Analytical Modeling and Correction of Distance Error in Homography-Based Ground-Plane Mapping
Authors: Mateusz Szulc, Marcin Iwanowski
Organizations: Institute of Control and Industrial Electronics, Faculty of Electrical Engineering, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
Accurate distance estimation from monocular cameras is essential for intelligent monitoring systems. In many deployments, image coordinates are mapped to ground positions using planar homographies initialized by manual selection of corresponding regions. Small inaccuracies in this initialization propagate into systematic distance distortions. This paper derives an explicit relationship between homography perturbations and the resulting distance error, showing that the error grows approximately quadratically with the true distance from the camera. Based on this model, two simple correction strategies are evaluated: regression-based estimation of the quadratic error function and direct optimization of the homography via coordinate-based gradient descent. A large-scale simulation study with more than 19 million test samples demonstrates that regression achieves higher peak accuracy when the model is reliably fitted, whereas gradient descent provides greater robustness against poor initial calibration. This suggests that improving geometric calibration may yield greater performance gains than increasing model complexity in many practical systems.
Figures & tables
Figure 1: Minimal geometry of homography-based depth mapping. A small perturbation of the effective horizon position changes the denominator (v−vh) in the projective depth model, which leads to increasing distance error at larger ranges.
Figure 2: Distribution of object distances from the camera in simulated pedestrian traffic, defining the operating range used for evaluation.
Figure 3: Example of distance-dependent calibration error, illustrating inconsistencies that may lead to overfitting on calibration points.
Metric
Regression
Grad. Descent
Positive path improvement
93.0%
98.8%
Positive calib. improvement
96.5%
100.0%
Mean path improvement
73.9%
76.5%
Median path improvement
90.0%
86.1%
Path–calib. correlation
0.84
0.56
Failure risk
higher
lower
Table 1: Summary comparison of correction methods.
Figure 4: Mean path improvement as a function of total distance sum and spread for the regression-based (a) and gradient-descent (b) methods.
While homographies are fundamental to many computer vision tasks, the majority of conventional estimation techniques provide only point estimates without directly quantifying uncertainty introduced by noisy observations. Uncertainty, though, propagates to subsequent processing steps such as camera calibration and 3D reconstruction and is particularly relevant in safety-critical and socially relevant fields including medical imaging, autonomous driving, and defense. We present a fast Bayesian formulation for homography estimation from point correspondences that explicitly incorporates measurement uncertainty and prior knowledge while providing a posterior distribution over the homography parameters. A closed-form solution of the posterior mean of the homography is derived in homogeneous coordinates and supplemented by an iterative Bayesian approach to handle non-linearities. Synthetic experiments demonstrate the applicability to projective transformations and show improved estimation accuracy over DLT under varying noise conditions. Image stitching experiments further demonstrate applicability to real image correspondences while additionally providing uncertainty information.
We present homographic navigation, a geometry-centric framework for guiding camera acquisition toward precise capture of planar regions. Rather than treating homography as an output, we use it as an organizing variable that unifies learning, alignment, and evaluation. From a single annotated reference image, we generate unlimited synthetic training data via homographic augmentation and train a single-shot model for joint recognition and localization of multiple artifacts (physical objects with a rectangular planar target) through sparse keypoint prediction. To address precision under limited model input resolution, we introduce a two-pass inference scheme with global detection followed by localized refinement, and a Stable Warp training strategy that significantly improves accuracy, particularly in the high-precision regime. The model also predicts confidence estimates per predicted keypoint and per the whole sample. Experimental results demonstrate that accurate planar alignment can be achieved from minimal supervision, providing a foundation for geometry-driven camera guidance and future learning from in-the-wild video data.
Dominik Kroupa, Marek Vaško, Muh Yuzril Ihza Baharuddin +1
Homography estimation, as one of the fundamental problems in computer vision, remains challenged by scale variation scenarios where image pairs potentially exhibit significant scale discrepancies. Existing deep learning frameworks frequently suffer from a significant performance degradation in such cases, as they rely on limited displacement assumptions and local feature consistency that might not hold under large scale gaps. In this paper, we propose SA-Homo, a novel scale-adaptive homography estimation framework designed to achieve robust alignment across a wide range of scale discrepancy ratios. We adopt a hierarchical scale alignment strategy that transitions from the global perspective with a heavy module to a local perspective with a light module. Specifically, we introduce the Scale-aware Discrepancy Bridging Module (SDBM) for initial alignment, which utilizes a Multi-scale Linear Attention Cascade (MLAC) to capture long-range dependencies and mitigate feature inconsistencies, along with a global Cross-scale Similarity Matrix Block (CSMB) for scale robust correlation representation. Once the initial scale gap is bridged, a lightweight Iterative Homography Estimation Refinement Module (IHERM) progressively polishes the result using local correlations. To facilitate this research, we contribute the HMSA dataset, a high-resolution, multi-modal satellite benchmark specifically tailored for scale-variant challenges. Extensive experiments demonstrate that SA-Homo maintains high precision even under 8× scale discrepancies, outperforming state-of-the-art methods in both conventional scale-similar scenarios and challenging scale variation scenarios. Code and collected datasets are available at https://github.com/shangxuanx330/SA_Homo
Shangxuan Xie, Haifeng Wu, Yuhang Wang +2
University of Electronic Science and Technology of China (UESTC), Chengdu, China · University of Chinese Academy of Sciences, Beijing, China · Beijing Institute of Technology, Beijing, China