Visual tracking is a basic task in many applications. However, the heavy computation and low speed of many recent trackers limit their applications in some computing power restricted scenarios. On the other hand, the simple update scheme of most correlation filter-based trackers restricts their robustness during target deformation and occlusion. In this paper, we explore the update scheme of correlation filter-based trackers and propose an efficient and adaptive training sample update scheme. The training sample extracted in each frame is updated to the training set according to its distance between existing samples measured with a difference hashing algorithm or discarded according to tracking result reliability. In addition, we expand our new tracker to long-term tracking. On the basis of the proposed model updating mechanism, we propose a new tracking state discrimination mechanism to accurately judge tracking failure, and resume tracking after the target is recovered. Experiments on OTB-2015, Temple Color 128 and UAV123 (including UAV20L) demonstrate that our tracker performs favorably against state-of-the-art trackers with light computation and runs over 100 fps on desktop computer with Intel i7-8700 CPU(3.2 GHz). Visual object tracking is a branch of video analysis as the foundation of high-level image understanding. The main task of visual tracking is to continuously estimate the state (position and size) of the object in the subsequent frames given the initial state in the first frame. Many challenging factors, such as illumination variation, occlusion, fast motion, deformation and distractor may lead to tracking failure. In recent years, visual tracking algorithms have made considerable progress in terms of tracking accuracy and robustness with the efforts of researchers, and constantly refresh the upper limit of performance in various benchmarks and challenges. However, in many applications, such as UAV  tracking and edge computing, performance of the tracker is not the most important factor limiting its practicability, efficiency is also very important in practice.In recent years, discriminative correlation filters (DCF) -based viIn this paper, to address the aforementioned issue, we focus on the update scheme of CRCF tracker and design an efficient and adaptive training sample update scheme. On this basis, we propose a multi-training samples DCF-based tracker with outstanding performance and light computation trained on the training set with representative appearances of the target, namely CRCF_ATU. In addition, we expand CRCF_ATU to long-term tracking considering practical applications, on the basis of CRCF_ATU and its model updating mechanism, we propose a new tracking state discrimination mechanism to accurately judge tracking failure, and then we perform global redetection, and resume tracking after the target is recovered. Experiments on OTB-2015 , Temple The main contributions of this paper are as follows: (1) We develop an efficient correlation filter-based tracker with outstanding performance, which is suitable for real-time applications. (2) We adaptively update training set and measure sample distance with difference hashing algorithm (DHA) . (3) We We note that a conference version of this paper appeared in reference . The worThe rest of this paper is organized as follows: Since Bolme et al.  firstly To solve the problem of scale estimation of correlation filtering with single template, the discriminant scale space tracker (DSST)  was propIn terms of feature representation, Danelljan et al.  brought For the boundary effects incurred by periodic assumption, spatially regularized discriminative correlation filter (SRDCF)  penalizeTo prevent model contamination and retain historical information, conventional DCF trackers ,15 updatShort-term tracking usually assumes that the target is always in the field of view, and does not care for target reposition after target tracking failure. Meanwhile, long-term tracking needs to deal with the situation that the target is completely blocked or out-of-view; it is required to accurately detect tracking failure or target disappearance, and reposition the target when the target appears again and resume tracking. Ma et al.  proposedIn this section, we first give a brief introduction of our previous CRCF tracker . Then, wIn order to make better use of target and background color information to improve the performance of tracker, we borrow the weight map idea of mean shift algorithm to propose CR features , and intm = 16 × 16 × 16 bins. Note that the target center is h and the target background bandwidth is s × h (s > 1). Then, the target model C and n and u, the weight can be calculated as follows:CR feature is obtained by comparing the target model with the target background model. Both the target model and the target background model are color histograms with s]. Thus, we can limit the value range of s to get the CR features map, which can be described as It has been proven in  that w[uThe algorithm flow of CRCF tracker is shown in The detailed description of the algorithm can be found in .Aiming at the problem of model drift in correlation tracking model updating, CRCF_ATU is proposed on the basis of the CRCF  and the N, we follow [N, the new sample is added into the training set.If the number of samples in the training set is less than N and the minimum weight is below the forgetting threshold, the sample with minimum weight is replaced by the new sample.If the number of training samples exceeds N and there is not any sample’s weight below the forgetting threshold, the closest two samples are merged into one sample.If the number of training samples exceeds For the training sample set of size e follow  to emploThe training set stores representative appearances of the target and the correlation filter is trained on the training set at sparser intervals. This update strategy enables the correlation filter with better generalization ability and reduces the computational burden at the same time.i-th row and j-th column of B and H. Each image patch can be represented with this hashing matrix and the distance between two image patches can be measured with the hamming distance between their corresponding hashing matrix However, in , the disd-th channel of the correlation filter h and the training sample y is the gaussian label and N is the number of samples, D is the number of feature channels, and Referring to the scheme of online maintenance of training sample set in ,19,27, wi-th row and j-th column on the response map, m and n are the width and height of the response map respectively. It is found that the product of peak value of response map and APCE can better reflect the reliability of response map, which can be calculated as R.To achieve robust target tracking, it is necessary to judge the reliability of tracking results, and stop updating the model when the tracking results are unreliable. The reliability of results in correlation, tracking can be measured by the peak value and fluctuation degree of the response map. For CRCF_ATU, the fluctuation degree of response map can be measured by APCE (average peak correlation energy).CRCF_ATU can deal with the situation of fast deformation or partial occlusion of the target; when the target is always in the field of view, it can achieve fast and robust target tracking. However, when the target is fully occluded for a long time or out of view and then returns to the field of view again, CRCF_ATU cannot successfully continue tracking. In order to address the above issues, we extend CRCF_ATU tracker to propose a long-term tracker named CRCF_LCT. One of the main problems in long-term tracking is how to judge tracking failure, and when to enable and stop global redetection. For our CRCF_LCT tracker, according to the criterion of the reliability of CRCF_ATU tracking results which is shown in Formula , trackinIn the Redetection Module of loy BACF  detectorP to cut the central target area of the filter, so that the filter can learn more real negative samples in the training process, which enhances the discrimination of the filter. Clipping operation can not only alleviate the edge effect, but also expand the search range of the tracker, and enhance the tracker’s ability to deal with rapid movement of the target.BACF   is an imK donate the number of channels, when detecting the target in the next frame, the trained filter Z are used to obtain the response map as follows:Let The coefficients of spatial filter trained by BACF algorithm are shown in While CRCF_ATU tracker maintains the training sample set, each sample retains a larger copy image block to train the BACF detector. When the Redetection Module is activated, the feature is extracted from large image blocks, and weighted sum of corresponding samples in the sample set are used to train the BACF filter. Finally, the target is detected globally by the BACF filter, which is expanded to the image size by zero padding. At the peak position of global detection, CRCF_ATU tracker is used again to optimize the position. If the response reliability of CRCF_ATU tracker  Formula ), the gl Formula .(11)τTCF, TCR  toolbox . For sca toolbox . The par toolbox . SpecifiTo validate the effectiveness of the update mechanism of our CRCF_ATU tracker, we compare the DHA-based samples update mechanism proposed in this paper and the update mechanism in Reference  on OTB-2OTB-2015  is one oTemple Color 128  consistsUAV monitoring is one of the most important applications of visual tracking. UAV123  consistsUAV20L is a subset of UAV123 , which cIn this paper, for the applications with restricted computing power, we propose an efficient training sample update scheme and adaptively maintain a training set using difference hashing algorithm. The filter with stronger discrimination and generalization ability is trained on the training set, which improves the performance of the CRCF_ATU tracker while maintaining high speed. For practical long-term tracking applications, a more accurate tracking state discrimination mechanism is proposed to judge the tracking state based on the update mechanism of our CRCF_ATU tracker. When the tracking failure is judged, the BACF with a wider search range is trained on the maintained sample set to conduct global redetection, and recover the short-term tracking after the target is found. Experimental results show that our tracker can achieve a close performance to state-of-the-art trackers, with relatively light computation and high speed. Moreover, the extended long-term tracker can accurately judge the tracking state and successfully retrieve the target when the target reappears. The balance between performance and efficiency enables our tracker to be practical in computation restricted applications.


