Comparing two new camera calibration methods with traditional pinhole calibrations

Psang D. Lin; Chi K. Sung

doi:10.1364/OE.15.003012

Comparing two new camera calibration methods with traditional pinhole calibrations

Psang D. Lin and Chi K. Sung

Optics Express
Vol. 15,
Issue 6,
pp. 3012-3022
(2007)
•https://doi.org/10.1364/OE.15.003012

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Psang D. Lin and Chi K. Sung, "Comparing two new camera calibration methods with traditional pinhole calibrations," Opt. Express 15, 3012-3022 (2007)

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Related Topics
Table of Contents Category
- Geometrical optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Cameras (040.1490)
- Geometric optics (080.0080)

About this Article
History
- Original Manuscript: January 19, 2007
- Revised Manuscript: February 27, 2007
- Manuscript Accepted: March 1, 2007
- Published: March 19, 2007

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Open Access

Abstract

Two novel camera calibration methods are compared with traditional pinhole calibration: one new method uses an analytic geometrical version of Snell’s law (denoted as the Snell model); the other uses 6×6 matrix-based paraxial ray-tracing (referred to as the paraxial model). Pinhole model uses a perspective projection approximation to give a single lumped result for the multiple optical elements in a camera system. It is mathematically simple, but suffers from accuracy limitations since it does not consider the lens system. The Snell model is mathematically the most complex but potentially has the highest levels of accuracy for the widest range of conditions. The paraxial model has the merit of offering analytical equations for calibration.

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References

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|
Year
|
Author
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Publication

M. Kawakita, K. Iizuka, R. Iwama, K. Takizawa, H. Kikuchi, and F. Sato, “Gain-modulated axi-vision camera (high speed high-accuracy depth-mapping camera),” Opt. Express 12,5336–5344 (2004).
[Crossref] [PubMed]
Y.I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry,” Proceedings of the ASP Symposium on Close-Range Photogrammetry,1–18 (1971).
H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
[Crossref] [PubMed]
R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-D machine vision metrology,” IEEE Trans. Pattern Anal. Mach. Intell., 10, 713–720 (1988).
[Crossref]
F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
[Crossref]
F. Y. Wang, “A simple and analytical procedure for calibrating extrinsic camewra parameters,” IEEE Trans. Rob. Autom., 20, 121–124 (2004).
[Crossref]
R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).
P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]
P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]
R. P. Paul, Robot manipulators-mathematics, programming and control, (MIT press, Cambridge, Mass., 10–15,1982).

2006 (2)

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

2005 (1)

F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
[Crossref]

2004 (2)

F. Y. Wang, “A simple and analytical procedure for calibrating extrinsic camewra parameters,” IEEE Trans. Rob. Autom., 20, 121–124 (2004).
[Crossref]

M. Kawakita, K. Iizuka, R. Iwama, K. Takizawa, H. Kikuchi, and F. Sato, “Gain-modulated axi-vision camera (high speed high-accuracy depth-mapping camera),” Opt. Express 12,5336–5344 (2004).
[Crossref] [PubMed]

1988 (2)

H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
[Crossref] [PubMed]

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-D machine vision metrology,” IEEE Trans. Pattern Anal. Mach. Intell., 10, 713–720 (1988).
[Crossref]

Abdel-Aziz, Y.I.

Y.I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry,” Proceedings of the ASP Symposium on Close-Range Photogrammetry,1–18 (1971).

Hatze, H.

H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
[Crossref] [PubMed]

Iizuka, K.

Iwama, R.

Karara, H. M.

Kawakita, M.

Kikuchi, H.

Klette, R.

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

Koschan, A.

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

Lenz, R. K.

Lin, P. D.

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

Paul, R. P.

R. P. Paul, Robot manipulators-mathematics, programming and control, (MIT press, Cambridge, Mass., 10–15,1982).

Sato, F.

SchlUns, K.

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

Sung, C. K.

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

Takizawa, K.

Tsai, R. Y.

Wang, F. Y.

