Closed-form solution for thin lens image irradiance under arbitrary solid angle

Robert D. Friedlander; Anthony J. Yezzi

doi:10.1364/JOSAA.377896

Journal of the Optical Society of America A
Vol. 37,
Issue 4,
pp. 568-578
(2020)
•https://doi.org/10.1364/JOSAA.377896

Closed-form solution for thin lens image irradiance under arbitrary solid angle

Robert D. Friedlander and Anthony J. Yezzi

Get PDF
Email
Share
Get Citation
Copy Citation Text
Robert D. Friedlander and Anthony J. Yezzi, "Closed-form solution for thin lens image irradiance under arbitrary solid angle," J. Opt. Soc. Am. A 37, 568-578 (2020)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Check for updates

Related Topics
Table of Contents Category
- Geometrical Optics
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: September 13, 2019
- Revised Manuscript: January 27, 2020
- Manuscript Accepted: February 2, 2020
- Published: March 11, 2020

Accepted Manuscript
Download Accepted Manuscript
Access to this article is made available via and is subject to OSA Publishing Terms of Use.

Abstract

Optical imaging systems are found everywhere in modern society. They are integral to computer vision, where the goal is often to infer geometric and radiometric information about a 3D environment given limited sensing resources. It is helpful to develop relationships between these real-world properties and the actual measurements that are taken, such as 2D images. To this end, we propose a new relationship between object radiance and image irradiance based on power conservation and a thin lens imaging model. The relationship has a closed-form solution for in-focus points and can be solved via numerical integration for points that are not focused. It can be thought of as a generalization of Horn’s commonly accepted irradiance equation. Through both ray tracing simulations and comparison to the intensity values of actual images, we believe our equation provides better accuracy than Horn’s equation. An improvement is most notable for large lenses and near-focused images where the pinhole imaging model implicit in Horn’s derivation breaks down. Outside of this regime, our model validates the use of Horn’s approximation through a more thorough theoretical foundation.

Full Article | PDF Article

More Like This

Passive non-line-of-sight imaging using plenoptic information

Di Lin, Connor Hashemi, and James R. Leger
J. Opt. Soc. Am. A 37(4) 540-551 (2020)

Light field reconstruction from scattered light using plenoptic data

Takahiro Sasaki and James R. Leger
J. Opt. Soc. Am. A 37(4) 653-670 (2020)

Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium

Harold T. Yura, Lars Thrane, and Peter E. Andersen
J. Opt. Soc. Am. A 17(12) 2464-2474 (2000)

Previous Article Next Article

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Figures (6)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Tables (5)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (54)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

