Abstract

For some time it has been known and recommended that the calculation of Zernike polynomials to radial orders higher than 8 to 10 should be performed using recurrence relations rather than explicit expressions due increasingly large cancellation errors. This paper presents a set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates as well. The recurrence relations are also well suited for the calculation of the Cartesian derivatives of the Zernike polynomials. The recurrence relations are easily extended to arbitrarily high orders. Assessments of the precision achievable with standard 64-bit floating point arithmetic show that Zernike polynomials up to radial order 30 can be calculated over the unit disc with errors not exceeding 5E-14, and up to radial order 50 with errors not exceeding 1.2E-13. Comparison with the Zernike capability in OpticStudio (Zemax) shows that the recurrence relations are superior in performance (both speed and precision) over the existing algorithm implemented in the software. General pseudo-code for the calculation of Zernike polynomials and their derivatives is also presented.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

E. T. Kvamme, D. M. Stubbs, and M. S. Jacoby, “The opto-mechanical design process: from vision to reality,” Proc. SPIE 10371, 103710P (2017).

J. Schwiegerling, “Review of Zernike polynomials and their use in describing the impact of misalignment in optical systems,” Proc. SPIE 10377, 103770D (2017).

2016 (2)

S. N. Bezdidko, “Orthogonal aberrations: theory, methods, and practical applications in computational optics,” J. Opt. Technol. 63(6), 351–359 (2016).
[Crossref]

T. Tanabe, M. Shibuya, and K. Maehara, “Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces,” Opt. Eng. 55(3), 035101 (2016).
[Crossref]

2014 (5)

2013 (3)

2012 (1)

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

2011 (1)

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58(7), 545–561 (2011).
[Crossref]

2010 (3)

C. Singh and E. Walla, “Fast and numerically stable methods for the computation of Zernike moments,” J. Patt. Recogn. 43(7), 2497–2506 (2010).
[Crossref]

K. M. Hosny, “A systematic method for efficient computation of full and subsets Zernike moments,” Inf. Sci. 180(11), 2299–2313 (2010).
[Crossref]

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010).
[Crossref] [PubMed]

2009 (1)

R. J. Mathar, “Zernike basis to Cartesian transformations,” Serb. Astron. J. 179(179), 107–120 (2009).
[Crossref]

2008 (1)

G. A. Papakostas, Y. S. Boutalis, C. N. Papaodysseus, and D. K. Fragoulis, “Numerical stability of fast computation algorithms of Zernike moments,” Appl. Math. Comput. 195(1), 326–345 (2008).
[Crossref]

2007 (3)

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. European Opt. Soc. 2, 07012 (2007).
[Crossref]

S. A. Comastri, L. I. Perez, G. D. Pérez, G. Martin, and K. Bastida, “Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations,” J. Opt. A, Pure Appl. Opt. 9(3), 209–221 (2007).
[Crossref]

C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials,” Opt. Express 15(26), 18014–18024 (2007).
[Crossref] [PubMed]

2006 (2)

Y.-H. Pang, A. B. J. Teoh, and D. C. L. Ngo, “A discriminant pseudo Zernike moments in face recognition,” J. Res. Pract. In Inform. Techn. 38(2), 197–211 (2006).

S.-K. Hwang and W.-Y. Kim, “A novel approach to the fast computation of Zernike moments,” Pattern Recognit. 39(11), 2065–2076 (2006).
[Crossref]

2004 (2)

D. A. Atchison, “Recent advances in representation of monochromatic aberrations of human eyes,” Clin. Exp. Optom. 87(3), 138–148 (2004).
[Crossref] [PubMed]

C. J. R. Sheppard, S. Campbell, and M. D. Hirschhorn, “Zernike expansion of separable functions of Cartesian coordinates,” Appl. Opt. 43(20), 3963–3966 (2004).
[Crossref] [PubMed]

2003 (2)

C.-W. Chong, R. Mukundan, and P. Raveendran, “An efficient algorithm for fast computation of pseudo-Zernike moments,” Int. J. Pattern Recognit. Artif. Intell. 17(6), 1011–1016 (2003).
[Crossref]

S. N. Bezdidko, “Study of the properties of Zernike’s orthogonal polynomials,” Proc. SPIE 5174, 227–234 (2003).
[Crossref]

2002 (3)

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, “Modeling of corneal surfaces with radial polynomials,” IEEE Trans. Biomed. Eng. 49(4), 320–328 (2002).
[Crossref] [PubMed]

V. L. Genberg, G. J. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002).
[Crossref]

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Patt. Recogn. 35(12), 2905–2911 (2002).
[Crossref]

2001 (1)

