Improved difference model applied in the Fourier-transform-based integration method based on Taylor theory

Xuanrui Gong; Zhuang Sun; Yaowen Lv; Yaowen Lv; Zhaoguo Jiang; Zhaoguo Jiang; Xiping Xu

doi:10.1364/AO.393949

Applied Optics
Vol. 59,
Issue 22,
pp. 6476-6483
(2020)
•https://doi.org/10.1364/AO.393949

Improved difference model applied in the Fourier-transform-based integration method based on Taylor theory

Xuanrui Gong, Zhuang Sun, Yaowen Lv, Zhaoguo Jiang, and Xiping Xu

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Xuanrui Gong, Zhuang Sun, Yaowen Lv, Zhaoguo Jiang, and Xiping Xu, "Improved difference model applied in the Fourier-transform-based integration method based on Taylor theory," Appl. Opt. 59, 6476-6483 (2020)

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Abstract

The two-dimensional Fourier-transform-based integration algorithm is widely used in shape or wavefront reconstruction from gradients. However, its reconstruction accuracy is limited by the truncation error of the difference model. The truncation error is affected by the distribution of the sampling points. It increases when the sampling points are unevenly distributed and arranged irregularly. For improving, a novel way to calculate the difference is proposed based on Taylor expansion theory of binary functions. The first-order partial derivative terms are used to estimate the second- and third-order partial derivative terms for reducing the truncation error. The proposed difference model is applied to Fourier-transform-based integration. The reconstruction results show that it can get better results when the sampling points are irregularly distributed.

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Unit: mm	Standard	Rotation	Barrel	Pillow
Our model	$7.71 \times 10^{- 7}$	$7.52 \times 10^{- 7}$	$8.88 \times 10^{- 7}$	$9.36 \times 10^{- 7}$
I-Southwell model	$4.70 \times 10^{- 8}$	$8.83 \times 10^{- 3}$	$4.85 \times 10^{- 4}$	$1.87 \times 10^{- 4}$
First-order model	$5.60 \times 10^{- 4}$	$5.50 \times 10^{- 4}$	$5.43 \times 10^{- 4}$	$5.54 \times 10^{- 4}$
Ren model	$2.15 \times 10^{- 5}$	$2.09 \times 10^{- 5}$	$2.04 \times 10^{- 5}$	$2.21 \times 10^{- 5}$

Unit: s	$64 \times 64$	$128 \times 128$	$256 \times 256$	$512 \times 512$
Our model	0.006	0.009	0.025	0.139
I-Southwell model	0.025	0.054	0.364	0.891
First-order model	0.021	0.049	0.286	0.798
Ren model	0.064	0.101	0.463	0.973

Unit: mm	Standard	Rotation	Barrel	Pillow
Our model	$5.53 \times 10^{- 9}$	$5.45 \times 10^{- 9}$	$2.88 \times 10^{- 7}$	$1.93 \times 10^{- 7}$
I-Southwell model	$1.79 \times 10^{- 12}$	$8.99 \times 10^{- 3}$	$9.99 \times 10^{- 4}$	$6.20 \times 10^{- 4}$
First-order model	$5.40 \times 10^{- 4}$	$5.39 \times 10^{- 4}$	$4.72 \times 10^{- 4}$	$5.81 \times 10^{- 4}$
Ren model	$3.70 \times 10^{- 7}$	$3.61 \times 10^{- 7}$	$1.10 \times 10^{- 6}$	$1.14 \times 10^{- 6}$

Unit: mm	Standard	Rotation	Barrel	Pillow
Our model	$9.75 \times 10^{- 8}$	$9.23 \times 10^{- 8}$	$9.62 \times 10^{- 8}$	$1.09 \times 10^{- 7}$
I-Southwell model	$6.28 \times 10^{- 9}$	$6.92 \times 10^{- 4}$	$2.06 \times 10^{- 5}$	$8.98 \times 10^{- 6}$
First-order model	$5.08 \times 10^{- 5}$	$5.01 \times 10^{- 5}$	$4.87 \times 10^{- 5}$	$5.20 \times 10^{- 5}$
Ren model	$4.83 \times 10^{- 6}$	$4.66 \times 10^{- 6}$	$4.60 \times 10^{- 6}$	$4.97 \times 10^{- 6}$

Unit: mm	Standard	Rotation	Barrel	Pillow
Our model	$1.05 \times 10^{- 10}$	$1.01 \times 10^{- 10}$	$1.15 \times 10^{- 8}$	$8.00 \times 10^{- 9}$
I-Southwell model	$1.24 \times 10^{- 13}$	$5.32 \times 10^{- 4}$	$4.75 \times 10^{- 5}$	$3.00 \times 10^{- 5}$
First-order model	$1.00 \times 10^{- 5}$	$1.00 \times 10^{- 5}$	$8.49 \times 10^{- 6}$	$1.10 \times 10^{- 5}$
Ren model	$3.10 \times 10^{- 8}$	$3.07 \times 10^{- 8}$	$9.54 \times 10^{- 8}$	$9.59 \times 10^{- 8}$

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Applied Optics