Radially Distorted Homographies, Revisited

TL;DR

This work addresses estimating a planar homography in the presence of radial lens distortion by unifying three configurations: one-sided ($λ'=0$), two-sided equal ($λ'=λ$), and two-sided independent ($λ,λ'$). It replaces the DLT with a closed-form homography expression to derive fast, numerically stable minimal solvers that jointly recover $\mathbf{H}$ and distortion parameters. The authors derive and compare solvers for all three cases, demonstrating speed-ups and comparable accuracy to prior methods, and validate them on fisheye benchmarks, HPatches, and the Grossmünster Church dataset, with an implementation released in HomLib. The results support the practicality of the unified approach for robust planar geometry estimation under distortion, enabling more reliable camera calibration and image stitching in distorted imagery.

Abstract

Homographies are among the most prevalent transformations occurring in geometric computer vision and projective geometry, and homography estimation is consequently a crucial step in a wide assortment of computer vision tasks. When working with real images, which are often afflicted with geometric distortions caused by the camera lens, it may be necessary to determine both the homography and the lens distortion-particularly the radial component, called radial distortion-simultaneously to obtain anything resembling useful estimates. When considering a homography with radial distortion between two images, there are three conceptually distinct configurations for the radial distortion; (i) distortion in only one image, (ii) identical distortion in the two images, and (iii) independent distortion in the two images. While these cases have been addressed separately in the past, the present paper provides a novel and unified approach to solve all three cases. We demonstrate how the proposed approach can be used to construct new fast, stable, and accurate minimal solvers for radially distorted homographies. In all three cases, our proposed solvers are faster than the existing state-of-the-art solvers while maintaining similar accuracy. The solvers are tested on well-established benchmarks including images taken with fisheye cameras. A reference implementation of the proposed solvers is made available as part of HomLib (https://github.com/marcusvaltonen/HomLib).

Figures (8)

• Figure 1: We propose three novel homography solvers simultaneously estimating a homography between two views and radial distortion coefficients. It is well-known that a homography $\mathbf{H}$ maps image point correspondences which lie on a planar surface (cf. the green plane in the scene); however, physical cameras may deviate significantly from the pinhole camera model due to e.g. radial distortion. In this figure of the Grossmünster church the input images are significantly distorted due to the camera being equipped with a fisheye lens. The reference 3D reconstruction was obtained using RadialSfM eccv/2020/larsson_etal.
• Figure 2: To each of the rectilinear image planes, there is a homography ($\mathbf{G}$ and $\mathbf{G}'$, respectively) from the scene plane (which itself is inherently rectilinear). There is also a homography between the two rectified image planes. For the distortion, three distinct cases are therefore of interest here: the one-sided case $\lambda'=0$,the two-sided equal case $\lambda'=\lambda$, andthe two-sided independent case (where $\lambda$ and $\lambda'$ are independent).The one-sided case is equivalent to just viewing the image plane with one camera.
• Figure 3: Numerical stability. Homography errors (left) and distortion coefficient error (right) for 10000.0 noise-free randomly generated problem instances.
• Figure 4: Sensitivity to noise. Homography errors (top) and distortion coefficient errors (bottom) for different noise levels. Each point reports the median error of 10000.0 randomly generated problem instances (per noise level). The left column shows the noise sensitivity for the two-sided independent solvers, the middle column shows the noise sensitivity for the two-sided equal solvers, and the right column shows the noise sensitivity for the one-sided solvers.
• Figure 5: RANSAC integration. Here we plot the cumulative number of inliers versus time for all the evaluated solver, for 20 outliers (top row), 40 outliers (mid row), and 60 outliers (bottom row). The left column shows the results for the one-sided solvers, the middle column shows the results for the two-sided equal solvers, and the right column shows the results for the two-sided independent solvers.