Optimal Warping Research Articles

A method for selecting the optimal warping path of dynamic time warping in gait analysis.

This study aims to demonstrate that when performing dynamic time warping (DTW) on gait data, multiple optimal warping paths (OWPs) with a minimum sum of local costs can occur and to propose an additional OWP selection method to address this problem. A 3-dimensional motion analysis experiment was conducted on 55 adult participants, including both males and females, to acquire gait data. This study analyzed 990 instances of DTW on gait data to examine the occurrence of multiple OWPs with the minimum sum of local costs. We subsequently applied an additional selection method to the multiple OWPs to determine the feasibility of identifying a single OWP. Multiple OWPs through DTW were observed 82 times, accounting for 8.28%. Notably, on the ankle joint of males, the rate was the highest at 11.11%. Cases with two multiple OWPs were the most prevalent at 56.10%, and cases with ten or more multiple OWPs accounted for 19.51%. The additional selection method proposed in this study was applied to the 82 instances in which multiple OWPs occurred. The results demonstrated the ability to identify a unique OWP in all cases. These results hold significance in identifying the shortcomings of conventional OWP selection methods previously employed and proposing solutions. It enhances the reliability, validity, and accuracy of studies utilizing DTW.

A stochastic process representation for time warping functions

Time warping function provides a mathematical representation to measure phase variability in functional data. Recent studies have developed various approaches to estimate optimal warping between functions. However, a principled, linear, generative representation on time warping functions is still under-explored. This is highly challenging because the warping functions are non-linear in the conventional L2 space. To address this problem, a new linear warping space is defined and a stochastic process representation is proposed to characterize time warping functions. The key is to define an inner-product structure on the time warping space, followed by a transformation which maps the warping functions into a sub-space of the L2 space. With certain constraints on the warping functions, this transformation is an isometric isomorphism. In the transformed space, the L2 basis in the Hilbert space is adopted for representation, which can be easily utilized to generate time warping functions by using different types of stochastic process. The effectiveness of this representation is demonstrated through its use as a new penalty in the penalized function registration, accompanied by an efficient gradient method to minimize the cost function. The new penalized method is illustrated through simulations that properly characterize nonuniform and correlated constraints in the time domain. Furthermore, this representation is utilized to develop a boxplot for warping functions, which can estimate templates and identify warping outliers. Finally, this representation is applied to a Covid-19 dataset to construct boxplots and identify states with outlying growth patterns.

Effect of Subsonic Compressibility and Constraints on Generation of 3-D Optimal Camber Shapes

A flat surface wing is optimized for minimum induced drag for low and high subsonic Mach numbers. Wing for transport airplane role is considered. Optimal warp is separated into twist and camber. Effect of compressibility is analysed on the camber shape generated for two different Mach numbers. Constraint of lift is firstly used for developing an objective function for drag minimization. Next the lift and wing root bending moment constraints are considered to form objective function. Analysis is made on the resulting camber and twist distributions.

A Reconstructive Approach for Design of a Planar Supercritical Wing

Effort is made to design an efficient planar wing that can support transonic pressure distribution. Wing for transport airplane role is considered. First optimal warp is generated for minimum induced drag for a Mach number that is just below transonic Mach number. Optimal warp thus generated is split into twist and camber. The resulting camber is used to form a planar wing. The camber of this aerofoil is varied along span. Twist is no more considered. Aerofoil thickness is superimposed on this camber. Mach number is increased till supercritical flow appears.The whole process is reconstructive in approach. Results obtained are very encouraging and are application oriented. Pattern of supercritical pressure distribution is studied from point of view of suction peaks, shock wave location, and aft loading intensity occurring towards trailing edge. Main effort of our exercise is to create shock free transonic wing.

