Incremental and Transitive Discrete Rotations

TL;DR

This paper presents an efficient, exact incremental algorithm for discrete image rotations that uses only integer arithmetic and analyzes the step function's discontinuities for optimal performance.

Contribution

It introduces a novel incremental discretized rotation algorithm that avoids trigonometric computations and leverages step function analysis for speed and accuracy.

Findings

Algorithm computes intermediate rotations efficiently.

Uses only integer arithmetic, no sine or cosine calculations.

Provides a complete description of the incremental rotation process.

Abstract

A discrete rotation algorithm can be apprehended as a parametric application $f\_\alpha$ from $\ZZ[i]$ to $\ZZ[i]$, whose resulting permutation ``looks like'' the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotate d copies of an image for angles in-between 0 and a destination angle. The di scretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm whic h computes incrementally a discretized rotation. The suggested method uses o nly integer arithmetic and does not compute any sine nor any cosine. More pr ecisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be th e key ingredient that will make the resulting procedure optimally fast and e xact. A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of defin itions for discrete geometry.