Homography (computer vision)

We have two cameras a and b, looking at points ${\displaystyle P_{i}}$ in a plane. Passing from the projection ${\displaystyle {}^{b}p_{i}=\left({}^{b}u_{i};{}^{b}v_{i};1\right)}$ of ${\displaystyle P_{i}}$ in b to the projection ${\displaystyle {}^{a}p_{i}=\left({}^{a}u_{i};{}^{a}v_{i};1\right)}$ of ${\displaystyle P_{i}}$ in a:

${\displaystyle {}^{a}p_{i}={\frac {{}^{b}z_{i}}{{}^{a}z_{i}}}K_{a}\cdot H_{ab}\cdot K_{b}^{-1}\cdot {}^{b}p_{i}}$

where ${\displaystyle {}^{a}z_{i}}$ and ${\displaystyle {}^{b}z_{i}}$ are the z coordinates of P in each camera frame and where the homography matrix ${\displaystyle H_{ab}}$ is given by

${\displaystyle H_{ab}=R-{\frac {tn^{T}}{d}}}$.

${\displaystyle R}$ is the rotation matrix by which b is rotated in relation to a; t is the translation vector from a to b; n and d are the normal vector of the plane and the distance to the plane respectively. K_a and K_b are the cameras' intrinsic parameter matrices.

The figure shows camera b looking at the plane at distance d. Note: From above figure, assuming ${\displaystyle n^{T}P_{i}+d=0}$ as plane model, ${\displaystyle n^{T}P_{i}}$ is the projection of vector ${\displaystyle P_{i}}$ along ${\displaystyle n}$, and equal to ${\displaystyle -d}$. So ${\displaystyle t=t\cdot 1=t\left(-{\frac {n^{T}P_{i}}{d}}\right)}$. And we have ${\displaystyle H_{ab}P_{i}=RP_{i}+t}$ where ${\displaystyle H_{ab}=R-{\frac {tn^{T}}{d}}}$.

This formula is only valid if camera b has no rotation and no translation. In the general case where ${\displaystyle R_{a},R_{b}}$ and ${\displaystyle t_{a},t_{b}}$ are the respective rotations and translations of camera a and b, ${\displaystyle R=R_{a}R_{b}^{T}}$ and the homography matrix ${\displaystyle H_{ab}}$ becomes

${\displaystyle H_{ab}=R_{a}R_{b}^{T}-{\frac {(-R_{a}*R_{b}^{T}*t_{b}+t_{a})n^{T}}{d}}}$

where d is the distance of the camera b to the plane.

The homography matrix can only be computed between images taken from the same camera shot at different angles. It doesn't matter what is present in the images. The matrix contains a warped form of the images.

When the image region in which the homography is computed is small or the image has been acquired with a large focal length, an affine homography is a more appropriate model of image displacements. An affine homography is a special type of a general homography whose last row is fixed to

${\displaystyle h_{31}=h_{32}=0,\;h_{33}=1.}$

• Direct linear transformation
• Epipolar geometry
• Feature (computer vision)
• Fundamental matrix (computer vision)
• Pose (computer vision)

• O. Chum and T. Pajdla and P. Sturm (2005). "The Geometric Error for Homographies" (PDF). Computer Vision and Image Understanding. 97 (1): 86–102. doi:10.1016/j.cviu.2004.03.004.

• homest is a GPL C/C++ library for robust, non-linear (based on the Levenberg–Marquardt algorithm) homography estimation from matched point pairs (Manolis Lourakis).
• OpenCV is a complete (open and free) computer vision software library that has many routines related to homography estimation (cvFindHomography) and re-projection (cvPerspectiveTransform).

• Serge Belongie & David Kriegman (2007) Explanation of Homography Estimation from Department of Computer Science and Engineering, University of California, San Diego.
• A. Criminisi, I. Reid & A. Zisserman (1997) "A Plane Measuring Device", §3 Computing the Plane to Plane Homography, from Visual Geometry Group, Department of Engineering Science, University of Oxford.
• Elan Dubrofsky (2009) Homography Estimation, Master's thesis, from Department of Computer Science, University of British Columbia.
• Richard Hartley & Andrew Zisserman (2004) Multiple View Geometry from Visual Geometry Group, Oxford. Includes Matlab Functions for calculating a homography and the fundamental matrix (computer vision).
• GIMP Tutorial – using the Perspective Tool by Billy Kerr on YouTube. Shows how to do a perspective transform using GIMP.
• Allan Jepson (2010) Planar Homographies from Department of Computer Science, University of Toronto. Includes 2D homography from four pairs of corresponding points, mosaics in image processing, removing perspective distortion in computer vision, rendering textures in computer graphics, and computing planar shadows.
• Plane transfer homography Course notes from CSE576 at University of Washington in Seattle.
• Etienne Vincent & Robert Laganiere (2000) Detecting Planar Homographies in an Image Pair from School of Information Technology and Engineering, University of Ottawa. Describes an algorithm for detecting planes in images, uses random sample consensus (RANSAC) method, describes heuristics and iteration.