6,714 research outputs found
Multi-directional Geodesic Neural Networks via Equivariant Convolution
We propose a novel approach for performing convolution of signals on curved
surfaces and show its utility in a variety of geometric deep learning
applications. Key to our construction is the notion of directional functions
defined on the surface, which extend the classic real-valued signals and which
can be naturally convolved with with real-valued template functions. As a
result, rather than trying to fix a canonical orientation or only keeping the
maximal response across all alignments of a 2D template at every point of the
surface, as done in previous works, we show how information across all
rotations can be kept across different layers of the neural network. Our
construction, which we call multi-directional geodesic convolution, or
directional convolution for short, allows, in particular, to propagate and
relate directional information across layers and thus different regions on the
shape. We first define directional convolution in the continuous setting, prove
its key properties and then show how it can be implemented in practice, for
shapes represented as triangle meshes. We evaluate directional convolution in a
wide variety of learning scenarios ranging from classification of signals on
surfaces, to shape segmentation and shape matching, where we show a significant
improvement over several baselines
TextureNet: Consistent Local Parametrizations for Learning from High-Resolution Signals on Meshes
We introduce, TextureNet, a neural network architecture designed to extract
features from high-resolution signals associated with 3D surface meshes (e.g.,
color texture maps). The key idea is to utilize a 4-rotational symmetric
(4-RoSy) field to define a domain for convolution on a surface. Though 4-RoSy
fields have several properties favorable for convolution on surfaces (low
distortion, few singularities, consistent parameterization, etc.), orientations
are ambiguous up to 4-fold rotation at any sample point. So, we introduce a new
convolutional operator invariant to the 4-RoSy ambiguity and use it in a
network to extract features from high-resolution signals on geodesic
neighborhoods of a surface. In comparison to alternatives, such as PointNet
based methods which lack a notion of orientation, the coherent structure given
by these neighborhoods results in significantly stronger features. As an
example application, we demonstrate the benefits of our architecture for 3D
semantic segmentation of textured 3D meshes. The results show that our method
outperforms all existing methods on the basis of mean IoU by a significant
margin in both geometry-only (6.4%) and RGB+Geometry (6.9-8.2%) settings
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