	$ρ = 0$		$ρ = d / 4$		$ρ = 4 d$
	$\| E_{m e a s}^{'} - E_{H o r n}^{'} \|$	$\| E_{m e a s}^{'} - E_{l e n s}^{'} \|$	$\| E_{m e a s}^{'} - E_{H o r n}^{'} \|$	$\| E_{m e a s}^{'} - E_{l e n s}^{'} \|$	$\| E_{m e a s}^{'} - E_{H o r n}^{'} \|$	$\| E_{m e a s}^{'} - E_{l e n s}^{'} \|$
$d = f / 32$
$z = 1.1 f$	$3.83 \times 10^{11}$	$7.10 \times 10^{6}$	$2.55 \times 10^{11}$	$1.35 \times 10^{8}$	$4.70 \times 10^{10}$	$3.08 \times 10^{8}$
$z = 10 f$	$6.11 \times 10^{12}$	$4.92 \times 10^{10}$	$5.09 \times 10^{12}$	$1.05 \times 10^{12}$	$3.30 \times 10^{12}$	$2.55 \times 10^{12}$
$z = 20 f$	$1.27 \times 10^{13}$	$2.05 \times 10^{11}$	$1.30 \times 10^{13}$	$4.71 \times 10^{12}$	$1.29 \times 10^{13}$	$1.14 \times 10^{13}$
$d = f / 5.6$
$z = 1.1 f$	$3.84 \times 10^{11}$	$4.70 \times 10^{6}$	$2.56 \times 10^{11}$	$1.25 \times 10^{8}$	$2.46 \times 10^{10}$	$6.90 \times 10^{6}$
$z = 10 f$	$6.05 \times 10^{12}$	$5.86 \times 10^{9}$	$5.05 \times 10^{12}$	$1.01 \times 10^{12}$	$3.26 \times 10^{12}$	$2.52 \times 10^{12}$
$z = 20 f$	$1.24 \times 10^{13}$	$3.92 \times 10^{10}$	$1.29 \times 10^{13}$	$4.63 \times 10^{12}$	$1.52 \times 10^{13}$	$1.37 \times 10^{13}$
$d = f / 0.7$
$z = 1.1 f$	$5.00 \times 10^{11}$	$2.59 \times 10^{8}$	$3.04 \times 10^{11}$	$1.67 \times 10^{8}$	$1.90 \times 10^{8}$	$7.37 \times 10^{6}$
$z = 10 f$	$6.15 \times 10^{12}$	$4.82 \times 10^{10}$	$1.91 \times 10^{13}$	$1.49 \times 10^{13}$	$2.15 \times 10^{12}$	$1.71 \times 10^{12}$
$z = 20 f$	$1.09 \times 10^{13}$	$1.72 \times 10^{12}$	$7.99 \times 10^{13}$	$7.16 \times 10^{13}$	$1.32 \times 10^{13}$	$1.18 \times 10^{13}$

Method	$C_{1, i}^{*}$	$C_{2, i}^{*}$	$‖ I - C_{1, i}^{} {\hat{E}}_{i}^{'} - C_{2, i}^{} ‖_{2}^{2}$
$f / 3.5$
Horn	$1.0429 \times 10^{5}$	−4522.7	$1.0819499 \times 10^{7}$
Lens	$1.3097 \times 10^{5}$	−4530.5	$1.0819465 \times 10^{7}$
$f / 1.4$
Horn	$2.8846 \times 10^{4}$	652.2722	$1.0934127 \times 10^{7}$
Lens	$3.6211 \times 10^{4}$	651.6485	$1.0934022 \times 10^{7}$

Aperture	$‖ I - E_{H o r n}^{'} ‖_{2}^{2} - ‖ I - E_{l e n s}^{'} ‖_{2}^{2}$	$max \| E_{H o r n}^{'} - E_{l e n s}^{'} \|$
$f / 3.5$	34.09	0.0144
$f / 1.4$	105.25	0.0672

Method	$N_{e s t}$	$‖ N - N_{e s t} ‖_{2}$	$\sum (\cos θ - \cos θ_{e s t})^{2}$
$z = 1 m$
Horn	$[0.6519, 0.0426, - 0.7571]^{T}$	$4.86 \times 10^{- 1}$	3.3832
Lens	$[0.6022, 0.2600, - 0.7549]^{T}$	$2.65 \times 10^{- 1}$	5.0956
$z = 100 m$
Horn	$[0.4997, 0.5001, - 0.7072]^{T}$	$2.92 \times 10^{- 4}$	0.0449
Lens	$[0.5000, 0.5000, - 0.7071]^{T}$	$1.62 \times 10^{- 5}$	0.0082

$e_{x} \cdot e_{r} = - \sin α$	$e_{z} \cdot e_{r} = \cos α$
$T \cdot e_{r} = \sin θ$	$N \cdot e_{r} = - \cos θ$
$T \cdot e_{x} = - \cos (θ - α)$	$N \cdot e_{x} = - \sin (θ - α)$
$T \cdot e_{z} = \sin (θ - α)$	$N \cdot e_{z} = - \cos (θ - α)$

Abstract

Cited By

Figures (6)

Tables (5)

Equations (54)

Journal of the Optical Society of America A