Curvature Information Research Articles

An Adaptive Memory Multi-Batch L-BFGS Algorithm for Neural Network Training

Motivated by the potential for parallel implementation of batch-based algorithms and the accelerated convergence achievable with approximated second order information a limited memory version of the BFGS algorithm has been receiving increasing attention in recent years for large neural network training problems. As the shape of the cost function is generally not quadratic and only becomes approximately quadratic in the vicinity of a minimum, the use of second order information by L-BFGS can be unreliable during the initial phase of training, i.e. when far from a minimum. Therefore, to control the influence of second order information as training progresses, we propose a multi-batch L-BFGS algorithm, namely MB-AM, that gradually increases its trust in the curvature information by implementing a progressive storage and use of curvature data through a development-based increase (dev-increase) scheme. Using six discriminative modelling benchmark problems we show empirically that MB-AM has slightly faster convergence and, on average, achieves better solutions than the standard multi-batch L-BFGS algorithm when training MLP and CNN models.

Navigation in Restricted Channels Under Environmental Conditions: Fast-Time Simulation by Asynchronous Deep Reinforcement Learning

This paper proposes an efficient method, based on reinforcement learning, to be used as ship controller in fast-time simulators within restricted channels. The controller must operate the rudder in a realistic manner in both time and angle variation so as to approximate human piloting. The method is well suited to scenarios where no previous navigation data is available; it takes into account, during training, both the effect of environmental conditions and also curves in channels. We resort to an asynchronous distributed version of the reinforcement learning algorithm Deep Q Network (DQN), handling channel segments as separate episodes and including curvature information as context variables (thus moving away from most work in the literature). We tested our proposal in the channel of Porto Sudeste, in the southern Brazilian coast, with realistic environment scenarios where wind and current incidence varies along the channel. The method keeps a simple representation and can be applied to any port channel configuration that respects local technical regulations.

Improved Canny Edge Detector Using Principal Curvatures

Canny edge detector is a very popular and effective edge feature detector that is used as a preprocessing step in many computer vision algorithms. It is a multi-step detector, which performs smoothing, filtering, non-maximum suppression, followed by a connected-component analysis stage to detect “true” edges, while suppressing “false” non-edge filter responses. Based on the literature, traditional Canny edge detector is sensitive to noise, hence it may lose the weak edge information after noise removal and show poor adaptability of fixed parameters like threshold values. In addition, Canny algorithm tends to over-smooth the noise, resulting in the loss of edge images or pseudo-edges, and the method of selecting thresholds is artificial, and the subjective factors are strong and computationally complex. This paper proposes an improvement to the traditional Canny algorithm by adding curvature information in the non-maximum suppression step (NMS) in order to obtain an accurate edge identification. Additionally, a set of tests and results is presented that show how by adding curvature characteristics to the NMS process, better results are obtained in the edge detection in Canny’s algorithm.

A Hierarchical Curve-Based Approach to the Analysis of Manifold Data.

One of the data structures generated by medical imaging technology is high resolution point clouds representing anatomical surfaces. Stereophotogrammetry and laser scanning are two widely available sources of this kind of data. A standardised surface representation is required to provide a meaningful correspondence across different images as a basis for statistical analysis. Point locations with anatomical definitions, referred to as landmarks, have been the traditional approach. Landmarks can also be taken as the starting point for more general surface representations, often using templates which are warped on to an observed surface by matching landmark positions and subsequent local adjustment of the surface. The aim of the present paper is to provide a new approach which places anatomical curves at the heart of the surface representation and its analysis. Curves provide intermediate structures which capture the principal features of the manifold (surface) of interest through its ridges and valleys. As landmarks are often available these are used as anchoring points, but surface curvature information is the principal guide in estimating the curve locations. The surface patches between these curves are relatively flat and can be represented in a standardised manner by appropriate surface transects to give a complete surface model. This new approach does not require the use of a template, reference sample or any external information to guide the method and, when compared with a surface based approach, the estimation of curves is shown to have improved performance. In addition, examples involving applications to mussel shells and human faces show that the analysis of curve information can deliver more targeted and effective insight than the use of full surface information.

