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Epipolar Rectification by Singular Value Decomposition of Essential Matrix

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Abstract

Image rectification is an important stage of applying a pair of projective transformations, or homographies, to a pair of stereo images, so that epipolar lines in the original images map to horizontally aligned lines in the rectified images. Considering that for some stereo rigs the intrinsic parameters of the cameras are known but their external parameters are unknown, in this paper, we present a novel method for stereo rectification based on the essential matrix which is derived from the fundamental matrix. Without any optimization process, closed-form analytical solutions of the projective transformations for epipolar rectification can be directly obtained by conducting SVD on the essential matrix. Experimental results show the proposed rectification method not only has higher efficiency and rectification precision, but also its scale invariance and robustness are superior to existing methods.

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Notes

  1. Available from http://perso.lcpc.fr/tarel.jean-philippe/syntim/paires.html

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Acknowledgements

This research was supported by National Natural Science Foundation of China (No. 61332015, No. 61402320);by Research project of Hubei Provincial Department of Education (B2017080),China.

Author information

Authors and Affiliations

  1. School of Automation and Information Engineering, Xi’an University of Technology, Xi’an, 710048, China

    Wenhuan Wu & Hong Zhu

  2. School of Electrical and Information Engineering, Hubei University of Automotive Technology, Shiyan, 442002, China

    Wenhuan Wu

  3. Department of Information Science and Technology, Taishan University, Tai’an, 271021, China

    Qian Zhang

Authors
  1. Wenhuan Wu
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  2. Hong Zhu
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  3. Qian Zhang
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Corresponding author

Correspondence to Hong Zhu.

Appendix A

Appendix A

Proposition

For any non-zero vector t = [tx, ty, tz]T, the symmetric matrix \( \mathrm{A}={\left[\mathbf{t}\right]}_{\times }{\left[\mathbf{t}\right]}_{\times}^{\mathrm{T}} \) has two equal eigenvalues which are equal to the square of the norm of vector t, and its third eigenvalue is 0.

Proof

Due to\( \mathrm{A}\mathbf{t}=\Big({\left[\mathbf{t}\right]}_{\times}\left[\mathbf{t}\Big]{}_{\times}^{\mathrm{T}}\right)\mathbf{t}=0 \), it can be concluded that zero is an eigenvalue of the symmetric matrix A as well as its eigenvector corresponding to zero is t. Furthermore, A can be written as

$$ \kern0.5em \mathbf{A}={\left[\mathbf{t}\right]}_{\times }{\left[\mathbf{t}\right]}_{\times}^{\mathrm{T}}=-{\left[\mathbf{t}\right]}_{\times }{\left[\mathbf{t}\right]}_{\times }=-\left[\begin{array}{ccc}\hfill -\left({\mathrm{t}}_{\mathrm{z}}^2+{\mathrm{t}}_{\mathrm{y}}^2\right)\hfill & \hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{y}}\hfill & \hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{z}}\hfill \\ {}\hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{y}}\hfill & \hfill -\left({\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{z}}^2\right)\hfill & \hfill {\mathrm{t}}_{\mathrm{y}}{\mathrm{t}}_{\mathrm{z}}\hfill \\ {}\hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{z}}\hfill & \hfill {\mathrm{t}}_{\mathrm{y}}{\mathrm{t}}_{\mathrm{z}}\hfill & \hfill -\left({\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{y}}^2\right)\hfill \end{array}\right] $$

Let λ 1 = 0 , λ 2 , λ 3 are the three eigenvalues of A, it follows that

$$ {\displaystyle \begin{array}{l}\left|\uplambda \mathbf{I}-\mathbf{A}\right|=\left|\begin{array}{ccc}\hfill \uplambda -\left({\mathrm{t}}_{\mathrm{z}}^2+{\mathrm{t}}_{\mathrm{y}}^2\right)\hfill & \hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{y}}\hfill & \hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{z}}\hfill \\ {}\hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{y}}\hfill & \hfill \uplambda -\left({\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{z}}^2\right)\hfill & \hfill {\mathrm{t}}_{\mathrm{y}}{\mathrm{t}}_{\mathrm{z}}\hfill \\ {}\hfill {\mathrm{t}}_{\mathrm{x}}{\mathrm{t}}_{\mathrm{z}}\hfill & \hfill {\mathrm{t}}_{\mathrm{y}}{\mathrm{t}}_{\mathrm{z}}\hfill & \hfill \uplambda -\left({\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{y}}^2\right)\hfill \end{array}\right|\hfill \\ {}=\uplambda \left(\uplambda -{\uplambda}_2\right)\left(\uplambda -{\uplambda}_3\right)={\uplambda}^3-\left({\uplambda}_2+{\uplambda}_3\right){\uplambda}^2+{\uplambda}_2{\uplambda}_3\uplambda =0\hfill \end{array}} $$

Then we have that

$$ {\displaystyle \begin{array}{l}{\uplambda}_2+{\uplambda}_3={\mathrm{A}}_{11}+{\mathrm{A}}_{22}+{\mathrm{A}}_{33}=2\left({\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{y}}^2+{\mathrm{t}}_{\mathrm{z}}^2\right)\hfill \\ {}{\uplambda}_2{\uplambda}_3={\mathrm{t}}_{\mathrm{x}}^2{\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{y}}^2{\mathrm{t}}_{\mathrm{y}}^2+{\mathrm{t}}_{\mathrm{z}}^2{\mathrm{t}}_{\mathrm{z}}^2+2{\mathrm{t}}_{\mathrm{x}}^2{\mathrm{t}}_{\mathrm{y}}^2+2{\mathrm{t}}_{\mathrm{y}}^2{\mathrm{t}}_{\mathrm{z}}^2+2{\mathrm{t}}_{\mathrm{x}}^2{\mathrm{t}}_{\mathrm{z}}^2={\left({\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{y}}^2+{\mathrm{t}}_{\mathrm{z}}^2\right)}^2\hfill \end{array}} $$

It can be easily obtained from above two equations that

$$ {\uplambda}_2={\uplambda}_3={\mathrm{t}}_{\mathrm{x}}^2+{\mathrm{t}}_{\mathrm{y}}^2+{\mathrm{t}}_{\mathrm{z}}^2={\mathbf{t}}^2 $$

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Wu, W., Zhu, H. & Zhang, Q. Epipolar Rectification by Singular Value Decomposition of Essential Matrix. Multimed Tools Appl 77, 15747–15771 (2018). https://doi.org/10.1007/s11042-017-5149-0

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