Chest compressions during cardiopulmonary resuscitation (CPR) induce artifacts in the ECG that may provoque inaccurate rhythm classification by the algorithm of the defibrillator. The objective of this study was to design an algorithm to produce reliable shock/no-shock decisions during CPR using convolutional neural networks (CNN). A total of 3319 ECG segments of 9  Out of hospital cardiac arrest (OHCA) is one of the leading causes of death worldwide ,2. The tThe mechanical activity of chest compressions during CPR induces artifacts in the ECG that impede a reliable shock/no-shock decision by the defibrillator . TherefoRecently, deep learning approaches have proven to be superior to classical machine learning algorithms in many biomedical signal applications ,28, inclThe objective of this study was to design the first deep learning solution to discriminate shockable from nonshockable rhythms during CPR. The method comprises two stages, an adaptive RLS filter to remove CPR artifacts from the ECG followed by a CNN to classify the filtered ECG. The paper is organized as follows: the study dataset is detailed in Data were extracted from a large prospective clinical trial designed to measure CPR quality during OHCA . The stu the ECG . FinallyThe rhythms in the OHCA episodes were originally annotated by two experienced resuscitation researchers/practitioners, a biomedical engineer and an anesthesiologist . For then is the sample index.The shock/no-shock decision algorithms proposed in this study are composed of two stages. First, an adaptive RLS filter was used to remove chest compression artifacts from the ECG. Then shock/no-shock decision algorithms were designed to classify the filtered ECG using CNNs. In what follows N terms (harmonics). The fundamental frequency of the artifact is that of the chest compressions [N-term Fourier series representation is then:N is the number of harmonics in the Fourier series and CPR artifacts were suppressed using a state-of-the-art method ,39 basedressions , which id tk see , the freThe RLS filter adaptively estimates the time-varying Fourier coefficients, lable in , and the λ=0.999 .The shock/no-shock algorithms trained and evaluated in this study comprise algorithms based on CNNs (core methods of the paper), and a state of the art algorithm based on classical machine learning techniques used as a baseline model for comparison. In both cases, the algorithms were designed to analyze the filtered ECG in the interval from 3–12 ansients . The anas [s see  and the N = 375 samples, b-th convolutional block consists of a convolutional layer with b), where n is the time index and m the filter index. In the first block the input is b can be expressed asK samples along the time index n to give the output of block b:Let us denote by b where Padding was applied before the convolutional and the max-pooling layers, so the only reduction of the dimensionality occurs at the max-pooling layers , and the minimum observations per leaf to 3 (to avoid growing excessively deep or overfit trees). To avoid class imbalance uniform priors were assigned and a cost function was introduced to penalize false shock classifications with a factor of 2.5 (similar to the shock/no-shock augmentation factor used in the CNN).The parameters of the RF classifier were fixed to those recommended in , that isAll the classification algorithms were trained/tested using 5-fold cross validation (CV). Folds were partitioned patient-wise to avoid training/test data leakage, and in a quasi-stratified way by ensuring that the shock/no-shock prevalences in all folds were at least 80% those of the whole dataset. The performance of the method was evaluated using the standard metrics for binary classification problems, taking the shockable class as positive and the nonshockable class as negative. For a The Balanced Accuracy (BAC) was used as target performance metric to ensure both shockable and nonshockable rhythms were accurately identified (as recommended by the AHA) despite the large class imbalance in the data.B), the size of the filters (I), and the number of filters  than the baseline model. As shown in The shock/no-shock decision algorithms using CNNs and the classical machine learning model were compared. CPR artifacts during chest compressions present very different noise levels in the ECG depending on variables like the position of the hands relative to the pads and cables, pad placement, or environmental conditions ,48. Thesp < 0.001, Mann-Whitney U test), and in approximately 15% of shockable and nonshockable cases the noise level was negligible  [For these experiments the 10 features at the output of the first fully connected layer were used as the features learned by the algorithm, these features will be named onducted , and theonducted . First, onducted . The resclusters . The exp t-test) . That isp < 0.05) than any of the handcrafted features. These results confirm the ability of the CNN to extract high quality discriminating features hidden in the signals.Second, the discriminating power of each feature was computed using the area under the receiver characteristics curve (AUC). The results were obtained over 500 bootstrap replicas to statistically characterize the AUCs and compare the AUC distributions for each feature (paired t-test). The results are shown in To further improve the BAC and accuracy of the CNN model three mixed architectures were also explored. First, the architecture of solution  in which−1. In these cases the filtered ECG still shows an organized activity locked to the compression frequency, incompatible with fast ventricular arrhythmia and thus classified as non-shockable. Interestingly, these errors can be related to the clustering analysis of To conclude the analyses, the classification errors for the CNN based algorithm were identified. Some typical patterns leading to errors are shown in This is, to the best of our knowledge, the first study that uses deep neural network models to discriminate between shockable and nonshockable rhythms during CPR. This algorithm consists of an adaptive RLS filter to remove CPR artifacts followed by a CNN to classify the filtered ECG. The algorithm designed for 9 One of the advantages of deep learning solutions is the capacity of the algorithms to learn discriminating features exploiting all information hidden in the ECG. This avoids the time-consuming feature extraction processes and, most importantly, improves the quality of the extracted features. The latter is well reflected by the AUCs on terature ,26.Two factors were key to improve the performance of the CNN based methods from the preliminary results communicated previously in . First, As This paper introduces the first shock/no-shock decision algorithm during CPR based on deep learning methods. This solution improves the accuracy of the best classical machine learning models based on handcrafted features, and is able to give a shock/no-shock diagnosis compliant with AHA recommendations for shockable and nonshockable rhythms. Moreover, deep learning algorithms have room for improvement if larger annotated datasets become available allowing the design of deeper networks. This may lead to the first practical solutions for rhythm analysis during CPR, eliminating the no-flow intervals for rhythm analysis and contributing to improve OHCA survival rates.


This paper proposes a beamforming method in the presence of coherent multipath arrivals at the array. The proposed method avoids the prior knowledge or estimation of the directions of arrival (DOAs) of the direct path signal and the multipath signals. The interferences are divided into two groups based on their powers and the interference-plus-noise covariance matrix (INCM) is reconstructed through the doubly covariance matrix reconstruction concept. The composite steering vector (CSV) that accounts for the direct path signal and multipath signals is estimated as the principal eigenvector of the sample covariance matrix with interferences and noise removed. The optimal weight vector is finally computed using the INCM and the CSV. The proposed method involves no spatial smoothing and avoids reduction in the degree of freedom. Simulation results demonstrate the improved performance of the proposed method. Adaptive beamforming is a well concerned research topic of array signal processing, with wide spread applications in radar, sonar signal processing and other fields ,3,4,5,6.Conventional adaptive beamforming techniques are effective under the assumption of the independence among the received signals. However, in real scenarios, the desired signal may be received as a group of coherent signals from multiple directions. For instance, in communication systems, due to the multipath effect, the desired signal may be overlapped by the reflected/scattered interferences from, say, ground or sea surface, which leads to the coherence of the received signals. In this case, the conventional adaptive beamformer may suppress the desired signal and degrade the performance of the beamformer. A class of pre-processing methods for reducing the influence of signal correlation are proposed. The well-known spatial smoothing method divides array sensors into subarrays and averages data covariance matrices of subarrays to decrease the coherence of signals . There aIn practical applications, the prior information about the desired signal is usually unavailable and the performance of the traditional approach may be significantly deteriorated. The blind beamforming technique can maintain satisfactory performance when the parameters are not precisely known, which dramatically improves the robustness and output performance of the beamforming algorithm.Blind identification was first applied on beamforming by using blind identification to estimate the steering vector without the prior knowledge of the array manifold . Based oWang et al.  proposedIn this paper, we propose the doubly covariance matrix reconstruction concept to achieve blind beamforming for coherent signals. The main problem is to reconstruct the interference-noise covariance matrix (INCM), and obtain the CSV of desired coherent signals, and then the weighting vector of MVDR beamformer can be calculated to eliminate interferences and noise which retains the desired signal sequence. Specifically, our approach consist of the following steps: First, the covariance matrix of comparable-to-desired-signals-in-power (CDSP) interferences is calculated after estimating their power and directions of arrivals (DOAs) through the spatial spectrum, then the covariance matrix of normal interferences is reconstructed using eigenvectors of a transformed form of sample covariance matrix (SCM). Second, with the INCM obtained by summing the covariance matrices of interference and noise, the CSV is estimated as a unblinding process, that is, through the eigenvalue decompostition of SCM with interference and noise components removed. Finally the weighting vector is determined by the INCM and CSV. Importantly, for the array, the degree of freedom is preserved since no smooth process is involved in the proposed approach. Simulation results demonstrate that output signal-to-interference-plus-noise ratio (SINR) of proposed approach attains satisfied performance in terms of blind beamforming for coherent signals and outperforms the compared approaches.The rest of this paper is organized as follows: In M radar receiving elements. Assume there are Q interference sources. In practice, due to multipath propagation, the desired signal arrives at