Abstract
Glaucoma is one of the leading causes of irreversible blindness in people over 40 years old. In [9] it was developed a computational tool for automatic glaucoma detection, which implements a novel method that has shown improvement in the accuracy of the detection compared to other classical methods. However, the method is sensitive to the quality of the acquired image, which is often contaminated by noise, and its quality can be poor. For this reason, automatic image restoration of the source images is needed to improve the quality of glaucoma detection. Partial differential equations to produce an image of much higher quality, enhance its sharpness, filter out the noise, extract shapes, etc. Here, we proposed the use of mimetic finite difference methods for the numerical solution of this kind of problems. The mimetic methods preserve the continuum properties of the mathematical operators often encountered in the image processing and analysis equations and ensuring better orders of convergence [5]. By ensuring these mathematical properties, the original structure of the source image is maintained, improving the diagnosis of the patient.
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References
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147. Springer, New York (2006)
Batista, E.D., Castillo, J.E.: Mimetic schemes on non-uniform structured meshes. Electron. Trans. Numer. Anal. 34(1), 152–162 (2009)
Benhamouda, B.: Parameter adaptation for nonlinear diffusion in image processing. Department of Mathematics, University of Kaiserslautern (1994)
Bredies, K., Lorenz, D.A.: Mathematical Image Processing. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham (2018)
Castillo, J.E., Grone, R.D.: A matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law. SIAM J. Matrix Anal. Appl. 25(1), 128–142 (2003)
Castillo, J.E., Yasuda, M.: Linear systems arising for second-order mimetic divergence and gradient discretizations. J. Math. Model. Algorithms 4(1), 67–82 (2005)
Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)
The International Agency for the Prevention of Blindness. Glaucoma. https://www.iapb.org/knowledge/what-is-avoidable-blindness/glaucoma/. Accessed 03 Aug 2019
Carrillo, J., Bautista, L., Villamizar, J., Rueda, J., Sanchez, M., Rueda, D.: Glaucoma detection using fundus images of the eye. In: 2019 XXII Symposium on Image, Signal Processing and Artificial Vision (STSIVA), Bucaramanga, Colombia, April 2019
Kwon, Y.H., Fingert, J.H., Kuehn, M.H., Alward, W.L.: Primary open-angle glaucoma. New Engl. J. Med. 11(360), 1113–1124 (2009)
Mikula, K.: Image processing with partial differential equations. In: Modern Methods in Scientific Computing and Applications, pp. 283–321. Springer (2002)
Nawaldgi, S.: Review of automated glaucoma detection techniques. In: 2016 International Conference on Wireless Communications, Signal Processing and Networking (WiSPNET), pp. 1435–1438. IEEE (2016)
American Academy of Ophthalmology: Angle-closure glaucoma. https://www.aao.org/munnerlyn-laser-surgery-center/angleclosure-glaucoma-19. Accessed 03 Aug 2019
World Health Organization: Blindness and vision impairment prevention. https://www.who.int/blindness/causes/priority/en/index6.html. Accessed 03 Aug 2019
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)
Rueda, J.C., Lesmes, D.P., Parra, J.C., Urrea, R., Rey, J.J., Rodríguez, L.A., Wong, C.A., Galvis, V.: Valores de paquimetría en personas sanas y con glaucoma en una población colombiana. MedUNAB 10(2), 81–85 (2007)
Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2006)
Solano Feo, F., Guevara Jordan, J.M., Rojas, O., Otero, B., Rodriguez, R.: A new mimetic scheme for the acoustic wave equation. J. Comput. Appl. Math. 295(1), 2–12 (2016)
Weickert, J., Romeny, B.M.T.H., Viergever, M.A., et al.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398–410 (1998)
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G. Calderón and J.C. Carrillo E. would like to acknowledge the financial support provided by the VIE-UIS under grant No 2415.
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Bautista, L., Villamizar, J., Calderón, G., Carrillo E., J.C., Rueda, J.C., Castillo, J. (2019). Mimetic Finite Difference Methods for Restoration of Fundus Images for Automatic Glaucoma Detection. In: Tavares, J., Natal Jorge, R. (eds) VipIMAGE 2019. VipIMAGE 2019. Lecture Notes in Computational Vision and Biomechanics, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-030-32040-9_12
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DOI: https://doi.org/10.1007/978-3-030-32040-9_12
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