Stochastic sub-sampled Newton method with variance reduction

Stochastic optimization on large-scale machine learning problems has been developed dramatically since stochastic gradient methods with variance reduction technique were introduced. Several stochastic second-order methods, which approximate curvature information by the Hessian in stochastic setting, have been proposed for improvements. In this paper, we introduce a Stochastic Sub-Sampled Newton method with Variance Reduction (S2NMVR), which incorporates the sub-sampled Newton method and stochastic variance-reduced gradient. For many machine learning problems, the linear time Hessian-vector production provides evidence to the computational efficiency of S2NMVR. We then develop two variations of S2NMVR that preserve the estimation of Hessian inverse and decrease the computational cost of Hessian-vector product for nonlinear problems.

Optimal design for adaptive smoothing splines

We consider the design problem of collecting temporal/longitudinal data. The adaptive smoothing spline is used as the analysis model where the prior curvature information can be naturally incorporated as a weighted smoothness penalty. The estimator of the curve is expressed in linear mixed model form, and the information matrix of the parameters is derived. The D-optimality criterion is then used to compute the optimal design points. An extension is considered, for the case where subpopulations exert different prior curvature patterns. We compare properties of the optimal designs with the uniform design using simulated data and apply our method to the Berkeley growth data to estimate the optimal ages to measure heights for males and females. The approach is implemented in an R package called “ODsplines”, which is available from github.com/jialiwang1211/ODsplines.

Accelerated Saddle Point Refinement through Full Exploitation of Partial Hessian Diagonalization.

Identification and refinement of first order saddle point (FOSP) structures on the potential energy surface (PES) of chemical systems is a computational bottleneck in the characterization of reaction pathways. Leading FOSP refinement strategies for modestly sized molecular systems require calculation of the full Hessian matrix, which is not feasible for larger systems such as those encountered in heterogeneous catalysis. For these systems, the standard approach to FOSP refinement involves iterative diagonalization of the Hessian, but this comes at the cost of longer refinement trajectories due to the lack of accurate curvature information. We present a method for incorporating information obtained by an iterative diagonalization algorithm into the construction of an approximate Hessian matrix that accelerates FOSP refinement. We measure the performance of our method with two established FOSP refinement benchmarks and find a 50% reduction on average in the number of gradient evaluations required to converge to a FOSP for one benchmark and a 25% reduction on average for the second benchmark.

A new hybrid conjugate gradient method for large-scale unconstrained optimization problem with non-convex objective function

The paper is aimed to employ a modified secant equation in the framework of the hybrid conjugate gradient method based on Andrei’s approach to solve large-scale unconstrained optimization problems. The CG parameter in the mentioned hybrid CG method is a convex combination of CG parameters corresponding to the Hestenes–Stiefel and Dai–Yuan algorithms. The main feature of these hybrid methods is that the search direction is the Newton direction. The modified secant equation is derived by means of the fifth-order tensor model to improve the curvature information of the objective function. Also, to achieve convergence for general function, the revised version of the method based on the linear combination of the mentioned secant equation and Li and Fukushima’s modified secant equation is suggested. Under proper conditions, globally convergence properties of the new hybrid CG algorithm even without convexity assumption on the objective function is studied. Numerical experiments on a set of test problems of the CUTEr collection are done; they demonstrate the practical effectiveness of the proposed hybrid conjugate gradient algorithm.

Machine learning-based multi-modal information perception for soft robotic hands

This paper focuses on multi-modal Information Perception (IP) for Soft Robotic Hands (SRHs) using Machine Learning (ML) algorithms. A flexible Optical Fiber-based Curvature Sensor (OFCS) is fabricated, consisting of a Light-Emitting Diode (LED), photosensitive detector, and optical fiber. Bending the roughened optical fiber generates lower light intensity, which reflecting the curvature of the soft finger. Together with the curvature and pressure information, multi-modal IP is performed to improve the recognition accuracy. Recognitions of gesture, object shape, size, and weight are implemented with multiple ML approaches, including the Supervised Learning Algorithms (SLAs) of K-Nearest Neighbor (KNN), Support Vector Machine (SVM), Logistic Regression (LR), and the unSupervised Learning Algorithm (un-SLA) of K-Means Clustering (KMC). Moreover, Optical Sensor Information (OSI), Pressure Sensor Information (PSI), and Double-Sensor Information (DSI) are adopted to compare the recognition accuracies. The experiment results demonstrate that the proposed sensors and recognition approaches are feasible and effective. The recognition accuracies obtained using the above ML algorithms and three modes of sensor information are higher than 85 percent for almost all combinations. Moreover, DSI is more accurate when compared to single modal sensor information and the KNN algorithm with a DSI outperforms the other combinations in recognition accuracy.

