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Accurate Computation of Orthogonal Fourier-Mellin Moments

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Abstract

Orthogonal Fourier-Mellin moments (OFMMs) suffer from geometric error and the numerical integration error. The geometric error arises when the square image is mapped into a unit disk and the mapping does not become perfect. The numerical integration error arises when the double integration is approximated by the zeroth order summation. In this paper, we propose methods which reduce these errors. The geometric error is reduced by considering the arc-grids lying on the boundary of the unit disk and the square grids lying completely inside the disk. The numerical integration error is reduced by Gaussian numerical integration, for which a simple computational framework is provided. The relative contributions of geometric error and numerical integration error to the total error are also analyzed. It is observed that the geometric error is significant only for the small images whereas the magnitude of numerical integration is significantly high for all image sizes, which increases with the order of moments. A simple computational framework which is similar to the conventional zeroth order approximation is also proposed which not only reduces numerical integration error but also reduces geometric error without considering arc-grids. The improved accuracy of OFMMs are shown to provide better image reconstruction, numerical stability and rotation and scale invariance. Exhaustive experimental results on a variety of real images have shown the efficacy of the proposed methods.

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Acknowledgements

We are thankful to the anonymous reviewers for their suggestions for raising the standard of the paper. The alternative recursive relations given in Sect. 3.3 for trigonometric functions are suggested by one of the reviewers which is highly appreciated.

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  1. Department of Computer Science, Punjabi University, Patiala, Punjab, India

    Chandan Singh & Rahul Upneja

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  1. Chandan Singh
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  2. Rahul Upneja
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Correspondence to Rahul Upneja.

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Singh, C., Upneja, R. Accurate Computation of Orthogonal Fourier-Mellin Moments. J Math Imaging Vis 44, 411–431 (2012). https://doi.org/10.1007/s10851-012-0335-1

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