The morphing for continuous generalization of linear features based on Dynamic Time Warping distance

Shape morphing has been used to generate arbitrary scale maps in the field of continuous generalization in recent years. The morphing approach consists of two main steps: shape characteristic matching and trajectory interpolation. Most of the shape characteristic matching methods are difficult to consider the local and global characteristics of spatial data at the same time, which will result in distortion of interpolation results. This paper proposes a Dynamic Time Warping (DTW) distance-based morphing method for continuous generalization of linear features. In this method, the DTW distance is employed as shape similarity distance to find the optimal correspondence by minimizing the total cost between each pair of vertices, which considers both the local and global structure of two linear features. Firstly, we build a matrix to record the distance between vertices in two linear features at two different scales. Then we use the DTW algorithm to find the optimal warping path to establish the corresponding relationship between the vertices of two linear features. Finally, the linear interpolation method is used to dynamically generate the geometric shape at any intermediate scale. Experiment results demonstrate that the proposed method can generate continuous and smooth geometric shapes with a gradual transformation effect, and can be used for the continuous generalization of linear features. The generalization results are consistent with map representation rules and human cognition.

Normalization and possibility of classification analysis using the optimal warping paths of dynamic time warping in gait analysis.

The purpose of this study was to verify classification performance and the difference analysis between gender using optimal warping paths of dynamic time warping (DTW) and to examine the usefulness of root mean square error (RMSE) represented by the perpendicular distance from the optimal warping path to the diagonal. A 3-dimensional motion analysis experiment was performed with 24 healthy adults (male=12, female=12) in their 20s of age without gait-related diseases or injuries for the past 6 months to collect gait data. This study performed a DTW 132 times in total (male=62, female=62) for the flexion angle of the right leg's hip, knee, and ankle joints. Then, the global cost and the RMSE of the optimal warping paths were calculated and normalized. The difference analysis was performed by independent t-test. Machine learning was performed to test the classification performance using the neural network, support vector machine, and logistic regression model among the supervised models. Results analyzed using global cost and RMSE for hip, knee, and ankle joints showed a statistically significant difference between genders in global cost and RMSE for hip and knee joints but not for ankle joints using RMSE. Considering both area under the receiver operating characteristic curve and F1-score, the logistic regression model has been evaluated as the most suitable for gender classification using the global cost or RMSE. This study demonstrated that optimal warping paths could be used for statistical difference analysis and classification analysis.

Feature Correspondences Increase and Hybrid Terms Optimization Warp for Image Stitching.

Feature detection and correct matching are the basis of the image stitching process. Whether the matching is correct and the number of matches directly affect the quality of the final stitching results. At present, almost all image stitching methods use SIFT+RANSAC pattern to extract and match feature points. However, it is difficult to obtain sufficient correct matching points in low-textured or repetitively-textured regions, resulting in insufficient matching points in the overlapping region, and this further leads to the warping model being estimated erroneously. In this paper, we propose a novel and flexible approach by increasing feature correspondences and optimizing hybrid terms. It can obtain sufficient correct feature correspondences in the overlapping region with low-textured or repetitively-textured areas to eliminate misalignment. When a weak texture and large parallax coexist in the overlapping region, the alignment and distortion often restrict each other and are difficult to balance. Accurate alignment is often accompanied by projection distortion and perspective distortion. Regarding this, we propose hybrid terms optimization warp, which combines global similarity transformations on the basis of initial global homography and estimates the optimal warping by adjusting various term parameters. By doing this, we can mitigate projection distortion and perspective distortion, while effectively balancing alignment and distortion. The experimental results demonstrate that the proposed method outperforms the state-of-the-art in accurate alignment on images with low-textured areas in the overlapping region, and the stitching results have less perspective and projection distortion.

Bipartite Ranking Fairness Through a Model Agnostic Ordering Adjustment.

Recently, with the applications of algorithms in various risky scenarios, algorithmic fairness has been a serious concern and received lots of interest in machine learning community. In this article, we focus on the bipartite ranking scenario, where the instances come from either the positive or negative class and the goal is to learn a ranking function that ranks positive instances higher than negative ones. We are interested in whether the learned ranking function can cause systematic disparity across different protected groups defined by sensitive attributes. While there could be a trade-off between fairness and performance, we propose a model agnostic post-processing framework xOrder for achieving fairness in bipartite ranking and maintaining the algorithm classification performance. In particular, we optimize a weighted sum of the utility as identifying an optimal warping path across different protected groups and solve it through a dynamic programming process. xOrder is compatible with various classification models and ranking fairness metrics, including supervised and unsupervised fairness metrics. In addition to binary groups, xOrder can be applied to multiple protected groups. We evaluate our proposed algorithm on four benchmark data sets and two real-world patient electronic health record repositories. xOrder consistently achieves a better balance between the algorithm utility and ranking fairness on a variety of datasets with different metrics. From the visualization of the calibrated ranking scores, xOrder mitigates the score distribution shifts of different groups compared with baselines. Moreover, additional analytical results verify that xOrder achieves a robust performance when faced with fewer samples and a bigger difference between training and testing ranking score distributions.

