Fast and accurate computation of polar harmonic Fourier moments for image description

Siyu Yang; Ansheng Deng

doi:10.1364/JOSAA.494299

Journal of the Optical Society of America A
Vol. 40,
Issue 9,
pp. 1714-1723
(2023)
•https://doi.org/10.1364/JOSAA.494299

Fast and accurate computation of polar harmonic Fourier moments for image description

Siyu Yang and Ansheng Deng

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Siyu Yang and Ansheng Deng, "Fast and accurate computation of polar harmonic Fourier moments for image description," J. Opt. Soc. Am. A 40, 1714-1723 (2023)

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Related Topics
Table of Contents Category
- Image Processing and Image Analysis
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: May 15, 2023
- Revised Manuscript: July 23, 2023
- Manuscript Accepted: July 23, 2023
- Published: August 15, 2023

Abstract

Continuous orthogonal moments are widely used in various image techniques due to their simplicity and good rotational invariance and stability. In recent years, numerous excellent continuous orthogonal moments have been developed, among which polar harmonic Fourier moments (PHFMs) exhibit strong image description capabilities. However, the numerical integration error is large in the calculation, which seriously affects the calculation accuracy, especially in higher-order calculation. In this paper, a continuous orthogonal moments-fast and accurate PHFM (FAPHFM) is proposed. It utilizes the polar pixel tiling technique to reduce numerical errors in the computation; this method particularly improves the accuracy of higher-order moments of traditional PHFMs. However, as accuracy increases, calculation complexity also increases. To address this issue, an eight-way symmetric/anti-symmetric calculation of the angular and radial functions was performed using the symmetry and anti-symmetry of traditional PHFMs, and clustering of pixels was performed as a way to improve the computational speed. The experimental results show that FAPHFMs perform better in image reconstruction (including noise), with higher computational accuracy, lower time complexity, and better image description ability.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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$k$	$η_{k}$	$z_{k}$
1	0.06667134	−0.97390652
2	0.14945134	−0.86506336
3	0.21908636	−0.67940956
4	0.26926671	−0.43339539
5	0.29552422	−0.14887433
6	0.29552422	0.14887433
7	0.26926671	0.43339539
8	0.21908636	0.67940956
9	0.14945134	0.86506336
10	0.06667134	0.97390652

Mod	$P_{1}$	$P_{2}$	$P_{3}$	$P_{4}$	$P_{5}$	$P_{6}$	$P_{7}$	$P_{8}$
$(m, 4)$	$m θ_{ik}$	$m (\frac{π}{2} - θ_{ik})$	$m (\frac{π}{2} + θ_{ik})$	$m (π - θ_{ik})$	$m (π + θ_{ik})$	$m (\frac{3 π}{2} - θ_{ik})$	$m (\frac{3 π}{2} + θ_{ik})$	$m (2 π - θ_{ik})$
0	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$
1	$\cos (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$\cos (m θ_{ik})$
2	$\cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$\cos (m θ_{ik})$
3	$\cos (m θ_{ik})$	$- \sin (m θ_{ik})$	$\sin (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$\cos (m θ_{ik})$

Mod	$P_{1}$	$P_{2}$	$P_{3}$	$P_{4}$	$P_{5}$	$P_{6}$	$P_{7}$	$P_{8}$
$(m, 4)$	$m θ_{ik}$	$m (\frac{π}{2} - θ_{ik})$	$m (\frac{π}{2} + θ_{ik})$	$m (π - θ_{ik})$	$m (π + θ_{ik})$	$m (\frac{3 π}{2} - θ_{ik})$	$m (\frac{3 π}{2} + θ_{ik})$	$m (2 π - θ_{ik})$
0	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$
1	$\sin (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \sin (m θ_{ik})$
2	$\sin (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$\sin (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$- \sin (m θ_{ik})$
3	$\sin (m θ_{ik})$	$- \cos (m θ_{ik})$	$- \cos (m θ_{ik})$	$\sin (m θ_{ik})$	$- \sin (m θ_{ik})$	$\cos (m θ_{ik})$	$\cos (m θ_{ik})$	$- \sin (m θ_{ik})$

Axis of Symmetry	Symmetry Point	Angle
$X = Y$	$S_{1} (j, i)$	$π / 2 - θ$
$Y$ axis	$S_{2} (- i, j)$	$π - θ$
Origin	$S_{3} (- i, - j)$	$π + θ$
$X$ axis	$S_{4} (i, - j)$	$- θ$

	Decomposition Time			Reconstruction Time
Order	PHFMs (s)	GIN Method (s)	FAPHFMs (s)	PHFMs (s)	GIN Method (s)	FAPHFMs (s)
5	0.0764	0.0694	0.0421	0.2550	0.2146	0.1905
20	0.5961	0.2162	0.1210	0.6391	0.3062	0.2574
40	2.5232	0.9181	0.6667	2.6853	1.1422	0.6774
60	6.0878	2.6429	1.3894	6.5154	2.5418	1.4025
80	9.4036	5.4155	2.7292	9.5575	6.1383	2.8188
100	13.7386	10.1631	4.2991	14.4481	11.3144	4.5748

Abstract

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Tables (5)

Equations (33)

Journal of the Optical Society of America A