P. Forney, “Integrated optical design,” Proc. SPIE 4441, 53–59 (2001).
[Crossref]

1997 (1)

H. H. van Brug, “Efficient Cartesian representation of Zernike polynomials in computer memory,” Proc. SPIE 3190, 382–392 (1997).
[Crossref]

1994 (1)

1992 (2)

1990 (1)

D. Malacara, J. M. Carpio-Valadéz, and J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

1989 (1)

1982 (1)

1980 (1)

1976 (2)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976).
[Crossref]

E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23(8), 679–680 (1976).
[Crossref]

1975 (1)

1954 (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[Crossref]

1951 (1)

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14(1), 95–120 (1951).
[Crossref]

1934 (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrast-methode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Atchison, D. A.

D. A. Atchison, “Recent advances in representation of monochromatic aberrations of human eyes,” Clin. Exp. Optom. 87(3), 138–148 (2004).
[Crossref] [PubMed]

Báez-Rojas, J. J.

Bastida, K.

S. A. Comastri, L. I. Perez, G. D. Pérez, G. Martin, and K. Bastida, “Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations,” J. Opt. A, Pure Appl. Opt. 9(3), 209–221 (2007).
[Crossref]

Bezdidko, S. N.

S. N. Bezdidko, “Orthogonal aberrations: theory, methods, and practical applications in computational optics,” J. Opt. Technol. 63(6), 351–359 (2016).
[Crossref]

S. N. Bezdidko, “Study of the properties of Zernike’s orthogonal polynomials,” Proc. SPIE 5174, 227–234 (2003).
[Crossref]

Bhatia, A. B.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[Crossref]

Born, M.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[Crossref]

Boutalis, Y. S.

G. A. Papakostas, Y. S. Boutalis, C. N. Papaodysseus, and D. K. Fragoulis, “Numerical stability of fast computation algorithms of Zernike moments,” Appl. Math. Comput. 195(1), 326–345 (2008).
[Crossref]

Braat, J. J. M.

Burge, J. H.

Camacho-Bello, C.

Cámara-Chávez, G.

K. C. Otiniano-Rodríguez, G. Cámara-Chávez, and D. Menotti, “Hu and Zernike moments for sign language recognition,” in 2012 International Conference on Image Processing, Computer Vision, and Pattern Recognition, IPCV'12 (2012), pp. 1–5.

Campbell, S.

Carpio-Valadéz, J. M.

D. Malacara, J. M. Carpio-Valadéz, and J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Chong, C.-W.

C.-W. Chong, R. Mukundan, and P. Raveendran, “An efficient algorithm for fast computation of pseudo-Zernike moments,” Int. J. Pattern Recognit. Artif. Intell. 17(6), 1011–1016 (2003).
[Crossref]

Collins, M. J.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, “Modeling of corneal surfaces with radial polynomials,” IEEE Trans. Biomed. Eng. 49(4), 320–328 (2002).
[Crossref] [PubMed]

Comastri, S. A.

S. A. Comastri, L. I. Perez, G. D. Pérez, G. Martin, and K. Bastida, “Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations,” J. Opt. A, Pure Appl. Opt. 9(3), 209–221 (2007).
[Crossref]

Davis, B.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, “Modeling of corneal surfaces with radial polynomials,” IEEE Trans. Biomed. Eng. 49(4), 320–328 (2002).
[Crossref] [PubMed]

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. European Opt. Soc. 2, 07012 (2007).
[Crossref]

Doyle, K. B.

V. L. Genberg, G. J. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002).
[Crossref]

Fleck, A.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58(7), 545–561 (2011).
[Crossref]

Forbes, G. W.

Forney, P.

P. Forney, “Integrated optical design,” Proc. SPIE 4441, 53–59 (2001).
[Crossref]

Fragoulis, D. K.

G. A. Papakostas, Y. S. Boutalis, C. N. Papaodysseus, and D. K. Fragoulis, “Numerical stability of fast computation algorithms of Zernike moments,” Appl. Math. Comput. 195(1), 326–345 (2008).
[Crossref]

Gavrielides, A.

Genberg, V. L.

V. L. Genberg, G. J. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002).
[Crossref]

Gu, J.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Patt. Recogn. 35(12), 2905–2911 (2002).
[Crossref]

Hertzberg, Y.

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Hirschhorn, M. D.

Honarvar Shakibaei, B.

Hosny, K.

K. Hosny, “Fast computation of accurate pseudo Zernike moments for binary and gray-level images,” Int. Arab J. Inf. Technol. 11(3), 243–249 (2014).

Hosny, K. M.

K. M. Hosny, “A systematic method for efficient computation of full and subsets Zernike moments,” Inf. Sci. 180(11), 2299–2313 (2010).
[Crossref]

Hwang, S.-K.

