In shape analysis, the concept of shape spaces has always been vague, requiring a case-by-case approach for every new type of shape. In this paper, we give a general definition for an abstract space of shapes in a manifold. This notion encompasses every shape space studied so far in the literature, and offers a rigorou…
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48 results for “Shape Analysis”
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.
A Python implementation of some geometric tools on Kendall 3D shape space for practical applicationsstat.CO
Python tools for 3D shape analysis on Kendall's space.
problem Lack of practical utilities for advanced 3D shape analysis.
method Developed Python tools for 3D shape analysis on Kendall's 3D Shape Space.
result Efficient, accessible software solutions for researchers.
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.
Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Sq…
Shape information is of great importance in many applications. For example, the oil-bearing capacity of sand bodies, the subterranean remnants of ancient rivers, is related to their cross-sectional shapes. The analysis of these shapes is therefore of some interest, but current classifications are simplistic and ad hoc.…
Optimizes shapes on non-standard manifolds.
problem Optimization on non-standard infinite-dimensional manifolds.
method Develops gradient descent on weak Riemannian manifolds.
result Establishes foundational properties for optimization on various weak Riemannian manifolds.
Paper analyzes shapes of brain arterial networks using statistical methods.
problem Quantifying and comparing shapes of brain arterial networks.
method Mathematical representation of BAN shapes as elastic shape graphs, development of Riemannian metrics and geometrical tools.
result Age has a clear, quantifiable effect on BAN shapes, with increased variance in shapes as age increases.
A new algorithm for parallel transport on shape spaces is presented and compared to existing methods.
problem Statistical analysis of shape data, especially in time series and optimization.
method Pole ladder algorithm for parallel transport on Kendall shape spaces, compared to integration methods.
result The pole ladder algorithm is a more efficient method for parallel transport.
Star-Shaped deviationsq-fin.RM
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
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…
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.
The paper explores metrics and models for analyzing biological shapes.
problem Analyzing biological shapes using mathematical metrics.
method Review of Riemannian metrics and evolution equations, focusing on diffeomorphic shape analysis.
result Introduction of a new class of metrics involving optimization of a growth tensor.
Paper connects surface shape analysis and unbalanced optimal transport.
problem Computing the SRNF shape distance on piecewise linear surfaces.
method Characterizes SRNF shape distance as WFR distance pullback, proposes new algorithm for WFR distance computation.
result Direct computation of SRNF shape distance on piecewise linear surfaces.
Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations exist, it is sometimes difficult to gain an overview how differential geometry and…
Hierarchical geodesic model for analyzing shapes on manifolds.
problem Analyzing temporal observations on manifold-valued data.
method Adapted functional-based metric for efficiency; variational time discretization of geodesics.
result Performed hypothesis tests and estimated mean trends in longitudinal analysis.
Extended orbit model theory for shape analysis using graded group action framework.
problem Limitations of standard orbit model theory in shape analysis.
method Developed graded group action (GGA) framework with regularity conditions.
result Uniqueness result for momentum map trajectory in multi-scale shape spaces.
In this paper, we describe a novel shape classification method which is embedded in the Bayesian paradigm. We discuss the modelling and the resulting shape classification algorithm for two and three dimensional data shapes. We conclude by evaluating the efficiency and efficacy of the proposed algorithm on the Kimia sha…
These are the proceedings of the workshop "Math in the Black Forest", which brought together researchers in shape analysis to discuss promising new directions. Shape analysis is an inter-disciplinary area of research with theoretical foundations in infinite-dimensional Riemannian geometry, geometric statistics, and geo…
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.
The paper uses 3D shapes to reveal sundial design adjustments based on latitude.
problem Identifying sundial design adjustments based on installation location.
method Shape analysis in a high-dimensional space, regression in shape space.
result Sundial design adjustments were latitude-dependent.
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed p…
Machine learning methods struggle with geometric data, but shape space analysis provides a framework for studying and analyzing geometric variability.
problem Machine learning methods struggle with geometric data
method Shape space analysis provides a mathematical and computational framework
result Characterizes shape variability, compares geometric objects, and analyzes structural trajectories
DNAMite creates interpretable, calibrated survival analysis models.
problem Limited interpretability in survival analysis models, especially for healthcare applications.
method Feature discretization and kernel smoothing in embedding module for flexible shape functions.
result DNAMite produces calibrated shape functions interpretable as contributions to cumulative incidence function.
In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT). We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on homogeneous manifolds. The main idea behind our approach is to exploit the geometr…
This work characterizes how data augmentation shapes neural representations.
problem Understanding the impact of data augmentation on neural network representations.
method Embedding neural network hidden representations into a metric space invariant to transformations, analyzing shape-space trajectories.
result Increasing data augmentation strength leads to well-behaved trajectories in the embedded space, and different augmentation types steer representations in distinct directions.
