Essential Matrix: Difference between revisions
Created page with "An essential matrix, denoted <math>\mathbf{E}</math>, is a <math>3 \times 3</math> matrix relating camera parameters.<br> You can compute the essential matrix based on feature..." |
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Given feature points <math>\mathbf{x}</math> and <math>\mathbf{x'}</math> from two images, | Given feature points <math>\mathbf{x}</math> and <math>\mathbf{x'}</math> from two images, | ||
the essential matrix satisfies the equation <math>\mathbf{x}'^T \mathbf{E} \mathbf{x} = 0</math> | the essential matrix satisfies the equation <math>\mathbf{x}'^T \mathbf{E} \mathbf{x} = 0</math> | ||
==Derivation== | |||
Given feature points <math>\mathbf{x}</math> and <math>\mathbf{x'}</math> from two images, | |||
we can relate them with a rotation <math>\mathbf{R}</math> and a translation <math>\mathbf{t}</math> | |||
such that <math>(\mathbf{R}\mathbf{x} + \mathbf{t}) = x'</math>. | |||
==Properties== | ==Properties== | ||
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==Calculating the Essential Matrix from two images== | ==Calculating the Essential Matrix from two images== | ||
===Planar Images=== | |||
===Spherical Images=== | |||
Here we assume an equirectangular projection. | |||
==Determining rotation <math>R</math> and translation <math>t</math>== | ==Determining rotation <math>\mathbf{R}</math> and translation <math>\mathbf{t}</math>== | ||
==3D points== | ==3D points== | ||
Revision as of 20:57, 29 April 2020
An essential matrix, denoted \(\displaystyle \mathbf{E}\), is a \(\displaystyle 3 \times 3\) matrix relating camera parameters.
You can compute the essential matrix based on features matches between two images.
Using the essential matrix, you can extract the relative rotation and translation between two cameras.
Given feature points \(\displaystyle \mathbf{x}\) and \(\displaystyle \mathbf{x'}\) from two images, the essential matrix satisfies the equation \(\displaystyle \mathbf{x}'^T \mathbf{E} \mathbf{x} = 0\)
Derivation
Given feature points \(\displaystyle \mathbf{x}\) and \(\displaystyle \mathbf{x'}\) from two images, we can relate them with a rotation \(\displaystyle \mathbf{R}\) and a translation \(\displaystyle \mathbf{t}\) such that \(\displaystyle (\mathbf{R}\mathbf{x} + \mathbf{t}) = x'\).
Properties
- A \(\displaystyle 3 \times 3\) matrix is an essential matrix iff two of its singular values are equal and the third value is \(\displaystyle 0\)
See Bartoli and Olsen[1].
Calculating the Essential Matrix from two images
Planar Images
Spherical Images
Here we assume an equirectangular projection.
Determining rotation \(\displaystyle \mathbf{R}\) and translation \(\displaystyle \mathbf{t}\)
3D points
See Wikipedia: Essential_matrix