May  2016, 10(2): 499-517. doi: 10.3934/ipi.2016009

A variational approach to edge detection

1. 

Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw

Received  May 2010 Revised  January 2013 Published  May 2016

In this paper, using the variational framework and elements of topological asymptotic analysis, we derive an algorithm for edge detection in a digital image in which an optimal value for the threshold is computed automatically. In order to examine this algorithm, we perform a simple experiment on synthetic images composed of two objects with different values of intensity and size. In this case, we are be able to find an exact condition which has to be satisfied so that an edge of object with lower contrast would be detected. At the end, we compare results of numerical experiments obtained by application of our algorithm and the algorithm proposed by Desolneux et al. [7,8]. We indicate some similarities between these two approaches to edge detection and discuss their differences.
Citation: Monika Muszkieta. A variational approach to edge detection. Inverse Problems and Imaging, 2016, 10 (2) : 499-517. doi: 10.3934/ipi.2016009
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.

[2]

S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property, Asymptotic Analysis, 49 (2006), 87-108.

[3]

E. Beretta, M. Grasmair, M. Muszkieta and O. Scherzer, A variational algorithm for the detection of line segments, Inverse Problems and Imaging, 8 (2014), 389-408. doi: 10.3934/ipi.2014.8.389.

[4]

J. Canny, A computational approach to edge detection, Readings in Computer Vision: Issues, Problem, Principles, and Paradigms, (1987), 184-203. doi: 10.1016/B978-0-08-051581-6.50024-6.

[5]

Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Mathematical Modeling and Numerical Analysis, 37 (2003), 159-173. doi: 10.1051/m2an:2003014.

[6]

D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595. doi: 10.1088/0266-5611/14/3/011.

[7]

A. Desolneux, L. Moisan and J.-M. Morel, Edge detection by Helmholtz principle, Journal of Mathematical Imaging and Vision, 14 (2001), 271-284.

[8]

A. Desolneux, L. Moisan and J.-M. Morel, From Gestalt Theory to Image Analysis, volume 34 of {Interdisciplinary Applied Mathematics}, Springer New York, 2008. doi: 10.1007/978-0-387-74378-3.

[9]

M. D. Donsker and J. L. Lions, Fréchet-Volterra variational equations, boundary value problems, and function space integrals, Acta Mathematica, 108 (1962), 147-228. doi: 10.1007/BF02545767.

[10]

R. A. Feijóo, A. Novotny, C. Padra and E. Taroco, The topological derivative for the Poisson problem, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1825-1844. doi: 10.1142/S0218202503003136.

[11]

S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: The elasticity case, SIAM Journal on Control and Optimization, 39 (2000), 1756-1778. doi: 10.1137/S0363012900369538.

[12]

M. Grasmair, M. Muszkieta and O. Scherzer, An approach to the minimization of the Mumford-Shah functional using $\Gamma$-convergence and topological asymptotic expansion, Interfaces and Free Boundaries, 15 (2013), 141-166. doi: 10.4171/IFB/298.

[13]

A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, 1989.

[14]

D. Marr and E. Hildreth, Theory of edge detection, Proceedings of the Royal Society of London, 207 (1980), 187-217. doi: 10.1098/rspb.1980.0020.

[15]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems, Communications on Pure and Applied Mathematics, 42 (1988), 577-685. doi: 10.1002/cpa.3160420503.

[16]

M. Muszkieta, Optimal edge detection by topological asymptotic analysis, Mathematical Models and Methods in Applied Science, 19 (2009), 2127-2143. doi: 10.1142/S0218202509004066.

[17]

M. Z. Nashed and E. P. Hamilton, Bivariational and singular variational derivatives, Journal of the London Mathematical Society, 41 (1990), 526-546. doi: 10.1112/jlms/s2-41.3.526.

[18]

W. K. Pratt, Digital Image Processing, Wiley, New York, 1991.

[19]

J. Sokołowski and A. Żochowski, On topological derivative in shape optimization, SIAM Journal on Control and Optimization, 37 (1999), 1251-1272. doi: 10.1137/S0363012997323230.

[20]

M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, Mathematical Modeling and Numerical Analysis, 34 (2000), 723-748. doi: 10.1051/m2an:2000101.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.

[2]

S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property, Asymptotic Analysis, 49 (2006), 87-108.

[3]

E. Beretta, M. Grasmair, M. Muszkieta and O. Scherzer, A variational algorithm for the detection of line segments, Inverse Problems and Imaging, 8 (2014), 389-408. doi: 10.3934/ipi.2014.8.389.

[4]

J. Canny, A computational approach to edge detection, Readings in Computer Vision: Issues, Problem, Principles, and Paradigms, (1987), 184-203. doi: 10.1016/B978-0-08-051581-6.50024-6.

[5]

Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Mathematical Modeling and Numerical Analysis, 37 (2003), 159-173. doi: 10.1051/m2an:2003014.

[6]

D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595. doi: 10.1088/0266-5611/14/3/011.

[7]

A. Desolneux, L. Moisan and J.-M. Morel, Edge detection by Helmholtz principle, Journal of Mathematical Imaging and Vision, 14 (2001), 271-284.

