# 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 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.

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##### 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.
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