S.-K. Hwang and W.-Y. Kim, “A novel approach to the fast computation of Zernike moments,” Pattern Recognit. 39(11), 2065–2076 (2006).
[Crossref]

Iskander, D. R.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, “Modeling of corneal surfaces with radial polynomials,” IEEE Trans. Biomed. Eng. 49(4), 320–328 (2002).
[Crossref] [PubMed]

Jacoby, M. S.

E. T. Kvamme, D. M. Stubbs, and M. S. Jacoby, “The opto-mechanical design process: from vision to reality,” Proc. SPIE 10371, 103710P (2017).

Janssen, A. J. E. M.

Kaye, E. A.

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Kim, W.-Y.

S.-K. Hwang and W.-Y. Kim, “A novel approach to the fast computation of Zernike moments,” Pattern Recognit. 39(11), 2065–2076 (2006).
[Crossref]

Kintner, E. C.

E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23(8), 679–680 (1976).
[Crossref]

Kvamme, E. T.

E. T. Kvamme, D. M. Stubbs, and M. S. Jacoby, “The opto-mechanical design process: from vision to reality,” Proc. SPIE 10371, 103710P (2017).

Lakshminarayanan, V.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58(7), 545–561 (2011).
[Crossref]

Levoy, M.

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Loomis, J. S.

Luo, L. M.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Patt. Recogn. 35(12), 2905–2911 (2002).
[Crossref]

Maehara, K.

T. Tanabe, M. Shibuya, and K. Maehara, “Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces,” Opt. Eng. 55(3), 035101 (2016).
[Crossref]

Mahajan, V. N.

Malacara, D.

D. Malacara, J. M. Carpio-Valadéz, and J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Martin, G.

S. A. Comastri, L. I. Perez, G. D. Pérez, G. Martin, and K. Bastida, “Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations,” J. Opt. A, Pure Appl. Opt. 9(3), 209–221 (2007).
[Crossref]

Marx, M.

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Mathar, R. J.

R. J. Mathar, “Zernike basis to Cartesian transformations,” Serb. Astron. J. 179(179), 107–120 (2009).
[Crossref]

Menotti, D.

K. C. Otiniano-Rodríguez, G. Cámara-Chávez, and D. Menotti, “Hu and Zernike moments for sign language recognition,” in 2012 International Conference on Image Processing, Computer Vision, and Pattern Recognition, IPCV'12 (2012), pp. 1–5.

Michels, G. J.

V. L. Genberg, G. J. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002).
[Crossref]

Mikš, A.

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014).
[Crossref]

Morelande, M. R.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, “Modeling of corneal surfaces with radial polynomials,” IEEE Trans. Biomed. Eng. 49(4), 320–328 (2002).
[Crossref] [PubMed]

Mukundan, R.

C.-W. Chong, R. Mukundan, and P. Raveendran, “An efficient algorithm for fast computation of pseudo-Zernike moments,” Int. J. Pattern Recognit. Artif. Intell. 17(6), 1011–1016 (2003).
[Crossref]

Navon, G.

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Ngo, D. C. L.

Y.-H. Pang, A. B. J. Teoh, and D. C. L. Ngo, “A discriminant pseudo Zernike moments in face recognition,” J. Res. Pract. In Inform. Techn. 38(2), 197–211 (2006).

Noll, R. J.

Novák, J.

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014).
[Crossref]

P. Novák and J. Novák, “Efficient and stable numerical method for evaluation of Zernike polynomials and their Cartesian derivatives,” Proc. SPIE 8789, 878913 (2013).
[Crossref]

Novák, P.

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014).
[Crossref]

P. Novák and J. Novák, “Efficient and stable numerical method for evaluation of Zernike polynomials and their Cartesian derivatives,” Proc. SPIE 8789, 878913 (2013).
[Crossref]

Otiniano-Rodríguez, K. C.

K. C. Otiniano-Rodríguez, G. Cámara-Chávez, and D. Menotti, “Hu and Zernike moments for sign language recognition,” in 2012 International Conference on Image Processing, Computer Vision, and Pattern Recognition, IPCV'12 (2012), pp. 1–5.

Padilla-Vivanco, A.

Pang, Y.-H.

Y.-H. Pang, A. B. J. Teoh, and D. C. L. Ngo, “A discriminant pseudo Zernike moments in face recognition,” J. Res. Pract. In Inform. Techn. 38(2), 197–211 (2006).

Papakostas, G. A.

