Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.
problem What conditions guarantee global existence of Gage's area-preserving flow for nonconvex initial curves?
method Using Dittberner's singularity analysis theory, constructed a ``flying wing'' curve to show limitations.
result Gage's area-preserving flow does not always preserve star-shapedness of evolving curves.
Extends shape analysis to framed space curves using quaternionic arithmetic.
problem Matching and classifying shapes of framed space curves.
method Extends square root transform to framed curves using quaternionic arithmetic and Hopf fibration properties. Describes geodesics in framed curve space explicitly.
result Explicit descriptions of geodesics in framed curve space and averages of collections of curves.
Reduces 3-body problem to shape and spherical geometry.
problem Investigates the motion of three bodies in a plane.
method Geometric reduction using equivariant Riemannian geometry.
result Time parametrization of moduli curve determined by shape curve and potential function.
We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape features in a visually intuitive manner. This method have implications in shape an…
Geomstats introduces shape module for analyzing shapes of objects.
problem Analyzing shapes of objects represented as landmarks, curves, and surfaces.
method Implementing shape spaces, group actions, fiber bundles, quotient spaces, and Riemannian metrics.
result Users can compare, average, and interpolate shapes inside shape spaces.
Classifies shapes of yield curves in the Svensson family.
problem Classifying shapes of yield curves in the Svensson family.
method Complete classification of shapes using mathematical analysis.
result Certain complex shapes cannot appear after a deterministic time horizon.
W-shaped vol curves in liquid options can be modeled with two variance-gamma models.
problem Reproducing W-shaped implied volatility curves in liquid option markets.
method Using a mixture of two variance-gamma models.
result W-shaped vol curves can be generated with fewer distributions (two) compared to lognormal models (at least three).
BézierGAN generates smooth curves from low-dimensional parameters.
problem Designing smooth curves for aerodynamic and hydrodynamic shapes.
method Generative model that maps low-dimensional latent representation to Bézier curve points.
result Generates diverse and realistic curves with consistent shape variation.
Flow turns star-shaped curves into circles.
problem Transforming star-shaped curves into circles.
method Gage's area-preserving flow.
result Curves evolve into circles over time.
Study classifies deformations of star-shaped curves in n-dimensional space.
problem Classifying deformations of star-shaped curves in n-dimensional space.
method Using connections on vector bundles and cyclic D-modules, defining integral curves and classifying them via iso-spectral flows.
result Iso-spectral flows are described by equations from the n-KdV hierarchy.
The paper explains the concave shape of yield curves from trading perspectives.
problem Lack of explanation for the concavity of yield curves from economics theory.
method Explains the concavity of yield curves from trading perspectives.
result Offers an explanation for the concave shape of yield curves.
The paper proposes a new method for modeling and quantifying uncertainty in multiple closed curves.
problem Modeling and uncertainty quantification of multiple closed curves.
method A multiple-output, multi-dimensional Gaussian process modeling framework.
result The proposed method provides meaningful uncertainty quantification for curve and shape-related tasks.
Shapes can roll downhill following any curve, but often return to initial orientation after crossing multiple copies.
problem How to design shapes that roll downhill along a given curve and its translations.
method Analyzing the geometric properties and motion of shapes on inclined planes.
result Most curves allow shapes to roll downhill following them and their translations, but some require crossing multiple copies.
Paper simplifies transforms for elastic metrics on curve shapes.
problem Improving the performance of shape analysis algorithms.
method Extending coordinate transformations to a family of isometries.
result Existence of optimal matchings over the diffeomorphism group.
The paper generalizes convex and star-shaped concepts to symplectic spaces and studies variational problems.
problem Generalizing convex and star-shaped concepts to symplectic vector spaces.
method Study of variational problems for symplectically convex and star-shaped curves.
result Extremal points of the variational problem are rigid multiply traversed conics for a range of parameters.
Unsupervised clustering of curves according to their shapes is an important problem with broad scientific applications. The existing model-based clustering techniques either rely on simple probability models (e.g., Gaussian) that are not generally valid for shape analysis or assume the number of clusters. We develop an…
A new FFT-based method for fast rigid alignment of 2D closed curves.
problem Rigid alignment of 2D closed curves with application to shape analysis.
method FFT-based algorithm for optimal rigid alignment of closed curves with O(N log N) complexity.
result Order of magnitude speed-up in curve alignment compared to previous methods.
Deep model predicts shapes of curves with multiple covariates.
problem Predicting shapes of planar curves with various covariates.
method Deep learning model using complex-valued functions, conditional covariance smoother with modality-specific encoders.
result Model accurately predicts shapes of curves with multimodal covariates.
A classical result in Riemannian geometry states that the absolutely continuous curves into a (finite-dimensional) Riemannian manifold form an infinite-dimensional manifold. In the present paper this construction and related results are generalised to absolutely continuous curves with values in a strong Riemannian mani…
The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.
problem Anisotropic length preservation in curve deformation.
method A curve flow that maintains anisotropic length, analyzed for convex closed curves.
result Convex curves evolve to homothetic limits of Wulff shapes as time approaches infinity.
Study the hanging chain shape around a circle.
problem Finding the shape of a curve extremizing potential energy to a circle.
method Analyzes curves minimizing potential energy to a circle, considering both inside and outside.
result Describes shapes of curves for different powers of distance to the circle.
