# Epipolar geometry between photogrammetry and computer vision: a computational guide

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## Abstract

Stereo image orientation is one of the major topics in computer vision, photogrammetry, and robotics. The stereo vision problem solution represents the basic element of the multi-view Structure from Motion SfM in computer vision and photogrammetry.
A successfully reconstructed stereo image geometry is based on solving the epipolar constraint using the fundamental matrix which is based on the projective geometry in computer vision. However, in photogrammetry, the problem is well known as relative orientation and there is a different solution that is based on the euclidean geometry using collinearity or coplanarity equations.
A lot of literature and discussions were found in the last decades to solve the epipolar geometry problem. However, there is still no clear description to compare between solutions introduced using both projective and euclidean solutions and which method of the relative image orientation is mostly preferred.
To the best of our knowledge, computing and plotting the epipolar lines using photogrammetric collinearity and coplanarity equations is not shown before in the educational litrature. In this paper, a detailed mathematical solution of the epipolar geometry will be shown using both photogrammetric and computer vision techniques. This is aimed to remove any confusion for new learners in using the current methods in both scientific fields and show that using any technique should lead to comparable results with advantages and disadvantages.
Original language English XXIV ISPRS Congress 2022 A.S. Kumar, P.L.N. Ranju, S.R. Reyes International Society for Photogrammetry and Remote Sensing (ISPRS) 25-32 8 5 https://doi.org/10.5194/isprs-annals-V-5-2022-25-2022 Published - 18 May 2022 XXIV ISPRS Congress 2022: Congress “Imaging today, foreseeing tomorrow” - Nice, FranceDuration: 6 Jun 2022 → 11 Jun 2022https://www.int-arch-photogramm-remote-sens-spatial-inf-sci.net/XLIII-B3-2022/index.html

### Publication series

Name ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences 5

### Conference

Conference XXIV ISPRS Congress 2022 ISPRS 2022 France Nice 6/06/22 → 11/06/22 https://www.int-arch-photogramm-remote-sens-spatial-inf-sci.net/XLIII-B3-2022/index.html

## Keywords

• Coplanarity
• Photogrammetry
• Relative orientation
• Essential matrix
• Fundamental matrix
• Collinearity
• ITC-GOLD

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