# Octahedron-Based Projections as Intermediate Representations for Computer Imaging: TOAST, TEA and More

## TL;DR

This paper introduces a set of all-sky projections, including TOAST, that serve as flexible, hierarchy-friendly intermediate representations for imaging data, optimized for GPU computation and dynamic format transformations.

## Contribution

It defines a new class of singularity-free all-sky projections, notably TOAST, and demonstrates their utility in hierarchical, GPU-friendly imaging data processing.

## Key findings

- TOAST is related to HTM pixelization and suitable for hierarchical image traversal.
- Projections can be computed recursively with simple algorithms.
- Useful for dynamic transformation of imaging data formats.

## Abstract

This paper defines and discusses a set of rectangular all-sky projections which have no singular points, notably the Tesselated Octahedral Adaptive Spherical Transformation (or TOAST) developed initially for the WorldWide Telescope (WWT). These have proven to be useful as intermediate representations for imaging data where the application transforms dynamically from a standardized internal format to a specific format (projection, scaling, orientation, etc.) requested by the user. TOAST is strongly related to the Hierarchical Triangular Mesh (HTM) pixelization and is particularly well adapted to the situations where one wishes to traverse a hierarchy of increasing resolution images. Since it can be recursively computed using a very simple algorithm it is particularly adaptable to use by graphical processing units.