Rectification of Stereo Images Милош Миловановић Урош Поповић.

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Presentation transcript:

Rectification of Stereo Images Милош Миловановић Урош Поповић

Camera Model c – optical center R – retinal plane F – focal plane f – focal length

Stereo Images

Epipolar Geometry m 1, m 2 – conjugated points m 1 c 1 c 2 – epipolar plane m 2 e 2 – eipiolar line conjugated to m 1 e 1 =c 1 c 2  R 1 – first epipole e 2 =c 1 c 2  R 2 – second epipole Rectification: e 1 , e 2 

Fundamental Matrix Am 1 m 1 =(x 1,y 1,1) m 2 =(x 2 y 2,1) m 2  line(Am 1,e 2 ) m 2 ∙(Am 1 X e 2 )=0 m 2 t F m 1 =0 m 1 t F t m 2 =0 Rank F=2 m 2 t Fe 1 =0 for all m 2 Fe 1 =0, F t e 2 =0

Estimating of Fundamental Matrix 8 points to estimate F up to a constant factor Singular value decomposition

Rectification x=0 – invariant points

Better solution H=GRT T – translation central point to the origin R – rotation epipole to be on X-axis G – projective transform

R. I. Hartley, Theory and practice of projective rectification, International Journal of Computer Vision 35 (2) (1999) 115–127. C. Loop, Z. Zhang, Computing rectifying homographies for stereo vision,in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Vol. 1, 1999, pp. 125–131. J. Mallon, P. F. Whelan: Projective rectification from the fundamental matrix. Image Vision Comput. 23(7): (2005)