1 Camera Internals for fisheye lens
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1 Camera Internals for fisheye lens • X = (X, Y, Z) 3D point in world coordinate system • x = (x, y) 2D projection into the image [pixel]
2 θ = arctan π
√
X2 + Y 2 Z
!
ρ = p0 + p1 θ + p2 θ2 + p3 θ3 with p1 = 1 (polynomial) ! √ ρX xh 2 +Y 2 X = √ ρX yh X 2 +Y 2 x c d xh c = + x y e f yh cy
(1) (2) (3) (4)
The paramaters c, d, e, f, cx , cy define the affine deformation of the circular image in pixel coordinates. For example, when using an 8mm Sigma lens (figure 1) on a Canon 6D camera with an image size of 5472 × 3648, these paramaters can be initialized as follows: 3648 • (cx , cy ) = ( 5472 2 , 2 ) pixel is the center of the projected image circle, • c = f = 1780 pixel is the radius of the image circle.
The diagonal elements of the affine matrix can be related to the focal length: F =
2c π
(5)
The off-diagonal elements are connected to the distortion of the projected image circle, which, in the most general case, can be a rotated ellipse.
1
2 Figure 1: The distortion of an 8mm Sigma lens on a Canon 6D.
1 Camera Internals for fisheye lens • X = (X, Y, Z) 3D point in world coordinate system • x = (x, y) 2D projection into the image [pixel]
2 θ = arctan π
√
X2 + Y 2 Z
!
ρ = p0 + p1 θ + p2 θ2 + p3 θ3 with p1 = 1 (polynomial) ! √ ρX xh 2 +Y 2 X = √ ρX yh X 2 +Y 2 x c d xh c = + x y e f yh cy
(1) (2) (3) (4)
The paramaters c, d, e, f, cx , cy define the affine deformation of the circular image in pixel coordinates. For example, when using an 8mm Sigma lens (figure 1) on a Canon 6D camera with an image size of 5472 × 3648, these paramaters can be initialized as follows: 3648 • (cx , cy ) = ( 5472 2 , 2 ) pixel is the center of the projected image circle, • c = f = 1780 pixel is the radius of the image circle.
The diagonal elements of the affine matrix can be related to the focal length: F =
2c π
(5)
The off-diagonal elements are connected to the distortion of the projected image circle, which, in the most general case, can be a rotated ellipse.
1
2 Figure 1: The distortion of an 8mm Sigma lens on a Canon 6D.
