Geometric operations implemented by conformal geometric algebra neural nodes
This work proposes a novel neural network architecture for geometric computing, potentially benefiting fields like computer graphics or robotics, but it appears incremental as an extension of existing geometric algebra concepts.
The paper introduces conformal geometric algebra neurons, which can process geometric objects like points, lines, and spheres as inputs and apply transformations such as reflections and scaling, leveraging the unified representation of conformal geometric algebra.
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this approach to elementary representations of arbitrary points, point pairs, lines, circles, planes and spheres. Apart from including curved objects, conformal geometric algebra has an elegant unified quaternion like representation for all proper and improper Euclidean transformations, including reflections at spheres, general screw transformations and scaling. Expanding the concepts of real and complex neurons we arrive at the new powerful concept of conformal geometric algebra neurons. These neurons can easily take the above mentioned geometric objects or sets of these objects as inputs and apply a wide range of geometric transformations via the geometric algebra valued weights.