7 research outputs found
Differential invariants for SE(2)-equivariant networks
Symmetry is present in many tasks in computer vision, where the same class of
objects can appear transformed, e.g. rotated due to different camera
orientations, or scaled due to perspective. The knowledge of such symmetries in
data coupled with equivariance of neural networks can improve their
generalization to new samples. Differential invariants are equivariant
operators computed from the partial derivatives of a function. In this paper we
use differential invariants to define equivariant operators that form the
layers of an equivariant neural network. Specifically, we derive invariants of
the Special Euclidean Group SE(2), composed of rotations and translations, and
apply them to construct a SE(2)-equivariant network, called SE(2) Differential
Invariants Network (SE2DINNet). The network is subsequently tested in
classification tasks which require a degree of equivariance or invariance to
rotations. The results compare positively with the state-of-the-art, even
though the proposed SE2DINNet has far less parameters than the compared models
Symbolic software for symmetry reduction and computation of invariant solutions of differential equations
Problems involving partial or ordinary differential equations arise in various fields of science. Therefore, the task of obtaining exact solutions of differential equations is of primary importance, and attracts high attention. The main purpose of the current thesis is the development of a Maple-based, symbolic software package for symmetry reduction of differential equations and computation of symmetry-invariant solutions. The package developed in the current thesis is compatible with and can be viewed as an extension of the package GeM for symbolic symmetry analysis, developed by Prof. Alexei Cheviakov. The reduction procedure is based on the Lie's classical symmetry reduction method involving canonical coordinates. The developed package is applicable for obtaining solutions arising from extension of Lie's method, in particular, nonlocal and approximate symmetries.
The developed software is applied to a number of PDE problems to obtain exact invariant solutions. The considered equations include the one-dimensional nonlinear heat equation, the potential Burgers' equation, as well as equations arising in nonlinear elastostatics and elastodynamics
Numerical invariantization for morphological PDE schemes
Based on a new, general formulation of the geometric method of moving frames, invariantization of numerical schemes has been established during the last years as a powerful tool to guarantee symmetries for numerical solutions while simultaneously reducing the numerical errors. In this paper, we make the first step to apply this framework to the differential equations of image processing. We focus on the Hamilton-Jacobi equation governing dilation and erosion processes which displays morphological symmetry, i.e. is invariant under strictly monotonically increasing transformations of gray-values. Results demonstrate that invariantization is able to handle the specific needs of differential equations applied in image processing, and thus encourage further research in this direction
Numerical invariantization for morphological PDE schemes, in: Scale Space and Variational Methods in Computer Vision
Abstract. Based on a new, general formulation of the geometric method of moving frames, invariantization of numerical schemes has been established during the last years as a powerful tool to guarantee symmetries for numerical solutions while simultaneously reducing the numerical errors. In this paper, we make the first step to apply this framework to the differential equations of image processing. We focus on the Hamilton–Jacobi equation governing dilation and erosion processes which displays morphological symmetry, i.e. is invariant under strictly monotonically increasing transformations of gray-values. Results demonstrate that invariantization is able to handle the specific needs of differential equations applied in image processing, and thus encourage further research in this direction.