Variational Approach for the Reconstruction of Damaged Optical Satellite Images Through Their Co-Registration with Synthetic Aperture Radar

Peter I. Kogut, Mykola V. Uvarov

Анотація


In this paper the problem of reconstruction of damaged multi-band optical
images is studied in the case where we have no information about brightness of such
images in the damage region. Mostly motivated by the crop field monitoring problem,
we propose a new variational approach for exact reconstruction of damaged multi-band
images using results of their co-registration with Synthetic Aperture Radar (SAR) images
of the same regions. We discuss the consistency of the proposed problem, give the scheme
for its regularization, derive the corresponding optimality system, and describe in detail
the algorithm for the practical implementation of the reconstruction procedure.


Ключові слова


Image reconstruction problem; synthetic aperture radar image; total variation; image processing; optimality conditions

Повний текст:

PDF (English)

Посилання


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

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

H. Antil, S. Bartels, Spectral approximation of fractional PDEs in image processing and phase field modeling, Computational Methods in Applied Mathematics, 17 (4) (2017), 661–678.

H. Antil, C.N. Rautenberg, Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications, SIAM Journal on Mathematical Analysis, 51 (3) (2019), 2479–2503.

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

R. Bergmann, J.H. Fitschen, J. Persch, G. Steidl, Priors with coupled first and second order differences for manifold-valued image processing, Journal of Mathematical Imaging and Vision, 60 (9) (2018), 1459–1481.

M. Bertalmio, G. Sapiro, V. Caselles, C. Ballester, Image Inpainting, Siggraph 2000, Computer Graphics Proceedings, 2000, 417–424.

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

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

T.F. Chan, S.H. Kang, J. Shen, Total variation denoising and enhancement of color images based on the CB and HSV color models, J. Visual Comm. Image Rep., 12 (4) (2001), 422-435.

T.F. Chan, J. Shen, Mathematical models for local deterministic inpaintings, SIAM Journal of Applied Math., 62 (3) (2001), 1019–1043.

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

R. Cristoferi, I. Fonseca, Piecewise constant reconstruction of damaged color images, arXiv:1712.10033v2, 2018, 1–28.

P.C. Doraiswamy, S. Moulin, P.W. Cook, A. Stern, Crop yield assessment from remote sensing, Photogramm. Eng. Remote Sens., 69 (2003), 665–674.

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

S. Esedoglu, J.-H. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13 (4) (2002), 353-370.

R. Ferreira, I. Fonseca, and M.L.S. Mascarenhas, Restoration of color images by vector valued BV functions and variational calculus, Calc. Var. Partial Differential Equations, 56 (2017), Art. 140, 53.

M. Fornasier, R. March, A chromaticity-brightness model for color images denoising in a Meyer’s u+v framework, Preprint (http://www.iac.cnr.it/march/), 2006.

I.N. Garkusha, V. V. Hnatushenko, V. V. Vasyliev, Research of influence of atmosphere and humidity on the data of radar imaging by Sentinel-1, 2017, IEEE 37th International Conference on Electronics and Nanotechnology (ELNANO), 2017, doi:10.1109/elnano.2017.7939787.

I.N. Garkusha, V. V. Hnatushenko, V. V. Vasyliev, Using Sentinel-1 data for monitoring of soil moisture, IEEE International Geoscience and Remote Sensing Symposium (IGARSS), 2017, doi:10.1109/igarss.2017.8127291.

R. Glowinski, T.W. Pan, X.C. Tai, Some facts about operator-splitting and alternating direction methods, UCLA CAM Reports, (16-10) (2016).

T. Goldstein, S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2) (2009), 323–343.

Y. Gousseau, J.-M. Morel, Are natural images of bounded variation?, SIAM J. Math. Anal., 33 (3) (2001), 634–648.

V.V. Hnatushenko, P.I. Kogut, M.V. Uvarow, On optimal 2-D domain segmentation problem via piecewise smooth approximation of a selective target mapping, J. of Optimization, Differential Equations and Their Applications (JODEA), 27 (2) (2019), 39–74.

M.R. Hestenes, Multiplier and gradient methods, J. Optim. Theory Appl., 4 (1969), 303–320.

V.V. Hnatushenko, K.Yu. Sierikova, I.Yu. Sierikov, Development of a Cloud-Based Web Geospatial Information System for Agricultural Monitoring Using Sentinel-2 Data, IEEE 13th International Scientific and Technical Conference on Computer Sciences and Information Technologies (CSIT), 1 (2018), 270–273.

R. Irony, D. Cohen-Or, D. Lischinski, Colorization by example, in Proceedings of the Sixteenth Eurographics Conference on Rendering Techniques, EGSR 2005, Switzerland, 2005, Eurographics Association, 201–210.

