Representation Learning on Unit Ball with 3D Roto-translational Equivariance
Date
2019
Authors
Ramasinghe, Sameera
Khan, Salman Hameed
Barnes, Nick
Gould, Stephen
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Publisher
Springer
Abstract
Convolution is an integral operation that defines how the shape of one function is modified by another function. This powerful
concept forms the basis of hierarchical feature learning in deep neural networks. Although performing convolution in Euclidean
geometries is fairly straightforward, its extension to other topological spaces—such as a sphere (S2) or a unit ball (B3)—
entails unique challenges. In this work, we propose a novel ‘volumetric convolution’ operation that can effectively model
and convolve arbitrary functions in B3. We develop a theoretical framework for volumetric convolution based on Zernike
polynomials and efficiently implement it as a differentiable and an easily pluggable layer in deep networks. By construction,
our formulation leads to the derivation of a novel formula to measure the symmetry of a function in B3 around an arbitrary
axis, that is useful in function analysis tasks. We demonstrate the efficacy of proposed volumetric convolution operation on
one viable use case i.e., 3D object recognition.
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Source
International Journal of Computer Vision
Type
Journal article
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Restricted until
2099-12-31
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