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Journals >Optoelectronic Technology >Volume 43 >Issue 4 >Page 332 > Article

Optoelectronic Technology
Vol. 43, Issue 4, 332 (2023)

An Image Pre‑distortion Algorithm Based on the Radial Basis Function

Xiaobo HUANG¹ and Limin YAN^1、2

Author Affiliations

¹School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, CHN

²Microelectronics Research and Development Center, Shanghai University, Shanghai 00444, CHN

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DOI: 10.19453/j.cnki.1005-488x.2023.04.010 Cite this Article

Xiaobo HUANG, Limin YAN. An Image Pre‑distortion Algorithm Based on the Radial Basis Function[J]. Optoelectronic Technology, 2023, 43(4): 332 Copy Citation Text

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Abstract

In order to correct distorted images in near-eye display， an image pre-distortion algorithm based on nonlinear fitting and forward mapping interpolation was proposed.The radial basis function fitting algorithm was used to get a more precise mapping relationship between the fitting original image and the pre-distorted image. The compactly supported radial basis function （CSRBF） interpolation algorithm based on forward mapping was used for image gray reconstruction to obtain better correction effect of image distortion. The experimental results showed that the fitting error of the radial basis function fitting algorithm could be reduced to less than 1 pixel， reaching to sub-pixel level. The pre-distortion algorithm had a good correction effect on the distortion produced by the near-eye optical system， which could make the display image clear and complete.

Keywords

image pre-distortion interpolation algorithm near-eye display radial basis function fitting algorithm

Q = F (P) = \sum_{i = 1}^{N} α_{i} φ_{i} (‖P - R_{N}‖) + O_{m} (P)

(1)

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φ (r) = (r^{2} + σ^{2})^{- 1 / 2}

(2)

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σ = \frac{d_{m a x}}{\sqrt[]{2 N}}

(3)

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σ_{j} = m i n_{j \neq i} ‖P_{j} - P_{i}‖

(4)

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Q = F (P) = \sum_{i = 1}^{N} α_{i} {({(‖P - R_{i}‖)}^{2} + σ_{i}^{2})}^{- 1 / 2} + O_{m} (P)

(5)

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\{\begin{matrix} Q_{x} - P_{x} = M α_{x} \\ Q_{y} - P_{y} = M α_{y} \end{matrix}

(6)

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M = [\begin{matrix} \frac{1}{\sqrt[]{(| | P_{1} - R_{1} {| |)}^{2} + {σ_{1}}^{2}}} & \frac{1}{\sqrt[]{(| | P_{1} - R_{2} {| |)}^{2} + {σ_{2}}^{2}}} & \dots & \frac{1}{\sqrt[]{(| | P_{1} - R_{N} {| |)}^{2} + {σ_{N}}^{2}}} \\ \frac{1}{\sqrt[]{(| | P_{2} - R_{1} {| |)}^{2} + {σ_{1}}^{2}}} & \frac{1}{\sqrt[]{(| | P_{2} - R_{2} {| |)}^{2} + {σ_{2}}^{2}}} & \dots & \frac{1}{\sqrt[]{(| | P_{2} - R_{N} {| |)}^{2} + {σ_{N}}^{2}}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \frac{1}{\sqrt[]{(| | P_{N} - R_{1} {| |)}^{2} + {σ_{1}}^{2}}} & \frac{1}{\sqrt[]{(| | P_{N} - R_{2} {| |)}^{2} + {σ_{2}}^{2}}} & \dots & \frac{1}{\sqrt[]{(| | P_{N} - R_{N} {| |)}^{2} + {σ_{N}}^{2}}} \end{matrix}]

(7)

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\{\begin{array}{l} α_{x} = M^{- 1} (Q_{x} - P_{x}) \\ α_{y} = M^{- 1} (Q_{y} - P_{y}) \end{array}

(8)

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ϕ (r) = \{\begin{matrix} {(1 - r)}^{α} p (r), \\ 0, \end{matrix} \begin{matrix} r < 1 \\ r \geq 1 \end{matrix}

(9)

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F = \sum_{i = 1}^{N} c_{i} ϕ_{i} (r)

(10)

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\begin{matrix} f_{i} = \sum_{i = 1}^{N} c_{i} ϕ_{i} ({r_{i}}_{j}), & i, j = 1,2, . . ., N \end{matrix}

(11)

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[\begin{matrix} ϕ (r_{11}) & ϕ (r_{12}) & \dots & ϕ (r_{1 N}) \\ ϕ (r_{21}) & ϕ (r_{22}) & \dots & ϕ (r_{2 N}) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ϕ (r_{N 1}) & ϕ (r_{N 2}) & \dots & ϕ (r_{N N}) \end{matrix}] [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{N} \end{matrix}] = [\begin{matrix} f_{1} \\ f_{2} \\ ⋮ \\ f_{N} \end{matrix}]

(12)

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A C = F

(13)

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\begin{matrix} f_{j} = \sum_{i = 1}^{N} c_{i} ϕ_{i} (r_{i j}), & j = 1,2, . . ., N \end{matrix}

(14)

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E_{m a x} = | | r_{m i} - r_{r i} | |_{m a x}

(15)

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E = \frac{1}{m \times n} \sum_{i = 1}^{m \times n} | | r_{m i} - r_{r i} | |

(16)

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Xiaobo HUANG, Limin YAN. An Image Pre‑distortion Algorithm Based on the Radial Basis Function[J]. Optoelectronic Technology, 2023, 43(4): 332

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