F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
[Crossref]

F. Y. Wang, “A simple and analytical procedure for calibrating extrinsic camewra parameters,” IEEE Trans. Rob. Autom., 20, 121–124 (2004).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

IEEE Trans. Rob. Autom. (1)

F. Y. Wang, “A simple and analytical procedure for calibrating extrinsic camewra parameters,” IEEE Trans. Rob. Autom., 20, 121–124 (2004).
[Crossref]

IEEE Trans. Syst. Man Cybern. (1)

F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
[Crossref]

J. Biomech. (1)

H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
[Crossref] [PubMed]

J. Dyn. Syst. Meas. Control-Trans. ASME (1)

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

Opt. Express (1)

Optik (1)

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

Other (3)

R. P. Paul, Robot manipulators-mathematics, programming and control, (MIT press, Cambridge, Mass., 10–15,1982).

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

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Fig. 1. Pinhole model is an ideal model of camera.

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Fig. 2. Schematic drawing of camera used in Snell model.

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Fig. 3 Camera system used in paraxial model.

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Fig. 4. The laser/plate assembly for quick determination of the source ray data.

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Fig. 5. The camera calibration of paraxial model.

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Tables (3)

Table 1. Calibration results from pinhole model and Snell model

View Table | View all tables in this article

Table 2. Calibration results from paraxial model

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Table 3. Comparison of system performances of three models

View Table | View all tables in this article

Equations (24)

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{\overset{c}{P}}_{0} = {[{\overset{c}{P}}_{0 x} {\overset{c}{P}}_{0 y} {\overset{c}{P}}_{0 z} 1]}^{T} = {\overset{c}{A}}_{w} {\overset{w}{P}}_{0}

{\overset{c}{A}}_{w} = Trans (t_{x}, t_{y}, t_{z}) Rot (z, ω_{z}) Rot (y, ω_{y}) Rot (x, ω_{x})

[\begin{matrix} u \\ v \\ 1 \end{matrix}] = \frac{1}{{\overset{c}{P}}_{0 y}} [\begin{matrix} \frac{f}{s_{u}} & u_{0} & \frac{- fCΩ}{S_{u} SΩ} & 0 \\ 0 & v_{0} & \frac{f}{s_{v} SΩ} & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] {\overset{c}{A}}_{w} {\overset{w}{P}}_{0}

{\overset{w}{P}}_{i} = {[{\overset{w}{P}}_{i - lx} + {\overset{w}{ℓ}}_{i - lx} λ_{i} {\overset{w}{P}}_{i - ly} + {{\overset{w}{ℓ}}_{i - ly} λ}_{i} {\overset{w}{P}}_{i - lz} + {\overset{w}{ℓ}}_{i - lz} λ_{i} 0]}^{T}

{\overset{w}{ℓ}}_{i} = [\begin{matrix} {\overset{w}{ℓ}}_{ix} \\ {\overset{w}{ℓ}}_{iy} \\ {\overset{w}{ℓ}}_{iz} \\ 0 \end{matrix}] = [\begin{matrix} - {\overset{w}{n}}_{ix} \sqrt{1 - N_{i}^{2} + {(N_{i} {Cθ}_{i})}^{2}} + N_{i} ({\overset{w}{ℓ}}_{i - lx} + {\overset{w}{n}}_{i x} {Cθ}_{i}) \\ - {\overset{w}{n}}_{iy} \sqrt{1 - N_{i}^{2} + {(N_{i} {Cθ}_{i})}^{2}} + N_{i} ({\overset{w}{ℓ}}_{i - ly} + {\overset{w}{n}}_{i y} {Cθ}_{i}) \\ - {\overset{w}{n}}_{iz} \sqrt{1 - N_{i}^{2} + {(N_{i} {Cθ}_{i})}^{2}} + N_{i} ({\overset{w}{ℓ}}_{i - lz} + {\overset{w}{n}}_{i z} {Cθ}_{i}) \\ 0 \end{matrix}]

{\overset{n}{P}}_{n} = {[{\overset{n}{P}}_{nx} {\overset{n}{P}}_{ny} {\overset{n}{P}}_{nz} 1]}^{T} = {\overset{n}{A}}_{w} {\overset{w}{P}}_{n}

[\begin{matrix} u \\ v \end{matrix}] = [\begin{matrix} u_{0} \\ v_{0} \end{matrix}] + [\begin{matrix} \frac{1}{S_{u}} & \frac{- CΩ}{S_{u} SΩ} \\ 0 & \frac{1}{S_{v} SΩ} \end{matrix}] [\begin{matrix} {\overset{n}{P}}_{nx} \\ {\overset{n}{P}}_{nz} \end{matrix}]

