Decoupled Geometric Parameterization and its Application in Deep Homography Estimation

TL;DR

This work addresses the lack of geometric interpretability in planar homography estimation by introducing a decoupled 8-DOF parameterization based on the similarity-kernel-similarity (SKS) decomposition. It splits the transformation into two independent 4-DOF sets, $\mathbf{H}_S$ (similarity) and $\mathbf{H}_K$ (kernel), with a linear relation between the similarity parameters and corner offsets and a geometric interpretation of the kernel through four angular offsets. The approach enables end-to-end neural estimation via matrix multiplication, eliminating the need for the DLT solver, while delivering performance comparable to traditional four-corner offset methods across multiple datasets and networks. The results demonstrate the viability of a unified, end-to-end 2D transformation estimation framework and introduce angular offsets as a new, interpretable metric for projective distortion. This parameterization also supports degeneration to affine and similarity cases and transfers well to estimating those degenerate transformations with existing networks.

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.

Figures (12)

• Figure 1: Parameterization in hierarchical 2D geometric transformations. Upper: shape distortion. Lower: matrix representation and parameterization. Our geometric parameterization is decoupled into two sets of four parameters: one for a similarity transformation $\mathbf{H}_S$ and the other for the kernel transformation $\mathbf{H}_K$.
• Figure 2: Our similarity estimation process utilizing translation normalization. Among the three sub-transformations, the translation transformation $\mathbf{H}_{T}$, which normalizes the square's center, is already known. The similarity transformation $\mathbf{H}_{S}$ serves to map $\{M_2, N_2\}$ to $\{M_1, N_1\}$, as well as to transform the normalized image $\mathcal{I}_2$ to $\mathcal{I}_1$.
• Figure 3: Improved SKS decomposition. Among the four sub-transformations, the similarity transformation $\mathbf{H}_{S_2}$ that normalizes $\{M, N\}$ to points $[\mp1,0]^\top$ is known. The unknown similarity transformation $\mathbf{H}_{S_1}$ plays the role of mapping $\{M, N\}$ to $\{\tilde{M}, \tilde{N}\}$. Additionally, the unknown kernel transformation $\mathbf{H}_{K}$ introduces the 4-DOF geometric distortion between two normalized images $\mathcal{I}_2$ and $\mathcal{I}_3$.
• Figure 4: Our geometric interpretation of kernel transformation. The proposed four angular offsets explicitly define the 4-DOF geometric distortion induced by the kernel transformation.
• Figure 5: Decoupled projective distortion and geometric parameters of homography. Under homography, a quadrangle (here is a square) undergoes an 8-DOF projective distortion, which we have decoupled into four positional and four angular offsets. The two sets of four geometric parameters we propose are demonstrated to be linearly transformable into their respective sets of offsets.