Volodymyr V. Hnatushenko, Peter I. Kogut, Mykola V. Uvarov


In this paper we propose a new technique for the solution of the image segmentation problem which is based on the concept of a piecewise smooth approximation of some target functional. We discuss in details the consistency of the new statement of segmentation problem and its solvability. We focus our main intension on the rigor mathematical substantiation of the proposed approach, deriving the corresponding optimality conditions, and show that the new optimization problem is rather flexible and powerful model to the study of variational image segmentation problems. We illustrate the accuracy and efficiency of the proposed algorithm by numerical experiences.

Ключові слова

Optimal segmentation problem; piecewise constant approximation; optimality conditions; variational problem

Повний текст:

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R. Acart, C.R. Vogel, Analysis of bounded variation penalty methods for illposed problems, Inverse Problems, 10 (1994), 1217–1229.

L. Alvarez, P.L Lions, J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. II, SIAM J. Numer. Anal., 29 (3) (1992), 845–866.

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (4) (1983), 293–360.

L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, New York, 2000.

H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, SIAM, Philadelphia, 2006.

G. Aubert, P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Second Edition, Series: Applied Mathematical Sciences, Vol.147, Springer, New York, 2006.

R. Boesch, Z. Wang, Segmentation optimization for aerial images with spacial constraints, The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XXXVII (Part B4) (2008), 285–289.

D. Bucur, G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhauser, Boston, 2005.

R. Caccioppoli, Misura e integrazione sugli insieme dimensionalmente orientali I.II. Rend. Acc. Naz. Lincei, 12 (8) (1952), 3–11.

F. Catt´e, T. Coll, P.L Lions, J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. I, SIAM J. Numer. Anal., 29 (1) (1992), 182–193.

A. Chambolle, P.L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167–188.

Y. Chen, Q. Chen, C. Jing, Multi-resolution segmentation parameters optimization and evaluation for VHR remote sensing image based on meanNSQI and discrepancy measure, Journal of Spatial Science, https://doi.org/10.1080/14498596.2019.1615011, (2019), 1–27.

G. Crasta, V. De Cicco, Anzellotti’s pairing theory and the Gauss-Green theorem, Advances in Mathematics, 343 (5) (2019), 935–970.

F. Demengel, R. Temam, Convex functions of a measure and applicatuions, Indiana Univ. Math. J., 33 (1984), 673–709.

I. Ekeland, R. Temam, Analyse Convexe et Probl`emes Variationnels, Dunod- Gauthier-Villars, Paris, 1974.

L. C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, Vol.74, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990.

H. Federer, Geometric measure theory, Springer, Berlin, 1969.

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkh¨auser, Boston, 1984.

V.V. Hnatushenko, D.K. Mozgovyi, V.V. Vasyliev, Satellite monitoring of deforestation as a result of mining, Scientific Bulletin of National Mining University, Dnipro, 161 (5) (2017), 94-99.

D. Hordiiuk, V. Hnatushenko, K. Maksymov, Ie. Oliinyk, Semantic segmentation for ships detection from satellite imagery, 2019 IEEE 39th International Conference on Electronics and Nanotechnology (ELNANO), (2019), 454–457.

B. Kawohl, F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem, Communications in Contemporary Mathematics, 9 (4) (2007), 515–543.

P.I. Kogut, G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Series: Systems and Control, Birkh¨auser Verlag, Boston, 2011.

U. Massari, M. Miranda, Minimal Surfaces of Codimension One, North- Holland, Amsterdam, 1984.

D. Mumford, J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Commun. Pure. Appl. Math., 42 (5) (1989), 577–685.

D.J. Mulla, Twenty five years of remote sensing in precision agriculture: key advances and remaining knowledge gaps, Biosystems Engineering, 114 (4) (2013), 358–371.

S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer, Milan, 2008.

C.-B. Schonlieb, Total variation minimization with an 1/H constraint, CRM Series 9, Singularities in Nonlinear Evolution Phenomena and Applications Proceedings, Scuola Normale Superiore Pisa, 2009, 201–232.

L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44 (2001), 131–161.

P. Xiao, X. Zhang, H. Zhang, R. Hu, X. Feng, Multiscale optimized segmentation of urban green cover in high resolution remote sensing image, Remote Sensing, 10 (Article 1813) (2018), 1–20.

J. Xue, B. Su, Significant remote sensing Vegetation Indices: A review of developments and applications, Hindawi Journal of Sensors, 2017 (Article ID 1353691) (2017), 1–17.

L. Yan, D.P. Roy, Automated crop field extraction from multi-temporal Web Enabled Landsat Data, Remote Sensing of Environment, 144 (2014), 42–64.

Ye. Yuan, C. He, Adaptive active contours without edges, Math. and Computer Modelling, 55 (2012), 1705–1721.

DOI: http://dx.doi.org/10.15421/141908


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