G. A. Papakostas, Y. S. Boutalis, C. N. Papaodysseus, and D. K. Fragoulis, “Numerical stability of fast computation algorithms of Zernike moments,” Appl. Math. Comput. 195(1), 326–345 (2008).
[Crossref]

Papaodysseus, C. N.

G. A. Papakostas, Y. S. Boutalis, C. N. Papaodysseus, and D. K. Fragoulis, “Numerical stability of fast computation algorithms of Zernike moments,” Appl. Math. Comput. 195(1), 326–345 (2008).
[Crossref]

Paramesran, R.

Pauly, K. B.

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Perez, L. I.

S. A. Comastri, L. I. Perez, G. D. Pérez, G. Martin, and K. Bastida, “Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations,” J. Opt. A, Pure Appl. Opt. 9(3), 209–221 (2007).
[Crossref]

Pérez, G. D.

S. A. Comastri, L. I. Perez, G. D. Pérez, G. Martin, and K. Bastida, “Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations,” J. Opt. A, Pure Appl. Opt. 9(3), 209–221 (2007).
[Crossref]

Prata, A.

Raveendran, P.

C.-W. Chong, R. Mukundan, and P. Raveendran, “An efficient algorithm for fast computation of pseudo-Zernike moments,” Int. J. Pattern Recognit. Artif. Intell. 17(6), 1011–1016 (2003).
[Crossref]

Rayces, J. L.

Rimmer, M. P.

Rusch, W. V. T.

Sánchez-Mondragón, J. J.

D. Malacara, J. M. Carpio-Valadéz, and J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Schwiegerling, J.

J. Schwiegerling, “Review of Zernike polynomials and their use in describing the impact of misalignment in optical systems,” Proc. SPIE 10377, 103770D (2017).

Sheppard, C. J. R.

Shibuya, M.

T. Tanabe, M. Shibuya, and K. Maehara, “Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces,” Opt. Eng. 55(3), 035101 (2016).
[Crossref]

Shu, H. Z.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Patt. Recogn. 35(12), 2905–2911 (2002).
[Crossref]

Silva, D. E.

Singh, C.

C. Singh and E. Walla, “Fast and numerically stable methods for the computation of Zernike moments,” J. Patt. Recogn. 43(7), 2497–2506 (2010).
[Crossref]

Stephenson, P. C. L.

Stubbs, D. M.

E. T. Kvamme, D. M. Stubbs, and M. S. Jacoby, “The opto-mechanical design process: from vision to reality,” Proc. SPIE 10371, 103710P (2017).

Tanabe, T.

T. Tanabe, M. Shibuya, and K. Maehara, “Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces,” Opt. Eng. 55(3), 035101 (2016).
[Crossref]

Teoh, A. B. J.

Y.-H. Pang, A. B. J. Teoh, and D. C. L. Ngo, “A discriminant pseudo Zernike moments in face recognition,” J. Res. Pract. In Inform. Techn. 38(2), 197–211 (2006).

Toumoulin, C.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Patt. Recogn. 35(12), 2905–2911 (2002).
[Crossref]

Toxqui-Quitl, C.

van Brug, H. H.

H. H. van Brug, “Efficient Cartesian representation of Zernike polynomials in computer memory,” Proc. SPIE 3190, 382–392 (1997).
[Crossref]

Walla, E.

C. Singh and E. Walla, “Fast and numerically stable methods for the computation of Zernike moments,” J. Patt. Recogn. 43(7), 2497–2506 (2010).
[Crossref]

Wang, J. Y.

Werner, B.

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Wolf, E.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[Crossref]

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14(1), 95–120 (1951).
[Crossref]

Wyant, J. C.

Zernike, F.

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrast-methode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Zhao, C.

Appl. Math. Comput. (1)

G. A. Papakostas, Y. S. Boutalis, C. N. Papaodysseus, and D. K. Fragoulis, “Numerical stability of fast computation algorithms of Zernike moments,” Appl. Math. Comput. 195(1), 326–345 (2008).
[Crossref]

Appl. Opt. (7)

Clin. Exp. Optom. (1)

D. A. Atchison, “Recent advances in representation of monochromatic aberrations of human eyes,” Clin. Exp. Optom. 87(3), 138–148 (2004).
[Crossref] [PubMed]

IEEE Trans. Biomed. Eng. (1)

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, “Modeling of corneal surfaces with radial polynomials,” IEEE Trans. Biomed. Eng. 49(4), 320–328 (2002).
[Crossref] [PubMed]

Inf. Sci. (1)

K. M. Hosny, “A systematic method for efficient computation of full and subsets Zernike moments,” Inf. Sci. 180(11), 2299–2313 (2010).
[Crossref]

Int. Arab J. Inf. Technol. (1)

K. Hosny, “Fast computation of accurate pseudo Zernike moments for binary and gray-level images,” Int. Arab J. Inf. Technol. 11(3), 243–249 (2014).