Quantum gradient descent and Newton’s method for constrained polynomial optimization

Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton’s method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.

Principal curvatures based local binary pattern for rotation invariant texture classification

Local binary patterns (LBP) have been proved to be an uncomplicated and high-efficiency descriptor for texture classification. However, the LBP and many variants are usually constantness only to restricted rotation angles. To address this limitation, we propose a new local texture descriptor, the principal curvatures based LBP (PCLBP). The proposed PCLBP descriptor is constructed by employing the complementary information of LBP and principal curvatures (PCs) of texture surface. The PCs, which consist of minimum and maximum curvatures, can capture the macro- and micro-structure texture information and have the capability of consecutive rotation invariance. To enhance the discriminativeness and the robustness of the PCLBP, a multiresolution scheme is developed by employing the strategy of cross-scale joint coding as well as adjusting three parameters accordantly. The proposed PCLBP method is simple, training-free, highly discriminative, invariant to illumination and rotation with advantage of computational simplicity. Experimental results on two well-known databases demonstrate that our PCLBP scheme achieves comparable or better performance than a large number of existing descriptors.

A three-dimensional magnetohydrodynamic equilibrium in an axial coordinate with a constant curvature

Motivated by the superior confinement observed in the relaxed three dimensional (3D) states in the reversed field pinch, 3D plasma equilibria in coordinate systems based on space curves with a constant curvature as the axial coordinates are studied by using the method of metric perturbation. Constancy of the curvature allows the development of magnetohydrodynamic equilibrium with asymptotic good 2D flux surfaces near the coordinate axis. The perturbation parameter is the product of the torsion variation along the coordinate axis and the distance from it. The lowest order equilibrium with good 2D flux surfaces is symmetric with respect to translation along the space curve. It embodies the approximate toroidal-helical symmetry and is determined by a generalized Grad–Shafranov equation which includes information of the constant curvature and the average torsion of the space curve. Based on this fundamental equilibrium, a formal scheme is developed that allows us to find the ideal MHD equilibrium taking into account the full metric variation of the torsion along the spatial axis. In this limit, the flux surfaces are shown to exist for the full plasma. Numerical examples are given for the lowest order equilibria. It is suggested that equilibria based on this type of coordinates can allow easier evaluation of plasma shapes and magnetic boundary conditions that are more compatible with the relaxed central core. This could be the needed requirement for strongly self-organized equilibria with a large good confinement region.

Determining the flow-related cap deformation of Taylor droplets at low Ca numbers using ensemble-averaged high-speed images

Droplets of microscopic Taylor flows can act as stirred batch microreactors, while the continuous phase compartments supply good mixing and mass transfer to the channel walls. The design of reliable reactors, however, requires a priori information about droplet formation frequency and droplet length, which is difficult to obtain. Here, we report about a new method to reliably describe the droplet deformation, which could act as a flow indicator. The flow-related interface shape deformation of liquid–liquid Taylor flows in square micro channels for $$Re < 5$$ and $$Ca < 0.02$$ is quantified by experimental studies and ensemble image averaging. We deduce an easy-to-access droplet cap deformation model, which allows reducing the complex interface curvature information to an ellipse and provide correlation functions to link this deformation ratio to the flow conditions quantitatively. In addition, we introduce an experimental approach using a ternary liquid–liquid material system that allows adjusting $$Re$$ and $$Ca$$ numbers independently without surfactant addition by changing the mass fraction of two liquids forming the continuous phase. In combination, we suggest an image processing method to overcome the optical issues arising from the poor image contrast of Taylor droplets with low differences of refractive index to the bulk material. The presented approach opens the possibility to benchmark numerical studies of the interface deformation and paves the way for improved flow modeling, e.g., pressure drop or relative velocity of Taylor droplets, since the flow-induced curvature could serve as an indicator of the forces acting on the droplet front and rear.

