Inverse Mapping Research Articles

A Note on Clarke’s Generalized Jacobian for the Inverse of Bi-Lipschitz Maps

Clarke’s inverse function theorem for Lipschitz mappings states that a bi-Lipschitz mapping f is locally invertible about a point x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document} if the generalized Jacobian ∂f(x0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial f(x_0)$$\\end{document} does not contain singular matrices. It is shown that under these assumptions the generalized Jacobian of the inverse mapping at f(x0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x_0)$$\\end{document} is the convex hull of the set of matrices that can be obtained as limits of sequences Jf(xk)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_f(x_k)^{-1}$$\\end{document} with f differentiable in xk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_k$$\\end{document} and xk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_k$$\\end{document} converging to x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document}. This identity holds as well if f is assumed to be locally bi-Lipschitz at x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document}.

Operators that are Weakly Coercive and a Compact Perturbation

ABSTRACT Let X, Y be two Banach spaces over K = ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C 1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

Data-driven prediction of αIIbβ3 integrin activation paths using manifold learning and deep generative modeling

The integrin heterodimer is a transmembrane protein critical for driving cellular process and is a therapeutic target in the treatment of multiple diseases linked to its malfunction. Activation of integrin involves conformational transitions between bent and extended states. Some of the conformations that are intermediate between bent and extended states of the heterodimer have been experimentally characterized, but the full activation pathways remain unresolved both experimentally due to their transient nature and computationally due to the challenges in simulating rare barrier crossing events in these large molecular systems. An understanding of the activation pathways can provide new fundamental understanding of the biophysical processes associated with the dynamic interconversions between bent and extended states and can unveil new putative therapeutic targets. In this work, we apply nonlinear manifold learning to coarse-grained molecular dynamics simulations of bent, extended, and two intermediate states of αIIbβ3 integrin to learn a low-dimensional embedding of the configurational phase space. We then train deep generative models to learn an inverse mapping between the low-dimensional embedding and high-dimensional molecular space and use these models to interpolate the molecular configurations constituting the activation pathways between the experimentally characterized states. This work furnishes plausible predictions of integrin activation pathways and reports a generic and transferable multiscale technique to predict transition pathways for biomolecular systems.

Inversion maps and torus actions on rational homogeneous varieties

Complex projective algebraic varieties with {{mathbb {C}}}^*-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a {{mathbb {C}}}^*-action with no proper isotropy subgroups.

Pose optimization for mobile manipulator grasping based on hybrid manipulability

PurposeProper platform pose is important for the mobile manipulator to accomplish dexterous manipulation tasks efficiently and safely, and the evaluation criterion to qualify manipulation performance is critical to support the pose decision process. This paper aims to present a comprehensive index to evaluate the manipulator’s operation performance from various aspects.Design/methodology/approachIn this research, a criterion called hybrid manipulability (HM) is proposed to assess the performance of the manipulator’s operation, considering crucial factors such as joint limits, obstacle avoidance and stability. The determination of the optimal platform pose is achieved by selecting the pose that maximizes the HM within the feasible inverse reachability map associated with the target object.FindingsA self-built mobile manipulator is adopted as the experimental platform, and the feasibility of the proposed method is experimentally verified in the context of object-grasping tasks both in simulation and practice.Originality/valueThe proposed HM extends upon the conventional notion of manipulability by incorporating additional factors, including the manipulator’s joint limits, the obstacle avoidance situation during the operation and the manipulation stability when grasping the target object. The manipulator can achieve enhanced stability during grasping when positioned in the pose determined by the HM.