the array from P different directions, and the DOAs of the direct path signal and multipath signals are Q interference signals are kth snapshot, the data vector received by the array can be expressed ask, and pth way. k and the power of qth interference signal is P propagation paths and Q interferences, respectively. Consider a uniform linear array (ULA) consists of With an infinite number of snapshots, the theoretical covariance matrix of data received by the array can be expressed ask can be written asThe beamforming technique can be applied to eliminate the interference and noise components in the received data while keep the desired signal unaffected through the weighted summation of data of different channels. The output of beamformer with interference and noise suppressed at instant The constraints of the well-known MVDR beamformer are given asK is the number of snapshots and kth snapshot. The beamformer above now turns into minimum power distortionless response (MPDR) beamformer, which is also called sample matrix inversion (SMI) beamformer. It is known that when desired signal is present in the covariance matrix, the performance of beamformer deteriorates.In practice, the INC matrix In this part, the proposed blind beamforming approach is introduced. From Q interferences are composed of In real application, the interferences we usually consider as “normal interferences” are those who obscure the desired signal due to their much larger power. On the other hand, there are also another kind of interferences (we named them as “CDSP interferences” in the later) with power comparable to that of the desired signal, that may also be regarded as false signals and influence the performance of beamforming. Therefore, we further extend the signal model in (1) to a more generalized situation, and give the following array signal model for better illustration. Assuming that We estimate the noise power e of R^x . Since tCapon spatial spectrum can be utilized to give the spatial spectrum distribution over the whole space, where the interference DOAs can be estimated from the location of the peaks ,26. MoreQ because there are some peaks formed by noise. Since the peaks formed by noise are far lower than the peaks corresponding to real interferences in the spectrum [Q. Considering the fact that there are M channels in the receiver array, the value of threshold can be set as The basis of the spatial spectrum is the ordinary steering vector spectrum , it is aAfter the selection, the remaining peaks contain both normal and CDSP interferences. The small value peaks are chosen out as peaks of CDSP interferences, i.e., the peaks of which the values are relatively larger are considered belonging to the normal interferences. Actually there is no precise limit to differ the relatively larger and relatively small peaks. Usually, we can give an experiential value of this limit through experimental measurements, which has been further explored in the Example 1 of simulation part. The number of CDSP interferences Before we proceed to the following content about the reconstruction of  used in , we now Theorem 1.Suppose where where erred to .mth diagonal entry of According to the above theorem, if n rank-one matrices This theorem can be extended to a generalized situation. For a Hermitian matrix Since rank-one matrices Suppose the eigenvalue decomposition of Assume the eigenvalue decomposition of According to the above theorem, if It should be noted that during the reconstruction of the Based on the reconstructed interference covariance matrices Therefore, the estimation of desired signal covariance matrix It is known that multiplying The weighting vector of the proposed blind beamforming approach can thereby be obtained and given asAlgorithm 1 Steps of doubly covariance matrix based blind beamforming approach.Part 1: doubly covariance matrix reconstruction1. eigendecomposing the SCM  2. 3.Q = number of spectrum peaks sufficiently larger than  4.Q peaks. 5. reconstructing covariance matrix for the   6. decomposing SCM with noise components removed as  7. reconstructing covariance matrix for the   Part 2: unblinding desired coherent signals.8. reconstructing the INC matrix  9. estimating the CSV of desired coherent signals  Final calculating the weighting vector  The detailed procedures of the proposed blind beamforming approach are summarized in Algorithm 1 shown as follows.In this section, simulations are provided to examine the performance of proposed approach via several examples. A ULA of 35 sensors is considered in the simulation for all examples, the covariance of the white Gaussian noise is assumed to be To better demonstrate the performance of the proposed approach, three representative blind beamformers—the EBB approach in , the MVDAs stated before, the normal interferences should be sufficiently stronger than the desired signal so that the proposed covariance matrix reconstruction method for the normal interferences can be applied. In the first example, simulation is performed to examine how large the interference-to-noise ratio (INR) should be to meet the restriction.To further investigate the performance of proposed approach, fixing the INR of Group 2 as 25 dB while keeping other conditions unchanged, in In the third example, with the interference-to-signal ratio (ISR) of two interference groups fixed and other conditions unchanged, we explore the performance of proposed approach at various SNRs. ISRs are set as 5 dB and 25 dB for Group 1 and Group 2, respectively. The SNR is varied from In the fourth example, to further validate the performance of the proposed method, the convergence ability of the proposed approach is tested. As illustrated in the figure, In the four examples concerned above, there exists an assumption that only two groups of interferences are employed, one for the normal interferences and one for the CDSP interferences. In this example, we further extend the scenario by adding one additional group of interferences A new blind beamforming approach for coherent signals is proposed in this paper. When the prior information about desired signals is unavailable, the proposed approach can effectively suppress the interferences and noise, and extract original signal sequence from the received data. The proposed approach is mainly based on the proposed doubly covariance matrix reconstruction concept. The covariance matrices are reconstructed respectively with two methods. Based on the reconstructed INCM, the unblinding procedure is implemented to obtain the CSV of desired coherent signals. Simulation examined the output SINR improvement of proposed approach compared to other approaches. The proposed approach avoids the freedom degree loss since no smooth procedure is involved. It has been verified through simulations that our proposed approach attains optimal when the ISR of normal interferences are around 10 dB, and this ISR can be easily met in many scenarios. In Example 5, we further explored a more complicated scenario, by adding an additional interference group. The simulation results illustrate that when ISR is stronger than 10 dB, the interferences can be considered and processed as normal interferences, and at ISR


Stress shielding and micromotions are the most significant problems occurring at the bone-implants interface due to a mismatch of their mechanical properties. Mechanical 3D metamaterials, with their exceptional behaviour and characteristics, can provide an opportunity to solve the mismatch of mechanical properties between the bone and implant. In this study, a new porous femoral hip meta-implant with graded Poisson’s ratio distribution was introduced and its results were compared to three other femoral hip implants  in terms of stress and micromotion distributions. For this aim, first, a well-known auxetic 3D re-entrant structure was studied analytically, and precise closed-form analytical relationships for its elastic modulus and Poisson’s ratio were derived. The results of the analytical solution for mechanical properties of the 3D re-entrant structure presented great improvements in comparison to previous analytical studies on the structure. Moreover, the implementation of the re-entrant structure in the hip implant provided very smooth results for stress and strain distributions in the lattice meta-implants and could solve the stress shielding problem which occurred in the solid implant. The lattice meta-implant based on the graded unit cell distribution presented smoother stress-strain distribution in comparison with the other lattice meta-implants. Moreover, the graded lattice meta-implant gave minimum areas of local stress and local strain concentration at the contact region of the implants with the internal bone surfaces. Among all the cases, the graded meta-implant also gave micromotion levels which are the closest to values reported to be desirable for bone growth (40 µm). There are two established methods for femoral implant fixation namely cementation  and presSome recent studies  have shoOne of the promising methods to reduce micromotion can be to manipulate Poisson’s ratio of an implant in different locations and directions. Depending on the extent of the load an implant receives and the extent of required in-situ relative lateral movement, an implant can be designed with the desired distribution of lateral movement. This can be accomplished not by normal materials, but by materials that show exotic properties, commonly known as metamaterials. Metamaterials are designer materials that exhibit properties not usually found in nature . Recent Designing a structure with a wide variety of positive and negative Poisson’s ratio at different locations can lead to implants which have lateral displacements in compliance with the surrounding bone. The most famous and applicable bio-metamaterial structures for this purpose are 3D re-entrant structures which could provide positive or negative values for their Poisson’s ratio depending on their internal struts angles . Such stIn recent years, many researchers have focused on the effects of geometry, porosity, material, micro-architecture, loading, coating, etc., on the micromotion extents in implants especially total hip replacements (THRs). Chen et al.  evaluateThe goal of this study is to tune the micromotion of hip implants to desirable ranges by tuning the micro-structural features of meta-implants. Gradient distribution of 3D re-entrant unit cells can be a good solution to mitigate micromotions and to improve stress shielding in the hip implant. For this aim, first, analytical relationships for mechanical properties of 3D re-entrant unit cell are derived and then four types of implants  are designed and studied using finite element analysis (FEA). As the porous implants are constructed by lattices of metamaterials (auxetics), they are referred to as meta-implants in the rest of the paper. The implants are then designed based on the anatomy of the femur bone to have better initial adjustment and fixation for hip implants. A geometrical model of the femur bone is created precisely by considering both the cancellous and cortical bones to obtain accurate results. Finally, the stress and strain distributions of the implants, as well as the micromotions at bone-implant interfaces (considering both gap and sliding distances), are analysed and compared between the four designs.In the first part of this section, analytical relationships for the main mechanical properties of 3D re-entrant unit cell, namely elastic modulus and Poisson’s