Investigation of warp tension on looms with a new automatic brake

In this article, the task was set to study the existing main planetary regulator on machines of the AT type. It is proposed to modernize the design of the planetary regulator, expediently providing for the features of the production of fabric from boiled natural silk. This will allow you to choose the optimal warp tension mode and reduce the breakage of the warp threads on the loom. Keywords: breakage, main threads, beam, planetary regulator, tension, influence, rock, friction, arises, brake, reverse, pulley, washer.

A general optimization framework for dynamic time warping

The goal of dynamic time warping is to transform or warp time in order to approximately align two signals. We pose the choice of warping function as an optimization problem with several terms in the objective. The first term measures the misalignment of the time-warped signals. Two additional regularization terms penalize the cumulative warping and the instantaneous rate of time warping; constraints on the warping can be imposed by assigning the value $$+\infty$$ to the regularization terms. Different choices of the three objective terms yield different time warping functions that trade off signal fit or alignment and properties of the warping function. The optimization problem we formulate is a classical optimal control problem, with initial and terminal constraints, and a state dimension of one. We describe an effective general method that minimizes the objective by discretizing the values of the original and warped time, and using standard dynamic programming to compute the (globally) optimal warping function with the discretized values. Iterated refinement of this scheme yields a high accuracy warping function in just a few iterations. Our method is implemented as an open source Python package GDTW.

Cycle-skipping mitigation using misfit measurements based on differentiable dynamic time warping

The dynamic time warping (DTW) misfit function has been used for full-waveform inversion to mitigate the local minima issue. However, it is not a smooth measurement and can yield strong discontinuities in the adjoint source. Such a weakness hampers the convergence of waveform inversion to informative minima. We have introduced a smooth DTW misfit function to remedy this shortcoming. The fundamental idea of the smooth DTW is to replace the min operator with its smooth relaxation. This replacement results in differentiable misfit measurements. Moreover, considering that the optimal warping plan indicates the traveltime difference between the observed and synthetic trace, we can construct a penalization term such that the penalized differentiable DTW distance further highlights the traveltime difference. Numerical examples demonstrate the advantage of the penalized differentiable DTW misfit function over the conventional nondifferentiable one.

Signal recognition method based on Mel frequency cepstral coefficients and fast dynamic time warping for optical fiber perimeter defense systems.

To improve the accuracy of signal recognition in optical fiber perimeter defense systems, a method based on Mel frequency cepstral coefficients (MFCCs) and a fast dynamic time warping (FastDTW) algorithm is proposed. Four kinds of sensing signals are acquired by employing an in-line Sagnac interferometer system. MFCC features of the signals are extracted. The FastDTW algorithm is utilized to calculate distances of the optimal warping paths between a test sample and all reference templates. According to the nearest neighbor criterion, signal recognition results are obtained. The average accuracy is over 96%. This proposed identification method is highly efficient and easy to implement.

Structure Preservation and Seam Optimization for Parallax-Tolerant Image Stitching

Parallax processing has long been a significant and challenging task in image stitching. In this paper, we study a new hybrid warping model based on multi-homography and structure preservation to achieve accurate alignment of regions at different depths while preserving local and global image structures. The homographies of different depth regions are estimated by dividing matching feature pairs into multiple layers. Then, layered warping is performed by determining the spatial relationships between the image mesh and these multi-homography, and then refining the local and global structural distortions through mesh optimization. Four constraints are considered during the local optimization process, including the local alignment error, global alignment error, and similarity error. In addition, we explore and introduce collinear structures into an objective function as a constraint for mesh optimization warping, which can preserve salient line structures while alleviating distortions in nonoverlapping areas. Furthermore, we develop an optimal seam search method based on seam error evaluation to improve the quality of the seams. Experimental results demonstrate that compared to existing methods, the proposed algorithm presents more accurate stitching results for images with large parallax and preserves salient image structures, and outperforming the existing methods both qualitatively and quantitatively.