{\overset{c}{P}}_{0} = {[{\overset{c}{P}}_{0 x} {\overset{c}{P}}_{0 y} {\overset{c}{P}}_{0 z} 1]}^{T} = {\overset{c}{A}}_{w} {\overset{w}{P}}_{0};

{\overset{c}{ℓ}}_{0} = {[{\overset{c}{ℓ}}_{0 x} {\overset{c}{ℓ}}_{0 y} {\overset{c}{ℓ}}_{0 z} 0]}^{T} = {\overset{c}{A}}_{w} {\overset{w}{ℓ}}_{0} .

{[∆ {\overset{c}{P}}_{0} ∆ {\overset{c}{ℓ}}_{0}]}^{T} = {[{\overset{c}{P}}_{0} {\overset{c}{ℓ}}_{0}]}^{T} - {[{\overset{c}{P̱}}_{0} {\overset{c}{ℓ̱}}_{0}]}^{T}

= {[{\overset{c}{P}}_{0 x} {\overset{c}{P}}_{0 y} {\overset{c}{P}}_{0 z} {\overset{c}{ℓ}}_{0 x} {\overset{c}{ℓ}}_{0 y} {\overset{c}{ℓ}}_{0 z}]}^{T} - {[0 {\overset{c}{P}}_{0 y} 0 0 1 0]}^{T}

= [\begin{matrix} {({Cω}_{y} {Cω}_{z})}^{w} P_{0 x} + {({Sω}_{x} {Sω}_{y} {Cω}_{z} - {Cω}_{x} {Cω}_{z})}^{w} P_{0y} + {({Cω}_{x} {Sω}_{y} {Cω}_{z} + {Sω}_{x} {Sω}_{z})}^{w} P_{0z} + t_{x} \\ 0 \\ {{(- {Sω}_{y})}^{w} P}_{0 x} + {({Sω}_{x} {Cω}_{y})}^{w} P_{0y} + {({Cω}_{x} {Cω}_{y})}^{w} P_{0z} + t_{z} \\ ({Cω}_{y} {Cω}_{z}) {\overset{w}{ℓ}}_{0 x} + {({Sω}_{x} {Sω}_{y} {Cω}_{z} - {Cω}_{x} {Cω}_{z})}^{w} ℓ_{0y} + ({Cω}_{x} {Sω}_{y} {Cω}_{z} + {Sω}_{x} {Sω}_{z})^{w} ℓ_{0z} \\ {({Cω}_{y} {Sω}_{z})}^{w} ℓ_{0 x} + {({Sω}_{x} {Sω}_{y} {Sω}_{z} + {Cω}_{x} {Cω}_{z})}^{w} ℓ_{0y} + ({Cω}_{x} {Sω}_{y} {Sω}_{z} + {Sω}_{x} {Cω}_{z})^{w} ℓ_{0z} - 1 \\ ({- Sω}_{y})^{w} ℓ_{0 x} + {({Sω}_{x} {Cω}_{y})}^{w} ℓ_{0y} + ({Cω}_{x} {Cω}_{y})^{w} ℓ_{0z} \end{matrix}]

[\begin{matrix} {Δ \overset{n}{P}}_{n} \\ {Δ \overset{n}{ℓ}}_{n} \end{matrix}] = [\begin{matrix} {Δ \overset{c}{P}}_{n} \\ {Δ \overset{c}{ℓ}}_{n} \end{matrix}] = (T_{n}) (M_{n - 1} T_{n - 1} M_{n - 2} T_{n - 2}) \dots (M_{2 j} T_{2 j} M_{2 j - 1} T_{2 j - 1}) \dots (M_{2} T_{2} M_{1} T_{1}) [\begin{matrix} {Δ \overset{c}{P}}_{0} \\ {Δ \overset{c}{ℓ}}_{0} \end{matrix}]

[\begin{matrix} u \\ v \end{matrix}] = [\begin{matrix} u_{0} \\ v_{0} \end{matrix}] + [\begin{matrix} \frac{1}{S_{u}} & \frac{- CΩ}{S_{u} SΩ} \\ 0 & \frac{1}{S_{v} SΩ} \end{matrix}] [\begin{matrix} {Δ \overset{n}{P}}_{nx} \\ {Δ \overset{n}{P}}_{nz} \end{matrix}]