ratio, are derived. The newly developed analytical relationships are more precise when compared to existing relationships available in the literature ,38. In tIn this subsection, analytical relationships for elastic modulus and Poisson’s ratio of the re-entrant unit cell are derived as functions of elastic properties of the constituent material  and one horizontal (xz) symmetry planes of the re-entrant structure and also two considered loading conditions in the x and y directions, the behaviour of the re-entrant unit cell can be modelled completely by 12 distinct degrees of freedom (DOF), see al plane . By consTo obtain the elements of The term ely . The main mechanical parameters required to create a meta-implant that is compliant with its surrounding femur bone of study are elastic modulus and Poisson’s ratio. The implant elastic modulus controls the stress flow and its distribution in the implant. While elastic modulus can affect the micromotion of the implant/bone interface in the longitudinal direction, Poisson’s ratio can affect the micromotion in the transverse direction. A lot of works in the literature have studied the mechanical properties of the human femur bone ,55,56,57According to our analytical solution, the re-entrant structure provides much larger elastic modulus ranges in the x-direction as compared to the y-direction (see Results Section). Therefore, to gain the elastic modulus of 16.7 GPa for the meta-implant, the analytical relationships for the elastic modulus of the unit cell in the x-direction (Equation (21)) has been used. Therefore, the x-direction of the re-entrant structure was aligned with the human femur bone shaft central axis at the lowest level of the hip implant. The low number of unit cells at the bottom part of the hip implant is the reason for this alignment as it provides better contact between unit cell struts and the bone internal surfaces. By considering the elastic modulus value of 3 have been used for generating the graded meta-implant.Based on the previous studies ,61 on thThe FE analyses of the implants under static compressive loading condition were performed using the ANSYS Workbench  software (Version 19.3). To better simulate the real human femur loading condition, the femur bones were angled at 12° flexion and 12° adduction, and the lower part of the femur bone (middle shaft) was fixed in all the directions. The femoral head was loaded by applying a 2000 N compressive load to the head of the femur implant. The loading and boundary conditions considered in this study are based on the basic experimental setups which have been used in several experimental studies of human femur hip implants ,62,63.The contact condition between the cortical and trabecular bones was considered as a bonded constraint. On the other hand, to track the micromotion between the implant external surfaces and bone internal surfaces, the contact condition between the bone and the implant was considered as frictional surface-to-surface contact allowing contact, sliding, and tensile separation to occur. The friction coefficient of 0.3 was applied for the frictional contact conditions . The 3D Variations in normalized elastic modulus m theory a. As for the Poisson’s ratio, the results show that even though ts value b. In facg et al.  do not hg et al.  is for νg et al. ’s theoryo −2.414 b. SimilaH = 0.16. As for the elastic modulus, the internal angle has an almost negligible effect in the y-direction. However, in the x-direction, the relative elastic modulus in The internal angles  plotted  is r/H = to θ=0° a.As for the Poisson’s ratio, the internal angle has a significant effect on both  and νyx b. νxy isle range b. νxy chlocal stress distribution in meta-implant does not change uniformly. In fact, different regions of each unit cell have different stresses. Some parts of each unit cell experience stress levels similar to that of the bone, while some other parts of it experience different stress levels. From a mechanical point of view, this was well expected as some struts mainly undergo flexural loading, while some other struts mainly undergo axial loading.The stress and strain distributions in femur bone filled by the solid implant, the meta-implant constructed by positive Poisson’s ratio unit cells, the meta-implant constructed by negative Poisson’s ratio unit cells, and the meta-implant with the graded distribution of unit cells are shown in In the case of structure constructed by negative unit cells, there are two improvements when compared to the structure constructed by positive unit cells. First, the stress and strain distributions in each unit cell become much more uniform locally. In other words, the stress or strain level throughout each unit cell remains almost constant. Secondly, and more importantly, the stress and strain levels in the case of structure constructed by negative unit cells is very close to that of the bone surrounding it at each location.As for the case of the graded meta-implant, even though the stress and strain distributions  in the graded meta-implant are much better than the structure constructed by positive unit cells, it performs worse than the structure constructed by negative unit cells.In addition to stress and strain distributions, the level of stress dominant in the implant is also important. Very high-stress levels can lead to early implant failure. As for the case of the solid implant, the maximum stress in the implant is x=68 MPa a which igap between the implant and the bone.Micromotion, another important factor in the durability of an implant, is compared between the four types of the implant in Except for one point, the solid implant demonstrates a near-zero gap in both back and front (the parts in compression and tension) of the implant a,b. AmonBased on the results of the gap and sliding distributions presented in th sides a,b. As sConsidering the behaviour of the re-entrant unit cell as the sum of the behaviour of individual DOFs for an analytical solution, which works as one of the most accurate approaches used for obtaining the analytical relationships for strut-based lattice structures and metamaterials. This technique, by considering all possible deformations of the struts at the joints in the lattice structure, provides comprehensive and accurate analytical solutions for the mechanical properties of lattice structures. The results of this study also confirmed that the obtained relationships for normalized elastic modulus and Poisson’s ratio of re-entrant structure based on this technique present satisfactory outcomes in comparison to other approaches presented in the literature . Moreoveg et al.  is indepHip implant geometry is one of the most important parameters which can affect the stress-strain and micromotion distribution at the bone-implant interface. There are lots of studies in the literature which have focused on the design and optimization of the hip implant geometry to improve the fixation and to reduce the micromotion at the implant surface ,40. UsinBy considering Ti-6Al-4V alloy for implants material, a large difference occurs between the value of elastic modulus of the femur bone and solid implant. This fact causes nonhomogeneous stress and strain distributions in the implant, especially at the implant-bone contact regions. For implantation with the cement-less condition (press-fit), sliding and gap distances occurring at the contact surface of the bone and solid implant during loading lead to local stress and strains at the implanted area  which caAlthough decreasing micromotions at the implant-bone interface can have positive influences on the bone ingrowth, eliminating the micromotion from the bone and implant interface altogether can have negative effects on bone ingrowth and fixation durability. Burke et al.  have demThanks to the anatomical geometry of implants and the adjusted elastic modulus of the meta-implants, all of the meta-implants based on the introduced unit cell gave acceptable stress and strain levels in the implantation area. Nevertheless, the meta-implant based on graded unit cell distribution presented smoother stress-strain distribution in comparison to the other meta-implants. Moreover, the graded meta-implant gave minimum areas of local stress and local strain concentration at the contact region of the implants with the bone internal surfaces. Among all the cases, the graded meta-implant also gave micromotion levels which were the closest to value reported to be desirable for bone growth (40 µm) . The maxOne of the most important parameters which affect the accuracy of analytical results is the effect of considering the effective lengths of struts in a unit cell. In higher values of relative densities, the struts get thick, and the overlapping of the struts at the joints should be taken into account in the analytical solution. By defining the effective length of the struts according to their orientation and thickness, the analytical solution could predict more accurate results for the actual condition of the lattice structure . AnotherIt must be also noted that there must be a balance between the micromotion, stress, and stiffness as the bone growth is driven by the mechanical load, which means that the elastic force acting on the bone must be somehow maintained. The effect of implant-bone interaction is usually termed as the primary stability of the implant and is closely related to the stiffness. Moreover, not only the hip stem is important, but more often, the acetabular component fails due to aseptic loosening. For real-life design, such limitations and considerations should also be taken into account, see ,75 for mAs mentioned before, the loading and boundary conditions considered in this study for hip bone are based on the loading and boundary conditions of commonly used experimental setups in several experimental studies of femur hip implants ,62,63. TA hip-joint meta-implant with an appropriate lattice structure can improve the stress and strain distributions in the implant and surrounding bone tissues. Moreover, designing an accurate and precise graded lattice structure for the implant can enhance implant deformation as well as provide suitable micromotion for bone ingrowth at the bone-implant interface. This study focused on designing a graded lattice hip meta-implant by implementing a well-known 3D re-entrant auxetic metamaterial to enhance the micromotions at the bone-implant contact surfaces. To reach this aim, analytical relationships for mechanical properties of the 3D re-entrant structure were obtained using a more accurate analytical approach when compared to methods previously used in the literature. The results demonstrated a huge improvement in the analytical solution which overlapped with FE results. One solid implant and three meta-implants were designed based on the anatomy of the human femur bone. The new exact relationships for mechanical properties of 3D re-entrant unit cells were implemented to create three meta-implants with positive, negative, and graded Poisson’s ratios. The numerical results of the implants under compressive loading conditions showed that the meta-implants could provide enhanced stress and strain distributions in the implant and its contact regions with the bone internal surface. Although the micromotion values for all the meta-implants were within appropriate ranges for bone ingrowth, the meta-implants with negative and graded Poisson’s ratios gave a more compatible and uniform distribution of micromotion at major regions of the bone-implant interface.


