Curvature Estimation Research Articles

In situ monitoring of quantum dot growth using a magnification inferred curvature method

The growth of InAs on a GaAs (001) substrate follows the Stranski–Krastanov (S–K) growth mode. Initially, the stress due to the lattice constant difference is small, resulting in two-dimensional growth. However, as the thickness of the growth layer increases, this stress accumulates, and upon reaching a critical film thickness, the growth transitions to three-dimensional, facilitating stress relaxation. Strain changes during crystal growth can be observed through variations in substrate curvature. However, in InAs/GaAs quantum dots (QDs), these changes are minimal, making observation challenging. Previously, the curvature of the substrate during InAs/GaAs QD formation could only be estimated with low precision, necessitating the use of very thin substrates with cantilever structures to achieve higher curvature. In this study, we applied the magnification inferred curvature (MIC) method, which allows for high-precision estimation of substrate curvature during molecular beam epitaxy growth. This method enabled us to observe strain changes during the formation and relaxation processes of QDs on standard-thickness GaAs (001) substrates. The results highlight the potential of the MIC method in investigating the complex interplay of strain and stress in semiconductor growth processes, emphasizing its suitability for fabricating next-generation QD devices.

Least Squares Fitting on a Segment of Ellipse and Its Application on Road Curvature Estimation

Least Squares Fitting on a Segment of Ellipse and Its Application on Road Curvature Estimation

Curvature estimates for spacelike stationary graphs in the 4-dimensional Lorentz space

Curvature estimates for spacelike stationary graphs in the 4-dimensional Lorentz space

Automatic bony structure segmentation and curvature estimation on ultrasound cervical spine images - a feasibility study

Abstract The loss of cervical lordosis is a common degenerative disorder known to be associated with abnormal spinal alignment. In recent years, ultrasound (US) imaging has been widely applied in the assessment of spine deformity and has shown promising results. The objectives of this study are to automatically segment bony structures from the 3D US cervical spine image volume and to assess the cervical lordosis on the key sagittal frames. In this study, a portable ultrasound imaging system was applied to acquire cervical spine image volume. The nnU-Net was trained on to segment bony structures on the transverse images and validated by 5-fold-cross-validation. The volume data were reconstructed from the segmented image series. An energy function indicating intensity levels and integrity of bony structures was designed to extract the proxy key sagittal frames on both left and right sides for the cervical curve measurement. The mean absolute difference (MAD), standard deviation (SD) and correlation between the spine curvatures of the left and right sides were calculated for quantitative evaluation of the proposed method. The DSC value of the nnU-Net model in segmenting ROI was 0.973. For the measurement of 22 lamina curve angles, the MAD±SD and correlation between the left and right sides of the cervical spine were 3.591±3.432° and 0.926, respectively. The results indicate that our method has a high accuracy and reliability in the automatic segmentation of the cervical spine and shows the potential of diagnosing the loss of cervical lordosis using the 3D ultrasound imaging technique.

Manifold-based approach for neural network robustness analysis

It is important to understand the mathematical foundations of neural networks and to include robustness in model evaluation. Here, we introduce algorithms based on manifold curvature estimation to assess neural network robustness. These algorithms rely solely on training data and do not require regular or adversarial test data. Initially, a metric is proposed to measure the curvature of discrete data manifolds by introducing weighted angles concept between subspaces. Following this, a robustness measure is introduced that is independent of network architecture or model parameters. Lastly, two additional methods are introduced, utilizing curvature estimation of special manifolds formed by using gradient vectors between output and input network layers, alongside manifold curvature estimation. A comprehensive evaluation is provided on multiple network models using the CIFAR-10 dataset. Manifold geometry-based robustness analysis may lead to the development of not only accurate but also robust neural network models.

Curvature Estimates for Stable Free Boundary Minimal Hypersurfaces in Locally Wedge-Shaped Manifolds

Abstract In this paper, we consider locally wedge-shaped manifolds, which are Riemannian manifolds that are allowed to have both boundary and certain types of edges. We define and study the properties of free boundary minimal hypersurfaces inside locally wedge-shaped manifolds. In particular, we show a compactness theorem for free boundary minimal hypersurfaces with curvature and area bounds in a locally wedge-shaped manifold. Additionally, using Schoen–Simon–Yau’s estimates, we also prove a Bernstein-type theorem indicating that, under certain conditions, a stable free boundary minimal hypersurface inside a Euclidean wedge must be a portion of a hyperplane. As our main application, we establish a curvature estimate for sufficiently regular free boundary minimal hypersurfaces in a locally wedge-shaped manifold with certain wedge angle assumptions. We expect this curvature estimate will be useful for establishing a min-max theory for the area functional in wedge-shaped spaces.

