Decoupled Geometric Parameterization and its Application in Deep Homography Estimation

2025-05-22

AI Summary

Planar homographies possess eight degrees of freedom, yet conventional four-corner offset parameterizations lack geometric interpretability and require solving an 8×9 linear system to recover the homography matrix. To address this, we propose a decoupled geometric parameterization based on the Similarity–Kernel–Similarity (SKS) decomposition, explicitly factoring the homography into two orthogonal four-dimensional parameter groups: similarity transformations and kernel transformations. Crucially, we establish, for the first time, an analytical linear mapping between kernel parameters and angular offsets, enabling direct, closed-form generation of the homography matrix without linear system solving. Evaluated on deep homography estimation tasks, our method achieves accuracy comparable to four-corner regression while significantly enhancing parameter interpretability and inference efficiency. This work introduces a novel paradigm for homography modeling that unifies geometric meaning with computational advantages.

Technology Category

• Machine Learning: Kernel Methods
• Computer Vision: Other Foundations of Computer Vision
• Planning, Routing, and Scheduling: Model-Based Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Lack of geometric interpretability in homography parameterization
• Need for solving linear systems to compute homography matrix
• Improving direct homography estimation via decoupled geometric parameters

Innovation

• Decouples homography into SKS geometric parameters
• Links kernel parameters to angular offsets linearly
• Estimates homography directly via matrix multiplication