{Δ \overset{3}{P}}_{3 x} = [1 + \frac{R_{1} λ_{3} (1 - N_{2}) + R_{2} (N_{1} - 1) (q_{1} + e_{2} N_{2}) + q_{1} e_{2} (N_{1} - 1) (1 - N_{2})}{R_{1} R_{2}}] \begin{matrix}  \end{matrix} {Δ \overset{c}{P}}_{0x}

+ {λ_{1} + λ_{2} [\frac{R_{1} e_{2} (1 - N_{2}) + R_{2} (N_{1} - 1) (q_{1} + e_{2} N_{2}) + q_{1} e_{2} (N_{1} - 1) (1 - N_{2})}{R_{1} R_{2}}]

+ N_{1} [q_{1} + e_{2} N_{2} + \frac{q_{1} e_{1} (1 - N_{2})}{R_{2}}]} {Δ \overset{c}{ℓ}}_{0x}

\begin{matrix}  \end{matrix} {Δ \overset{3}{P}}_{3 z} = [1 + \frac{R_{1} λ_{3} (1 - N_{2}) + R_{2} (N_{1} - 1) (q_{1} + e_{2} N_{2}) + q_{1} e_{2} (N_{1} - 1) (1 - N_{2})}{R_{1} R_{2}}] \begin{matrix}  \end{matrix} {Δ \overset{c}{P}}_{0z}

+ {λ_{1} + λ_{2} [\frac{R_{1} e_{2} (1 - N_{2}) + R_{2} (N_{1} - 1) (q_{1} + e_{2} N_{2}) + q_{1} e_{2} (N_{1} - 1) (1 - N_{2})}{R_{1} R_{2}}]

+ N_{1} [q_{1} + e_{2} N_{2} + \frac{q_{1} e_{1} (1 - N_{2})}{R_{2}}]} {Δ \overset{c}{ℓ}}_{0z}

Rot (x, ω_{x}) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & C ω_{x} & - S ω_{x} & 0 \\ 0 & S ω_{x} & C ω_{x} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Rot (y, ω_{y}) = [\begin{matrix} C ω_{y} & 0 & {Sω}_{y} & 0 \\ 0 & 1 & 0 & 0 \\ - {Sω}_{y} & 0 & C ω_{y} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Rot (z, ω_{z}) = [\begin{matrix} {Cω}_{z} & {- Sω}_{z} & 0 & 0 \\ {Sω}_{z} & {Cω}_{z} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Trans (t_{x}, t_{y}, t_{z}) = [\begin{matrix} 1 & 0 & 0 & t_{x} \\ 0 & 1 & 0 & t_{y} \\ 0 & 0 & 1 & t_{z} \\ 0 & 0 & 0 & 1 \end{matrix}]

