Decoupled Geometric Parameterization and its Application in Deep Homography Estimation

TL;DR

This paper introduces a new geometric parameterization for homographies that simplifies estimation by avoiding linear system solving, improving interpretability and maintaining competitive performance in deep homography tasks.

Contribution

It proposes a decoupled geometric parameterization using SKS decomposition, enabling direct homography estimation with clear geometric meaning.

Findings

Achieves performance comparable to traditional corner-based methods

Eliminates the need for solving linear systems in homography estimation

Provides a linear relation between parameters and angular offsets

Abstract

Planar homography, with eight degrees of freedom (DOFs), is fundamental in numerous computer vision tasks. While the positional offsets of four corners are widely adopted (especially in neural network predictions), this parameterization lacks geometric interpretability and typically requires solving a linear system to compute the homography matrix. This paper presents a novel geometric parameterization of homographies, leveraging the similarity-kernel-similarity (SKS) decomposition for projective transformations. Two independent sets of four geometric parameters are decoupled: one for a similarity transformation and the other for the kernel transformation. Additionally, the geometric interpretation linearly relating the four kernel transformation parameters to angular offsets is derived. Our proposed parameterization allows for direct homography estimation through matrix multiplication, eliminating the need for solving a linear system, and achieves performance comparable to the four-corner positional offsets in deep homography estimation.