noise density Ratio (C/N The stochastic model of observations describes the dispersion of the measurement random errors . Together with a functional model, expressing deterministic relationships between observations and unknown parameters, it forms the mathematical model of observation. The stochastic model is expressed by a variance-covariance (VC) matrix that defines the precision of observations through variance elements and their physical correlations through covariance elements. It is an essential element in the parameter estimation process for almost all measurement systems wherein the minimum variance estimator condition is assumed. Furthermore, the stochastic model is of particular importance in the case of Global Navigation Satellite Systems (GNSS) data adjustment, for which precise definition of a fully propagated VC matrix is difficult to obtain due to a multi-dimensional dependency of GNSS observations quality and the complex correlations existing between the measurements. In recent years, the joint processing of multi-system observations from Global Positioning System (GPS), GLONASS, Galileo and BeiDou (referred to as Multi-GNSS solution) has become the expected standard in precision positioning applications. The stochastic properties of the Multi-GNSS solution can vary significantly, which increase the importance of the correct definition of the VC matrix. Nevertheless, adopting homoscedastic or simplified models of variance and neglecting various correlations between the GNSS signals leads to a reduction in the positioning solution accuracy and reliability. These are crucial issues in relative positioning models using carrier-phase observations, in which correct integer carrier-phase ambiguity resolution (AR) is a necessary condition for precise positioning, since it defines the shape and size of the search space for the ambiguity estimation and contributes to the significance level and power of the validation tests. Unfortunately, nowadays the most common approach is the simplified definition of the stochastic model, which ignores the differences in accuracy and correlation between different GNSS systems.The existing approaches to model the GNSS observation variance components (mostly adapted from the GPS system) use, among others, the following methods: satellite elevation-dependent variance exponential function ,2 and coed model ,9 with e model ; s; sτ, is The measurement VC matrix must be presented as a function of white (uncorrelated) measurement noise VC matrix  done as :(25)Mt=ΨThe solution for the updated state is then computed using a gain matrix The modified KF was used to obtain the float solution  with its VC matrix. The final baseline solution was obtained after fixed integer ambiguity using Integer Least-Squares (ILS) estimates and updating float solution. Note that the presented algorithm taking into account the time-correlation of observations can be simply reduced to a standard Kalman Filter when a time-correlation coefficient is assumed to be zero. A detailed description of the data set used to evaluate the empirical stochastic model impact on the positioning model solution is presented in To determine and test the empirical stochastic model of GNSS observations, measurements from two reference stations WUT1 and WUT2 located on the roof of Warsaw University of Technology Main Building  the empirical model +C.”). The 3D position error statistics are shown in the form of a box plot, where a central mark of the box indicates the median, the top and bottom edges of the box mark the 75th and 25th percentiles of errors, and the whiskers are extended to the most extreme values of errors not considered outliers for a confidence level of approximately 99%. A red dot marks the maximum position error for each scenario. These statistics confirm that the positioning accuracy is increasing when considering the empirical stochastic parameters of measurement properties. The greatest accuracy characterizes the GPS and GPS + BeiDou solutions , while for the GPS + Galileo solution very low maximum errors were obtained (16.9 mm). The optimal compromise is the Multi-GNSS solution, where high accuracy (with median 4.3 mm) and the lowest maximum error (15.2 mm) are achieved.the C/Nthe realistic empirical stochastic model increases the accuracy and reliability of positioning solution, in particular the effectiveness of ambiguity resolution, while including the correlation parameters increases this efficiency;the highest efficiency of applying the empirical model is noticeable for Multi-GNSS solutions, for which the adoption of uniform stochastic parameters for all systems leads to significantly lower reliability of the solution;the optimal solution in the context of AR, positioning accuracy and maximum errors was obtained for a four-system GNSS solution using a fully populated empirical model individual for satellite blocks.This contribution presents research on individual empirical stochastic models of Multi-GNSS measurements. Based on the theoretical analysis of the GNSS receiver tracking errors, which depend on the carrier-to-noise density ratio parameter, a variance model was proposed. The empirical parameters of the model were determined utilizing the measurement data from the zero-length and very short baselines. Comparing the nominal tracking errors for Septentrio receivers with those determined using the empirical model confirmed the high compliance of theoretical assumptions with the modeling results. Additionally, the zero-length baseline data time series analysis was used to determine the parameters of cross- and time-correlation of GNSS signals. Using the weekly solution for the empirical stochastic parameters, a fully populated variance-covariance matrix, individual for GNSS system, signal and satellite blocks, was created. The impact of empirical stochastic modeling on ambiguity resolution as well as on position accuracy was analyzed with respect to the standard elevation-dependent model for different positioning scenarios. Positioning models were derived using modified Kalman Filter in which time-correlation is taken into account. Based on the stochastic modeling and positioning results, we can conclude that:It should be noted that the determined stochastic parameters, apart from the signal properties, are also closely related to the tested receivers and may differ depending on the equipment used. However, the presented methodology for creating empirical stochastic models and including them in the positioning model is comprehensive and can be used for all types of GNSS receivers and measurements. On the other hand, the bottleneck of the presented methodology may be the fact that the C/N


