Abstract
Implicit polynomials (IPs) are considered as a powerful tool for object curve fitting tasks due to their simplicity and fewer parameters. The traditional linear methods, such as 3L, MinVar, and MinMax, often achieve good performances in fitting simple objects, but usually work poorly or even fail to obtain closed curves of complex object contours. To handle the complex fitting issues, taking the advantages of deep neural networks, we designed a neural network model continuity-sparsity constrained network (CSC-Net) with encoder and decoder structure to learn the coefficients of IPs. Further, the continuity constraint is added to ensure the obtained curves are closed, and the sparseness constraint is added to reduce the spurious zero sets of the fitted curves. The experimental results show that better performances have been obtained on both simple and complex object fitting tasks.
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References
BLANE 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 & machine intelligence, 2000, 22: 298–313.
FORSYTH D, MUNDY J, ZISSERMAN A, et al. Invariant descriptors for 3D object recognition and pose[J]. IEEE transactions on pattern analysis & machine intelligence, 1991, 13(10): 971–991.
TAREL J P, COOPWE D B. The complex representation of algebraic curves and its simple exploitation for pose estimation and invariant recognition[J]. IEEE transactions on pattern analysis & machine intelligence, 2000, 22(7): 663–674.
HELZER A, BARZOHAR M, MALAH D. Using implicit polynomials for image compression[C]//Proceedings of 21st IEEE Convention of the Electrical & Electronic Engineers in Israel, April 11–12, 2000, Tel-Aviv, Israel. New York: IEEE, 2000: 384–388.
KEREN D. Topologically faithful fitting of simple closed curves[J]. IEEE transactions on pattern analysis and machine intelligence, 2004, 26(1): 118–123.
MAROLA G. A technique for finding the symmetry axes of implicit polynomial curves under perspective projection[J]. IEEE transactions on pattern analysis and machine intelligence, 2005, 27(3): 465–470.
SAHIN T, UNEL M. Fitting globally stabilized algebraic surfaces to range data[C]//10th IEEE International Conference on Computer Vision (ICCV’05), October 17–21, 2005, Beijing, China. New York: IEEE, 2005, 2: 1083–1088.
TASDIZEN T, TAREL J P. Improving the stability of algebraic curves for applications[J]. IEEE transactions on image processing, 2000, 9(3): 405–416.
ROUHANI M, SAPPA A D, BOYER E. Implicit B-spline surface reconstruction[J]. IEEE transactions on image processing, 2015, 24(1): 22–32.
MAJEED A, ABBAS M, QAYYUM F, et al. Geometric modeling using new cubic trigonometric B-spline functions with shape parameter[J]. Mathematics, 2020, 8: 2102.
PATRIZI F, DOKKEN T. Linear dependence of bivariate minimal support and locally refined B-splines over LR-meshes[J]. Computer aided geometric design, 2022, 77: 101803.
WU G, ZHANG Y C. A novel fractional implicit polynominal approach for stable representation of complex shapes[J]. Journal of mathematical imaging and vision, 2016, 55: 89–104.
HELZER A, BARZOHAR M, MALAH D. Stable fitting of 2D curves and 3D surfaces by implicit polynomials[J]. IEEE transactions on pattern analysis & machine intelligence, 2004, 26(10): 1283–1294.
VOULODIMOS A, DOULAMIS N, DOULAMIS A, et al. Deep learning for computer vision: a brief review[J]. Computational intelligence and neuroscience, 2018, 2018: 1–13.
DEVLIN J, CHANG M W, LEE K, et al. BERT: pre-training of deep bidirectional transformers for language understanding[C]//Proceedings of the 2019 Conference of the North American Chapter of the As-sociation for Computational Linguistics, June, 2019, Minneapolis, Minnesota, USA. Stroudsburg: Association for Computational Linguistics, 2019: 4171–4186.
VASWANI A, SHAZEER N, PARMAR N, et al. Attention is all you need[C]//31st Conference on Neural Information Processing Systems (NIPS 2017), December 4–9, 2017, Long Beach, CA, USA. New York: Curran Associates Inc, 2017.
WANG G J, LI W J, ZHANG L P, et al. Encoder-X: solving unknown coefficients automatically in polynomial fitting by using an autoencoder[J]. IEEE transactions on networks and learning systems, 2022, 33(8): 3264–3276.
ZHENG B, TAKAMATSU J, IKEUCHI K. An adaptive and stable method for fitting implicit polynomial curves and surfaces[J]. IEEE transactions on pattern analysis & machine intelligence, 2010, 32(3): 561–568.
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The authors declare that there are no conflicts of interest related to this article.
This work has been supported by the Key-Area Research and Development Program of Guangdong Province (No.2019B010107001), and the Fund of Guangdong Support Program (No.2019TY05X071).
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Liu, J., Yu, L., Sun, L. et al. Fitting objects with implicit polynomials by deep neural network. Optoelectron. Lett. 19, 60–64 (2023). https://doi.org/10.1007/s11801-023-2065-6
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DOI: https://doi.org/10.1007/s11801-023-2065-6