Variable-lag Granger Causality and Transfer Entropy for Time Series Analysis

Granger causality is a fundamental technique for causal inference in time series data, commonly used in the social and biological sciences. Typical operationalizations of Granger causality make a strong assumption that every time point of the effect time series is influenced by a combination of other time series with a fixed time delay. The assumption of fixed time delay also exists in Transfer Entropy, which is considered to be a non-linear version of Granger causality. However, the assumption of the fixed time delay does not hold in many applications, such as collective behavior, financial markets, and many natural phenomena. To address this issue, we develop Variable-lag Granger causality and Variable-lag Transfer Entropy, generalizations of both Granger causality and Transfer Entropy that relax the assumption of the fixed time delay and allow causes to influence effects with arbitrary time delays. In addition, we propose methods for inferring both Variable-lag Granger causality and Transfer Entropy relations. In our approaches, we utilize an optimal warping path of Dynamic Time Warping to infer variable-lag causal relations. We demonstrate our approaches on an application for studying coordinated collective behavior and other real-world casual-inference datasets and show that our proposed approaches perform better than several existing methods in both simulated and real-world datasets. Our approaches can be applied in any domain of time series analysis. The software of this work is available in the R-CRAN package: VLTimeCausality.

A Methodology to Handle Spectral Smearing in Gearboxes Using Adaptive Mode Decomposition and Dynamic Time Warping

Tacho-less order tracking is an effective tool for fault detection in gearboxes operating under speed fluctuations. This technique's performance depends on extracting instantaneous shaft speed information from a complex multicomponent gearbox signal, which is first preprocessed by a bandpass filter. However, spectral overlap resulting from the time-varying frequency components makes it challenging to isolate the shaft speed harmonic mono-component using this approach. In order to overcome such issues, an adaptive tacho-less order-tracking (OT) method combining the variational mode decomposition (VMD) and the fast dynamic time warping (FDTW) is proposed in this article. The proposed method first decomposes the measured gearbox vibration signal using VMD to estimate the instantaneous shaft speed profile to construct the shaft vibration signal. The gearbox vibration signal is then resampled based on the optimal warping path obtained by FDTW, which performs an “elastic” stretching and compression along the time axis of the extracted shaft vibration signal with respect to a sinusoidal reference signal with a constant shaft rotational frequency. Finally, the presence of gear fault is detected by constructing the order spectrum of the resampled vibration signal. The effectiveness of the proposed algorithm is demonstrated using both simulation analysis and experimental validation using measurements from two different wind-turbine gearboxes collected under test and field conditions, respectively.

Simultaneous Video Stabilization and Rolling Shutter Removal.

Due to the delay in the row-wise exposure and the lack of stable support when a photographer holds a CMOS camera, video jitter and rolling shutter distortion are closely coupled degradations in the captured videos. However, previous methods have rarely considered both phenomena and usually treat them separately, with stabilization approaches that are unable to handle the rolling shutter effect and rolling shutter removal algorithms that are incapable of addressing motion shake. To tackle this problem, we propose a novel method that simultaneously stabilizes and rectifies a rolling shutter shaky video. The key issue is to estimate both inter-frame motion and intra-frame motion. Specifically, for each pair of adjacent frames, we first estimate a set of spatially variant inter-frame motions using a neighbor-motion-aware local motion model, where the classical mesh-based model is improved by introducing a new constraint to enhance the neighbor motion consistency. Then, different from other 2D rolling shutter removal methods that assume the pixels in the same row have a single intra-frame motion, we build a novel mesh-based intra-frame motion calculation model to cope with the depth variation in a mesh row and obtain more faithful estimation results. Finally, temporal and spatial motion constraints and an adaptive weight assignment strategy are considered together to generate the optimal warping transformations for different motion situations. Experimental results demonstrate the effectiveness and superiority of the proposed method when compared with other state-of-the-art methods.