Smartphone-based optical spectrometers allow the development of a new generation of portable and cost-effective optical sensing solutions that can be easily integrated into sensor networks. However, most commonly the spectral calibration relies on the external reference light sources which have known narrow spectral lines. Such calibration must be repeated each time the fiber and diffraction grating holders are removed from the smartphone and reattached. Moreover, the spectrometer wavelength scale can drift during the measurement because of the smartphone temperature fluctuations. The present work reports on a novel spectral self-calibration approach, based on the correspondence between the light wavelength and the hue features of the spectrum measured using a color RGB camera. These features are caused by the nonuniformity of camera RGB filters’ responses and their finite overlap, which is a typical situation for RGB cameras. Thus, the wavelength scale should be externally calibrated only once for each smartphone spectrometer and can further be continuously verified and corrected using the proposed self-calibration approach. An ability of the plug-and play operation and the temperature drift elimination of the smartphone spectrometer was experimentally demonstrated. Conducted experiments involved interrogation of optical fiber Fabry-Perot interferometric sensor and demonstrated a nanometer-level optical path difference resolution. Optical fiber sensors (OFS) are an emerging research topic, gaining a lot of attention from both academia and industry. During the last three decades they have progressed from simple laboratory prototypes to high-end commercial products, forming a several billion dollar market. OFS are an ideal choice for various sensing tasks, including biomedical, oil and gas, avionics, structure health monitoring and others. Their advantages are small size, biocompatibility, chemical neutrality, ability to perform remote measurements and multiplex several sensors, electromagnetic neutrality and absence of electric currents or radiofrequency fields in the sensing element. Another advantage is the ability to either make the sensing element out of a section of a fiber, or to form a sensing element with free-space propagation of light, whilst fiber acts as a light feeding element.One of the most commonly used interrogation approaches for OFS is spectral interrogation, which includes measurement of a spectral function of an OFS (either reflective or transmissive) and its further processing. This approach is used to acquire signals from OFS based on surface plasmon resonance (SPR) , gratingSpectral interferometry is widely used for interrogation of interferometric OFS with small  optical path differences (OPD), typically extrinsic Fabry-Perot interferometers (EFPI)  or in-fiShortly after the smartphones became widespread and deeply integrated into our everyday routines, it was proposed to use them as the core elements of portable optical spectrometers ,15, whicSpectral calibration methods are of particular importance for smartphone-interrogated optical fiber sensors with spectral interrogation since the additional constructions holding fibers and diffraction gratings usually can not be robustly fixed on a smartphone and are typically fixed on it with some pegs or clamps ,28. TherTo date, we are aware of an attempt to use the hue (H) information of the spectrum captured with a color smartphone camera to evaluate the wavelength scale of the measured spectrum , where tfound in , where ifound in  onto theActually, the use of color camera for optical spectrum measurement is quite intuitive and logical, since color camera already contains spectral resolving elements, which, however, are not suited for high-precision spectral discrimination, in contrast with diffraction gratings. Nevertheless, a way of exploiting these spectral elements and the spectrum features they cause for the purpose of spectral calibration can be found. In the current work, we have developed a spectral calibration approach similar to ; yet, inThe concept of the smartphone-based interrogation system, similar to most of the already known smartphone-based spectrometers (SPBS) ,23,25, ious work . The onlous work .The smartphone used in the proposed OFS interrogation setup was Huawei P20 Pro, featuring a 10-megapixel color camera. Most of the experiments were conducted using this model. However, several other widespread smartphone models were also tested and turned out to be suitable for realization of the proposed spectral self-calibration approach as well.Since spatially incoherent light irradiated by the smartphone LED cannot be efficiently coupled to singlemode fibers due to fundamental reasons ,34, it iThe sensing element used in the performed experiments was an extrinsic fiber Fabry-Perot interferometer, formed by the end faces of two multimode fibers. The reflectivities of the fiber ends are about 3%, which is due to the Fresnel effect. Thus the interferometer was low-finesse and a two-beam approximation was used for signal processing. The fibers were terminated with FC-PC connectors, which were fixed inside a standard mating sleeve. The air gap An example of image of the smartphone LED spectrum measured with the smartphone camera is shown in v axis at R, G and B. As follows from the Equation , which takes place for all spectrometers . The secHowever, for finite width IF  it will not be the case—the hue value of each pixel will be affected not only by the input signal intensity at a particular wavelength, but rather by the signal intensity in some spectral neighborhood. The wider the IF, the more the Hue distribution will depend on the input signal spectrum. In the context of our work, this is an undesirable effect as it will cause smoothing of the hue distribution features, discussed above. In order to avoid this detrimental effect, we have assembled a smartphone spectrometer with a fairly high spectral resolution of 4 nm.When the hue distribution is independent of the input signal spectrum and possesses clearly detectable features, these features can be unambiguously associated with the corresponding wavelength points. In other words, the wavelength scale of the measured spectrum will be linked to the spectral responses of the RGB optical filters, which are quite stable. A vital advantage of such correspondence is that for each smartphone it must be performed only once, after that the self-calibration can be performed for all subsequent measurements. The most crucial condition for robust operation of the proposed spectral self-calibration approach is that the analyzed optical spectrum must be sufficiently broad  and sufficiently bright for proper calculation of hue distribution of the spectrum image.Below, we describe the consecutive stages of the proposed spectral self-calibration approach, based on the spectra features described in the previous subsection.(1)x is the analyzed array, t is the indexing variable (row number in our case). The rows falling outside of the [After acquisition of a spectrum image, it was transformed from RGB to HSV format. Then it was cropped in order to remove blank areas—columns which RMS values of the value channel were less than half of the maximal column’s RMS were removed, after that RMS position (2)M is the number of pixels in the cropped image, H values. At the next step, histogram of all hue channel values of the cropped spectrum image was calculated. The number of histogram bins nis rule 4)Nb=(mNb=(mNb w(3)After calculating the histogram, the coordinates of its bins quations  and 3)..hr, hg aa and t is the indexing variable (row number in our case) and a, This allowed the red-to-green and green-to-blue transition areas to be identified in each column as the pixels, whose hue values lie within [(4)u coordinate (spectrum column number) were approximated by second-order polynomial fits, allowing to eliminate some noise effects. The fits are shown in The obtained dependencies of hue transition areas’ coordinates on (5)Finally, knowing the positions of two reference points, corresponding to certain wavelengths for each spectrum column, it was possible to interpolate all the columns and resample them to a true wavelength scale. The interpolation based on the fast Fourier transform  was applReproducibility of the proposed spectral self-calibration approach was tested using 7 smartphone models. The test results are presented in Calibration of the wavelengths corresponding to the red-to-green and green-to-blue transition areas was performed by measuring spectrum of a thermally and mechanically isolated Fabry-Perot interferometer with our developed SPBS and commercial spectrometer Hamamatsu C10082CAH. Smartphone LED was used as a light source for both measurements. Reference shape of the LED spectrum was previously recorded by directly irradiating the LED light to the spectrometers. After proper normalization, accounting for the reflection coefficients of the fiber ends, forming the interferometer, the quasi-static component of the both spectra was removed. This allowed to accurately estimate the positions of the FPI spectrum fringes by fitting the spectrum peaks by Gaussian functions. The spectra obtained with the commercial spectrometer and SPBS are shown in The correspondence between the peaks positions is shown in Wavelength sampling interval of the SPBS  could be evaluated after this calibration and turned out to be 0.22 nm/pixel. Spectral resolution of the assembled smartphone spectrometer was also experimentally evaluated by observing the spectra of 3 lasers: green and red semiconductor lasers with spectral linewidths essed as (6)ΔλMEAThis calibration operation needs to be performed only once prior to the use of the SPBS for optical fiber sensor interrogation if the spectrometer IF and the spectral transfer functions of the RGB channels of the smartphone camera do not change over time.Due to nonuniformity of the LED spectrum, the measured interferometer’s spectra have a quasi-constant additive component and parasitic amplitude modulation. The most convenient way to eliminate these signal distortions is their cancellation based on the a-priori information on the LED spectral shape. This way, preceded by proper normalization, the LED spectrum can be subtracted from the measured interferometer spectrum, thus canceling the quasi-static component. Parasitic amplitude modulation can be suppressed by normalizing the interferometer spectra on the LED spectrum.L is interferometer OPD and is the optimization parameter, After the quasi-static component and parasitic amplitude modulation were suppressed, the interferometer spectra were processed, resulting in demodulation of the interferometer OPD etail in  was usedpagation  and modepagation . It must2 errors . In our The whole signal processing workflow, whose stages were described above, including spectrum image cropping, hue transition areas identification, spectra interpolation, averaging and normalization, as well as the wavelength scale calculation and interferometer OPD demodulation is shown schematically in The performance of the developed smartphone-based OFS interrogation system and the proposed spectral self-calibration approach was verified experimentally, for which we have carried out two experiments. The first one was aimed at proving the ability of plug-and-play operation, i.e., provide accurate interferometer OPD readings after