A layer potential approach to inverse problems in brain imaging

Abstract We study the inverse source localisation problem using the electric potential measured point-wise inside the head with stereo-ElectroEncephaloGraphy (sEEG), the electric potential measured point-wise on the scalp with ElectroEncephaloGraphy (EEG) or the magnetic flux density measured point-wise outside the head with MagnetoEncephaloGraphy (MEG). We present a method that works on a wide range of models of primary currents; in particular, we give details for primary currents that are assumed to be smooth vector fields that are supported on and normally oriented to the grey/white matter interface. Irrespective of the data used, we also solve the transmission problem of the electric potential associated with a recovered source; hence we solve the cortical mapping problem. To ensure that the electric potential and normal currents are continuous in the head, the electric potential is expressed as a linear combination of double layer potentials and the magnetic flux density is expressed as a linear combination of single layer potentials. Numerically, we solve the problems on meshed surfaces of the grey/white matter interface, cortical surface, skull and scalp. A main feature of the numerical approach we take is that, on the meshed surfaces, we can compute the double and single layer potentials exactly and at arbitrary points. Because we explicitly study the transmission of the electric potential in head when using magnetic data, the coupling of electric and magnetic data in the source recovery problem is made explicit and shows the advantage of using simultaneous electric and magnetic data. We provide numerical examples of the source recovery and inverse cortical mapping using synthetic data.

Real-time lane departure warning with cascade lane segmentation

<span>Lane departure warning (LDW) is one of the safety innovations in autonomous cars that provides vehicle position monitoring. This technology will alarm if the vehicle moves out of the lane. Lane detection and lane measurement is the main part of LDW. The novelty of this research is the method can measure different lane marks. Very important to know the lane mark that can and cannot be passed. We use semantic segmentation to segment lane mark solid, lane mark dashed, and road. After extracting road and lane marks, we use inverse perspective mapping (IPM) to help calculate the measurement between the car and lane mark. The data is that 374 images were collected from several roads in Makassar City. The model was evaluated using intersection over union (IoU), reaching 79.8% accuracy. The developed system also estimates the measure between the vehicle and lane marks. <br /> The lane measurement estimation system’s test results were evaluated using the root mean square error (RMSE) method to reach between 0.025254 and 0.134345.</span>

Insensitizing control problem for the Hirota–Satsuma system of KdV–KdV type

This paper is concerned with the existence of insensitizing controls for a nonlinear coupled system of two Korteweg–de Vries (KdV) equations, typically known as the Hirota–Satsuma system. The idea is to look for controls such that some functional of the states (the so-called sentinel) is insensitive to the small perturbations of initial data. Since the system is coupled, we consider a sentinel in which we observe both components of the system in a localized observation set. By some classical argument, the insensitizing problem is then reduced to a null-control problem for an extended system where the number of equations is doubled. We study the null-controllability for the linearized model associated to that extended system by means of a suitable Carleman estimate which is proved in this paper. Finally, the local null-controllability of the extended (nonlinear) system is obtained by applying the inverse mapping theorem, and this implies the required insensitizing property for the concerned model.

Optimization Technology of the Microwave Diplexer Based on Spatial Mapping Algorithm

An optimization technology of the microwave diplexer is proposed in this paper. The spatial mapping algorithm has been adopted to analyze and design the microwave diplexer, and the mapping relationship between the physical circuit (accurate model) and the schematic circuit (rough model) has been established, and then the least square method has been applied to fit it. Moreover, the inverse mapping method has been applied to guide the design of the accurate model by the optimization of the rough model. To verify the presented optimization technology of the microwave diplexer, a diplexer with the center frequencies of 1.85 GHz and 2.4 GHz respectively is designed, optimized, and fabricated. The measured results show that the insertion loss is about 0.8/1.5 dB. The measured output return loss is greater than 20 dB. Moreover, the measured isolation between two channel is greater than 35 dB. The total size of the fabricated diplexer is only 0.25λg 0.2λg. The measured results are in good agreement with the simulated ones, which was able to demonstrate the feasibility and advantage of this algorithm in designing microwave diplexer circuits.