Representation of Surfaces with Normal Cycles and Application to Surface Registration

In this paper, we present a framework for computing dissimilarities between surfaces which is based on the mathematical model of normal cycle from geometric measure theory. This model allows to take into account all the curvature information of the surface without explicitly computing it. By defining kernel metrics on normal cycles, we define explicit distances between surfaces that are sensitive to curvature. This mathematical framework also has the advantage of encompassing both continuous and discrete surfaces (triangulated surfaces). We then use this distance as a data attachment term for shape matching, using the large deformation diffeomorphic metric mapping for modelling deformations. We also present an efficient numerical implementation of this problem in PyTorch, using the KeOps library, which allows both the use of auto-differentiation tools and a parallelization of GPU calculations without memory overflow. We show that this method can be scalable on data up to a million points, and we present several examples on surfaces, comparing the results with those obtained with the similar varifold framework.

A Novel Tip-positioning Control of a Magnetically Steerable Guidewire in Sharply Curved Blood Vessel for Percutaneous Coronary Intervention

This paper suggests a novel method for the positioning control of a magnetically steerable guidewire without needing to detect a tip angle, which can pass through sharply curved blood vessels with potentially reduced contact between the guidewire end tip and the blood vessel for Percutaneous Coronary Intervention (PCI). The proposed tip-positioning control consists of four parts: 1) Path-regeneration, 2) Feeding control, 3) EMA (Electromagnetic Actuator) control, and 4) Vision tracking. Path-regeneration provides an improved path based on the properties of the guidewire and the size of the blood vessel when the guidewire cannot follow the pre-planned path. Based on the desired path, the feeding control drives the translational motion of the guidewire while the EMA system steers the guidewire’s end tip, which is made of a permanent magnet. A pair of Helmholtz coils is employed for the EMA system, and a vision-tracking algorithm is used to detect the position of the magnet tip in real time. The system was integrated for the preliminary study, and the magnet tip could be controlled so as to track a desired path without any shape or curvature information, which requires the use of an additional catheter model or angle estimation algorithm. The proposed catheter tip position control method was verified by experiments on both smooth and sharply curved blood vessel phantoms. The results showed that the proposed method could accomplish the tip position control of the guidewire with high accuracy in real time.

Identifying Centromere Position of Human Chromosome Images using Contour and Shape based Analysis

The most significant information of the shape of any image/object is concentrated in curvature regions along the contour and objects boundaries rather than uniformly distributed contour. The points belonging to greater magnitude of curvature gives more meaningful information about the shape of an object. The sign of the curvature can be positive (convex) and negative (concave), the negative curvature information is most significant for segmentation. The contour and region based geometry gives a better visual representation of the shape of an object and helps in identifying the centromere position in chromosomes. Centromere of a chromosome is the constriction point which divides the chromosome into two sections or arms. The two arms are p arm (short arm) and q arm (long arm). The size of the arms are calculated with respect to the position of the centromere. The centromere is identified using boundary concavity method which helps in detecting the dominant points (centromere points) in chromosomes. The method uses the concave function and weighted shortest path calculation for centromere detection. SVM classifier is used for improving the accuracy in detecting the centromere of the chromosomes. As the classifier is binary classifier, it helps in recognizing the centromere and non-centromere regions in chromosomes. Comparative analysis is performed with two other methods (i) Medial Axis Transform (MAT) and (ii) Projection Vector. Boundary concavity proves to be efficient for straight, bent and severely bent chromosomes.

High-Precision Extraction of Segment Points of 2D Profiles Based on a Dynamic Grid Method

In order to improve the reconstruction precision and performance of 2D profiles, this paper presents a new high-precision extraction method of segment points of 2D profile features. Firstly, we extract the intervals which include initial segment points based on the curvature information of data points. Secondly, a grid is dynamically constructed in the interval and all the nodes of the grid are considered as candidate segment points for an optimization process. Thirdly, the optimization model of each feature curve is constructed on the basis of boundary constraints. Selection of the optimal segment point is related to two factors: the number of control points of B-spline and the total approximation error of all collected points to the fitted curves. Numerical experiments were conducted and the results demonstrate the efficacy of our method in capturing design intent and in industrial applications, compared to existing methods.

