Optical design using image distortion for orthorectification

Tim P. Johnson; Tim P. Johnson; Jose Sasian; Lacy G. Cook

doi:10.1364/AO.392302

Applied Optics
Vol. 59,
Issue 22,
pp. G175-G184
(2020)
•https://doi.org/10.1364/AO.392302

Optical design using image distortion for orthorectification

Tim P. Johnson, Jose Sasian, and Lacy G. Cook

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Tim P. Johnson, Jose Sasian, and Lacy G. Cook, "Optical design using image distortion for orthorectification," Appl. Opt. 59, G175-G184 (2020)

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Abstract

Imaging the curved Earth from above typically results in a distorted image with reduced spatial resolution near the edge of the field. The effect is proportional to the field of view (FOV) and altitude. It is similar to the negative (barrel) distortion common in fish-eye lenses, but is due here to the convex object and not the optical system. Although image processing methods exist to partially correct for negative distortion, the reduced spatial resolution near the edge of the field is unrecoverable. Instead this can be corrected for optically by inducing the right amount of positive distortion into the optical design. The amount of positive distortion required to counter the negative distortion from the curved Earth is calculated as a function of FOV and altitude. An optical system with positive distortion is more challenging to design than with negative distortion as this increases the FOV in image space and requires a larger focal plane array. An off-axis, all-reflective design with $f/{2.5}$, 2.5 in. effective focal length, ${70}^\circ \times {4}^\circ$ FOV, ${+}{13}\%$ distortion is shown that could be used as a push-broom satellite sensor in a polar orbit with constant spatial resolution.

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Wavefront Aberration Function	Name	Axial Symmetry	Causes Blurring
W_20000 ( $H \cdot H$ )	Quadratic piston	✓	✗
W_02000 ( $ρ \cdot ρ$ )	Defocus	✓	✓
W_11100 ( $H \cdot ρ$ )	Magnification	✓	✗
W_10010 ( $H \cdot i$ )	Linear piston	✗	✗
W_00001 ( $ρ \cdot i$ )	Field displacement	✗	✗
W_22000 ( $H \cdot H$ )( $ρ \cdot ρ$ )	Field curvature	✓	✓
W_31100 ( $H \cdot H$ )( $H \cdot ρ$ )	Cubic distortion	✓	✗
W_13100 ( $ρ \cdot ρ$ )( $H \cdot ρ$ )	Linear coma	✓	✓
W_30010 ( $H \cdot H$ )( $H \cdot i$ )	Cubic piston	✗	✗
W_12010 ( $ρ \cdot ρ$ )( $H \cdot i$ )	Field tilt	✗	✓
W_21110 ( $H \cdot ρ$ )( $H \cdot i$ )	Quadratic distortion 2	✗	✗
W_20001 ( $H \cdot H$ )( $ρ \cdot i$ )	Quadratic distortion 1	✗	✗
W_02001 ( $ρ \cdot ρ$ )( $ρ \cdot i$ )	Constant coma	✗	✓
W_11101 ( $H \cdot ρ$ )( $ρ \cdot i$ )	Linear astigmatism	✗	✓
W_10011 ( $H \cdot i$ )( $ρ \cdot i$ )	Anamorphism	✗	✗
W_40000 ( $H \cdot H$ )( $H \cdot H$ )	Quartic piston	✓	✗
W_04000 ( $ρ \cdot ρ$ )( $ρ \cdot ρ$ )	Spherical aberration	✓	✓
W_22200 ( $H \cdot ρ$ )( $H \cdot ρ$ )	Quadratic astigmatism	✓	✓
W_20020 ( $H \cdot i$ )( $H \cdot i$ )	Quadratic piston	✗	✗
W_00002 ( $ρ \cdot i$ )( $ρ \cdot i$ )	Constant astigmatism	✗	✓

FOV (Full)	$70 \times 4$ deg
EPD	1 in.
$f / #$	2.5
FPA size	$4 \times 0.2$ in.
Mirror powers	N, 0, N, P
Wavefront field avg (RMS at 1 µm)	0.07 waves
Primary mirror height	27 in.
Secondary mirror height	13 in.
Pupil mag (XP/EP)	2.8
Pupil circularity, on-axis	90.8%
Smile angle, output	0.6 deg
Max distortion	13%
Max Zernike term used	26

Wavefront Aberration Function	Name	Axial Symmetry	Causes Blurring
W_20000 ( $H \cdot H$ )	Quadratic piston	✓	✗
W_02000 ( $ρ \cdot ρ$ )	Defocus	✓	✓
W_11100 ( $H \cdot ρ$ )	Magnification	✓	✗
W_10010 ( $H \cdot i$ )	Linear piston	✗	✗
W_00001 ( $ρ \cdot i$ )	Field displacement	✗	✗
W_22000 ( $H \cdot H$ )( $ρ \cdot ρ$ )	Field curvature	✓	✓
W_31100 ( $H \cdot H$ )( $H \cdot ρ$ )	Cubic distortion	✓	✗
W_13100 ( $ρ \cdot ρ$ )( $H \cdot ρ$ )	Linear coma	✓	✓
W_30010 ( $H \cdot H$ )( $H \cdot i$ )	Cubic piston	✗	✗
W_12010 ( $ρ \cdot ρ$ )( $H \cdot i$ )	Field tilt	✗	✓
W_21110 ( $H \cdot ρ$ )( $H \cdot i$ )	Quadratic distortion 2	✗	✗
W_20001 ( $H \cdot H$ )( $ρ \cdot i$ )	Quadratic distortion 1	✗	✗
W_02001 ( $ρ \cdot ρ$ )( $ρ \cdot i$ )	Constant coma	✗	✓
W_11101 ( $H \cdot ρ$ )( $ρ \cdot i$ )	Linear astigmatism	✗	✓
W_10011 ( $H \cdot i$ )( $ρ \cdot i$ )	Anamorphism	✗	✗
W_40000 ( $H \cdot H$ )( $H \cdot H$ )	Quartic piston	✓	✗
W_04000 ( $ρ \cdot ρ$ )( $ρ \cdot ρ$ )	Spherical aberration	✓	✓
W_22200 ( $H \cdot ρ$ )( $H \cdot ρ$ )	Quadratic astigmatism	✓	✓
W_20020 ( $H \cdot i$ )( $H \cdot i$ )	Quadratic piston	✗	✗
W_00002 ( $ρ \cdot i$ )( $ρ \cdot i$ )	Constant astigmatism	✗	✓

FOV (Full)	$70 \times 4$ deg
EPD	1 in.
$f / #$	2.5
FPA size	$4 \times 0.2$ in.
Mirror powers	N, 0, N, P
Wavefront field avg (RMS at 1 µm)	0.07 waves
Primary mirror height	27 in.
Secondary mirror height	13 in.
Pupil mag (XP/EP)	2.8
Pupil circularity, on-axis	90.8%
Smile angle, output	0.6 deg
Max distortion	13%
Max Zernike term used	26

Abstract

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Equations (28)

Applied Optics