[ ^wP₀ ](mm)	[u v] (pixel)	Pinhole model		Snell model
[ ^wP₀ ](mm)	[u v] (pixel)	Parameters	[u-u v-v] (pixel)	Parameters	[u-u v-v] (pixel)
[11.444 79.645 5.751]	[319 361]	t_x= -51.2623 (mm)	[3 -3]	t_3x=0.000 (mm)	[4 0]
[7.308 75.613 9.114]	[384 308]	t_y= -307.097 (mm)	[ 5 0]	t_3y=-635.747 (mm)	[6 0]
[3.125 71.705 12.391]	[455 260]	t_z= -43.2148 (mm)	[-1 2]	t_3z=0.0000 (mm)	[2 -2]
[-2.271 73.566 8.799]	[542 317]	ω_x=9.1948 (deg.)	[-3 -2]	ω_3x=0.0002 (deg.)	[-1 -3]
[-7.474 76.182 4.201]	[625 390]	ω_y=0.9872 (deg.)	[-1 -5]	ω_3y=-0.295 (deg.)	[-2 -2]
[17.867 78.153 -2.476]	[224 487]	ω_z= -8.7406 (deg.)	[-4 3]	ω_3z=-0.0002 (deg.)	[-3 6]
[18.983 73.666 -6.545]	[207 553]	u₀= -112 (pixel)	[-1 3]	t_1x=-0.0001 (mm)	[-1 4]
[19.901 68.758 -11.192]	[193 628]	v₀= -41 (pixel)	[4 0]	t_1y=-646.575 (mm)	[1 1]
[15.038 71.164 -14.676]	[270 687]	f/s_u = 3669.876 (pix.)	[-1 -3]	t_1z=0.000 (mm)	[-2 -1]
[9.467 73.680 -18.961]	[360 757]	f/s_v=3697.456 (pix.)	[-6 -1]	ω_1x=0.000 (deg.)	[-6 -1]
[-12.072 73.053 -4.333]	[696 520]	Ω=89.0405 (deg.)	[1 4]	ω_1y=0.000 (deg.)	[-2 4]
[-10.706 69.016 -8.422]	[675 585]		[-3 4]	ω_1z=0.000 (deg.)	[-4 3]
[-9.716 63.204 -13.050]	[653 660]		[-1 0]	u₀=504(pixel)	[1 -1]
[-5.331 67.523 -16.293]	[583 715]		[4 -3]	v₀=454(pixel)	[3 -4]
[-0.875 71.984 -19.497]	[513 766]		[5 0]	s_u=0.0066(mm/pix.)	[4 -2]
				sv=0.0065(mm/pix.)
				Ω=90.0156 (deg.)

[ ^wP₀ ^wℓ₀ ](mm)						[u v] (pixel)	Parameters	[u-u v-v] (pixel)
[0.000	0.000	0.000	0.000	1.000	-0.000]	[650 514]	t_x = -0.3908 (mm)	[-1 1]
[-30.911	1.227	1.024	0.057	0.998	-0.012]	[1092 498]	t_y = -588.2050 (mm)	[-1 0]
[31.952	-8.278	1.343	-0.055	0.998	-0.010]	[203 494]	t_z = 2.0405 (mm)	[-3 -3]
[4.188	-10.891	13.122	-0.014	0.999	-0.033]	[591 333]	ω_x=-0.0005 (deg.)	[-1 -2]
[4.128	-12.038	6.024	-0.008	1.000	-0.015]	[593 435]	ω_y=-0.3004 (deg.)	[ 2 1]
[-20.747	-6.994	13.908	0.038	1.000	-0.030]	[942 322]	ω_z= 0.0028 (deg.)	[ 2 3]
[13.083	5.629	12.392	-0.020	1.000	-0.030]	[465 332]	e₂=49.8448 (mm)	[-1 2]
[-3.083	2.201	8.368	0.003	1.000	-0.026]	[694 390]	u₀= 643 (pixel)	[1 2]
[-4.739	-0.231	1.474	0.012	1.000	-0.003]	[718 498]	v₀= 547 (pixel)	[-1 3]
[25.111	4.457	-9.627	-0.040	0.999	0.009]	[292 648]	s_u= 0.0065 (mm/pix.)	[0 -1]
[-21.328	-0.186	-11.670	0.047	0.999	0.009]	[956 681]	s_v=0.0065 (mm/pix.)	[-2 2]
[-15.908	6.598	8.468	0.030	0.999	-0.027]	[835 503]	Ω=89.2834 (deg.)	[ 0 2]
[12.673	3.083	9.243	-0.022	1.000	-0.019]	[881 387]		[4 1]
[12.599	1.259	-2.097	-0.026	1.000	-0.004]	[466 384]		[-4 3]
[7.746	-4.313	-7.617	-0.008	1.000	0.004]	[469 540]		[1 1]

