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March  2021, 14(3): 1123-1132. doi: 10.3934/dcdss.2020389

Segmentation of color images using mean curvature flow and parametric curves

Petr Pauš 1,, and  Shigetoshi Yazaki 2,

1. 

Czech Technical University in Prague, Trojanova 13,120 00 Prague, Czech Republic
2. 

Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan

* Corresponding author: Petr Pauš

Received  January 2019 Revised  February 2020 Published  March 2021 Early access  June 2020

Automatic detection of objects in photos and images is beneficial in various scientific and industrial fields. This contribution suggests an algorithm for segmentation of color images by the means of the parametric mean curvature flow equation and CIE94 color distance function. The parametric approach is enriched by the enhanced algorithm for topological changes where the intersection of curves is computed instead of unreliable curve distance. The result is a set of parametric curves enclosing the object. The algorithm is presented on a test image and also on real photos.

Keywords: Image segmentation, mean curvature flow, parametric curves, finite volume method, color images.
Mathematics Subject Classification: Primary: 68U05, 65D18; Secondary: 14H50.
Citation: Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389
References:
[1]

M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 2008 (2008), 509-523. 

[2]

I. C. Consortium and et al., Specification icc. 1: 2004-10, (profile version 4.2. 0.0): Image technology colour management, 2004.

[3]

K. Deckelnick and G. Dziuk, Discrete anisotropic curvature flow of graphs, ESAIM: Mathematical Modelling and Numerical Analysis, 33 (1999), 1203-1222.  doi: 10.1051/m2an:1999141.

[4]

K. Deckelnick and G. Dziuk, Mean curvature flow and related topics, Frontiers in Numerical Analysis, Universitext, Springer, Berlin, (2002), 63–108.

[5]

R. McDonald and K. J. Smith, Cie94-a new colour-difference formula, Journal of the Society of Dyers and Colourists, 111 (1995), 376-379.  doi: 10.1111/j.1478-4408.1995.tb01688.x.

[6]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 153. Springer-Verlag, New York, 2003. doi: 10.1007/b98879.

[7]

P. Pauš and M. Beneš, Direct approach to mean-curvature flow with topological changes, Kybernetika (Prague), 45 (2009), 591-604. 

[8]

P. Pauš and M. Beneš, Algorithm for topological changes of parametrically described curves, Proceedings of ALGORITMY, (2009), 176–184.

[9]

P. Pauš and S. Yazaki, Exact solution for dislocation bowing and a posteriori numerical technique for dislocation touching-splitting, JSIAM Letters, 7 (2015), 57-60.  doi: 10.14495/jsiaml.7.57.

[10]

D. Ševčovič, Qualitative and quantitative aspects of curvature driven flows of planar curves, Topics on Partial Differential Equations, Jindřich Nečas Cent. Math. Model. Lect. Notes, MatFyzPress, Prague, 2 (2007), 55–119.

[11]

D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan Journal of Industrial and Applied Mathematics, 28 (2011), 413-442.  doi: 10.1007/s13160-011-0046-9.

show all references

References:
[1]

M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 2008 (2008), 509-523. 

[2]

I. C. Consortium and et al., Specification icc. 1: 2004-10, (profile version 4.2. 0.0): Image technology colour management, 2004.

[3]

K. Deckelnick and G. Dziuk, Discrete anisotropic curvature flow of graphs, ESAIM: Mathematical Modelling and Numerical Analysis, 33 (1999), 1203-1222.  doi: 10.1051/m2an:1999141.

[4]

K. Deckelnick and G. Dziuk, Mean curvature flow and related topics, Frontiers in Numerical Analysis, Universitext, Springer, Berlin, (2002), 63–108.

[5]

R. McDonald and K. J. Smith, Cie94-a new colour-difference formula, Journal of the Society of Dyers and Colourists, 111 (1995), 376-379.  doi: 10.1111/j.1478-4408.1995.tb01688.x.

[6]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 153. Springer-Verlag, New York, 2003. doi: 10.1007/b98879.

[7]

P. Pauš and M. Beneš, Direct approach to mean-curvature flow with topological changes, Kybernetika (Prague), 45 (2009), 591-604. 

[8]

P. Pauš and M. Beneš, Algorithm for topological changes of parametrically described curves, Proceedings of ALGORITMY, (2009), 176–184.

[9]

P. Pauš and S. Yazaki, Exact solution for dislocation bowing and a posteriori numerical technique for dislocation touching-splitting, JSIAM Letters, 7 (2015), 57-60.  doi: 10.14495/jsiaml.7.57.

[10]

D. Ševčovič, Qualitative and quantitative aspects of curvature driven flows of planar curves, Topics on Partial Differential Equations, Jindřich Nečas Cent. Math. Model. Lect. Notes, MatFyzPress, Prague, 2 (2007), 55–119.

[11]

D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan Journal of Industrial and Applied Mathematics, 28 (2011), 413-442.  doi: 10.1007/s13160-011-0046-9.

Figure 1.  Algorithm for topological changes of a closed curve $ \Gamma $ which overlaps itself under the external force. The intersections are computed and the overlapping segments of the curve are removed. The resulting two closed curves continue evolution in time
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Figure 2.  Original image with white background (left), gray-scale intensity image from a red color (middle), and simple conversion to gray-scale and inversion (right)
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Figure 3.  Artificial color image segmentation with the red reference color
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Figure 4.  Comparison of the color distance segmentation (left) and simple gray-scale conversion segmentation (right)
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Figure 5.  Original yellow flower photo (left), the distance image from a yellow color (middle), and a simple conversion to gray-scale (right)
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Figure 6.  Comparison of the color distance segmentation (left) and simple gray-scale conversion segmentation (right) for a yellow flower
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Figure 7.  The photo of a cloud and its distance image from the almost white (very light light blue) color
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Figure 8.  Segmentation of the original cloud image
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Figure 9.  Segmentation of the sunflower with different shades of yellow
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