Fast and Interpretable 2D Homography Decomposition: Similarity-Kernel-Similarity and Affine-Core-Affine Transformations

TL;DR

This paper introduces two fast, interpretable decompositions for 2D homography under the minimal 4-point condition: Similarity-Kernel-Similarity (SKS) and Affine-Core-Affine (ACA). SKS and ACA decompose a homography into stratified sub-transformations driven by anchor points, enabling division-free computation up to scale and dramatically reducing FLOPs (approximately $169$ FLOPs for SKS complete and about $85$ FLOPs for simplified SKS; ACA around $85$ FLOPs with a total near $97$ FLOPs). The methods offer clear geometric interpretation, unify with SAP in special cases, and are tensorizable for deep-learning pipelines (TensorACA). Extensive experiments on CPU/GPU show substantial speedups over state-of-the-art solvers and compatibility with traditional RANSAC and deep-homography frameworks. The results suggest strong practical impact for real-time geometric vision tasks, including image stitching, pose estimation, and robust homography estimation in the presence of outliers.

Abstract

In this paper, we present two fast and interpretable decomposition methods for 2D homography, which are named Similarity-Kernel-Similarity (SKS) and Affine-Core-Affine (ACA) transformations respectively. Under the minimal $4$-point configuration, the first and the last similarity transformations in SKS are computed by two anchor points on target and source planes, respectively. Then, the other two point correspondences can be exploited to compute the middle kernel transformation with only four parameters. Furthermore, ACA uses three anchor points to compute the first and the last affine transformations, followed by computation of the middle core transformation utilizing the other one point correspondence. ACA can compute a homography up to a scale with only $85$ floating-point operations (FLOPs), without even any division operations. Therefore, as a plug-in module, ACA facilitates the traditional feature-based Random Sample Consensus (RANSAC) pipeline, as well as deep homography pipelines estimating $4$-point offsets. In addition to the advantages of geometric parameterization and computational efficiency, SKS and ACA can express each element of homography by a polynomial of input coordinates ($7$th degree to $9$th degree), extend the existing essential Similarity-Affine-Projective (SAP) decomposition and calculate 2D affine transformations in a unified way. Source codes are released in https://github.com/cscvlab/SKS-Homography.

Figures (8)

• Figure 1: Sketch of homography computation methods under the minimal condition. According to various primitive configurations and underlying mathematical principles, solving methods may differ considerably. The proposed SKS and ACA can deal with three categories of primitive configurations in a unified way, while is suitable to be tensorized in deep homography pipeline.
• Figure 2: Comparison of solving process between previous $4$-point homography methods and ours. (a) The previous $4$-point homography methods (NDLT-SVD Hartley2003Multiple, HO-SVD HO_BMVC05, GPU-LU OpenCV_GPT, RHO-GE Bazargani2015Fast) indiscriminately utilize all four point correspondences (drawn with red dots) to construct a linear system (Eq. \ref{['equ:intro_3']}) and solves it using well-established matrix factorization methods. (b) The proposed SKS decomposition includes three stratified sub-transformations, which are the first similarity transformation $\mathbf{H}_{S_1}$, the middle kernel transformation $\mathbf{H}_{K}$ and the last similarity transformation $\mathbf{H}_{S_2}$. Two corresponding points on source plane (drawn with yellow) and target plane (drawn with green) are used to compute $\mathbf{H}_{S_1}$ and $\mathbf{H}_{S_2}$ respectively. The residual two point correspondences (drawn with blue) on the two intermediate planes are used to compute $\mathbf{H}_{K}$ with $4$ DOF. (c) The proposed ACA decomposition is similar to SKS, except three point correspondence is used to compute the affine transformation $\mathbf{H}_{A_1}$ and $\mathbf{H}_{A_2}$, while one point correspondence is used to compute the core transformation $\mathbf{H}_{C}$ with $2$ DOF.
• Figure 3: Sub-transformations in SKS decomposition.
• Figure 4: Sub-transformations in ACA decomposition.
• Figure 5: Affine transformation mapping a rectangle (with four parameters) in source image to the canonical square.