MetaHDR: single shot high-dynamic range imaging and sensing using a multifunctional metasurface

We present MetaHDR, which is a single-shot high-dynamic range (HDR) imaging and sensing system using a multifunctional metasurface. The metasurface is capable of splitting an incident beam into multiple focusing beams with different amounts of power, simultaneously forming multiple low dynamic range (LDR) images with distinct irradiance on a photosensor. Then, the LDR images are jointly processed using a gradient-based HDR fusion algorithm, which is shown to be effective in attenuating the residual light artifacts incurred by the metasurface and the lens flare. MetaHDR achieves single-shot HDR photography and videography that increases the dynamic range by at least 50 dB compared to the original dynamic range of the photosensor. It can also perform single-shot HDR sensing, including reflectance calibration and surface curvature estimation of reflective materials. MetaHDR’s demonstrated functionalities could be broadly applied in surveillance and security, microscopic imaging, advanced manufacturing, etc.

The impact of diffusion tensor imaging tractography settings on muscle fascicle architecture and diffusion parameter estimates: Tract length, completion, and curvature are most sensitive to tractography settings.

Diffusion-tensor (DT)-MRI tractography provides information about properties relevant to muscle health and function, including estimates of architectural properties such as fascicle length, pennation angle, and curvature and diffusion properties such as mean diffusivity (MD) and fractional anisotropy (FA). Tractography settings, including integration algorithms, thresholds for early tract termination, and tract smoothing approaches, impact the accuracy of the muscle property estimates. However, muscle DT-MRI tractography is performed using a variety of these settings, complicating comparisons between different studies. The effects of different tractography settings on muscle architecture estimates have not been fully explored, and optimized settings for muscle tractography have not yet been determined. We examined the influence of integration algorithm and termination check settings combined with a range of step sizes, termination criteria, and smoothing polynomial orders on tract characteristics, completion/reason for termination, and goodness of fit between fiber tracts and smoothing polynomials using 3-T DT-MR images of the lower leg muscles of seven healthy adults. We found that tract length and completion were highly sensitive to strict FA and intersegment angle thresholds (25%-69% reduction in complete fiber tracts from lowest to highest minimum FA threshold and 11%-36% reduction from highest to lowest intersegment angle threshold). Higher order polynomials (third and fourth order vs. second order) better fit the muscle fiber trajectories, but curvature estimates were highly sensitive to smoothing polynomial order (3.9-6.6 m-1 increase for second- vs. fourth-order fitting polynomials). Step size impacted curvature estimates, albeit to a lesser degree. Integration algorithm had little impact, and mean pennation angle, and tract-based FA and MD, were relatively insensitive to all parameters. The results demonstrate which muscle diffusion measures and architectural estimates are most sensitive to varying tractography settings and support the need for consistent reporting of tractography details to aid interpretation and comparison of results between studies.

LiDAR point cloud simplification algorithm with fuzzy encoding-decoding mechanism

With the explosive growth in the density of acquired point cloud data, point cloud processing tasks will face tremendous challenges. LiDAR point cloud simplification is a key phase in addressing this issue, which effectively promotes the development of LiDAR technology in many engineering fields. In this study, an innovative point cloud simplification algorithm with the fuzzy encoding-decoding mechanism is proposed. In the developed scheme, an approach for curvature estimation is first designed on the basis of the k-neighbor searching and principal component analysis. Then, a collection of feature point sets is set up with the ordered curvatures. Subsequently, a Fuzzy C-Means clustering based encoding mechanism is employed to capture the level point cloud structures in depth and establish a reasonable and streamlined strategy for point clouds. Each feature point set and non-feature point set are encoded into a prototype matrix and a partition (membership) matrix. The membership degree of each feature point to its prototype becomes the basis for the simplification strategy. Finally, the simplification result of the point cloud is formed through merging the simplification results of all subsets. The method proposed in this study effectively preserves the point cloud features and ensures a uniform distribution of the simplified point cloud. A comparative analysis of the point cloud simplification is conducted. The experimental results demonstrate that the developed algorithm outperformed other point cloud simplification algorithms.