A new method to realize cyclic time averaging for change detection in dynamical machine status

Detection of changes in dynamical machine status is of great importance to enhance the function of industrial machineries. Cyclic time averaging (CTA) is a powerful tool to achieve this end, but extraction of the so-called average time waveform (ATW) is still challenging. This paper presents a new method to derive the ATW based on optimal warping path which is coupled with adaptive online cycle segmentation. The method has obvious advantages such as no requirement of sensor for calibration and signal pre-processing algorithms in its computation. Taking merit of this method, an automatic analysis of continuous monitoring of vibration signal is proposed by means of a null hypothesis testing to monitor the machine status. The proposed framework is applied to various engineering applications where excellent performance is demonstrated outperforming state-of-the-art methods. Comprehensive experimental results, along with theoretical interpretation and explanation, suggest the great potential of the framework in real applications.

Order Tracking Using Variational Mode Decomposition to Detect Gear Faults under Speed Fluctuations

Fault detection in gearboxes plays a significant role in ensuring their reliability. Vibration signals collected during gearbox operation contain a wealth of valuable condition information that can be exploited for fault detection. However, in an industrial environment machine operating speed always fluctuates around its nominal value, which causes smearing of the gearbox vibration spectrum. Considering operating speed fluctuation and multi-component nature of measured gearbox vibration signals, an order-tracking method combining the variational mode decomposition (VMD) and the fast dynamic time warping (FDTW) is proposed in this paper. Firstly, the multi-component vibration signal is decomposed into several intrinsic mode functions (IMFs) using VMD in order to extract a signal component with higher signal-to-noise ratio (SNR). Then, the sensitive fault information carrying IMF is exploited to estimate the instantaneous speed profile in order to construct the shaft rotational vibration signal. The measured vibration signal is then resampled based on the optimal warping path obtained by FDTW, which performs an “elastic” stretching and compression along the time axis of the extracted shaft vibration signal with respect to a sinusoidal reference signal of constant shaft rotational frequency. Finally, the gear fault is detected by constructing the order spectrum of the resampled vibration signal. The effectiveness of the proposed algorithm is demonstrated using simulation results.

A Two-Step Geometric Framework For Density Modeling

We introduce a novel two-step approach for estimating a probability density function (pdf) given its samples, with the second and important step coming from a geometric formulation. The procedure involves obtaining an initial estimate of the pdf and then transforming it via a warping function to reach the final estimate. The initial estimate is intended to be computationally fast, albeit suboptimal, but its warping creates a larger, flexible class of density functions, resulting in substantially improved estimation. The search for optimal warping is accomplished by mapping diffeomorphic functions to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. Using a truncated basis expansion, we estimate the optimal warping under a (penalized) likelihood criterion and, thus, the optimal density estimate. This framework is introduced for univariate, unconditional pdf estimation and then extended to conditional pdf estimation. The approach avoids many of the computational pitfalls associated with classical conditional-density estimation methods, without losing on estimation performance. We derive asymptotic convergence rates of the density estimator and demonstrate this approach using both synthetic datasets and real data, the latter relating to the association of a toxic metabolite on preterm birth.

Optimizing parameters of warp fatigue life tester by response surface methodology

The aim of this article was to provide an effective measurement method of the warp dynamic load capacity based on a warp fatigue life tester for sized yarns. Five parameters which affected the fatigue life of warp, respectively, namely, load frequency, warp speed, pre-tension weight, heald frame stroke, and static tension of the warp fatigue life tester, were studied. First, the influences of the five parameters on warp fatigue life were analyzed through single-factor experiments, and then, the Box–Behnken design based on the response surface methodology was adopted to optimize warp fatigue life tester parameters. The results showed that the optimum parameter conditions of the warp fatigue life tester for warp fatigue life were as follows: when heald frame stroke at 77.16 mm, load frequency of 205.49 times/min, and static tension as 6.96 cN, the optimal warp fatigue life was 1394.05 times. The theoretical and experimental values were in reasonable agreement, and the relative error was less than 1%. Verification and repeated trial results showed that the optimal parameters had good reproducibility, and the warp fatigue life tester with the optimal parameters provided an effective and reliable evaluation to the warp frictional, stretching, and buckling effects.