Lattice structures have shown great potential in that mechanical properties are customizable without changing the material itself. Lattice materials could be light and highly stiff as well. With this flexibility of designing structures without raw material processing, lattice structures have been widely used in various applications such as smart and functional structures in aerospace and computational mechanics. Conventional methodologies for understanding behaviors of lattice materials take numerical approaches such as FEA  and high-fidelity computational tools including ANSYS and ABAQUS. However, they demand a high computational load in each geometry run. Among many other methodologies, homogenization is another numerical approach but that enables to model behaviors of bulk lattice materials by analyzing either a small portion of them using numerical regression for rapid processing. In this paper, we provide a comprehensive survey of representative homogenization methodologies and their status and challenges in lattice materials with their fundamentals. Lattice material is a cellular material consisting of a periodic network of structural elements such as rods or beams. This network of lattices exists over a wide spectrum of scale from the nanoscale to macroscale and has been applied in a wide area of applications. In the nanoscale spectrum, most of the CNT (Carbon Nano Tube) based sensors are made using lattice materials  as shownLattice structures or materials could be also classified into several parameters, namely, geometry, deformation properties, and rigidity. These determine a proper approach for understanding dynamics of lattice accurately extend to design. Geometry-based classification is widely received in mathematics and solid-state physics and especially in 2-D, two main categories are considered: regular and semi-regular . RepreseIn engineering applications, spatially periodic patterns of lattices can be viewed as a material or a structure depending on its length scale. When the deformation is at a much larger length scale than the individual beam length, such a network of a lattice is defined as “lattice material”. The other key parameter that determines a suitable approach for understanding lattices is relative density. The relative density is defined as the density ratio of lattice material to the solid material  has been extensively employed to obtain the mechanical properties of lattice materials ,34,35. Ai et al. ,37. Theyi et al. .Some homogenization approaches introduced here come from micro-polar theory ,38,39,40Recently, Machine Learning has been adopted to study lattice materials ,43,44,45As has been briefly summarized above, many varieties of homogenization methods exist to analyze the behavior of lattice materials. All the methods mentioned came from various areas of discipline such as elasticity, solid-state physics, and even computer science. This implies the applications of lattice materials are substantial in many areas of science and engineering disciplines. The objective of this paper is to thoroughly review all the existing method that is relevant according to the author’s knowledge and interest. Its foundation, methodology, strength, and limitation will be discussed comprehensively here in a concise form. The final goal that this paper wants to achieve is for the reader to be able to carefully select their homogenization method based on its characteristic so that it could be applied optimally to each particular research.RVE should include the main microstructural characteristic of the heterogonous material and expand to the global medium when uniform strain or stress is applied as boundary condition [The fundamentals of the homogenization method are the properties of the heterogeneous material could be obtained from the analysis of a small portion of it . The limondition ,46,47. TRVE is applied to a square unit cell. A body RVE should be determined in such a way that the macroscopic behavior of The concept of homogenization of lattice materials is illustrated in uivalent . Below iRVE. Over the years, analytical closed-form formula of the mechanical properties of lattice materials for different shapes and geometry has been derived [The Beam Theory approach is also known as the force-based approach . It is e derived ,14,16. C derived  also givGibson and Ashby are pioneers in the analysis of cellular materials, in particular of honeycomb shapes . They anlinear-elastic behavior is predominantly caused by the bending of the cell walls and edges, with minor contributions from shear and axial deformation. For stretching dominated lattice, the cell walls are treated as trusses/columns where the structure is capable of sustaining residual stresses that make equilibrium equations are insufficient to determine the state of internal forces on the cell walls. Additionally, compatibility equations should be used to find the effective elastic properties of the lattice. If the residual stress is assumed zero, then simple truss analysis suffices.Different treatments are expected for a different type of unit cell. For bending dominated lattice, the cell walls are treated as beams. Standard beam theory is employed here to calculate effective stiffness. In this case, the The main advantage of this approach is that the obtained mechanical properties are closed-form analytical formulas and they are useful to generate a chart. Assuming cell wall as a beam limits the applications of this method as follows: First, this formulation can be applied to only cases with low relative density  It could only be applied to unit cells with a certain shape that contains a single joint at the center or the unit cell, and (2) the newly introduced micropolar variable acts as an additional degree of freedom. Hence, an additional step is required to solve the governing equations.Due to its similarity, the concept of solid-state physics can be adapted into solid mechanics to investigate the characteristics of lattice materials. The lattice, in solid-state physics, is defined as an infinitely periodic arrangement of points. When periods of the unit cell are perfectly stacked in two or three dimensions, the space is considered to be tessellated. The bases are the mathematical formulation for the physical quantities that are repeated in every cell translation . In contBloch wave analysis and the Cauchy–Born hypothesis, in particular, are methods for solid-state physics that can be adapted into solid mechanics to investigate