Star-shaped p-convex hypersurfaces with prescribed curvature in space forms

In this paper, we establish curvature estimates for p-convex hypersurfaces with prescribed curvature for n2≤p≤n−1 in space forms Nn+1(K) with K=−1,0,1. Then the existence of a closed star-shaped p-convex hypersurface with prescribed curvature is obtained by the method of continuity.

Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three

abstract: We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in $\mathbb{R}^{3}$. The method is motivated by the integral method of Warren and Yuan. The new observation here is that we construct a ``Lagrangian'' graph which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation.

Directionally‐split volume‐of‐fluid technique for front propagation under curvature flow

AbstractA directionally‐split volume‐of‐fluid (VOF) methodology for evolving interfaces under curvature‐dependent speed is devised. The interface is reconstructed geometrically and the volume fraction is advected with a technique to incorporate a topological volume conservation constraint. The proposed approach uses the idea that the role of curvature in a speed function is analogous to the role of viscosity in the corresponding hyperbolic conservation law to propagate complex interfaces where singularities may exist. The approach has the advantage of simple implementation and straightforward extension to more complex multiphase systems by formulating the interface evolution problem using energy functionals to derive an expression for the interface‐advecting velocity. The numerical details of the volume‐of‐fluid based formulation are discussed with emphasis on the importance of curvature estimation. Finally, canonical curves and surfaces traditionally investigated by the level set (LS) method are tested with the devised approach and the results are compared with existing work in LS.

Minimal graphs in Riemannian spaces

In this treatise, we discuss existence and uniqueness questions for parametric minimal surfaces in Riemannian spaces, which represent minimal graphs. We choose the Riemannian metric suitably such that the variational solution of the Riemannian Dirichlet integral under Dirichlet boundary conditions possesses a one-to-one projection onto a plane. At first we concentrate our considerations on existence results for boundary value problems. Then we study uniqueness questions for boundary value problems and for complete minimal graphs. Here we establish curvature estimates for minimal graphs on discs, which imply Bernstein-type results.

Exploring Quaternion Neural Network Loss Surfaces

This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface curvature estimates. The paper presents a comprehensive comparison of the average surface curvature of a tuned QMLP model and a tuned real-valued MLP model on both a regression task and a classification task. The results provide strong support for the improved optimization performance observed in QMLPs across various problem domains.

Curvature estimates for a class of Hessian quotient type curvature equations

Curvature estimates for a class of Hessian quotient type curvature equations

Shape Sensing for Continuum Robotics Using Optoelectronic Sensors with Convex Reflectors

Three-dimensional shape sensing in soft and continuum robotics is a crucial aspect for stable actuation and control in fields such as minimally invasive surgery, engine repairs and search and rescue operations, as the estimation of complex curvatures while using continuum robotic tools is required to manipulate through fragile paths. This challenge has been addressed using a range of different sensing techniques, for example, Fibre Bragg grating (FBG) technology, inertial measurement unit (IMU) sensor networks, or stretch sensors. Previously, an optics-based method using optoelectronic sensors was explored, offering a simple and cost-effective solution for shape sensing in a flexible tendon-actuated manipulator in two orientations. This was based on proximity-modulated angle estimation and has been the basis for the shape sensing method addressed in this paper. The improved and miniaturised technique demonstrated in this paper is based on the use of a convex shaped reflector with optoelectronic sensors integrated into a tendon-actuated robotic manipulator. Upgraded sensing capability is achieved using optimisation of the convex reflector shape in terms of sensor range and resolution, and improved calibration is achieved through the integration of spherical bearings for friction-free motion. Shape estimation is achieved in two orientations upon calibration of sensors, with a maximum Root-Mean-Square Error (RMS) of 3.37°.

B-spine: Learning B-spline Curve Representation for Robust and Interpretable Spinal Curvature Estimation

Spinal curvature estimation is important to the diagnosis and treatment of the scoliosis. Existing methods face several issues such as the need of expensive annotations on the vertebral landmarks and being sensitive to the image quality. It is challenging to achieve robust estimation and obtain interpretable results, especially for low-quality images which are blurry and hazy. In this paper, we propose B-Spine, a novel deep learning pipeline to learn B-spline curve representation of the spine and estimate the Cobb angles for spinal curvature estimation from low-quality X-ray images. Given a low quality input, a novel SegRefine network which employs the unpaired image-to-image translation is proposed to generate a high quality spine mask from the initial segmentation result. Next, a novel mask-based B-spline prediction model is proposed to predict the B-spline curve for the spine centerline. Finally, the Cobb angles are estimated by a hybrid approach which combines the curve slope analysis and a curve based regression model. We conduct quantitative and qualitative comparisons with the representative and SOTA learning-based methods on the public AASCE2019 dataset and our new proposed JLU-CJUH dataset which contains more challenging low-quality images. The superior performance on both datasets shows our method can achieve both robustness and interpretability for spinal curvature estimation.

