Definition

DLT normalisation is a similarity preconditioning of point sets before solving any Direct Linear Transform (DLT) problem — fundamental matrix, homography, projective camera, Moving-DLT cell. The recipe (Hartley 1997) is two lines:

1. Translate so the centroid of the points is at the origin.
2. Isotropically scale so the average distance from the origin is $\sqrt{2}$.

Apply the same recipe independently to each image's point set. The transform is encoded as a $3 \times 3$ similarity matrix $T$:

where $(\bar u, \bar v)$ is the centroid and $\bar d$ the mean distance to the centroid. After normalisation, a typical point is of the form $(1, 1, 1)^T$ — every entry of the DLT design matrix is $O(1)$, and the condition number of $A^T A$ drops by approximately $10^8$ (Graph 1 of Hartley 1997).

The normalisation is exact and invertible: solve in normalised coordinates, then recover the original-coordinate solution by denormalisation. For a fundamental matrix $F = T'^T \hat F\,T$; for a homography $H = T'^{-1} \hat H\,T$.

Mathematical Description

Why the design matrix needs conditioning

For a fundamental-matrix DLT, each correspondence $(\mathbf{u}_i, \mathbf{u}'_i)$ contributes one row

For typical $\sim 1000$-pixel image coordinates, the entries of $A_i$ span $\sim 1$ to $\sim 10^6$. The condition number of $A^T A$ scales with the square of the coordinate range, so $\kappa(A^T A) \sim 10^{12}$ on un-normalised data — well beyond the $\sim 10^7$ resolution of single-precision floating-point. The smallest right singular vector of $A$ (the DLT solution $\mathbf{f}$) is dominated by floating-point error.

A homography DLT is structurally identical: each row of the $2N \times 9$ design matrix has entries up to $\sim u_i u'_i$, so $\kappa(A^T A) \sim u_{\max}^4$ on un-normalised data. The same conditioning argument and the same fix apply.

Why the rank-2 SVD truncation is the most-perturbed step

Hartley's §5 insight: the entries of $F$ multiplied by the largest point coordinates correspond to the smallest singular values of $A^T A$. These are exactly the entries most corrupted by the rank-2 SVD truncation when $A^T A$ is ill-conditioned. Normalisation makes all entries of $F$ comparable in magnitude, so the rank-2 projection perturbs all of them equally instead of catastrophically perturbing the load-bearing ones. This is the structural reason normalisation matters for the F-matrix specifically, not just a generic conditioning argument.

Isotropic vs anisotropic normalisation

Hartley's §6.2 tested an alternative non-isotropic normalisation — translate to origin, scale to unit principal moments via an affine transformation — and reported the results "were little different from those obtained using the isotropic scaling method." The isotropic version is one parameter ($s$) instead of three (an affine matrix), simpler to implement, and gives equivalently good conditioning. Use isotropic.

Reuse in Moving-DLT (APAP)

APAP (Zaragoza 2013) reuses Hartley normalisation once per image pair, before its per-cell weighted SVDs. Without this, every cell's local DLT inherits the same $\kappa \sim 10^{12}$ catastrophe on each per-pixel SVD — which would amplify per-cell across the grid. With normalisation, each per-cell SVD operates on a numerically well-scaled $A$ and the only remaining numerical concern is the floor $\gamma$ on the Gaussian weights (which prevents data-poor cells from going singular).

What normalisation is not

• Not Cholesky / QR. Those are linear-system solver preconditioners; they don't change the data. Hartley normalisation changes the data so the resulting linear system is well-conditioned regardless of solver.
• Not RANSAC. RANSAC handles outliers by sampling. Normalisation handles noise by conditioning. Use both: RANSAC sample → normalise inliers → DLT-solve → denormalise.
• Not iterative refinement. The normalised linear DLT is almost the gold-standard estimator. For calibration-grade accuracy (sub-pixel epipolar error), follow with Levenberg-Marquardt minimisation of the Sampson or gold-standard error using the normalised DLT result as the initial guess.

Numerical Concerns

Floating-point precision. The normalised DLT operates on coordinates of magnitude $\sim 1$. SVD in 64-bit double-precision is essential — single-precision can lose accuracy in the smallest singular vector even after normalisation when $N$ is large. The 32-bit / 64-bit choice matters more for the SVD step than for the data conditioning itself.

Centroid degeneracy. If all points coincide ($\bar d = 0$), the normalisation transform is undefined. This is a trivial degeneracy — no estimator works on coincident points — and the implementation should reject it explicitly rather than divide by zero.

Mean vs median distance. The recipe says "mean distance from origin." Implementations that use the median (more robust to outliers) work equally well in Hartley's experiments; the precise choice is not load-bearing. The $\sqrt{2}$ target is what matters: it ensures a typical homogeneous point is $(1, 1, 1)^T$.

Independent per-image normalisation. In multi-view problems, each image gets its own $T_k$. Do not apply the same normalisation across views — the conditioning gain comes from each design-matrix block having $O(1)$ entries, which requires per-image scaling.

Denormalisation order. For the F-matrix, denormalise as $F = T'^T \hat F\,T$. For the H-matrix, as $H = T'^{-1} \hat H\,T$. Mixing up transpose vs inverse is a silent bug — both forms are valid invertible $3 \times 3$ matrices, so the result still "looks like" a matrix but encodes wrong geometry.

Composability. Normalisation can be composed with other preconditioners (e.g., a unit conversion on millimetre-coordinate data). The composition is a $3 \times 3$ similarity matrix; apply once, denormalise once.

Where it appears

• fundamental-matrix-eight-point — the canonical use case; the algorithm is essentially "DLT + Hartley normalisation + rank-2 enforcement."
• homography — same recipe applied to the 9-vector homography DLT. The reuse is direct: same $T$, same denormalisation pattern with $H = T'^{-1} \hat H\,T$.
• apap-image-stitching — Hartley pre-conditioning is applied once before the Moving-DLT per-cell solves so every per-cell SVD operates on a numerically well-scaled design matrix.
• epipolar-geometry — the concept page that documents the F-matrix estimation pipeline; the "Numerical Concerns" section there describes Hartley normalisation as non-optional for reliable F-matrix recovery.

References

1. R. Hartley. In Defense of the Eight-Point Algorithm. IEEE TPAMI 19(6)
   –593, 1997. pdf (The original normalisation argument; §5 specifically on why F-matrix entries are differentially perturbed.)
2. J. Zaragoza, T.-J. Chin, M. S. Brown, D. Suter. As-Projective-As-Possible Image Stitching with Moving DLT. IEEE CVPR 2013. (Hartley pre-conditioning reused before per-cell SVD.)
3. R. Hartley, A. Zisserman. Multiple View Geometry in Computer Vision, 2nd ed. Cambridge University Press, 2004. Chapters 4 and 11. (Standard reference; §4.4 explicitly recommends normalisation as a default for any DLT problem.)