Filter construction using Ronchi masks and Legendre polynomials to
analyze the noise in aberrations by applying the irradiance transport
equation

J. A. Arriaga Hernández; B. T. Cuevas Otahola; J. Oliveros; A. Jaramillo Núñez; M. Morín Castillo

doi:10.1364/AO.389716

Applied Optics
Vol. 59,
Issue 13,
pp. 3851-3860
(2020)
•https://doi.org/10.1364/AO.389716

Filter construction using Ronchi masks and Legendre polynomials to analyze the noise in aberrations by applying the irradiance transport equation

J. A. Arriaga Hernández, B. T. Cuevas Otahola, J. Oliveros, A. Jaramillo Núñez, and M. Morín Castillo

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J. A. Arriaga Hernández, B. T. Cuevas Otahola, J. Oliveros, A. Jaramillo Núñez, and M. Morín Castillo, "Filter construction using Ronchi masks and Legendre polynomials to analyze the noise in aberrations by applying the irradiance transport equation," Appl. Opt. 59, 3851-3860 (2020)

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- Imaging Systems, Image Processing, and Displays
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About this Article
History
- Original Manuscript: February 4, 2020
- Revised Manuscript: March 12, 2020
- Manuscript Accepted: March 25, 2020
- Published: April 21, 2020

Abstract

We tested different optical elements placed in three different positions by applying the irradiance transport equation (ITE), obtaining the wavefront $[ W(x,y) ]$ and aberration surface [$ \textit{AS}(r,\theta ) $]. The existing noise in the captures $ I $ as well as in the $ W $ and $ AS $ were analyzed applying several filters: first a filter based on Legendre polynomials (LP), generating the most probable points increasing the data resolution; second, a filter based on a 50 deg 2D-LP was used as a multilineal fit (multiple linear regression); and third, an ideal bandpass filter in the Fourier space after inducing a periodicity using Ronchi simulated masks with periods in $ x,y,xy $ was used to perform data scanning (similar to the four-step phase-shifting method). Signal-to-noise ratio values were obtained for each proposed filter along with the most probable image free from noise, determined from a linear combination of the original data and the applied filters.

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Corrections

15 June 2020: [Code 1] and [Code 2] were added to this paper. See [Supplement 1] for supporting content.

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Supplementary Material (3)

Name	Description
Code 1	The Legrendre 2D code is intended to remove the dirt on the measurement surface as well as in the capture elements (camera or CCD), allowing to characterize such imperfections as noise.The code obtains 1-dimensional Legendre polynomials from which 2-dimensional polynomials are built. These 2D Legendre polynomials are fitted using a least-square method, giving as a result the set of Legendre coefficients.
Code 2	The i-2DLP was developed with the aim of increasing the resolution of a given image. This code increases the resolution of a rm mxm image to a size of (m-1)x Npoints+m with Npoints the number of intermediate points between every pair of the original matrix points.
Supplement 1	Supplemental document

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Equations (18)

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$Z_{m, s} (r, θ)$	Aberr	EO L	Mirror	L1	L2
0,0	Piston	0,0033	0,0002	0,0015	0,0001
2,0	Defocus	0,192	1,3091	0,0234	0,8745
1,−1	$T i l t_{y}$	0,2301	0,4009	0,3007	0,2903
1,1	$T i l t_{x}$	0,3589	0,6028	0,4921	0,4013
2,−2	45ºA1	0,2365	0,3339	0,2918	0,2509
2,2	0ºA1	0,2098	0,3092	0,2503	0,2315
4,−2	45ºA2	0,0596	0,0931	0,0623	0,0601
4,0	Spherical	0,0015	0,0565	0,0107	0,0092
4,2	0ºA2	0,0926	0,1009	0,0953	0,0905

$Z_{m, s} (r, θ)$	Aberr	EO L	Mirror	L1	L2
0,0	Piston	0.2193	0.4310	1.2443	7.3581
2,0	$T i l t_{y}$	0.1586	0.3083	1.9355	4.1612
1,−1	$T i l t_{x}$	0.5849	0.6295	1.3091	1.1434
1,1	45°A	0.2365	0.6821	0.7298	1.3092
2,−2	Defocus	0.1352	5.1378	10.3871	16.1104
2,2	0°A	0.0037	0.1968	1.1331	5.5379
4,−2	45°A	0.3402	0.0348	0.5753	1.0239
4,0	Spherical	0.2248	0.7064	0.2545	1.2571
4,2	0°A	0.2639	0.0102	0.6808	1.6198
6,0	2Spherical	0.1085	0.0508	0.3682	5.6895
8,0	3Spherical	4.2547	49.4753	3.2794	1.1863
10,0	4Spherical	0.5867	1.2153	0.5594	1.9346

$Z_{m, s} (r, θ)$	Aberr	EO L	Mirror	L1	L2
0,0	Piston	0.4971	0.0714	0.8333	3.2704
2,0	$T i l t_{y}$	43.5191	0.4531	1.8093	2.4718
1,−1	$T i l t_{x}$	0.3614	0.4781	1.3025	1.0555
1,1	45°A	0.0363	0.0037	0.8492	1.2278
2,−2	Defocus	0.2647	0.1453	23.7354	20.2250
2,2	0°A	0.0835	0.3210	1.0666	5.7047
4,−2	45°A	3.6107	0.4797	0.7745	1.0187
4,0	Spheric	8.2730	1.5199	0.2988	0.4406
4,2	0°A	0.8060	0.5298	0.4493	1.5788
6,0	2Spheric	0.0270	0.1582	0.4999	4.5691
8,0	3Spheric	0.4848	0.1815	0.6826	3.0004
10,0	4Spheric	2.7378	0.3207	0.1046	1.6562

Filter	EO Lens	Mirror	Thick Lens	Thin Lens
I-2DLP	19.852	62.540	72.910	8.500
Legendre2D	2.702	9.057	8.693	1.084
bRM	1.561	9.918	5.311	0.899

$Z_{m, s} (r, θ)$	Aberr	EO L	Mirror	L1	L2
0,0	Piston	0,0033	0,0002	0,0015	0,0001
2,0	Defocus	0,192	1,3091	0,0234	0,8745
1,−1	$T i l t_{y}$	0,2301	0,4009	0,3007	0,2903
1,1	$T i l t_{x}$	0,3589	0,6028	0,4921	0,4013
2,−2	45ºA1	0,2365	0,3339	0,2918	0,2509
2,2	0ºA1	0,2098	0,3092	0,2503	0,2315
4,−2	45ºA2	0,0596	0,0931	0,0623	0,0601
4,0	Spherical	0,0015	0,0565	0,0107	0,0092
4,2	0ºA2	0,0926	0,1009	0,0953	0,0905

J. A. Arriaga Hernández	https://orcid.org/0000-0001-8059-9399
B. T. Cuevas Otahola	https://orcid.org/0000-0002-1046-1500
J. Oliveros	https://orcid.org/0000-0002-1715-3631
A. Jaramillo Núñez	https://orcid.org/0000-0002-1995-9743
M. Morín Castillo	https://orcid.org/0000-0002-9121-5917

Abstract

Corrections

Supplementary Material (3)

Cited By

Figures (9)

Tables (4)

Equations (18)

Applied Optics