the behavior of lattice materials ,55. Blocd joints ,58.Bloch’s theorem defines the deformation mechanism corresponding to periodic joint displacement fields. The Cauchy–Born hypothesis is needed to analyze the macroscopic strain field generated by periodic condition ,60. FromEquation (16) is the kinematic boundary condition of the Cauchy–Born hypothesis. Applying this boundary condition to the unit cell joint displacement vector,Substituting Equation (17) into the kinematic system of the unit cell results in:Equations (17) and 18) describe the application of the Cauchy–Born kinematic boundary condition to the continuum kinematic system of the lattice microstructure to express the relation between the microscopic displacements and a macroscopic strain field. The Cauchy–Born hypothesis cannot be applied to the kinematic compatibility relation of the unit cell without resorting to the Dummy node scheme  describe,60,61 anAnalytical solutions have shown some limitations in the applications of more general cases. Hence, one of the well-developed theories, with a sound mathematical foundation, that has been successfully used to predict mechanical properties in porous materials  is the ARVE size and the size of the macroscopic medium means that stress/strain will vary faster by The pivot assumption of AH is that each physical quantity depends on two different scales: one on the macroscopic level RVE, ic scale . Take thRVE volume  in 1-D can be rewritten as:Consider a composite bar consist of two materials that interchange periodically with Young’s moduli Applying integration by partsThe strong form of Equation (27) isyIntegrating gives the solutionSince the displacements Hence, the effective stiffness can be expressed asEvaluating the above integral gives us:Thus, for one-dimensional case, the effective stiffness obtained using AH method and the standard mechanics approach is equal.The notable advantage of AH is that the stress distribution in microscale can be modeled accurately and thus give us a detailed analysis of the periodic materials. Furthermore, AH has neither limitation on the cell topology nor the range of the relative density which is a substantial gain of this method . The majRVE. This method is developed based on the earlier work done by Eshelby [RVE features are somewhat similar to the ones that Elsheby has studied. It consists of a bounded area of the domain that contains the main microstructural properties of the material and behaves as an infinite medium if boundary conditions are imposed.This approach is often called global-local analysis as it involves a two-scale process. This method is originally applied to heterogeneous material in order to create constitutive relationships from the analysis of the  Eshelby  which inIn general, this method utilizes a two-scale approach. One is the macroscopic FE model of the homogeneous continuum where boundary conditions are defined by the problem. The other is the microscopic level which numerically investigates the stress-strain relationship where boundary conditions are generated by the macroscopic scale. This approach allows the macroscopic stress to be determined as the gradient of the strain energy density involving the components of the macroscopic gradient. This approach results in a compact matrix formulation for the macroscopic stress as a function of the macroscopic displacement gradient.The method that is described here is the application multi-scale homogenization method to develop non-linear constitutive models for lattice materials . This hoRVE, the corresponding array of the nodal forces, RVE. The distribution of the strain energy, due to macroscopic strain, can be obtained by employing the principle of the virtual work:RVE, the stress tensor can be obtained:Let Solving the equation above will introduce the boundary conditions for the microscopic model. Once the microscopic boundary value problem is solved, the components of RVE’s size might influence the equilibrium equation of the lattice especially in the presence of bifurcations [RVE.The main advantages of this method are that it accounts for geometrical material nonlinearity as have shown above and this approach has no restrictions in terms of relative density and unit cell shape. This model is capable to capture the local bucking of cell struts under multiple loading conditions and thus can predict the points where bifurcation occurs. However, unlike the AH approach, the choice of the rcations . Hence, In recent years, there has been significant development of homogenization methods using machine learning algorithms ,43,44,45RVE simulations can be utilized for training process [RVE simulation method and then used the results as the training data which is shown in The fundamental initial phase of using machine learning algorithms, in this case, neural network approaches, is to generate training data. Either experimental data ,68 or RV process ,66,67. S process  used theThe other approach is to use finite element simulation (FEA) as the training data . HoweverAnother implementation strategy done by Koeppe et al.  is to coWe try to point out each method’s main characteristic, its advantages, and limitation to give a concise comparison for the reader. The summary can be seen in This paper has provided a concise review of several homogenization methods that can be applied to the analysis and design of lattice materials. These methods came from various areas of discipline such as elasticity, solid-state physics, and even computer science. Relative density, cell geometry, lattice category , and cell element assumptions have important roles in the behavior of lattice materials. Hence, it is critical to employ a proper model for the lattice regarding those parameters. A summary of each strength and weakness of each method has been shown in Out of all methods, due to its efficiency and accuracy, there has been a growing interest in the homogenization method using machine learning algorithms recently as it has proven to be a reliable computational tool and has been employed in constitutive modeling. Furthermore, it has been shown in the previous section how the machine learning approaches can overcome some major limitations that are posted by the classical homogenization technique.Other than increasing efficiency, the recent and future works of homogenization are directed more towards the area of structure optimization. Homogenization coupled with optimization method has proven to increase both the efficiency of the optimization procedure and the overall performance of a lattice structure ,71,72,73