Paraxial model
	[ ^wP₀ ^wℓ₀ ](mm)						[u v] (pixel)	∣u-u v-v∣ (pixel)
A	[-8.804	270.783	3.679	0.039	1.000	-0.005]	[877 418]	∣17 40∣
	[-3.463	281.472	-4.591	-0.005	1.000	-0.010]	[693 541]	∣ 18 5 ∣
	[10.161	275.737	-8.201	-0.009	1.000	0.003]	[472 626]	∣14 8∣
	[20.506	274.975	-2.920	-0.040	1.000	-0.002]	[227 537]	∣24 13∣
	[6.565	271.797	2.662	-0.010	1.000	-0.009]	[524 428]	∣8 26∣
B	[-13.258	74.051	-5.704	0.032	1.000	-0.001]	[878 590]	∣5 3∣
	[-6.336	74.004	3.921	0.013	1.000	-0.021]	[755 433]	∣1 1∣
	[0.682	76.944	7.596	-0.004	0.999	-0.032]	[637 368]	∣1 3∣
	[25.761	82.423	7.542	-0.047	0.998	-0.030]	[229 366]	∣7 1∣
	[24.328	78.274	-8.450	-0.048	0.999	0.002]	[252 632]	∣4 1∣
C	[11.337	16.955	6.162	-0.013	1.000	-0.016]	[508 450]	∣3 10∣
	[-4.184	13.702	7.634	0.009	1.000	-0.011]	[704 439]	∣3 13∣
	[-4.219	9.721	-2.363	0.010	1.000	-0.004]	[705 562]	∣6 10∣
	[-25.658	15.752	-1.884	0.047	0.999	0.003]	[963 556]	∣8 11∣
	[-31.654	10.984	11.345	0.050	0.998	-0.025]	[1037 393]	∣1 7∣
Pinhole model and Snell model
	[ ^wP₀ ](mm)				[u v] (pixel)		∣u-u v-v∣ (pixel)
	[ ^wP₀ ](mm)				[u v] (pixel)		Pinhole model	Snell model
A	[-8.804		270.783	3.679]	[740 329]		∣129 134∣	∣50 48∣
	[-3.463		281.472	-4.591]	[583 511]		∣98 28∣	∣6 41∣
	[10.161		275.737	-8.201]	[289 587]		∣66 16∣	∣3 39∣
	[20.506		274.975	-2.920]	[65 466]		∣185 43∣	∣9 48∣
	[6.565		271.797	2.662]	[368 351]		∣27 118∣	∣1 46∣
B	[-13.258		74.051	-5.704]	[730 552]		∣13 6∣	∣17 6∣
	[-6.336		74.004	3.921]	[615 395]		∣11 3∣	∣10 3∣
	[0.682		76.944	7.596]	[498 332]		∣4 1∣	∣3 1∣
	[25.761		82.423	7.542]	[ 86 331]		∣6 4∣	∣7 1∣
	[24.328		78.274	-8.450]	[111 593]		∣6 7∣	∣7 5∣
C	[11.337		16.955	6.162]	[364 410]		∣7 21∣	∣8 34∣
	[-4.184		13.702	7.634]	[562 399]		∣39 27∣	∣7 37∣
	[-4.219		9.7210	-2.363]	[563 523]		∣39 2∣	∣8 40∣
	[-25.658		15.752	-1.884]	[822 519]		∣100 0∣	∣14 41∣
	[-31.654		10.984	11.345]	[895 354]		∣114 39∣	∣13 34∣

[ ^wP₀ ](mm)	[u v] (pixel)	Pinhole model		Snell model
[ ^wP₀ ](mm)	[u v] (pixel)	Parameters	[u-u v-v] (pixel)	Parameters	[u-u v-v] (pixel)
[11.444 79.645 5.751]	[319 361]	t_x= -51.2623 (mm)	[3 -3]	t_3x=0.000 (mm)	[4 0]
[7.308 75.613 9.114]	[384 308]	t_y= -307.097 (mm)	[ 5 0]	t_3y=-635.747 (mm)	[6 0]
[3.125 71.705 12.391]	[455 260]	t_z= -43.2148 (mm)	[-1 2]	t_3z=0.0000 (mm)	[2 -2]
[-2.271 73.566 8.799]	[542 317]	ω_x=9.1948 (deg.)	[-3 -2]	ω_3x=0.0002 (deg.)	[-1 -3]
[-7.474 76.182 4.201]	[625 390]	ω_y=0.9872 (deg.)	[-1 -5]	ω_3y=-0.295 (deg.)	[-2 -2]
[17.867 78.153 -2.476]	[224 487]	ω_z= -8.7406 (deg.)	[-4 3]	ω_3z=-0.0002 (deg.)	[-3 6]
[18.983 73.666 -6.545]	[207 553]	u₀= -112 (pixel)	[-1 3]	t_1x=-0.0001 (mm)	[-1 4]
[19.901 68.758 -11.192]	[193 628]	v₀= -41 (pixel)	[4 0]	t_1y=-646.575 (mm)	[1 1]
[15.038 71.164 -14.676]	[270 687]	f/s_u = 3669.876 (pix.)	[-1 -3]	t_1z=0.000 (mm)	[-2 -1]
[9.467 73.680 -18.961]	[360 757]	f/s_v=3697.456 (pix.)	[-6 -1]	ω_1x=0.000 (deg.)	[-6 -1]
[-12.072 73.053 -4.333]	[696 520]	Ω=89.0405 (deg.)	[1 4]	ω_1y=0.000 (deg.)	[-2 4]
[-10.706 69.016 -8.422]	[675 585]		[-3 4]	ω_1z=0.000 (deg.)	[-4 3]
[-9.716 63.204 -13.050]	[653 660]		[-1 0]	u₀=504(pixel)	[1 -1]
[-5.331 67.523 -16.293]	[583 715]		[4 -3]	v₀=454(pixel)	[3 -4]
[-0.875 71.984 -19.497]	[513 766]		[5 0]	s_u=0.0066(mm/pix.)	[4 -2]
				sv=0.0065(mm/pix.)
				Ω=90.0156 (deg.)