In a recent paper (arXiv:math-ph/0609076) the authors investigated the basic global geometry of congruence moduli curves and shape curves of 3-body motions with vanishing angular momentum. Here the study is extended to the case of planary 3-body motions in general. In particular, the results on the separation of the si…
The study classifies term structure shapes in the two-factor Vasicek model using total positivity.
problem Classifying all possible term structure shapes in the two-factor Vasicek model of interest rates.
method Total positivity theory pioneered by Samuel Karlin.
result Four additional shapes can be produced in certain parameter regimes.
Paper computes optimal matching between curves on manifolds.
problem Matching curves on infinite-dimensional manifolds.
method Geodesic computation using Riemannian metric and quotient structure.
result Algorithm for computing geodesics in shape space.
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.
problem Comparing and analyzing shapes of curves.
method Construction and theoretical properties of quotient elastic metrics, special case of square root velocity metric, numerical approaches for estimation.
result Simplified expression for the square root velocity metric distance.
Optimizes shapes of curves using Möbius energy gradients.
problem Finding optimal shapes of curves within isotopy classes.
method Gradient-based optimization with Sobolev inner products.
result Significantly more efficient and robust optimization methods.
Transforms curves and surfaces for efficient geometric analysis.
problem Efficiently analyzing and comparing curves and surfaces.
method Square root velocity transformation for curves and intrinsic comparison for surfaces.
result Fundamental geometric properties of curves under the transformation.
The paper encodes local shapes of polynomial curves using permutations.
problem Measuring non-convexity of real algebraic plane curves.
method Generic projections avoiding specific tangencies.
result Local shapes of curves can be encoded in alternating permutations.
Second order Sobolev metrics are a useful tool in the shape analysis of curves. In this paper we combine these metrics with varifold-based inexact matching to explore a new strategy of computing geodesics between unparametrized curves. We describe the numerical method used for solving the inexact matching problem, appl…
Corrected an error in yield curve behavior models.
problem Error in the boundary expression for yield curve shapes.
method Revised the mathematical expression for yield curve behavior.
result Corrected the boundary expression for normal and humped yield curves.
A new method for analyzing shapes using FDA techniques.
problem Statistical shape analysis of deformed contours.
method Functional Data Analysis (FDA) with basis expansion and principal component analysis.
result Successfully identifies deformation parameters and captures contour distributions.
Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects. In …
Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a continuation of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens spac…
The paper introduces various canonical parameterizations for 2D-curved shapes.
problem Comparing unparameterized simple curves in the plane.
method Proposes diverse canonical parameterizations, including arc-length and curvature-based.
result Natural parameterizations correspond to physical movements and are geometric invariants.
Paper tackles shape graph registration using neural networks.
problem Constrained registration of shape graphs with varying nodes and edges.
method Shape-Graph Matching Network (SGM-net) with an elastic shape metric loss function.
result State-of-the-art matching performance and reduced computational cost.
A new method combines metrics for shape registration, inheriting advantages from both curve and surface approaches.
problem Designing effective Riemannian metrics for diffeomorphic shape registration.
method Combining parametrization-invariant metrics on immersions with metrics from Riemannian submersions from diffeomorphism groups.
result The hybrid approach inherits advantages from both methods and provides additional flexibility.
Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.
problem Characterizing and comparing convex shapes using length measures.
method Developed length measures for curves and convex shapes, derived properties, and introduced a new distance metric.
result Unique convex curve maximizes signed area among curves with same length measure.
A new algorithm computes elastic shape distances between curves efficiently.
problem Computing elastic shape distances between curves in high dimensions.
method Dynamic Programming for optimal diffeomorphisms and Kabsch-Umeyama algorithm for optimal rotation matrices.
result Efficient computation of elastic shape distances with improved efficiency for closed curves.
Study efficient geodesics in curve complex using dot graphs.
problem Characterize efficient geodesics in curve complexes.
method Introduced dot graphs to record intersection patterns and used them to prove existence and properties of efficient geodesics.
result The shape of dot graphs for efficient geodesics is contained within a spindle shape region, controlling curve coordinates.
We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order n≥2 are metrically complete on the space In(S1,Rd) of Sobolev immersions of the same regularity and that any two curves i…
Bayesian nonparametric method partitions shapes using curves.
problem Capturing complex shapes in multi-dimensional data.
method Proposes a novel spline partitioning approach using curves.
result Demonstrates improved shape modeling compared to existing methods.
Functional BART adds shape priors to Bayesian tree regression for better curve fitting.
problem Regression with function-on-scalar data and shape constraints.
method Bayesian tree structure with spline representations, customized Bayesian backfitting algorithm, shape priors.
result Improved estimation and prediction accuracy with shape priors.
Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
New metrics for surface shapes incorporating curve properties.
problem Developing metrics for surface shape spaces.
method Incorporates geodesic and normal curvatures of curves on surfaces.
result Explicitly defined 6-parameter family of metrics.
New model for shape graph registration with partial matching constraints.
problem Shape graph registration with topological inconsistencies and partial matching.
method Higher order invariant Sobolev metrics, varifolds, inexact variational formulation, SFISTA algorithm.
result Existence of minimizers for variational problem with TV regularization.
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
A new method for pricing options with flexible volatility shapes.
problem Parameterizing risk-neutral distributions for accurate option pricing.
method Parsimonious and interpretable parameters for direct control over implied volatility curves.
result Accurate calibration across a large dataset of option curves.