S.H. Kang, R. March, Variational Models for Image Colorization via Chromaticity and Brightness Decomposition, IEEE Transactions on Image Processing, 16 (9) (2007), 2251–2261, DOI: 10.1109/TIP.2007.903257.

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, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Discrete and Continuous Dynamical Systems, Series A, 34 (5) (2014), 2105–2133.

P.I. Kogut, Variational S-convergence of minimization problems. Part I. Definitions and basic properties, Problemy Upravleniya i Informatiki (Avtomatika), (5) (1996), 29–42.

P.I. Kogut, O.P. Kupenko, Approximation Methods in Optimization of Nonlinear Systems, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 32, Walter de Gruyter GmbH, Berlin, Boston, 2019.

P.I. Kogut, G. Leugering, On S-homogenization of an optimal control problem with control and state constraints, Zeitschrift fur Analysis und ihre Anwendung, 20 (2) (2001), 395–429.

P.I. Kogut, R. Manzo, A.O. Putchenko, On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type, Boundary Value Problems, 2016(1)(2016), 1–32.

A. Levin, D. Lischinski, Y. Weiss, Colorization using optimization, ACM Trans. Graph., 23 (2004), 689–694.

L.H. Lieu, L. Vese, Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces, Appl. Math. Optim., 58 (2008), 167–193.

T.M. Lillesand, R.W. Keifer, Remote Sensing and Image Interpretation, John Willey and Sons: New York, NY, USA, 1994.

A. Linderhed, Image EMD: A new tool for image processing, Advances in Adaptive Data Analysis, 1 (2) (2009), Id: 265294.

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

S. Masnou, J.M. Morel, Level Lines based Disocclusion, 5th IEEE Int?l Conf. on Image Processing, Chicago, IL, Oct. 4–7, 1998, 259–263.

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

M.U. Muller, J.D. Shepherd, J.R. Dymond, Support vector machine classification of woody patches in New Zealand from synthetic aperture radar and optical data, with LiDAR training, J. Appl. Remote Sens, 9 (1) (2015), Id: 095984.

R.E. Plant, D.S. Munk, B.R. Roberts, R.L. Vargas, D.W. Rains, R.L. Travis, R.B. Hutmacher, Relationship between remotely sensed reflectance data and cotton growth and yield, Trans. ASAE 43 (2000), 535–546.

L. Roncal, P.R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math., 18 (3) (2016), Id:1550033.

L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica, 60(D) (1992), 259–268.

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

M.R. Saradjian, M. Hosseini, Comparison of optical, radar and hybrid soil moisture estimation models using experimental data, J. Appl. Remote Sens, 5 (1) (2011), Id: 053524.

G. Sapiro, Inpainting the colors, ICIP 2005. IEEE International Conference on Image Processing, 60 (2005), 698–701.

G. Sapiro, L. Yatziv, Fast image and video colorization using chrominance blending, IEEE Trans. Image Process., 15 (2006), 1120–1129.

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

D. S´ykora, J. Buri´anek, and J. Z´ara, Unsupervised colorization of blackand-white cartoons, Proceedings of the 3rd International Symposium on Nonphotorealistic animation and Rendering, NPAR 2004, New York, NY, USA, 2004, ACM, 121–127.

L. Tartar, An introduction to Sobolev spaces and interpolation spaces, in Lecture Notes of the Unione Matematica Italiana, Vol.3, Springer, Berlin; UMI,Bologna, 2007.

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

A. Tsai, Jr.A. Yezzi, A.S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation and magnification, IEEE Trans. Image Process., 10 (8) (2001), 1169–1186.

A. Weinmann, L. Demaret, M. Storath, Total variation regularization for manifold-valued data, SIAM J. Imag. Sci., 7 (4) (2014), 2226–2257.

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.

M. Yashtini, S.H. Kang, W. Zhu, Efficient alternating minimization methods for variational edge-weighted colorization models, Advances in Computational Mathematics, 45 (2019), 1735–1767.




DOI: http://dx.doi.org/10.15421/1420O4

Посилання

  • Поки немає зовнішніх посилань.



Індексування журналу

Журнал розміщено у наукометричних базах, репозитаріях та пошукових системах:

                 


Адреса редколегії: 49050, Україна, Дніпровський національний університет імені Олеся Гончара, вул. Козакова 18, корп. 14, механіко-математичний факультет, д-р фіз.-мат. наук, проф. Когут П.І. 

email: p.kogut@i.ua

www.dnu.dp.ua


Free counters! 

Лицензия Creative Commons
Це видання має доступ за ліцензією Creative Commons «Attribution» («Атрибуция») 4.0 Всемирная.


Open Science in Ukraine - website development