Real-time risk assessment of road vehicles based on inverse perspective mapping

Pan/Tilt/Zoom (PTZ) cameras play an important role in traffic scenes due to their wide monitoring fields and high flexibility. However, since the focal length and angle of PTZ cameras change irregularly with the monitoring needs, it is difficult to obtain accurate physical information about the real world from the image information of PTZ cameras. Aiming to address the need for real-time risk assessment of road vehicles in traffic monitoring scenarios, a vehicle position and velocity measurement scheme based on camera inverse perspective transformation is proposed, along with a method for real-time risk assessment based on the position and velocity. Specifically, Firstly, the vehicle target in the video is detected and tracked by deep learning YOLO detection algorithm and optical flow tracking algorithm. According to the obtained trajectory set, the vanishing points in the road direction are calculated by Cascade Hough Transform and the road marking lines are detected. Then, according to the vanishing point and marking line, the camera calibration task is accomplished via exploratory focal length. After camera calibration, the camera-to-road inverse perspective transformation is applied to project the image plane onto the road surface and obtain, the actual position information of vehicles. Finally, the vehicle speed measurement and real-time road risk assessment are achieved by calculating the average of instantaneous velocities across multiple frames. Simulation experiment results in a traffic monitoring scenario demonstrate that this perspective-based method for real-time road vehicle risk assessment achieves good stability and practicality, which meets the requirements for vehicle speed measurement and real-time road risk assessment.

An embedding observer for nonlinear dynamical systems with global convergence

AbstractState observers for nonlinear systems are often designed for a canonical form of this system. However, this form may possess singular points, where the vector field is not defined or a Lipschitz condition is not fulfilled. This unpleasant behavior can possibly be avoided using an embedding into a higher dimensional space. A construction of such an embedding and the corresponding inverse map is discussed for polynomial systems using methods from algebraic geometry.

Joint Inversion of Gravity and Magnetic Anomalies to Image Salt–Basement Structures Offshore Abu Dhabi, UAE, Using Deep Neural Networks

Summary By using a deep neural network (DNN), a novel technique is developed for a 2.5D joint inversion of gravity and magnetic anomalies to model subsurface salts and basement structures. The joint application of gravity and magnetic anomalies addresses the inherent nonuniqueness problem of geophysical inversions. Moreover, DNN is used to conduct the nonlinear inverse mapping of gravity and magnetic anomalies to depth-to-salt and depth-to-basement. To create the training data set, a three-layer forward model of the subsurface is designed indicating sediments, salts, and the basement. The length and height of the model are determined based on the dimensions of the target area to be investigated. Several random parameters are set to create different representations of the forward model by altering the depth and shape of the layers. Given the topography of the salts and basement layers as well as their predefined density and susceptibility values, the gravity and magnetic anomalies of the forward models are calculated. Using multiprocessing algorithms, thousands of training examples are simulated comprising gravity and magnetic anomalies as input features and depth-to-salt and depth-to-basement as labels. The application of the proposed technique is evaluated to interpret the salt–basement structures over hydrocarbon reservoirs in offshore United Arab Emirates (UAE). Correspondingly, a DNN model is trained using the simulated data set of the target region and is assessed by making predictions on the random actual and noise-added synthetic data. Finally, gravity-magnetic anomalies are fed into the DNN inverse model to estimate the salts and basement structures over three profiles. The results proved the capability of our technique in modeling the subsurface structures.

Research on the design and presentation of traditional ethnic costumes based on virtual reality technology

Abstract In this paper, the 3D model of the human body is drawn, and the 3D transformation in the 3D garment piece design is realized by using the inverse mapping of 2D screen coordinates to 3D model coordinates. Then, the technique of virtual presentation of traditional ethnic clothing is studied, alternating the left and in-eye views and using the overlap of calculation to optimize the calculation to achieve a three-dimensional display. The virtual design and presentation effects of traditional ethnic costumes were assessed through a comprehensive comparison, sensory experience, interactive experience, emotional experience, and cognitive experience. The results showed that the regression coefficients of cognitive experience, sensory experience, interactive experience, and emotional experience of the virtual design and presentation of costumes were 0.98, 0.875, 0.993, and 0.996, respectively, which were 0.506, 0.36, 0.11, 0.161, and 0.048 more than the regression coefficients of the traditional display method in each dimension. This study provides the theoretical basis for combining traditional costumes with virtual technology.

