# Inverse Analytical Formula for the Correction of Severe Barrel Lens Distortion Modelled by a Depressed Radial Distortion Polynomial

## TL;DR

This paper introduces a new analytical method to correct severe lens distortion in images, achieving higher accuracy and stability than traditional iterative approaches.

## Contribution

A non-iterative analytical framework for inverse radial distortion modeling using constrained algebraic polynomials.

## Key findings

- The analytical method achieved sub-pixel accuracy in image rectification with reduced reprojection errors.
- The proposed approach outperformed iterative methods in numerical stability and accuracy for high-order distortion models.
- The method is suitable for high-precision imaging systems requiring robust and repeatable distortion correction.

## Abstract

Accurate correction of radial lens distortion is a fundamental requirement in computer vision and photogrammetry, as geometric inaccuracies directly affect 3D reconstruction, mapping, and geospatial measurements, particularly in high-precision imaging systems. In this study, we propose a fully analytical, non-iterative method for truncated inverse modeling of radial lens distortion, applicable to general radial distortion polynomials that contain constant terms. Unlike classical truncated Lagrange series reversion, which relies on recursive expansion and combinatorial series construction, the proposed formulation determines inverse distortion coefficients directly through a system of constrained algebraic inverse polynomials. This enables deterministic computation of inverse parameters without iterative refinement, numerical root finding, or combinatorial complexity. The method was evaluated using ultra-wide-angle smartphone camera imagery exhibiting severe barrel distortion modeled by an eighth-degree depressed radial distortion polynomial. Its performance was compared with a commonly used iterative inverse modeling approach. The analytical formulation demonstrated improved numerical stability and substantially reduced reprojection errors when correcting highly nonlinear distortion profiles, achieving sub-pixel accuracy in image rectification. In contrast, the iterative approach exhibited instability and significantly larger reprojection errors under identical conditions. These results demonstrate that the proposed framework provides a general, robust, and repeatable solution for inverse radial distortion modeling, particularly for high-order polynomial models. The method offers clear practical advantages for camera calibration pipelines in photogrammetry, remote sensing, robotics, and other applications requiring high-fidelity imaging.