Int. J. Pattern Recognit. Artif. Intell. (1)

C.-W. Chong, R. Mukundan, and P. Raveendran, “An efficient algorithm for fast computation of pseudo-Zernike moments,” Int. J. Pattern Recognit. Artif. Intell. 17(6), 1011–1016 (2003).
[Crossref]

J. European Opt. Soc. (1)

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. European Opt. Soc. 2, 07012 (2007).
[Crossref]

J. Mod. Opt. (1)

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58(7), 545–561 (2011).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

S. A. Comastri, L. I. Perez, G. D. Pérez, G. Martin, and K. Bastida, “Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations,” J. Opt. A, Pure Appl. Opt. 9(3), 209–221 (2007).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Opt. Technol. (1)

S. N. Bezdidko, “Orthogonal aberrations: theory, methods, and practical applications in computational optics,” J. Opt. Technol. 63(6), 351–359 (2016).
[Crossref]

J. Patt. Recogn. (1)

C. Singh and E. Walla, “Fast and numerically stable methods for the computation of Zernike moments,” J. Patt. Recogn. 43(7), 2497–2506 (2010).
[Crossref]

J. Res. Pract. In Inform. Techn. (1)

Y.-H. Pang, A. B. J. Teoh, and D. C. L. Ngo, “A discriminant pseudo Zernike moments in face recognition,” J. Res. Pract. In Inform. Techn. 38(2), 197–211 (2006).

Math. Proc. Camb. Philos. Soc. (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Camb. Philos. Soc. 50(01), 40–48 (1954).
[Crossref]

Med. Phys. (1)

E. A. Kaye, Y. Hertzberg, M. Marx, B. Werner, G. Navon, M. Levoy, and K. B. Pauly, “Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound,” Med. Phys. 39(10), 6254–6263 (2012).
[Crossref] [PubMed]

Opt. Acta (Lond.) (1)

E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23(8), 679–680 (1976).
[Crossref]

Opt. Eng. (2)

D. Malacara, J. M. Carpio-Valadéz, and J. J. Sánchez-Mondragón, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

T. Tanabe, M. Shibuya, and K. Maehara, “Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces,” Opt. Eng. 55(3), 035101 (2016).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014).
[Crossref]

Opt. Lett. (2)

Patt. Recogn. (1)

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Patt. Recogn. 35(12), 2905–2911 (2002).
[Crossref]

Pattern Recognit. (1)

S.-K. Hwang and W.-Y. Kim, “A novel approach to the fast computation of Zernike moments,” Pattern Recognit. 39(11), 2065–2076 (2006).
[Crossref]

Physica (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrast-methode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Proc. SPIE (7)

P. Forney, “Integrated optical design,” Proc. SPIE 4441, 53–59 (2001).
[Crossref]

E. T. Kvamme, D. M. Stubbs, and M. S. Jacoby, “The opto-mechanical design process: from vision to reality,” Proc. SPIE 10371, 103710P (2017).

H. H. van Brug, “Efficient Cartesian representation of Zernike polynomials in computer memory,” Proc. SPIE 3190, 382–392 (1997).
[Crossref]

V. L. Genberg, G. J. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002).
[Crossref]

P. Novák and J. Novák, “Efficient and stable numerical method for evaluation of Zernike polynomials and their Cartesian derivatives,” Proc. SPIE 8789, 878913 (2013).
[Crossref]

S. N. Bezdidko, “Study of the properties of Zernike’s orthogonal polynomials,” Proc. SPIE 5174, 227–234 (2003).
[Crossref]

J. Schwiegerling, “Review of Zernike polynomials and their use in describing the impact of misalignment in optical systems,” Proc. SPIE 10377, 103770D (2017).

Rep. Prog. Phys. (1)

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14(1), 95–120 (1951).
[Crossref]

Serb. Astron. J. (1)

R. J. Mathar, “Zernike basis to Cartesian transformations,” Serb. Astron. J. 179(179), 107–120 (2009).
[Crossref]

Other (12)

K. C. Otiniano-Rodríguez, G. Cámara-Chávez, and D. Menotti, “Hu and Zernike moments for sign language recognition,” in 2012 International Conference on Image Processing, Computer Vision, and Pattern Recognition, IPCV'12 (2012), pp. 1–5.

D. Malacara, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (John Wiley & Sons, 1992), Appendix 2, pp 489–503.

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley-Interscience, 2007).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

E. H. Linfoot, Recent Advances in Optics (Clarendon, 1955).

K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis, 2nd ed. (SPIE, 2012).