Machine learning for rapid inference of critical dimensions in optical metrology of nanopatterned surfaces

Scatterometry-based metrology has the capability to perform high-throughput inspection of large-area nanopatterned surfaces. It utilizes physics-based dependencies between reflectance spectra of light scattered from nanopatterned surfaces and the geometric parameters, or so-called Critical Dimensions (CDs), of such nanopatterns. This paper proposes a novel method for rapid and accurate inference of CDs of complex nanopatterned surfaces through the use of a Machine Learning (ML)-based inverse problem mapping, coupled with the use of a strategic feature selection algorithm, which also originated from the ML domain. Specifically, Latin Hypercube Sampling was utilized to create a multi-dimensional grid of CDs which are used as inputs for physics-based simulations of scattered light spectra for nanopatterned surfaces with those CDs. Simulation results are used to build the training data for the ML model relating simulated reflectance spectra with the corresponding CDs. The Extreme Gradient Boosting (XGBoost) algorithm was utilized to realize the aforementioned mapping. In addition, most informative spectral wavelengths were selected using Recursive Feature Elimination (RFE) algorithm based on the feature importance scores obtained from XGBoost. Capabilities of the newly proposed approach were evaluated through inspection of a semiconductor wafer sample with hourglass patterns, which are characterized by eight CDs. It was observed that the proposed method is capable of real-time CD metrology of large-area nanostructured surfaces with complex nanopatterns, with accuracy and repeatability comparable to that of Scanning Electron Microscopy.

Innovation technology opportunity identification of civil aircraft mechanical connections based on generative topographic mapping.

In order to advance civil aircraft manufacturing to higher levels, there is an urgent need to identify technological innovation opportunities to help new technology development. This paper first analyses the current state of the research field and determines the topic. It preprocesses papers and patents within the research topic to obtain a base database. Then, the database is analyzed using the LDA (Latent Dirichlet Analysis) cluster analysis method. The TF-IDF (Term Frequency-Inverse Document Frequency) algorithm processes the data to obtain critical technical words. The abstracts of patents and papers are processed to construct a binary-based vector of technical keywords. The papers and patents are visualized in a two-dimensional space technology map by generative topographic mapping (GTM) to create a technology map to identify technology blank dots. The combination of technologies characterized by each technology blank dot is obtained by GTM inverse mapping. Finally, technology opportunities with a high probability of development are identified to achieve innovation opportunity identification. It also provides countermeasures for the research institution, enterprise, sector, and industry. After research and analysis, the future in the mechanical connection technology of civil aircraft is necessary to strengthen basic technology development and improve the study of intelligence, integration, and flexibility. Technology such as sensors and lasers can improve the precision and efficiency of mechanical connections.

On mappings with an analog of the hydrodynamical normalization in the Euclidean space

Spatial mappings that satisfy some spatial analog of the hydrodynamic condition in the vicinity of an infinitely remote point have been studied. It has been proved that homeomorphisms of the indicated class form equicontinuous families under certain conditions on their quasi-conformity characteristic. The issue concerning the closure of those classes with respect to the locally uniform convergence has also been considered. Relevant results have been obtained for mappings with integral restrictions, as well as for classes of corresponding inverse mappings.

