# American Institute of Mathematical Sciences

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 & 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, (2003).   Google Scholar [2] S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property,, Asymptotic Analysis, 49 (2006), 87.   Google Scholar [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.  doi: 10.3934/ipi.2014.8.389.  Google Scholar [4] J. Canny, A computational approach to edge detection,, Readings in Computer Vision: Issues, (1987), 184.  doi: 10.1016/B978-0-08-051581-6.50024-6.  Google Scholar [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.  doi: 10.1051/m2an:2003014.  Google Scholar [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.  doi: 10.1088/0266-5611/14/3/011.  Google Scholar [7] A. Desolneux, L. Moisan and J.-M. Morel, Edge detection by Helmholtz principle,, Journal of Mathematical Imaging and Vision, 14 (2001), 271.   Google Scholar [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.  Google Scholar [9] M. D. Donsker and J. L. Lions, Fréchet-Volterra variational equations, boundary value problems, and function space integrals,, Acta Mathematica, 108 (1962), 147.  doi: 10.1007/BF02545767.  Google Scholar [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.  doi: 10.1142/S0218202503003136.  Google Scholar [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.  doi: 10.1137/S0363012900369538.  Google Scholar [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.  doi: 10.4171/IFB/298.  Google Scholar [13] A. K. Jain, Fundamentals of Digital Image Processing,, Prentice Hall, (1989).   Google Scholar [14] D. Marr and E. Hildreth, Theory of edge detection,, Proceedings of the Royal Society of London, 207 (1980), 187.  doi: 10.1098/rspb.1980.0020.  Google Scholar [15] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems,, Communications on Pure and Applied Mathematics, 42 (1988), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar [16] M. Muszkieta, Optimal edge detection by topological asymptotic analysis,, Mathematical Models and Methods in Applied Science, 19 (2009), 2127.  doi: 10.1142/S0218202509004066.  Google Scholar [17] M. Z. Nashed and E. P. Hamilton, Bivariational and singular variational derivatives,, Journal of the London Mathematical Society, 41 (1990), 526.  doi: 10.1112/jlms/s2-41.3.526.  Google Scholar [18] W. K. Pratt, Digital Image Processing,, Wiley, (1991).   Google Scholar [19] J. Sokołowski and A. Żochowski, On topological derivative in shape optimization,, SIAM Journal on Control and Optimization, 37 (1999), 1251.  doi: 10.1137/S0363012997323230.  Google Scholar [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.  doi: 10.1051/m2an:2000101.  Google Scholar

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, Academic Press, (2003).   Google Scholar [2] S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property,, Asymptotic Analysis, 49 (2006), 87.   Google Scholar [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.  doi: 10.3934/ipi.2014.8.389.  Google Scholar [4] J. Canny, A computational approach to edge detection,, Readings in Computer Vision: Issues, (1987), 184.  doi: 10.1016/B978-0-08-051581-6.50024-6.  Google Scholar [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.  doi: 10.1051/m2an:2003014.  Google Scholar [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.  doi: 10.1088/0266-5611/14/3/011.  Google Scholar [7] A. Desolneux, L. Moisan and J.-M. Morel, Edge detection by Helmholtz principle,, Journal of Mathematical Imaging and Vision, 14 (2001), 271.   Google Scholar [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.  Google Scholar [9] M. D. Donsker and J. L. Lions, Fréchet-Volterra variational equations, boundary value problems, and function space integrals,, Acta Mathematica, 108 (1962), 147.  doi: 10.1007/BF02545767.  Google Scholar [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.  doi: 10.1142/S0218202503003136.  Google Scholar [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.  doi: 10.1137/S0363012900369538.  Google Scholar [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.  doi: 10.4171/IFB/298.  Google Scholar [13] A. K. Jain, Fundamentals of Digital Image Processing,, Prentice Hall, (1989).   Google Scholar [14] D. Marr and E. Hildreth, Theory of edge detection,, Proceedings of the Royal Society of London, 207 (1980), 187.  doi: 10.1098/rspb.1980.0020.  Google Scholar [15] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems,, Communications on Pure and Applied Mathematics, 42 (1988), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar [16] M. Muszkieta, Optimal edge detection by topological asymptotic analysis,, Mathematical Models and Methods in Applied Science, 19 (2009), 2127.  doi: 10.1142/S0218202509004066.  Google Scholar [17] M. Z. Nashed and E. P. Hamilton, Bivariational and singular variational derivatives,, Journal of the London Mathematical Society, 41 (1990), 526.  doi: 10.1112/jlms/s2-41.3.526.  Google Scholar [18] W. K. Pratt, Digital Image Processing,, Wiley, (1991).   Google Scholar [19] J. Sokołowski and A. Żochowski, On topological derivative in shape optimization,, SIAM Journal on Control and Optimization, 37 (1999), 1251.  doi: 10.1137/S0363012997323230.  Google Scholar [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.  doi: 10.1051/m2an:2000101.  Google Scholar
 [1] Grégory Faye, Pascal Chossat. A spatialized model of visual texture perception using the structure tensor formalism. Networks & 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 & 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 & 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 & Imaging, 2015, 9 (2) : 551-578. doi: 10.3934/ipi.2015.9.551 [5] Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023 [6] Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 [7] Marissa Condon, Jing Gao, Arieh Iserles. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4813-4837. doi: 10.3934/dcds.2016008 [8] Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543 [9] 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 & Imaging, 2016, 10 (2) : 327-367. doi: 10.3934/ipi.2016003 [10] Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661 [11] Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043 [12] Jie Huang, Marco Donatelli, Raymond H. Chan. Nonstationary iterated thresholding algorithms for image deblurring. Inverse Problems & Imaging, 2013, 7 (3) : 717-736. doi: 10.3934/ipi.2013.7.717 [13] 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 [14] 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 & Management Optimization, 2016, 12 (2) : 781-810. doi: 10.3934/jimo.2016.12.781 [15] Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015 [16] Michael Dellnitz, O. Junge, B Thiere. The numerical detection of connecting orbits. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 125-135. doi: 10.3934/dcdsb.2001.1.125 [17] Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433 [18] Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420 [19] David Henry, Octavian G. Mustafa. Existence of solutions for a class of edge wave equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1113-1119. doi: 10.3934/dcdsb.2006.6.1113 [20] Elena Beretta, Markus Grasmair, Monika Muszkieta, Otmar Scherzer. A variational algorithm for the detection of line segments. Inverse Problems & Imaging, 2014, 8 (2) : 389-408. doi: 10.3934/ipi.2014.8.389

2018 Impact Factor: 1.469