D. A. Atchison, D. H. Scott, and M. J. Cox, “Mathematical treatment of ocular aberrations: a user’s guide,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., 35, 110–130 (2000).

J. Loomis, “A computer program for analysis of interferometric data,” in Optical Interferograms – Reduction and Interpretation, American Society for Testing and Materials Special Technical Publication 666, A. H. Guenther and D. H. Liebenberg, ed., (Philadelphia, 1978), 71–86.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, (Dover Publications, 1972).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, (Academic Press, 1980).

OpticStudio is a registered trademark of Zemax LLC, www.zemax.com

T. B. Andersen, “Zernike_pol_algorithm.txt”, figshare (2018) [retrieved 02 April 2018], https://doi.org/10.6084/m9.figshare.6073898 .

Supplementary Material (1)

NameDescription
» Code 1       Pseudo-code to calculate unit-normalized Zernike polynomials and their x,y derivatives

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Figures (7)

Fig. 1
Fig. 1 Estimated maximum error in calculating Zernike polynomials over the unit disc. Solid blue: conservative estimate using R -polynomials. Solid red: conservative estimate using A -polynomials. Dashed blue: rss-estimate using R -polynomials. Dashed red: rss-estimate using A -polynomials. Green: using A -polynomials (Sec. 4). Purple: using Cartesian recurrence relations (Sec. 5).
Fig. 2
Fig. 2 Maximum error over the unit disc for individual Zernike polynomials with radial orders up to 30. The separate curves show the trends for the even-ordered rotationally symmetric polynomials (rad. symm.), and for the odd-ordered sine and cosine polynomials where the maximum errors are the largest.
Fig. 3
Fig. 3 Maximum error over the unit disc for individual Zernike polynomials with radial orders up to 50. The separate curve shows the trends for the even-ordered rotationally symmetric polynomials (rad. symm.).
Fig. 4
Fig. 4 Errors in individual Zernike Standard polynomials evaluated by OpticStudio at selected points in the unit disc. a) at the points (0.663,-0.396), (0.5,0.5). b) at the two points in a) plus the point (−0.873,0.485). The values are for σ -normalized polynomials, using the linear numbering scheme by Noll [9].
Fig. 5
Fig. 5 Surface profile and sag errors for Zernike surface in OpticStudio. Left: the surface profile with values between −14.4 and + 26.8. Right: 10-based logarithm of the absolute value of the difference between the Standard Surface sag and the user-defined surface sag. The log-values are in the range −25 to −9. The differences themselves were between −9.62E-10 and + 7.47E-10. The grid-size for both plots is 501x501 data points.
Fig. 6
Fig. 6 Comparison of performance in speed between OpticStudio Zernike Standard Sag surface and user-defined surface using Zernike recurrence relations. Left: The surface profile generated in OpticStudio (same as Fig. 5, left). Right: CPU-times for generating 2D grids of surface sag data for different sizes of the square grid-side. The ratio between the CPU times for the Zernike Standard Sag surface and the user-defined surface is also shown.
Fig. 7
Fig. 7 Recursive formulas for unnormalized Zernike polynomials and their derivatives

Equations (54)