reattachment of the fiber and diffraction grating holder from the smartphone without any calibration. The second experiment was aimed at demonstrating an improvement of the temperature stability of the smartphone-based OFS interrogation system.Obviously, the wavelength-to-pixel number calibration of a smartphone-based spectrometer with conventional pre-calibration is lost after the fiber and diffraction grating holders are shifted or removed and reattached. Given that the size of the state-of-the-art smartphone camera pixels is on the order of 1–2 Our hue-based spectral self-calibration approach allows us to overcome this limitation and enables correct wavelength scale reconstruction and accurate signal demodulation in case of reasonably large offsets of the fiber and diffraction grating holders. A series of experimental trials was performed to verify this ability. The interferometric sensing element was stabilized by being put inside a thermally and vibrationally isolated chamber, thus ensuring that its true OPD remained constant, while fluctuation of the measured OPD value can characterize the robustness and accuracy of the proposed spectral self-calibration approach. Prior to each measurement of the interferometer spectrum, the holder was removed from the smartphone and then reattached without any special alignment. The only condition for the calibration and the measurement procedures to be performed is that the light from the LED is coupled to the fiber and interferometer spectrum is visible at some place of the camera image. Six spectra images are shown in However, despite noticeable misalignment of the measured spectra images, the developed spectral self-calibration approach enables to reduce the detrimental effect of misalignment and provide an accurate demodulated sensor signal. This is shown in It is logical to anticipate that thermal expansion of the 3D-printed holders, which provide the alignment of all optical elements  will cause the change of the initial elements alignment. In turn, this will lead to the shift of the optical spectrum projected onto the smartphone camera and change of the corresponding wavelength scale. Therefore, the initial spectral calibration  will be lost, resulting in incorrect demodulation of sensors interrogated in spectral domain, such as interferometric and fiber grating-based sensors.Our proposed spectral self-calibration approach is aimed at solving this problem. The verification experiment was carried out in the following manner—the smartphone-based OFS interrogation system was turned on and the temperature on the smartphone surface near the fiber and diffraction grating holder was monitored with an external electronic thermometer with a small footprint. The smartphone itself produces some heat even in idle mode; moreover, when the interrogation system is operating and there is an increased workload on the processor and the LED is switched on, it results in heating of the holder, causing it to expand, which results in a shift of the measured spectra images.On the other hand, if the interrogated interferometer is placed in a thermally and mechanically isolated chamber, its real OPD remains constant, while the change of the demodulated OPD value will clearly indicate the influence of the interrogation system heating on the measurement outcome.During the conducted experiment, the flash LED was constantly turned on, resulting in gradual change of temperature on the smartphone back panel near the 3D-printed fiber holders from nearly 33 Celsius to almost 41 Celsius over about an hour, which is shown in Demodulated OPD values obtained with and without the developed hue-based spectral self-calibration approach as well as the shifts of the wavelength scale are shown in The paper presents a novel spectral self-calibration approach for smartphone spectrometers. Thanks to the mutual overlap between the spectral ranges covered by the red and green and green and blue channels of a smartphone camera, the hue distribution of the measured spectrum has two distinctive features, whose corresponding wavelengths can be calibrated with a reference spectrometer. Once the wavelengths corresponding to the hue distribution features are known, these features can be further used to restore the wavelength scale of the measured spectrum.The use of the developed spectral self-calibration approach allowed to eliminate temperature dependence of the smartphone spectrometer readings and make the system plug-and-play, in contrast to the most other known smartphone spectrometers, which spectral accuracy relies upon wavelength-to-pixel calibration performed with external reference light sources and is lost if the fiber and diffraction grating holders are removed and reattached to the smartphone, for example, during transportation. However, the fallout of this improved accuracy is about 10 times reduced OPD measurement resolution, which is a result of finite accuracy of fitting the slopes of hue distribution. However, in practical situations self-calibration can be performed not continuously, but upon user’s request or periodically in order to detect deviations of the wavelength scale. This will enable to achieve a trade-off between the sensing resolution and accuracy.The performed experiments validated the efficiency of the developed spectral self-calibration approach and its ability to be used for interrogation of optical fiber Fabry-Perot interferometers. Further work must be directed towards estimating the accuracy limitations of the proposed approach  and ways to improve the accuracy. It should be noted that the developed spectral self-calibration approach can be applied not only to interferometric sensors, but also to other sensors with spectral interrogation, such SPR and some grating-based sensors, as well as to spectroscopic tasks. The main requirement for robust operation of the proposed spectral self-calibration approach is relatively high spectrum intensity in red-to-green and green-to-blue transition areas, finite mutual overlap of red and green and green and blue camera channels’ spectral functions, resulting in nonuniform hue distribution of the measured spectrum image and relatively high spectrometer spectral resolution ensuring correct capture of the spectra images’ hue distribution.Obtained experimental data, Matlab source codes for spectra images’ processing and models for 3D-printing the fiber and diffraction grating holder are available from the authors on reasonable request.


Chromatin communities stabilized by protein machinery play essential role in gene regulation and refine global polymeric folding of the chromatin fiber. However, treatment of these communities in the framework of the classical network theory  does not take into account intrinsic linear connectivity of the chromatin loci. Here we propose the polymer block model, paving the way for community detection in polymer networks. On the basis of this new model we modify the non-backtracking flow operator and suggest the first protocol for annotation of compartmental domains in sparse single cell Hi-C matrices. In particular, we prove that our approach corresponds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking operator resolves the true compartmental structure up to the theoretical detectability threshold, while all commonly used operators fail above it. We test various operators on real data and conclude that the sizes of the non-backtracking single cell domains are most close to the sizes of compartments from the population data. Moreover, the found domains clearly segregate in the gene density and correlate with the population compartmental mask, corroborating biological significance of our annotation of the chromatin compartmental domains in single cells Hi-C matrices. Cellular networks demonstrate modular organization, which optimizes crucial biological processes and relationships, such as synchronization of neurons in the connectome6, efficiency of metabolic pathways8], genes specialization9 or interaction between enhancers and promoters10.Many real-world stochastic networks split into self-organized communities. Social networks feature circles of friends13 provide the genome-wide colocalization data of chromatin loci. As the main outcome of the experiment, large genome-wide matrices of contacts from each individual cell or from the population are produced. Analyses of these matrices has revealed that the eukaryotic genome is organized in various and biologically relevant communities, whose main function is to insulate some regions of DNA and to provide easy access to the others. In particular, the data collected from a population of cells suggest that transcribed (“active”) chromatin segregates from the, “inactive” one, forming two compartments in the bulk of the nucleus14. Within compartments chromatin is organized further as a set of topologically-associated domains (TADs)17 that regulate chromatin folding at finer scales. However, interpretation and validation of communities in individual cells remains vaguely defined due to sparsity of respective data.Interest to polymer modular networks has appeared recently in the context of genome spatial folding. Proximity of chromatin loci in space is believed to be deeply connected with gene regulation and function. Hi-C experiments22. Typically, leading eigenvectors of these operators positively correlate with the true community structure or with the underlying core-periphery organization of the network23. These algorithms, along with the majority of theoretical results in the field, are derived for the stochastic block model (SBM)22 as an extension of Erdös-Rényi graphs24 to graphs with explicitly defined communities. One of the strongest limitations of the SBM is that edges between vertices belonging to the same cluster inevitably attain equal weights. At the same time, biological networks typically have several levels of organization within their communities25. In particular, identification of several hierarchical levels in the network becomes tremendously important in the case of polymer networks, where different pairs of loci have marginally different probabilities to form a contact in space26, caused by the frozen linear connectivity along the chain.The broad field of applications of stochastic modular networks has initiated the boost development of community detection methods. Spectral algorithms exploit the spectrum of various operators  defined on a network to identify the number of communities and to infer the optimal network partition27. In this work we propose the “polymer stochastic block model” which reflects a specific global polymer network organization with explicit structuring into communities. The main new ingredient of the model under consideration is the average contact probability i, j) which is constant for standard non-polymeric networks, however cannot be neglected for polymers.Even for simplest polymer systems the contact probability demonstrates a power-law behavior with the dimensional-dependent scaling exponent characterizing universal long-ranged behavior of polymer folding28. It is known that upon reduction of the total number of edges in the network, there is a fundamental resolution limit for all community detection methods29. Furthermore, traditional operators  fail far above this resolution limit, i.e. their leading eigenvectors become uncorrelated with the true community structure above the threshold30. That