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A homography is a 3×33\times 3 projective transformation that maps points on one plane to points on another plane. Homographies arise naturally whenever a camera views a planar surface, making them indispensable for image rectification, panoramic mosaics, and augmented reality.

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Definition

Given two planes π1\pi_1 and π2\pi_2, a homography HH maps a point m1\mathbf{m}_1 on π1\pi_1 to its corresponding point m2\mathbf{m}_2 on π2\pi_2: m2∼H m1,H∈R3×3,det⁡(H)≠0\mathbf{m}_2 \sim H\,\mathbf{m}_1, \qquad H \in \mathbb{R}^{3\times 3}, \quad \det(H) \neq 0 HH is defined up to scale (8 degrees of freedom). It encodes the full projective relationship between the two planes: straight lines map to straight lines, but parallel lines and distances in general do not.

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Physical interpretations

  • Two images of the same planar surface are related by a homography.
  • Two images taken by a camera undergoing pure rotation (no translation) are related by a homography.
  • A fronto-parallel rectification maps a tilted planar surface to a canonical view.

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Estimating a homography from correspondences

Each point correspondence m1(i)↔m2(i)\mathbf{m}_1^{(i)} \leftrightarrow \mathbf{m}_2^{(i)} gives 2 linear equations in the 9 entries of HH (after eliminating the scale). With 4 or more correspondences the system is: A h=0,h=vec(H)A\,\mathbf{h} = \mathbf{0}, \qquad \mathbf{h} = \text{vec}(H) The Direct Linear Transform (DLT) solves for h\mathbf{h} as the right singular vector of AA corresponding to its smallest singular value (SVD).
Normalise input coordinates before running DLT: translate centroid to origin and scale so the mean distance to the origin is 2\sqrt{2}. Denormalise HH afterwards.

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RANSAC for robust estimation

In practice, SIFT/ORB feature matching produces many outlier correspondences (mismatches). Applying DLT directly to all matches gives a poor homography. RANSAC solves this:
1

Sample 4 correspondences

Randomly draw 4 point pairs — the minimum needed to determine HH uniquely.
2

Estimate H with DLT

Compute the candidate homography from the 4 sampled pairs.
3

Count inliers

For each remaining correspondence compute the symmetric transfer error: d(m2, Hm1)2+d(m1, H−1m2)2<ϵ2d(\mathbf{m}_2,\, H\mathbf{m}_1)^2 + d(\mathbf{m}_1,\, H^{-1}\mathbf{m}_2)^2 < \epsilon^2 Count how many pairs satisfy this threshold.
4

Keep best and refit

After NN iterations, refit DLT using all inliers of the best hypothesis.

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Applications

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Image rectification

Given 4 known world points on a planar surface and their image projections, the homography HH warps the image so the plane appears fronto-parallel — useful for reading text on a tilted book, correcting perspective distortion in a photo of a sign, or “unrolling” a clock face.

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Panoramic mosaics

When a camera rotates about its optical centre, consecutive frames are related by a homography. SIFT keypoints provide dense correspondences across frames; RANSAC filters mismatches; and the estimated homographies are composed to stitch the frames into a seamless panorama.

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MATLAB code example

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Python resources

Homographies example (Colab)

Interactive notebook: compute and apply homographies using DLT and RANSAC in Python.

E02 — Clock face rectification

Practical exercise: use a homography to rectify a tilted clock face to a frontal view.

Panoramic mosaics with SIFT (Colab)

Build panoramic mosaics by chaining homographies estimated from SIFT feature matches.

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Video lecture

Lecture: Homographies and geometric rectification (2021)

Recorded class covering homography estimation, RANSAC, and applications to rectification and mosaics.

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Frequently asked questions

A homography has 8 degrees of freedom (9 entries up to scale), so 4 point correspondences are sufficient for an exact solution (if noise-free). For a least-squares estimate over more points, use DLT with all inlier correspondences.
The symmetric transfer error measures how well HH maps points in both directions: it sums the squared distance of Hm1H\mathbf{m}_1 from m2\mathbf{m}_2 and of H−1m2H^{-1}\mathbf{m}_2 from m1\mathbf{m}_1. This is more robust than a one-sided error when both images have noise.
A homography relates two views only when the scene is planar OR the camera translates without any rotation between frames. If the scene is 3D (non-planar) and the camera translates, the correct model is the fundamental matrix (see Multiple View Geometry).
The required number of iterations is N=log⁡(1−p)/log⁡(1−(1−ϵ)s)N = \log(1-p) / \log(1-(1-\epsilon)^s), where pp is the desired success probability (e.g., 0.99), ϵ\epsilon is the outlier fraction, and s=4s=4 is the minimal sample size. For 50 % outliers and p=0.99p=0.99, about 1177 iterations are needed.