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x 2 + y 2 1                       .
x = r cos θ   , y = r sin θ
U n m ( r , θ ) = { R n | n 2 m | ( r ) sin ( n 2 m ) θ   ,   f o r   n 2 m > 0 R n | n 2 m | ( r ) cos ( n 2 m ) θ   ,   f o r   n 2 m 0                       ,
R n μ ( r ) = k = 0 n μ 2 ( 1 ) k ( n k ) ! k ! ( n μ 2 k ) ! ( n + μ 2 k ) ! r n 2 k , n 0   ,   0 μ n   ,   n μ   e v e n
max   C | U n m | = 1                       .
Z n m ( r , θ ) = N n m U n m ( r , θ )                       ,
N n m = ( 2 δ n , ( 2 m ) ) ( n + 1 )
W ( r , θ ) = n = 0 n m a x m = 0 n a n m U n m ( r , θ )
var C W = n = 0 n m a x m = 0 n a n m 2 N n m 2                       .
R n μ ( r ) = 2 π ( 1 ) ( n μ ) / 2 0 J n + 1 ( 2 π k ) J μ ( 2 π r k ) d k = ( 1 ) ( n μ ) / 2 0 J n + 1 ( x ) J μ ( r x ) d x                                                                   ,
J n 1 ( x ) J n + 1 ( x ) = 2 J n ' ( x )                       ,
R n μ ( r ) + R n 2 μ ( r ) = ( 1 ) ( n μ ) / 2 0 J n + 1 ( x ) J μ ( r x ) d x + ( 1 ) ( n 2 μ ) / 2 0 J n 1 ( x ) J μ ( r x ) d x = ( 1 ) ( n μ ) / 2 0 ( J n + 1 ( x ) J n 1 ( x ) ) J μ ( r x ) d x = ( 1 ) ( n μ ) / 2 0 2 J n ' ( x ) J μ ( r x ) d x = 2 ( 1 ) ( n μ ) / 2 [ J n ( x ) J μ ( r x ) ] 0 + 2 ( 1 ) ( n μ ) / 2 0 J n ( x ) r J μ ' ( r x ) d x = ( 1 ) ( n μ ) / 2 0 J n ( x ) r ( J μ 1 ( r x ) J μ + 1 ( r x ) ) d x = ( 1 ) ( n μ ) / 2 0 J n ( x ) r J μ 1 ( r x ) d x ( 1 ) ( n μ ) / 2 0 J n ( x ) r J μ + 1 ( r x ) d x = ( 1 ) n 1 ( μ 1 ) 2 0 J n ( x ) r J μ 1 ( r x ) d x + ( 1 ) n 1 ( μ + 1 ) 2 0 J n ( x ) r J μ + 1 ( r x ) d x = r R n 1 μ 1 ( r ) + r R n 1 μ + 1 ( r )                       .
R n 0 ( r ) + R n 2 0 ( r ) = ( 1 ) n / 2 0 J n + 1 ( x ) J 0 ( r x ) d x + ( 1 ) ( n 2 ) / 2 0 J n 1 ( x ) J 0 ( r x ) d x = ( 1 ) n / 2 0 ( J n + 1 ( x ) J n 1 ( x ) ) J 0 ( r x ) d x = ( 1 ) n / 2 0 2 J n ' ( x ) J 0 ( r x ) d x = 2 ( 1 ) n / 2 [ J n ( x ) J 0 ( r x ) ] 0 + 2 ( 1 ) n / 2 0 J n ( x ) r J 0 ' ( r x ) d x = 2 ( 1 ) n / 2 0 J n ( x ) r J 1 ( r x ) d x = 2 ( 1 ) ( n 1 1 ) / 2 0 J n ( x ) r J 1 ( r x ) d x = 2 r R n 1 1 ( r )         .
R n 2 n ( r ) = 0 J n 1 ( x ) J n ( r x ) d x = { 0 , f o r 0 r < 1 1 2 , f o r r = 1                             ,
R n μ ( r ) =   r R n 1 | μ 1 | ( r ) + r R n 1 μ + 1 ( r ) R n 2 μ ( r )
R 0 0 ( r ) = 1       ,         R 1 1 ( r ) = r               ,
( cos m θ sin m θ ) = ( cos θ sin θ sin θ cos θ ) ( cos ( m 1 ) θ sin ( m 1 ) θ )               .
r = x 2 + y 2     ,     cos θ = x r     ,     sin θ = y r
μ = n 2 m               .
R n μ ( r ) =   r ( R n 1 μ 1 ( r ) + R n 1 μ + 1 ( r ) ) R n 2 μ ( r )
R n n ( r ) = r R n 1 n 1 ( r )       ,   f o r   μ = n               ,
R n 0 ( r ) = 2 r R n 1 1 ( r ) R n 2 0 ( r )     ,   f o r   μ = 0               .
R n μ ( r ) = r μ A n μ ( r )               .
U n m ( r , θ ) = { A n | n 2 m | ( r ) r | n 2 m | sin ( n 2 m ) θ   ,   f o r   n 2 m > 0 A n | n 2 m | ( r ) r | n 2 m | cos ( n 2 m ) θ   ,   f o r   n 2 m 0               ,
( r m cos m θ r m   sin m θ ) = ( r cos θ r sin θ r sin θ r cos θ ) ( r m 1 cos ( m 1 ) θ r m 1 sin ( m 1 ) θ ) = ( x y y x ) ( r m 1 cos ( m 1 ) θ r m 1 sin ( m 1 ) θ )                                                                                 .