Towards intelligent design assistants for planar multibody mechanisms

AbstractGeneral‐purpose mechanisms can perform a broad range of tasks but are usually rather heavy and expensive. If only particular movements need to be executed, more efficient special‐purpose mechanisms can be employed. However, they typically require an expert to design the system based on manual inspection of simulations and experimental results. This procedure is not only time‐consuming, but the outcome also depends on the expert's experience. Hence, the design process stems from subjective criteria while only a limited number of structurally different mechanisms can be considered. In contrast, a design assistant can consider a broad range of mechanisms and leverage multi‐objective optimization to retrieve optimal designs for the given task. Due to the systems being synthesized based on mathematical functions rather than individual experience, the assistant allows a more transparent development of optimal problem‐specific mechanisms compared to the conventional process. Experts can then fine‐tune and analyze the proposed designs to compose the final system. In recent years, neural networks have been utilized to directly learn the inverse mapping from a trajectory to a mechanism design. This requires some parameterization of the trajectory to be fed into the network. In this work, we evaluate various preprocessing methods for the trajectory on a simple mechanism design model problem. We assess multiple configurations such as different neural network sizes, applying input‐output normalization, and varying the number of features. Consequently, we investigate and compare the trends and robustness of the implemented methods.

Bayesian optimization of multiscale kernel principal component analysis and its application to model Gas-to-liquid (GTL) process data

Kernel methods map the data from original space into a higher-dimensional space in which linear methods are applied. In many applications, the inverse mapping is also important, and the pre-image of a feature vector must be found in the original space. Kernel principal component analysis (KPCA) based kernel density estimation (KDE) has been developed to solve this problem. However, the performance of the KPCA technique greatly depends on the choice of some parameters which can lead to poor modeling performance when these parameters are not well identified. Thus, fully Bayesian optimization KPCA (BOKPCA) is proposed to enhance the performance of the KPCA model. BOKPCA method aims to automatically select the best parameters of the KPCA model. Generally, kernel methods struggle to handle nonlinear data contaminated with high levels of noise. This is because the noise affects every principal component, making it challenging to mitigate its influence during the reconstruction step. Consequently, to further enhance the ability of KPCA and BOKPCA models, we propose to integrate multi-scale filtering with these two models. The efficiency of the proposed methods are evaluated using a simulated nonlinear process and real data generated from a bench-scale Fischer–Tropsch (FT) process.

Real-time 360 degrees view for the operator of milliAmpere 2

In the evolving domain of autonomous marine operations, accurate perception and representation of the surrounding environment are crucial for safe and effective execution. This paper addresses this issue by developing and testing a near real-time 360-degree bird’s eye view system for the situations where the ferry, milliAmpere 2, has to be manually controlled by a local operator onboard. The goal was to aid the operator during the critical phase of docking, by displaying the surrounding area of the ferry from a bird’s eye view. The bird’s eye view was made by using inverse perspective mapping on the undistorted images from the 8 cameras onboard. The system was implemented in Python, and aimed to reach a run-time of less than 200ms. This goal was reached during the initial phase of the work. However, during live testing, only near real-time performance was achieved. Despite some shortcomings, the operators found the system to be a “useful additional assistance” during the docking process.

Robust Accurate Lane Detection and Tracking for Automated Rubber-Tired Gantries in a Container Terminal

Lane detection and tracking technique is the autonomous driving basis for Rubber-Tired Gantries (RTGs), vital to the automation and intelligence updating of man-driven container terminals. However, the existing lane detection methods developed for common road scenarios cannot meet the high-precision and robust all-weather requirements of RTG autonomous driving. In this article, we propose an Adaptive Edge-based Lane Detection and Tracking method considering RTG lanes’ characteristics in this paper. First, the candidate edges of lane lines are detected and paired based on the enhanced gradient features. Next, inverse perspective mapping is employed to search the right edges, followed by an adaptive sliding-window method. Ultimately, we develop an adaptive Kalman filter to track lane lines robustly, detecting confidence weighting by relaxing the constraint of lane line width. The proposed method is tested in an actual container yard, the lane centerline’s average position error is 2.051 pixels, and the detection success rate is close to 100%.