Home > Archive>Volume 19, Issue 4, 2023 >252-256. DOI:https://doi.org/10.1007/s11801-023-2199-6
PDF HTML XML Export Cite reminder
Mathematical representation of 2D image boundary contour using fractional implicit polynomial
CSTR:
[cstr]
Author:
Affiliation:

1. Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China;2. School of Materials Science and Optoelectronic Technology & School of Integrated Circuits, University of Chinese Academy of Sciences, Beijing 100049, China;3. Beijing Key Laboratory of Semiconductor Neural Network Intelligent Sensing and Computing Technology, Beijing 100083, China;4. DapuStor Corporation, Shenzhen 518100, China

  • Article
    • | |
  • Metrics
    • |
  • Reference [23]
    • |
  • Related
    • |
  • Cited by
    • | |
  • Comments
    Abstract:

    Implicit polynomial (IP) fitting is an effective method to quickly represent two-dimensional (2D) image boundary contour in the form of mathematical function. Under the same maximum degree, the fractional implicit polynomial (FIP) can express more curve details than IP and has obvious advantages for the representation of complex boundary contours. In existing studies, algebraic distance is mainly used as the fitting objective of the polynomial. Although the time cost is reduced, there are problems of low fitting accuracy and spurious zero set. In this paper, we propose a two-stage neural network with differentiable geometric distance, which uses FIP to achieve mathematical representation, called TSEncoder. In the first stage, the continuity constraint is used to obtain a rough outline of the fitting target. In the second stage, differentiable geometric distance is gradually added to fine-tune the polynomial coefficients to obtain a contour representation with higher accuracy. Experimental results show that TSEncoder can achieve mathematical representation of 2D image boundary contour with high accuracy.

    Reference
    [1]TAUBIN G, CUKIERMAN F, SULLIVAN S, et al. Parameterized families of polynomials for bounded algebraic curve and surface fitting[J]. IEEE transactions on pattern analysis and machine intelligence, 1994, 16(3)：287-303.
    [2] HUAMIN T, JIANJUN Y, CHUNLEI Z. Polarization radar target recognition based on optimal curve fitting[C]//Proceedings of the IEEE 1998 National Aerospace and Electronics Conference, August 31- September 4, 1998, Dayton, USA. New York： IEEE, 1998：434-437.
    [3] JI C, YANG X, WANG W. A novel method for image recognition based on polynomial curve fitting[C]//2015 8th International Symposium on Computational Intelligence and Design, December 12-13, 2015, Hangzhou, China. New York：IEEE, 2015：354-357.
    [4] ODEN C, ERCIL A, BUKE B. Combining implicit polynomials and geometric features for hand recognition[J]. Pattern recognition letters, 2003, 24(13)：2145-2152.
    [5] ODRY á, FULLéR R, RUDAS I J, et al. Kalman filter for mobile-robot attitude estimation：novel optimized and adaptive solutions[J]. Mechanical systems and signal processing, 2018, 110：569-589.
    [6] TAREL J P, COOPER D B. A new complex basis for implicit polynomial curves and its simple exploitation for pose estimation and invariant recognition[C]//1998 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, June 25, 1998, San Diego, CA, USA. New York：IEEE, 1998：111-117.
    [7] WU G, ZHANG Y. A novel fractional implicit polynomial approach for stable representation of complex shapes[J]. Journal of mathematical imaging and vision, 2016, 55(1)：89-104.
    [8] WU G, YANG J. A representation of time series based on implicit polynomial curve[J]. Pattern recognition letters, 2013, 34(4)：361-371.
    [9] TAUBIN G. Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation[J]. IEEE transactions on pattern analysis & machine intelligence, 1991, 13(11)：1115-1138.
    [10] TAUBIN G, CUKIERMAN F, SULLIVAN S, et al. Parameterized families of polynomials for bounded algebraic curve and surface fitting[J]. IEEE transactions on pattern analysis and machine intelligence, 1994, 16(3)：287-303.
    [11] KEREN D, GOTSMAN C. Fitting curves and surfaces with constrained implicit polynomials[J]. IEEE transactions on pattern analysis and machine intelligence, 1999, 21(1)：31-41.
    [12] KEREN D, COOPER D, SUBRAHMONIA J. Describing complicated objects by implicit polynomials[J]. IEEE transactions on pattern analysis and machine intelligence, 1994, 16(1)：38-53.
    [13] GUILLAUME P, SCHOUKENS J, PINTELON R. Sensitivity of roots to errors in the coefficient of polynomials obtained by frequency-domain estimation methods[J]. IEEE transactions on instrumentation and measurement, 1989, 38(6)：1050-1056.
    [14] BLANE M M, LEI Z, CIVI H, et al. The 3L algorithm for fitting implicit polynomial curves and surfaces to data[J]. IEEE transactions on pattern analysis and machine intelligence, 2000, 22(3)：298-313.
    [15] HELZER A, BARZOHAR M, MALAH D. Stable fitting of 2D curves and 3D surfaces by implicit polynomials[J]. IEEE transactions on pattern analysis and machine intelligence, 2004, 26(10)：1283-1294.
    [16] WANG G, LI W, ZHANG L, et al. Encoder-X：solving unknown coefficients automatically in polynomial fitting by using an autoencoder[J]. IEEE transactions on neural networks and learning systems, 2021.
    [17] HU M, ZHOU Y, LI X. Robust and accurate computation of geometric distance for Lipschitz continuous implicit curves[J]. The visual computer, 2017, 33(6)：937-947.
    [18] TASDIZEN T, TAREL J P, COOPER D B. Improving the stability of algebraic curves for applications[J]. IEEE transactions on image processing, 2000, 9(3)：405-416.
    [19] WANG W, POTTMANN H, LIU Y. Fitting B-spline curves to point clouds by curvature-based squared distance minimization[J]. ACM transactions on graphics, 2006, 25(2)：214-238.
    [20] SONG X, JüTTLER B. Modeling and 3D object reconstruction by implicitly defined surfaces with sharp features[J]. Computers & graphics, 2009, 33(3)：321-330.
    [21] UPRETI K, SONG T, TAMBAT A, et al. Algebraic distance estimations for enriched isogeometric analysis[J]. Computer methods in applied mechanics and engineering, 2014, 280：28-56.
    [22] PAPARI G, PETKOV N. Edge and line oriented contour detection：state of the art[J]. Image and vision computing, 2011, 29(2-3)：79-103.
    [23] ROUHANI M, SAPPA A D. Implicit polynomial representation through a fast fitting error estimation[J]. IEEE transactions on Image Processing, 2011, 21(4)：2089-2098.
    Related
    Cited by
Get Citation

TONG Yuerong, YU Lina, LI Weijun, LIU Jingyi, WU Min, YANG Yafei. Mathematical representation of 2D image boundary contour using fractional implicit polynomial[J]. Optoelectronics Letters,2023,19(4):252-256

Copy
Share
Article Metrics
  • Abstract:514
  • PDF: 481
  • HTML: 0
  • Cited by: 0
History
  • Received:October 09,2022
  • Revised:November 26,2022
  • Online: April 19,2023
Article QR Code
Phone：86-22-60214471
URL：http://www.oelett.com E-mail：oel@263.net
Supported by:Beijing E-Tiller Technology Development Co., Ltd.
® 2025 All Rights Reserved