[8]

A. Desolneux, L. Moisan and J.-M. Morel, From Gestalt Theory to Image Analysis, volume 34 of {Interdisciplinary Applied Mathematics}, Springer New York, 2008. doi: 10.1007/978-0-387-74378-3.

[9]

M. D. Donsker and J. L. Lions, Fréchet-Volterra variational equations, boundary value problems, and function space integrals, Acta Mathematica, 108 (1962), 147-228. doi: 10.1007/BF02545767.

[10]

R. A. Feijóo, A. Novotny, C. Padra and E. Taroco, The topological derivative for the Poisson problem, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1825-1844. doi: 10.1142/S0218202503003136.

[11]

S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: The elasticity case, SIAM Journal on Control and Optimization, 39 (2000), 1756-1778. doi: 10.1137/S0363012900369538.

[12]

M. Grasmair, M. Muszkieta and O. Scherzer, An approach to the minimization of the Mumford-Shah functional using $\Gamma$-convergence and topological asymptotic expansion, Interfaces and Free Boundaries, 15 (2013), 141-166. doi: 10.4171/IFB/298.

[13]

A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, 1989.

[14]

D. Marr and E. Hildreth, Theory of edge detection, Proceedings of the Royal Society of London, 207 (1980), 187-217. doi: 10.1098/rspb.1980.0020.

[15]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems, Communications on Pure and Applied Mathematics, 42 (1988), 577-685. doi: 10.1002/cpa.3160420503.

[16]

M. Muszkieta, Optimal edge detection by topological asymptotic analysis, Mathematical Models and Methods in Applied Science, 19 (2009), 2127-2143. doi: 10.1142/S0218202509004066.

[17]

M. Z. Nashed and E. P. Hamilton, Bivariational and singular variational derivatives, Journal of the London Mathematical Society, 41 (1990), 526-546. doi: 10.1112/jlms/s2-41.3.526.

[18]

W. K. Pratt, Digital Image Processing, Wiley, New York, 1991.

[19]

J. Sokołowski and A. Żochowski, On topological derivative in shape optimization, SIAM Journal on Control and Optimization, 37 (1999), 1251-1272. doi: 10.1137/S0363012997323230.

[20]

M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, Mathematical Modeling and Numerical Analysis, 34 (2000), 723-748. doi: 10.1051/m2an:2000101.

[1]

Grégory Faye, Pascal Chossat. A spatialized model of visual texture perception using the structure tensor formalism. Networks and Heterogeneous Media, 2013, 8 (1) : 211-260. doi: 10.3934/nhm.2013.8.211

[2]

Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems and Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51

[3]

Liming Zhang, Tao Qian, Qingye Zeng. Edge detection by using rotational wavelets. Communications on Pure and Applied Analysis, 2007, 6 (3) : 899-915. doi: 10.3934/cpaa.2007.6.899

[4]

Yuying Shi, Ying Gu, Li-Lian Wang, Xue-Cheng Tai. A fast edge detection algorithm using binary labels. Inverse Problems and Imaging, 2015, 9 (2) : 551-578. doi: 10.3934/ipi.2015.9.551

[5]

Yuying Shi, Zijin Liu, Xiaoying Wang, Jinping Zhang. Edge detection with mixed noise based on maximum a posteriori approach. Inverse Problems and Imaging, 2021, 15 (5) : 1223-1245. doi: 10.3934/ipi.2021035

[6]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations and Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023

[7]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

[8]

Marissa Condon, Jing Gao, Arieh Iserles. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4813-4837. doi: 10.3934/dcds.2016008

[9]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543

[10]

Fabien Caubet, Carlos Conca, Matías Godoy. On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives. Inverse Problems and Imaging, 2016, 10 (2) : 327-367. doi: 10.3934/ipi.2016003

[11]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661

[12]

Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043

[13]

Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157

[14]

Zixiao Liu, Jiguang Bao. Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022081

[15]

Andreas Klein. How to say yes, no and maybe with visual cryptography. Advances in Mathematics of Communications, 2008, 2 (3) : 249-259. doi: 10.3934/amc.2008.2.249

[16]

Jie Huang, Marco Donatelli, Raymond H. Chan. Nonstationary iterated thresholding algorithms for image deblurring. Inverse Problems and Imaging, 2013, 7 (3) : 717-736. doi: 10.3934/ipi.2013.7.717

[17]

Audun D. Myers, Firas A. Khasawneh, Brittany T. Fasy. ANAPT: Additive noise analysis for persistence thresholding. Foundations of Data Science, 2022, 4 (2) : 243-269. doi: 10.3934/fods.2022005

[18]

Feng Yang, Kok Lay Teo, Ryan Loxton, Volker Rehbock, Bin Li, Changjun Yu, Leslie Jennings. VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems. Journal of Industrial and Management Optimization, 2016, 12 (2) : 781-810. doi: 10.3934/jimo.2016.12.781

[19]

Hui Xu, Guangbin Cai, Xiaogang Yang, Erliang Yao, Xiaofeng Li. Stereo visual odometry based on dynamic and static features division. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2109-2128. doi: 10.3934/jimo.2021059

[20]

Michael Dellnitz, O. Junge, B Thiere. The numerical detection of connecting orbits. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 125-135. doi: 10.3934/dcdsb.2001.1.125

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (219)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]