Riemannian Geometry Approach for Minimizing Distortion and its Applications

TL;DR

This paper introduces a Riemannian geometry framework for affine transformations to minimize Fisher distortion, providing algorithms for computing mean transformations, with applications in rendering affine panoramas.

Contribution

It establishes a Riemannian metric structure on Fisher distortion and proposes an algorithm to find mean transformations minimizing overall distortion.

Findings

The Fisher distortion has a Riemannian metric structure.

An algorithm for computing mean distorting transformations is provided.

Application demonstrated in rendering affine panoramas.

Abstract

Given an affine transformation $T$, we define its Fisher distortion $Dist_F(T)$. We show that the Fisher distortion has Riemannian metric structure and provide an algorithm for finding mean distorting transformation -- namely -- for a given set $\{T_{i}\}_{i=1}^N$ of affine transformations, find an affine transformation $T$ that minimize the overall distortion $\sum_{i=1}^NDist_F^{2}(T^{-1}T_{i}).$ The mean distorting transformation can be useful in some fields -- in particular, we apply it for rendering affine panoramas.