Numerically exact distortion estimation
and numerically exact undistortion
validated against known ground-truth distortion fields —
available in MATLAB and Python.
The distortion engine provides a controlled ground-truth test field:
a known analytical radial distortion model with known coefficients and known
radial behavior. The estimator is then evaluated against that ground truth.
It does not need to recover the same symbolic structure; it can use only the
required degrees and terms to reproduce the radial mapping across the covered field.
Ground truth: 6-term polynomial with mixed odd and even degrees [2, 3, 5, 6, 9, 11].
Estimated model: automatically selected degrees, converged when RMSE dropped below threshold.
Five checkerboard boards at different field positions.
The coefficient table shows near-machine-epsilon recovery within the covered radial range.
Automatic Degree Selection
Starting from r², the estimator adds polynomial terms one by one and
re-fits after each addition. It stops when RMSE drops below a user-defined
threshold — no manual model specification required.
Mixed Odd & Even Terms
Odd, even, sparse, and mixed-degree terms are all supported.
The model is not restricted to Brown–Conrady-style r³, r⁵, r⁷ terms.
Deterministic Estimation
Each fit is a direct linear least-squares solve —
no iterative optimizer, no convergence loop, no initialization sensitivity.
Only the model structure grows; the fitting step stays deterministic.
Numerically Exact Undistortion
After the distortion function is estimated, the same analytical model is
inverted through a deterministic LUT-based inverse. For valid monotonic
mappings, the recovered image matches the original up to interpolation-grid
and resampling effects. For invalid non-monotonic mappings, folding and
irreversible information loss are detected before inversion is trusted.
Four design choices distinguish the engine from standard calibration and
undistortion workflows.
01 — Free Polynomial Model
Arbitrary Degree & Term Count
Polynomial degree and term count are fully user-defined.
Odd, even, sparse, and mixed-degree terms are all supported.
Standard tools commonly use fixed distortion families
and limited radial coefficient structures.
02 — Deterministic Calibration
Linear Least-Squares, Not Iterative Optimisation
Estimation is a direct linear solve for the distortion constants.
No Levenberg–Marquardt loop, no gradient descent, no local minima.
Standard tools typically minimize reprojection error
through iterative optimisation.
03 — Deterministic Inverse
LUT-Based Undistortion
The inverse mapping uses a pre-computed lookup table with interpolation.
There is no Newton–Raphson convergence loop at each pixel.
Standard inverse workflows often rely on iterative
coordinate inversion.
04 — Physical Validity Control
Hard Loss / Soft Loss Analysis
Every distortion model is evaluated for monotonicity before inversion.
Folding, low-slope regions, and irreversible information loss are
quantified explicitly.
Standard tools usually do not report whether the radial
mapping is physically invertible over the effective image domain.
Estimate → Validate → Undistort
The workflow is intentionally explicit. The distortion function is not hidden
inside a black-box calibration result; it remains visible, inspectable, and
testable against known ground truth throughout the pipeline.
Checkerboard Images
Corner Detection
Polynomial Recovery
Monotonicity Validation
LUT-Based Inverse
Safe Undistortion
Automatic Model Selection
Starting from r², polynomial terms are added one by one and re-fitted
after each addition. The process stops when RMSE drops below a user-defined
threshold — the model grows only as complex as the data requires.
Multi-Board Pooling
Multiple checkerboards at different field positions can be combined
to improve radial coverage and estimation stability.
Monotonicity Validation
Before inversion, the estimated model is evaluated for folding and
information loss. Hard loss and soft loss ratios are reported explicitly.
Unified Distortion Engine
The same analytical forward-first framework is available as a MATLAB toolbox
and as a standalone Python package. The forward engine is used both as a
practical modeling tool and as a ground-truth generator for validation.