Can Climate Shocks Make Vulnerable Subjects More Willing to Take Risks?

While economists in the past tended to assume that individual preferences, including risk preferences, are stable over time, a recent literature has developed and indicates that risk preferences respond to shocks, with mixed evidence on the direction of the responses. This paper utilizes a natural experiment with covariate (drought) and idiosyncratic shocks in combination with an independent field risk experiment. The risk experiment uses a Certainty Equivalent-Multiple Choice List approach and is played 1–2 years after the subjects were (to a varying degree) exposed to a covariate drought shock or idiosyncratic shocks for a sample of resource-poor young adults living in a risky semi-arid rural environment in Sub-Saharan Africa. The experimental approach facilitates a comprehensive assessment of shock effects on experimental risk premiums for risky prospects with varying probabilities of good and bad outcomes. The experiment also facilitates the estimation of the utility curvature in an Expected Utility (EU) model and, alternatively, separate estimation of probability weighting and utility curvature in three different Rank Dependent Utility models with a two-parameter Prelec probability weighting function. Our study is the first to comprehensively test the theoretical predictions of Gollier and Pratt (Econom J Econom Soc 64:1109–1123, 1996) versus Quiggin (Econ Theor 22(3):607–611, 2003). Gollier and Pratt (1996) build on EU theory and state that an increase in background risk will make subjects more risk averse while Quiggin (2003) states that an increase in background risk can enhance risk-taking in certain types of non-EU models. We find strong evidence that such non-EU preferences dominate in our sample.

Testing the spatial geometry of the Universe with TianQin: Prospect of using supermassive black hole binaries

Context. The determination of the spatial geometry of the Universe plays an important role in modern cosmology. Any deviation from the cosmic curvature ΩK = 0 would have a profound impact on the primordial inflation paradigm and fundamental physics. Aims. In this paper, we carry out a systematic study of the prospect of measuring the cosmic curvature with the inspiral signal of supermassive black hole binaries (SMBHBs) that could be detected with TianQin. Methods. The study is based on a method that is independent of cosmological models. It extended the application of gravitational wave (GW) standard sirens in cosmology. By comparing the distances from future simulated GW events and simulated H(z) data, we evaluated whether TianQin produced robust constraints on the cosmic curvature parameter Ωk. More specifically, we considered three-year to ten-year observations of supermassive black hole binaries with total masses ranging from 103 M⊙ to 107 M⊙. Results. Our results show that in the future, with the synergy of ten-year high-quality observations, we can tightly constrain the curvature parameter at the level of 1σ Ωk = −0.002 ± 0.061. Moreover, our findings indicate that the total mass of SMBHB does influence the estimation of cosmic curvature, as implied by the analysis performed on different subsamples of gravitational wave data. Conclusions. Therefore, TianQin is expected to provide a more powerful and competitive probe of the spatial geometry of the Universe, compared to future spaced-based detectors such as DECIGO.

Comparison of methods for curvature estimation from volume fractions

This paper evaluates and compares the accuracy and robustness of curvature estimation methods for three-dimensional interfaces represented implicitly by discrete volume fractions on a Cartesian mesh. The height function (HF) method is compared to three paraboloid fitting methods: fitting to the piecewise linear interface reconstruction centroids (PC), fitting to the piecewise linear interface reconstruction volumetrically (PV), and volumetrically fitting (VF) the paraboloid directly to the volume fraction field. A shape-independent test of the curvature estimation is introduced through the use of randomly oriented and shaped paraboloids as reference interfaces. The coupling of the curvature estimation, interface transport, and Navier–Stokes surface tension force is evaluated through the measurement of spurious velocities in simulations of stationary and translating droplets. These studies find that while the curvature error from the VF method converges with second-order accuracy as with the HF method for static interfaces represented by exact volume fractions, the PV method best balances low curvature errors with low computational cost when errors are introduced in the volume fraction field either artificially or through the interface advection and reconstruction steps.