Paper introduces a method to generate stable shapes using Grassmann manifolds.
problem Generating stable shapes with minimal extraneous transformations.
method Continuous normalization flows on Grassmann manifolds to eliminate extraneous transformations.
result The method significantly outperforms state-of-the-art methods in generating high-quality samples.
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…
In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Book…
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.
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 …
The paper proposes a deep learning approach to efficiently approximate diffeomorphisms for shape alignment.
problem Finding optimal reparameterizations of shapes for computing geodesic distances.
method The authors develop a neural network-based algorithm to construct approximations of diffeomorphisms using PyTorch.
result The proposed method achieves universal approximation properties and bounds on Lipschitz constants for the constructed diffeomorphisms.
The paper presents a method for analyzing shape graphs using specific features.
problem Analyzing geometric and topological variations in shape graphs.
method Curated set of topological, geometric, and directional features for shape graph analysis.
result The feature representation is effective for tasks like group comparison and classification.
A Wide and Deep Neural Network for Survival Analysis from Anatomical Shape and Tabular Clinical Datacs.LG
We introduce a wide and deep neural network for prediction of progression from patients with mild cognitive impairment to Alzheimer's disease. Information from anatomical shape and tabular clinical data (demographics, biomarkers) are fused in a single neural network. The network is invariant to shape transformations an…
The paper explores three methods to assign a metric to shape spaces.
problem Assigning a Riemannian metric to shape spaces without parameterization.
method Three methods to put a Riemannian metric on shape spaces.
result Methods provide a way to measure deformations independent of parameterization.
Generative modeling of 3D shapes has become an important problem due to its relevance to many applications across Computer Vision, Graphics, and VR. In this paper we build upon recently introduced 3D mesh-convolutional Variational AutoEncoders which have shown great promise for learning rich representations of deformab…
A new method for analyzing shapes and forms using additive models on manifolds.
problem Analyzing shapes and forms under geometric transformations.
method Extending generalized additive regression to models for shapes/forms using squared geodesic distance and Riemannian -Boosting algorithm.
result Automated model selection and intuitive visualization of covariate effects in shape/form space.
SVarM uses varifold representations for shape classification and regression.
problem Challenges in analyzing geometric data due to non-Euclidean shape spaces.
method Develops a neural network-based framework for varifold representations of shapes.
result Demonstrates strong performance and robustness in shape classification and regression.
A novel 3D shape registration method using spectral graph embedding and probabilistic matching.
problem Challenges in 3D shape analysis and registration, especially with large variability.
method Combining spectral graph matching with Laplacian embedding for large graphs, using commute-time embedding and PCA.
result A method to register shapes with different samplings and isometric deformations.
Paper controls shape stability in infinite Riemannian manifolds.
problem Characterizing optimal shapes in infinite-dimensional Riemannian manifolds.
method Uses Riemannian manifold framework and mean curvature analysis.
result Control on shape stability depends only on mean curvature.
New method for surface analysis using restricted deformation bases.
problem Surface registration and comparison without pre-registered data.
method Elastic Riemannian metrics with basis-restricted transformations.
result Effective implementation on human body and face scans.
Efficient method for shape modeling invariant to rigid motion.
problem Statistical shape modeling for rigidly moving shapes.
method Non-Euclidean Lie group analysis of metric distortion and curvature.
result Outperforms state-of-the-art classifiers in sparse data.
Enhanced 3D shape analysis using information geometry.
problem Challenges in comparing 3D point clouds due to their unstructured nature and complex geometry.
method Information geometric framework for 3D point cloud shape analysis using Gaussian Mixture Models (GMMs) on a statistical manifold. Proposed MSKL divergence with upper and lower bounds.
result MSKL provides stable and monotonically varying values that directly reflect geometric variation, outperforming traditional distances and existing KL approximations.
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.
Paper develops a new method to analyze 3D tree-like objects.
problem Analyzing complex geometrical and topological variations in 3D tree-like objects.
method Extended SRVF representation and new metric for tree-shaped 3D objects.
result Captures full elasticity and topological variations of branches.
A neural network learns efficient parametrizations of product shape spaces.
problem Efficiently parametrize complex shape spaces with high computational costs.
method Developed a neural network architecture that separately learns approximations for low-dimensional factors and combines them.
result Demonstrated the effectiveness of the approach on synthetic and real data.
Geometric shape matching for recovering protein conformations from single-particle Cryo-EM dataq-bio.BM
Researchers use shape analysis to recover protein structures from Cryo-EM data.
problem Recovering the three-dimensional backbone structure of single polypeptide proteins from noisy tomographic projections.
method Shape analysis and matrix Lie group actions to deform point clouds to match 2D tomography data.
result Optimal deformations are computed to recover the three-dimensional backbone structure of proteins.
New method learns shape correspondences robustly from raw geometry.
problem Inaccurate and poor generalization of shape correspondences.
method Learning-based approach with feature-extraction network and functional map representation.
result Robust and accurate shape correspondence learning with less training data.