is explained by emergence of tree-like subgraphs (hubs) overlapping with true clusters in the isolated part of the spectrum for these operators. The edge of the spectral density of sparse networks is universal and demonstrates the so-called “Lifshitz tail”34. Localization on hubs, but not on true communities is a drawback of all conventional spectral methods in the sparse regime.Chromatin single cell networks are not only polymeric, but also sparse30. The crucial property of non-backtracking walks35 is that they do not concentrate on hubs. It has been shown that the non-backtracking operator is able to resolve the community structure in sparse stochastic block model up to the theoretical resolution limit. Typically, the majority of eigenvalues of the non-backtracking operator  are located inside a disc in a complex plane, and a number of isolated eigenvalues lie on the real axis.To prevent the effect of localization on hubs and to make spectral methods useful in sparse regime, Krzakala et al. proposed to deal with non-backtracking random walks on a directed graph that cannot revisit the same node on the subsequent stepFor the sake of community detection in sparse polymer networks we construct the polymer-type non-backtracking walks, appropriate for community detection in graphs with hidden linear memory (“polymeric background”). We establish the connection between this operator and the generalized polymer modularity, thus, bridging a gap with the maximum entropy principle. We test the performance of different spectral methods (with and without polymer background) on sparse artificial benchmarks of polymer networks that mimic compartmentalization in single cell Hi-C graphs. We show that polymer non-backtracking walks resolve the structure of communities up to the detectability threshold, while all other operators fail above it. In order to demonstrate efficiency of the method on real data, we partition a set of single cell Hi-C contact maps of mouse oocytes into active (A) and inactive (B) compartments by different operators. Found domains are shown to have similar sizes to the compartmental domains and correlate with the compartmental mask from the population-averaged data. Analyses of the GC content within the domains demonstrates enrichment and depression of the genes density in the two clusters, thus, corroborating their biological significance.The structure of the paper is as follows. In Section “N-bead polymer chain by coordinates 12. A graph V. The edge between a pair of nodes  is defined by the condition 28. Summation over realizations of adjacency matrices i, j) is less than Characterize a P(s) is probability to find two beads of a linear chain, separated by a chemical distance s, close to each other in space. The critical exponent, s in a D-dimensional space27. Such a memory can arise due to some equilibrium topological state of chromatin, or could be a result of partial relaxation of mitotic chromosomes36. Notable examples of 39.Already for the simplest configurations, such as a conformation of ideal polymer chain isomorphic to the random walk, the matrix 22 superimposed over the background. Stochastic block model is a network model in which N nodes of a network are split into q different groups i, j)  readsA and B type (painted in red and blue), whose lengths are Poisson-distributed with the mean length 28. Since segments of the same type are surrounded in space by segments of the other type, they form local “blob-like” clusters along the main diagonal of the adjacency matrix reminiscent to topologically-associated domains15. However, they are likely formed by a different mechanism and have an order of magnitude larger size than TADs40. Such a multi-domain blob structure in Fig. Communities of folded chromatin refine the background (polymeric) contact probability at small scales and are biologically significant. We treat communities as canonical stochastic blocksmentclass2pt{minimmentclass2pt{minimmentclass2pt{minimmentclass2pt{minimmentclass2pt{minimi and j coming from a population of single cells. Thus, after proper normalization, P and group labels of the nodes Suppose that a population-averaged matrix tated by , \documeentclass1pt{minimad is the vector of degrees,  in the population-averaged data, or as a probability of the link in the single cell graph. Indeed, in absence of the stochastic blocks, this value equals The entropic functional , up to ndegrees,  reduces rding to . We real22. However, as it was first noted by Krzakala et al.30, for sparse networks leading eigenvectors become uncorrelated with true community structure well above the theoretical threshold. As a result, all conventional operators such as adjacency, Laplacian and modularity fail to find communities in rather sparse networks.Search for the global maximum to the modularity functional is a very hard computational problem. One of most promising approaches which avoids a brute force, is to suggest that if the community structure is significantly strong, there is an operator whose eigenvectors encode the network partitioning in these communities35. It is defined on the edges of the directed graph, 31. Real eigenvalues of 19 M. Newman suggested a normalized operator, that conserves the probability flow at each step of the walker.To overcome this difficulty, it was proposed to exploit the spectrum of the Hashimoto matrix For the sake of community detection in sparse polymer graphs, we propose a conceptually similar operator that describes the evolution of the non-backtracking probability flow on a graph with intrinsic linear memory19. For a pure polymeric graph without contamination by communities, the spectrum of  underscores the importance of taking into account the contact probability (polymer background) when dealing with polymer graphs, and (ii) recapitulates efficiency of non-backtracking walks in resolving communities in sparse networks.In Fig. 42 . Due to the overall small number of contacts in the network, the polymer non-backtracking flow ends up rationalizing them as separate clusters.For the stochastic block model the number of isolated eigenvalues of mentclass2pt{minimbegin with the initial value use the amount of inner and outer edges for estimating recalculate rding to ;repeat the procedure iteratively until Results of this procedure are demonstrated in the Fig. The value of vided by . A drawb28. From the public repository we have taken the single cells Hi-C data on cis-contacts of 20 chromosomes from 13 single cells . While single cells matrices with sufficiently large number of contacts are not sparse and can be split into compartments using conventional methods largely used for the bulk data , here we take the cells with low to moderate amount of contacts for the sake of comparative analyses of clustering performance of different spectral methods on sparse polymer graphs.To check robustness of the polymer non-backtracking flow operator on real Hi-C data we run it on a set of individual oocyte cells of mouseA and B compartments. We note, that, in principle, the method is applicable at higher resolution as well. However, there are two important considerations. The non-backtracking operator is defined on the edges, therefore, the leading eigenvectors need to be computed for much larger matrix than in case of traditional operators, which are defined on the nodes . This means that the computation time of the method is very sensitive to the resolution. Furthermore, one needs to be very careful with the overall network density: it decreases by several times upon decreasing of the bin size, so that one can occasionally cross the detectability limit  is encoded within the coarse-grained blobs and does not communicate with two large-scale ty limit . In eachMost of the contacts in the cells have degeneracy 1 at the chosen resolution, however, several pairs of bins have more than 1 contact. To preserve this feature of enhanced connectivity, we consider the counts of contacts between the pairs as weights of the corresponding edges. Furthermore, the single-cell maps are noisy and some of really existing contacts get lost due to technical shortcomings of the experimental protocol. As long as the neighboring blobs in the chromatin chain are connected with probability 1, all lost contacts 28, human lymphoblastoid cells12 and Drosophila cells43. As it was shown in previous Section, in order to extract compartmentalization profile overlaying a specific long-ranged folding, it is crucial to incorporate the respective background contact probability into the polymer model of the stochastic blocks.To determine the background model for our analyses we calculate the contact probability Having the background model determined, we construct the polymer non-backtracking flow operator with the variable parameter A compartment, thus, the fraction of GC letters in bins of active compartmental domains needs to be larger than in inactive domains. To validate that the clusters found in single cells respond to the transcriptional domains and are biologically significant, we calculate the GC content profiles around the centers of all A and B domains separately and then take the average of these profiles in each group. The types of the domains were phased in accordance with the leading eigenvector of the bulk data . We also plot analogous profiles for the leading eigenvector of the bulk data. In absence of direct annotation methods for single cells due to their sparsity, these two measures have been of use to approximate positions of the compartmental domains in single cell Hi-C data28.The partition of the single cells in two compartments has been performed in the leading eigenvector approximation of the different operators. The boundaries of active and inactive domains are determined according to the sign of the respective compartmental signal see Fig.  online. A domains and symmetrical dips at the center of B domains with the z-score amplitude equal to 0.4 (GC) and 0.7 (E1), correspondingly. Single cells profiles demonstrate notably lower amplitudes  of the polymer non-backtracking flow fall symmetrically to zero at the same genomic distance, around P(s) for the set of single cells, Fig. As expected, the GC content for the population-averaged map and the bulk E1 vector both have pronounced peaks at the center of see Fig.  online. Note that the partitions of the polymeric operators  are visibly much more adequate to apparent clustering of contacts in a particular cell (Supplementary Fig. In this paper we have developed theoretical grounds for spectral community detection in sparse polymer networks. On the basis of suggested polymeric extension of the stochastic block model, we have proposed the polymer non-backtracking flow operator and have proven that its leading eigenvector performs partitioning of a polymeric network into two clusters according the maximum entropy principle. The established connection with the modularity functional provides a computationally efficient tool for the network partitioning and search for the optimal resolution parameter of the partition in polymer networks, which, however, is inferior to the non-backtracking in efficiency for sparse networks.The proposed theoretical framework is verified by extensive numerical simulations of polymer benchmarks, constructed in order to emulate compartmentalization in sparse chromatin networks. Comparative analyses of different operators on the benchmark has suggested