A n μ ( r ) =   A n 1 μ 1 ( r ) + r 2 A n 1 μ + 1 ( r ) A n 2 μ ( r )
A n n ( r ) = A n 1 n 1 ( r )       ,   f o r   μ = n             ,
A n 0 ( r ) = 2 r 2 A n 1 1 ( r ) A n 2 0 ( r )     ,   f o r   μ = 0               .
A 0 0 ( r ) = 1       ,         A 1 1 ( r ) = 1                         .
U n m = R n μ ( r ) sin μ θ = r R n 1 μ 1 ( r ) sin μ θ + r R n 1 μ + 1 ( r ) sin μ θ R n 2 μ ( r ) sin μ θ = r R n 1 μ 1 ( r ) sin ( ( μ 1 ) θ + θ ) + r R n 1 μ + 1 ( r ) sin ( ( μ + 1 ) θ θ ) R n 2 μ ( r ) sin μ θ = r R n 1 μ 1 ( r ) ( sin ( μ 1 ) θ cos θ + cos ( μ 1 ) θ sin θ ) + r R n 1 μ + 1 ( r ) ( sin ( μ + 1 ) θ cos θ cos ( μ + 1 ) θ sin θ ) R n 2 μ ( r ) sin μ θ = x U n 1 , m + y U n 1 , n 1 m + x U n 1 , m 1 y U n 1 , n m U n 2 , m 1
U n m = R n μ ( r ) cos μ θ = r R n 1 μ 1 ( r ) cos μ θ + r R n 1 μ + 1 ( r ) cos μ θ R n 2 μ ( r ) cos μ θ = r R n 1 μ 1 ( r ) cos ( ( μ 1 ) θ + θ ) + r R n 1 μ + 1 ( r ) cos ( ( μ + 1 ) θ θ ) R n 2 μ ( r ) cos μ θ = r R n 1 μ 1 ( r ) ( cos ( μ 1 ) θ cos θ sin ( μ 1 ) θ sin θ ) + r R n 1 μ + 1 ( r ) ( cos ( μ + 1 ) θ cos θ + sin ( μ + 1 ) θ sin θ ) R n 2 μ ( r ) cos μ θ = x U n 1 , m + y U n 1 , n 1 m + x U n 1 , m 1 y U n 1 , n m U n 2 , m 1
U n , m = x U n 1 , m + y U n 1 , n 1 m + x U n 1 , m 1 y U n 1 , n m U n 2 , m 1
U n , 0 = x U n 1 , 0 + y U n 1 , n 1   ,   f o r   m = 0
U n , n = x U n 1 , n 1 y U n 1 , 0   ,   f o r   m = n
U n , m = y U n 1 , n 1 m + x U n 1 , m 1 y U n 1 , n m U n 2 , m 1 ,   f o r   n   o d d   a n d   m = n 1 2
U n , m = x U n 1 , m + y U n 1 , n 1 m + x U n 1 , m 1 U n 2 , m 1 ,   f o r   n   o d d   a n d   m = n 1 2 + 1
U n , m = 2 x U n 1 , m + 2 y U n 1 , m 1 U n 2 , m 1   ,   f o r   n   e v e n   a n d   m = n 2
U 00 = 1     ,     U 10 = y     ,     U 11 = x                 .
Z n μ ( x , y ) = R n μ ( cos μ θ + i sin μ θ )
Z n μ x = Z n 2 μ x + n ( Z n 1 μ 1 + Z n 1 μ + 1 )
Z n μ y = Z n 2 μ y + i n ( Z n 1 μ 1 Z n 1 μ + 1 )               .
Z n μ = U n , n + μ 2 + i U n , n μ 2
U n , n m x = n U n 1 , n m 1 + n U n 1 , n m + U n 2 , n m 1 x
U n , m x = n U n 1 , m + n U n 1 , m 1 + U n 2 , m 1 x                               .
U n , m y = n U n 1 , n m 1 n U n 1 , n m + U n 2 , m 1 y                               .
U n , 0 x = n U n 1 , 0 U n , 0 y = n U n 1 , n 1 }   f o r   m = 0                     ,
U n , n x = n U n 1 , n 1 U n , n y = n U n 1 , 0 }   f o r   m = n                     ,
U n , m x = n U n 1 , m 1 + U n 2 , m 1 x U n , m y = n U n 1 , n m 1 n U n 1 , n m + U n 2 , m 1 y }   f o r   n   o d d   a n d   m = n 1 2           ,
U n , m x = n U n 1 , m + n U n 1 , m 1 + U n 2 , m 1 x U n , m y = n U n 1 , n m 1 + U n 2 , m 1 y }   f o r   n   o d d   a n d   m = n 1 2 + 1   ,
U n , m x = 2 n U n 1 , m + U n 2 , m 1 x U n , m y = 2 n U n 1 , n m 1 + U n 2 , m 1 y }   f o r   n   e v e n   a n d   m = n 2             .
U 00 x = U 00 y = 0   ,     U 10 x = U 11 y = 0   ,     U 11 x = U 10 y = 1         .
ε a n g = 1.281 10 15 m                                   .
f ( x , y ) = n = 0 20 m = 0 n a n m U n m ( x , y )
a n m = sin ( 100 m n 2 + 0.1 n + 1 )                     .

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