110,097 research outputs found
Geodesic analysis in Kendallâs shape space with epidemiological applications
We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendallâs shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application, we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone
Geometric Projectors: Geometric Constraints based Optimization for Robot Behaviors
Generating motion for robots that interact with objects of various shapes is
a complex challenge, further complicated when the robot's own geometry and
multiple desired behaviors are considered. To address this issue, we introduce
a new framework based on Geometric Projectors (GeoPro) for constrained
optimization. This novel framework allows for the generation of task-agnostic
behaviors that are compliant with geometric constraints. GeoPro streamlines the
design of behaviors in both task and configuration spaces, offering diverse
functionalities such as collision avoidance and goal-reaching, while
maintaining high computational efficiency. We validate the efficacy of our work
through simulations and Franka Emika robotic experiments, comparing its
performance against state-of-the-art methodologies. This comprehensive
evaluation highlights GeoPro's versatility in accommodating robots with varying
dynamics and precise geometric shapes. For additional materials, please visit:
https://www.xueminchi.com/publications/geoproComment: 9 pages, 5 figure
On the optimality of shape and data representation in the spectral domain
A proof of the optimality of the eigenfunctions of the Laplace-Beltrami
operator (LBO) in representing smooth functions on surfaces is provided and
adapted to the field of applied shape and data analysis. It is based on the
Courant-Fischer min-max principle adapted to our case. % The theorem we present
supports the new trend in geometry processing of treating geometric structures
by using their projection onto the leading eigenfunctions of the decomposition
of the LBO. Utilisation of this result can be used for constructing numerically
efficient algorithms to process shapes in their spectrum. We review a couple of
applications as possible practical usage cases of the proposed optimality
criteria. % We refer to a scale invariant metric, which is also invariant to
bending of the manifold. This novel pseudo-metric allows constructing an LBO by
which a scale invariant eigenspace on the surface is defined. We demonstrate
the efficiency of an intermediate metric, defined as an interpolation between
the scale invariant and the regular one, in representing geometric structures
while capturing both coarse and fine details. Next, we review a numerical
acceleration technique for classical scaling, a member of a family of
flattening methods known as multidimensional scaling (MDS). There, the
optimality is exploited to efficiently approximate all geodesic distances
between pairs of points on a given surface, and thereby match and compare
between almost isometric surfaces. Finally, we revisit the classical principal
component analysis (PCA) definition by coupling its variational form with a
Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can
handle cases that go beyond the scope defined by the observation set that is
handled by regular PCA
Shape-matching soft mechanical metamaterials
Architectured materials with rationally designed geometries could be used to create mechanical metamaterials with unprecedented or rare properties and functionalities. Here, we introduce "shape-matching" metamaterials where the geometry of cellular structures comprising auxetic and conventional unit cells is designed so as to achieve a pre-defined shape upon deformation. We used computational models to forward-map the space of planar shapes to the space of geometrical designs. The validity of the underlying computational models was first demonstrated by comparing their predictions with experimental observations on specimens fabricated with indirect additive manufacturing. The forward-maps were then used to devise the geometry of cellular structures that approximate the arbitrary shapes described by random Fourier's series. Finally, we show that the presented metamaterials could match the contours of three real objects including a scapula model, a pumpkin, and a Delft Blue pottery piece. Shape-matching materials have potential applications in soft robotics and wearable (medical) devices
Spectral Generalized Multi-Dimensional Scaling
Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches
An aerothermodynamic design optimization framework for hypersonic vehicles
In the aviation field great interest is growing in passengers transportation at hypersonic speed. This requires, however, careful study of the enabling technologies necessary for the optimal design of hypersonic vehicles. In this framework, the present work reports on a highly integrated design environment that has been developed in order to provide an optimization loop for vehicle aerothermodynamic design. It includes modules for geometrical parametrization, automated data transfer between tools, automated execution of computational analysis codes, and design optimization methods. This optimization environment is exploited for the aerodynamic design of an unmanned hypersonic cruiser flying at Mâ=8 and 30 km altitude. The original contribution of this work is mainly found in the capability of the developed optimization environment of working simultaneously on shape and topology of the aircraft. The results reported and discussed highlight interesting design capabilities, and promise extension to more challenging and realistic integrated aerothermodynamic design problems
The State of the Art in Cartograms
Cartograms combine statistical and geographical information in thematic maps,
where areas of geographical regions (e.g., countries, states) are scaled in
proportion to some statistic (e.g., population, income). Cartograms make it
possible to gain insight into patterns and trends in the world around us and
have been very popular visualizations for geo-referenced data for over a
century. This work surveys cartogram research in visualization, cartography and
geometry, covering a broad spectrum of different cartogram types: from the
traditional rectangular and table cartograms, to Dorling and diffusion
cartograms. A particular focus is the study of the major cartogram dimensions:
statistical accuracy, geographical accuracy, and topological accuracy. We
review the history of cartograms, describe the algorithms for generating them,
and consider task taxonomies. We also review quantitative and qualitative
evaluations, and we use these to arrive at design guidelines and research
challenges
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