[ ^wP₀ ^wℓ₀ ](mm)						[u v] (pixel)	Parameters	[u-u v-v] (pixel)
[0.000	0.000	0.000	0.000	1.000	-0.000]	[650 514]	t_x = -0.3908 (mm)	[-1 1]
[-30.911	1.227	1.024	0.057	0.998	-0.012]	[1092 498]	t_y = -588.2050 (mm)	[-1 0]
[31.952	-8.278	1.343	-0.055	0.998	-0.010]	[203 494]	t_z = 2.0405 (mm)	[-3 -3]
[4.188	-10.891	13.122	-0.014	0.999	-0.033]	[591 333]	ω_x=-0.0005 (deg.)	[-1 -2]
[4.128	-12.038	6.024	-0.008	1.000	-0.015]	[593 435]	ω_y=-0.3004 (deg.)	[ 2 1]
[-20.747	-6.994	13.908	0.038	1.000	-0.030]	[942 322]	ω_z= 0.0028 (deg.)	[ 2 3]
[13.083	5.629	12.392	-0.020	1.000	-0.030]	[465 332]	e₂=49.8448 (mm)	[-1 2]
[-3.083	2.201	8.368	0.003	1.000	-0.026]	[694 390]	u₀= 643 (pixel)	[1 2]
[-4.739	-0.231	1.474	0.012	1.000	-0.003]	[718 498]	v₀= 547 (pixel)	[-1 3]
[25.111	4.457	-9.627	-0.040	0.999	0.009]	[292 648]	s_u= 0.0065 (mm/pix.)	[0 -1]
[-21.328	-0.186	-11.670	0.047	0.999	0.009]	[956 681]	s_v=0.0065 (mm/pix.)	[-2 2]
[-15.908	6.598	8.468	0.030	0.999	-0.027]	[835 503]	Ω=89.2834 (deg.)	[ 0 2]
[12.673	3.083	9.243	-0.022	1.000	-0.019]	[881 387]		[4 1]
[12.599	1.259	-2.097	-0.026	1.000	-0.004]	[466 384]		[-4 3]
[7.746	-4.313	-7.617	-0.008	1.000	0.004]	[469 540]		[1 1]

Abstract

References

2006 (2)

2005 (1)

2004 (2)

1988 (2)

Abdel-Aziz, Y.I.

Hatze, H.

Iizuka, K.

Iwama, R.

Karara, H. M.

Kawakita, M.

Kikuchi, H.

Klette, R.

Koschan, A.

Lenz, R. K.

Lin, P. D.

Paul, R. P.

Sato, F.

SchlUns, K.

Sung, C. K.

Takizawa, K.

Tsai, R. Y.

Wang, F. Y.

IEEE Trans. Pattern Anal. Mach. Intell. (1)

IEEE Trans. Rob. Autom. (1)

IEEE Trans. Syst. Man Cybern. (1)

J. Biomech. (1)

J. Dyn. Syst. Meas. Control-Trans. ASME (1)

Opt. Express (1)

Optik (1)

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