128,581 research outputs found
Shape-from-intrinsic operator
Shape-from-X is an important class of problems in the fields of geometry
processing, computer graphics, and vision, attempting to recover the structure
of a shape from some observations. In this paper, we formulate the problem of
shape-from-operator (SfO), recovering an embedding of a mesh from intrinsic
differential operators defined on the mesh. Particularly interesting instances
of our SfO problem include synthesis of shape analogies, shape-from-Laplacian
reconstruction, and shape exaggeration. Numerically, we approach the SfO
problem by splitting it into two optimization sub-problems that are applied in
an alternating scheme: metric-from-operator (reconstruction of the discrete
metric from the intrinsic operator) and embedding-from-metric (finding a shape
embedding that would realize a given metric, a setting of the multidimensional
scaling problem)
Steklov Spectral Geometry for Extrinsic Shape Analysis
We propose using the Dirichlet-to-Neumann operator as an extrinsic
alternative to the Laplacian for spectral geometry processing and shape
analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator,
cannot capture the spatial embedding of a shape up to rigid motion, and many
previous extrinsic methods lack theoretical justification. Instead, we consider
the Steklov eigenvalue problem, computing the spectrum of the
Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable
property of this operator is that it completely encodes volumetric geometry. We
use the boundary element method (BEM) to discretize the operator, accelerated
by hierarchical numerical schemes and preconditioning; this pipeline allows us
to solve eigenvalue and linear problems on large-scale meshes despite the
density of the Dirichlet-to-Neumann discretization. We further demonstrate that
our operators naturally fit into existing frameworks for geometry processing,
making a shift from intrinsic to extrinsic geometry as simple as substituting
the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde
OperatorNet: Recovering 3D Shapes From Difference Operators
This paper proposes a learning-based framework for reconstructing 3D shapes
from functional operators, compactly encoded as small-sized matrices. To this
end we introduce a novel neural architecture, called OperatorNet, which takes
as input a set of linear operators representing a shape and produces its 3D
embedding. We demonstrate that this approach significantly outperforms previous
purely geometric methods for the same problem. Furthermore, we introduce a
novel functional operator, which encodes the extrinsic or pose-dependent shape
information, and thus complements purely intrinsic pose-oblivious operators,
such as the classical Laplacian. Coupled with this novel operator, our
reconstruction network achieves very high reconstruction accuracy, even in the
presence of incomplete information about a shape, given a soft or functional
map expressed in a reduced basis. Finally, we demonstrate that the
multiplicative functional algebra enjoyed by these operators can be used to
synthesize entirely new unseen shapes, in the context of shape interpolation
and shape analogy applications.Comment: Accepted to ICCV 201
Surface Networks
We study data-driven representations for three-dimensional triangle meshes,
which are one of the prevalent objects used to represent 3D geometry. Recent
works have developed models that exploit the intrinsic geometry of manifolds
and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants,
which learn from the local metric tensor via the Laplacian operator. Despite
offering excellent sample complexity and built-in invariances, intrinsic
geometry alone is invariant to isometric deformations, making it unsuitable for
many applications. To overcome this limitation, we propose several upgrades to
GNNs to leverage extrinsic differential geometry properties of
three-dimensional surfaces, increasing its modeling power.
In particular, we propose to exploit the Dirac operator, whose spectrum
detects principal curvature directions --- this is in stark contrast with the
classical Laplace operator, which directly measures mean curvature. We coin the
resulting models \emph{Surface Networks (SN)}. We prove that these models
define shape representations that are stable to deformation and to
discretization, and we demonstrate the efficiency and versatility of SNs on two
challenging tasks: temporal prediction of mesh deformations under non-linear
dynamics and generative models using a variational autoencoder framework with
encoders/decoders given by SNs
Is shape in the eye of the beholder? Assessing landmarking error in geometric morphometric analyses on live fish
Geometric morphometrics is widely used to quantify morphological variation between biological specimens, but the fundamental influence of operator bias on data reproducibility is rarely considered, particularly in studies using photographs of live animals taken under field conditions. We examined this using four independent operators that applied an identical landmarking scheme to replicate photographs of 291 live Atlantic salmon (Salmo salar L.) from two rivers. Using repeated measures tests, we found significant inter-operator differences in mean body shape, suggesting that the operators introduced a systematic error despite following the same landmarking scheme. No significant differences were detected when the landmarking process was repeated by the same operator on a random subset of photographs. Importantly, in spite of significant operator bias, small but statistically significant morphological differences between fish from the two rivers were found consistently by all operators. Pairwise tests of angles of vectors of shape change showed that these between-river differences in body shape were analogous across operator datasets, suggesting a general reproducibility of findings obtained by geometric morphometric studies. In contrast, merging landmark data when fish from each river are digitised by different operators had a significant impact on downstream analyses, highlighting an intrinsic risk of bias. Overall, we show that, even when significant inter-operator error is introduced during digitisation, following an identical landmarking scheme can identify morphological differences between populations. This study indicates that operators digitising at least a sub-set of all data groups of interest may be an effective way of mitigating inter-operator error and potentially enabling data sharing
A simple model for the quenching of pairing correlations effects in rigidly deformed rotational bands
Using Chandrasekhar's S-type coupling between rotational and intrinsic
vortical modes one may simply reproduce the HFB dynamical properties of
rotating nuclei within Routhian HF calculations free of pairing correlations
yet constrained on the relevant so-called Kelvin circulation operator. From the
analogy between magnetic and rotating systems, one derives a model for the
quenching of pairing correlations with rotation, introducing a critical angular
velocity -- analogous to the critical field in supraconductors -- above which
pairing vanishes. Taking stock of this usual model, it is then shown that the
characteristic behavior of the vortical mode angular velocity as a function of
the global rotation angular velocity can be modelised by a simple two parameter
formula, both parameters being completely determined from properties of the
band-head (zero-spin) HFB solution. From calculation in five nuclei, the
validity of this modelised Routhian approach is assessed. It is clearly shown
to be very good in cases where the evolution of rotational properties is only
governed by the coupling between the global rotation and the pairing-induced
intrinsic vortical currents. It therefore provides a sound ground base for
evaluating the importance of coupling of rotation with other modes (shape
distortions, quasiparticle degrees of freedom).Comment: 10 pages, 8 figures. Submited to PR
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