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A Novel Fractional Implicit Polynomial Approach for Stable Representation of Complex Shapes

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Abstract

Implicit polynomials (IPs) are applied to represent 2D object shapes in image processing and computer vision. However, it is difficult for IPs to represent complex object shapes due to high computational cost and high instability. In this work, we present a new representation model based on IPs, which is called fractional implicit polynomial (FIP). Firstly, the general formula for FIP and a definition of base are given; secondly, we investigate the properties of FIPs and conclude that the FIP representation exhibits higher stability and higher power than IP representation due to the presence of the base. Thirdly, we develop an algorithm for determination of a moderate degree for an FIP to represent a given shape, which can be obtained by only computing the number of stationary points on the shapes. We compare FIPs with IPs by test on various object shapes and the results show that the FIP is indeed sufficiently powerful to represent the complex object shapes.

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Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grants 71372188 and National Center for International Joint Research on E-Business Information Processing under Grant 2013B01035.

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Authors and Affiliations

  1. Institute of Multimedia Computing Technology, Nanjing University of Finance and Economics, Nanjing, 210003, China

    Gang Wu

  2. Centre for Applied Informatics, Victoria University, Melbourne, VIC, 8001, Australia

    Yanchun Zhang

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  1. Gang Wu
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  2. Yanchun Zhang
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Correspondence to Gang Wu.

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Wu, G., Zhang, Y. A Novel Fractional Implicit Polynomial Approach for Stable Representation of Complex Shapes. J Math Imaging Vis 55, 89–104 (2016). https://doi.org/10.1007/s10851-015-0615-7

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  • DOI: https://doi.org/10.1007/s10851-015-0615-7

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