Segmentation of Overlapping Cytoplasm in Cervical Smear Images via Adaptive Shape Priors Extracted From Contour Fragments

We present a novel approach for segmenting overlapping cytoplasm of cells in cervical smear images by leveraging the adaptive shape priors extracted from cytoplasm's contour fragments and shape statistics. The main challenge of this task is that many occluded boundaries in cytoplasm clumps are extremely difficult to be identified and, sometimes, even visually indistinguishable. Given a clump where multiple cytoplasms overlap, our method starts by cutting its contour into a set of contour fragments. We then locate the corresponding contour fragments of each cytoplasm by a grouping process. For each cytoplasm, according to the grouped fragments and a set of known shape references, we construct its shape and, then, connect the fragments to form a closed contour as the segmentation result, which is explicitly constrained by the constructed shape. We further integrate the intensity and curvature information, which is complementary to the shape priors extracted from contour fragments, into our framework to improve the segmentation accuracy. We propose to iteratively conduct fragments grouping, shape constructing, and fragments connecting for progressively refining the shape priors and improving the segmentation results. We extensively evaluate the effectiveness of our method on two typical cervical smear datasets. The experimental results demonstrate that our approach is highly effective and consistently outperforms the state-of-the-art approaches. The proposed method is general enough to be applied to other similar microscopic image segmentation tasks, where heavily overlapped objects exist.

Constraints on cosmic curvature with lensing time delays and gravitational waves

Assuming the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model, the CMB and BAO observations indicate a very flat Universe. Model-independent measurements are therefore worth studying. Time delays measured in lensed quasars provide the time delay distances. When compared with the luminosity distances from the supernova Ia (SNe Ia) observation, the measurements can provide the curvature information under the Distance Sum Rule of the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) metric. This method is limited by the low redshifts of SNe Ia. In this work, we propose that gravitational waves from the Einstein Telescope, as standard sirens which reach higher redshifts covering the redshift range of lensed quasars from the Large Synoptic Survey Telescope, could provide much more stringent constraints on the curvature. We first consider a conservative case where only 100 gravitational waves with electromagnetic counterparts are available; the $1\ensuremath{\sigma}$ uncertainty for the curvature parameter ${\mathrm{\ensuremath{\Omega}}}_{k}$ is 0.057. In an optimistic case with 1000 signals available, then ${\mathrm{\ensuremath{\Omega}}}_{k}$ uncertainty is 0.027. Combining with SNe Ia from the Dark Energy Survey, ${\mathrm{\ensuremath{\Omega}}}_{k}$ can be further constrained to 0.027 and 0.018, respectively.

A robust and efficient framework for fast cylinder detection

In this work, a complete solution is provided for detecting and identifying cylindrical shapes, which are commonly found in household and industrial environments, using consumer-grade RGB-D cameras. Most standard approaches to detect and identify cylinders are not robust to outliers (e.g. points on other objects in the scene), which limits their applicability in realistic scenes. In addition, these methods fail to benefit from environmental constraints, e.g. the fact that cylinders often lie or stand on flat surfaces. To tackle the aforementioned limitations, we introduce three main novelties: (i) a point cloud soft voting scheme with curvature information that reduces the influence of outliers and noise, (ii) a selective sampling of the orientation space that favorsorientations known a priori, and (iii) a deep-learning based classifier to filter out objects with non-cylindrical appearance in the 2D images, thus further improving robustness to outliers.A set of experiments with synthetically generated data are used to assess the robustness of our fitting method to different levels of outliers and noise. The results demonstrate that incorporating the principal curvature direction within the orientation voting process allows for large improvements on cylinders parameters estimation. Furthermore, we demonstrate that combining the 2D deep-learning cylinder classifier with the 3D orientation voting scheme allows for large speed-up and accuracy improvements on cylinder identification. The qualitative and quantitative results with real data acquired from a consumer RGB-D camera, confirm the advantages of the proposed framework.