that the polymer flow detects the communities up to the theoretical detectability limit, while all other operators fail above it. At the same time, the amount of isolated eigenvalues of the polymer flow operator can be larger than amount of true communities present in the network, due to frozen linear connectivity that forces the chain to form “blobs” along the chain contour. This result distinguishes the polymer system with thespect the canonical stochastic block model, where the number of isolated eigenvalues of the non-backtracking exactly matches the number of communities.Analyses of the single cell Hi-C data of mouse oocytes suggests that the non-backtracking walks efficiently split experimental sparse networks into biologically significant communities, characterized by enrichment and depression of the genes density. The sizes of the compartmental domains are fairly close to the sizes of the population-averaged domains. Comparison with characteristics of the domains, inferred by other operators, underscores superiority of the non-backtracking walks in partitioning sparse polymer networks.46. In the context of the gene interactions the spectral statistics has been discussed in47 for the matrices with the real spectrum. The non-backtracking matrices enjoy complex spectrum hence the special means are required to analyze the level spacing in this case. The corresponding tool has been invented recently49, therefore, the spectral statistics of the polymer non-backtracking flow operator certainly deserves a separate study.In this study we have exploited for the polymer network analysis only the simplest spectral characteristics. More involved ones, e.g. spectral correlators and the level spacing distribution, carry additional information about the propagation of excitations in network. The spectral statistics and non-ergodicity have been discussed in clustered networks inSupplementary information


An unsteady convection-radiation interaction flow of power-law non-Newtonian nanofluids using the time-fractional derivative is examined. The flow domain is an enclosure that has a free surface located at the top boundaries. Also, the geometry is filled by aluminum foam as a porous medium and the overall thermal conductivity as well as the heat capacity are approximated using a linear combination of the properties of the fluid and porous phases. Additionally, the dynamic viscosity and thermal conductivity of the mixture are expressed as a function of velocity gradients with a fractional power. Marangoni influences are imposed to the top free surface while the bottom boundaries are partially heated. Steps of the solution methodology are consisting of approximation of the time fractional derivatives using the conformable definition, using the finite differences method to discretize the governing system and implementation the resulting algebraic system. The main outcomes reveled that as the fractional order approaches to one, the maximum values of the stream function, the bulk-averaged temperature and cup-mixing temperature are reduces, regardless values of the time. The power-law non-Newtonian nanofluids are new category of the nanomixtures which contains a non-Newtonain base fluid (For example Carboxymethyl Cellulose (CMC) solutions) and one or more type of the nanoparticles. The purpose of these compounds is to support the thermal transfer of conventional fluids. An enormous number of studies have been presented using Newtonian nanofluids9 while the interest in the non-Newtonian nanofluids is still weak. The FVM (finite volume method) was applied by Zhuang and Zhu10 to study the double diffusive within the cubic enclosure filled with a heterogeneous porous medium. The Marangoni effects together with the CMC-Cuo nanofluids were considered. The findings revealed that the decrease of the power-law index causes an increase in the heat and mass transfer rate as well as intensification in the fluid motion. The heat transfer due to the Marangoni convection using the power –law fluid over a surface saturated porous media has been presented by Jiao et al.11. The surface tension is considered as a quadratic function of the gradients of the temperature. It is noticeable that the velocity is reduced while the temperature is increased as the porosity parameter is grown. Lin et al.12 investigated the radiation- Marangoni interaction flow using the pseudo-plastic nanofluids having a variable thermal conductivity. The results indicating that the shear stress of CMC-TiO2 nanofluid is diminished as the power-law index is altered.Svey of the heat transport using the non-Newtonian fluid within domains filled by porous elements is a major topic in the CFD  filed due to the important practical applications in the real life. These enforcements include oil recovery, material processing, polymer, synthetic lubricants, liquid films, cosmetics, oil–water emulsion, paints, jellies, etc.15. Non- Newtonian fluids dynamics are one of these fields that was modeled by fractional derivatives models because the non-locality of fractional calculus which gives long-term memory25. However, it is very difficult to achieve the exact solution for nonlinear fractional problems. It is possible to use some numerical and approximate methods to find numerical solutions for most nonlinear fractional problems30. The natural convection along a vertical wall and cylinder using Caputo time-fractional derivatives are presented in32. Also, several concepts of fractional derivatives such as fractional logistic models, fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems, simulating of COVID-19 using the fractional derivatives and natural convection flow of a fluid using Atangana and Baleanu fractional model are presented in37. There are many definitions for fractional derivatives and fractional integrals are defined in different ways such as Riemann Liouville, Caputo and others15. Khalil et al.38 presented a new operator called “conformable derivative” which satisfied new conventional properties. Several authors effectively used the conformable operators of fractional order in modelling several models47. Some recent works use the conformable fractional operator to discuss fractional Newtonian mechanics48.Tabulated in48, the conformable fractional operator Over the years ago, fractional calculus theory is generalizing the integer order of differentiation to non-integer order. It has many advantageous in several real life fields such as fluid mechanics, optics, plasma, electromagnetism physics, engineering, biology and economics because it shows the new properties of these problems47:This fractional derivative has the following properties as it is stated inwhere 59. Further, the Marangoni convection, non-Newtonian second grade nanofluid flow and non-Newtonian ferrofluid flow are presented in the valuable investigations64.In addition to the previous survey, the fractional derivatives are used to simulate the convective flow in various published works65.This paper aims to use the fractional derivative approaches to examine the radiation and Marangoni influences on the power-law non-Newtonian nanofluid flow within an inclined domains. The geometry has a free surface where the surface tension is a function of the temperature gradients and is filled by a porous medium. The worked liquid is consisting of carboxymethyl cellulose (CMC) as a non-Newtonian base fluid while CuO elements are considered as nanoparticles. Also, the aluminum foam is considered as porous elements while the radiation flux is considered in the normal direction. The conformable operator is used for estimating the time fractional derivatives while the dimensionless governing system is solved numerically using an implicit FDM. The novelty of this work appears in simulating important impacts such as Marangoni influences on the flow of unused nanofluids frequently using the fractional partial differential equations that is did not presented before, and is more attractive for the researchers. Also, the results of the current simulations can be effective in various industrial practices such as oil recovery and materials processing. Further, a good survey on applications of the fractional calculus to oil industry is presented in Martínez-Salgado et al.Height of the enclosure is H and the inclination angle is Length of the heated section is A low temperature condition  has a heat transfer based on the Newton's low cooling.The surface tension The flow domain is illustrated in Fig. The nanofluid flow is unsteady, laminar and two dimensional.The non-Newtonian power law nanofluids are represented by the single-phase model.The base fluid is carboxymethyl cellulose (CMC) while CuO is assumed as nanoparticles.The thermophysical properties of the components of the nanofluid are given in Table The domain is filled by homogeneous aluminum foam and the Darcy model is applied.The thermal conductivity of the porous medium is considered variable and the thermal radiation is taken in The thermal equilibrium state is satisfied between the porous and nanofluid phase.where 67:The mathematical formulations of the present case are modeled using the continuity, momentum and energy equations based on the previous assumptions; those are written as, see10.In Eq. , \documeAlso, 42) as:The overall thermal conductivity is depending on the features of the power-law  are are20) aFinally, the following algebraic system is obtained:Select a suitable grid. It is recommended to start with All dependent variables are initialized to zero.The new boundary conditions at the first iteration are calculatedThe new temperature at the current iteration is calculated from previous values at all internal grid points.The velocities (The same procedure is followed by starting with step (c)to obtain the solution at the next iteration.The iteration process is terminated if the following condition satisfies:The Nusselt and entropy generation are then calculated.Here, the following algorithm is used to implement the obtained discretized equations:67. Also, Fig. 67. All these validation tests show that there are excellent agreements between the outcomes.The alternating direction implicit (ADI) is applied to solve the resulting system while the time step is selected to be entclass1pt{minimaA comprehensive discussion of the obtained outcomes is presented in this section. Here, it is interested with the influences of the time parameter (Using the fractional derivatives basics, the unsteady convective-radiation flow confined an enclosure filled with CMC-water power-law non-Newtonian nanofluids was investigated. The fractional derivatives were taken on the time while the conformable definitions were used to approximate the calculations. The Marangoni effects are imposed to the top-free surface of the domain while the bottom boundaries are partially heated. The one-phase model in which the overall dynamic viscosity and thermal conductivity are functions of the power-law index is presented while the Rosseland approximation is used for the thermal radiation. Beside the cup-mixing temperature and the bulk-averaged temperature, the entropy of the system is examined for the variations of the controlling parameter. The main outcomes of this study revealed that the increase in the fractional order enhances the average Nusselt number while the maximum values of the stream function, the cup-mixing temperature and the bulk-averaged temperature are reduced as