Inteval Analysis for two spherical functions arising from robust Perspective-n-Lines problem

TL;DR

The paper studies interval analysis for two spherical functions arising from a robust Perspective-n-Lines problem, aiming to compute global solutions via a dimension-reduction strategy. It develops an Accelerating Consensus Maximization framework that reduces consensus maximization to interval stabbing and provides provable lower and upper bounds to prune the search. Focusing on rotation search in $SO(3)$, it parameterizes rotations by a distinguished angle and axis, derives residual decompositions $f_1$ and $f_2$ with boundable components $h_1$ and $h_2$, and proves extreme-point theorems guiding efficient boundary searches. The results are supported by rigorous proofs and extensive numerical experiments, enabling robust, globally optimal solutions for PnL under noise and outliers, with potential impact on 3D reconstruction and camera-motion estimation tasks.

Abstract

This report presents a comprehensive interval analysis of two spherical functions derived from the robust Perspective-n-Lines (PnL) problem. The study is motivated by the application of a dimension-reduction technique to achieve global solutions for the robust PnL problem. We establish rigorous theoretical results, supported by detailed proofs, and validate our findings through extensive numerical simulations.

Theorems & Definitions (8)

• Theorem 1: Extreme Point Theorem for $h_2(\vec{\mathbf{u}}|\mathbf{s}_k)$
• proof
• Theorem 2: Extreme Point Theorem for $h_1(\vec{\mathbf{u}}|\mathbf{s}_k)$
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1