ORIGINAL PAPERS

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

  • Ciro D'Apice ,
  • Peter I. Kogut ,
  • Rosanna Manzo ,
  • Antonino Parisi
Expand
  • 1. Dipartimento di Scienze Aziendali-Management and Innovation Systems, University of Salerno, 84084 Fisciano, SA, Italy;
    2. Department of Differential Equations, Oles Honchar Dnipro National University, Dnipro 49010, Ukraine;
    3. EOS Data Analytics Ukraine, Dnipro 49010, Ukraine;
    4. Department of Political and Communication Sciences, University of Salerno, 84084 Fisciano, SA, Italy;
    5. Dipartimento di Matematica e Informatica “Ulisse Dini”, University of Firenze, Viale Morgagni, 67/a, 50134 Florence, Italy

Received date: 2023-08-25

  Revised date: 2023-12-26

  Online published: 2025-12-24

Supported by

Open access funding provided by Università degli Studi di Salerno within the CRUI-CARE Agreement. This work was supported by Visiting Professors Program-UNISA Call 2022.

Abstract

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

Cite this article

Ciro D'Apice , Peter I. Kogut , Rosanna Manzo , Antonino Parisi . Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement[J]. Communications on Applied Mathematics and Computation, 2025 , 7(6) : 2385 -2419 . DOI: 10.1007/s42967-024-00382-1

References

[1] Alaa, H., Alaa, N.E., Bouchriti, A., Charkaou, A.: An improved nonlinear anisotropic PDE with $ p(x) $-growth conditions applied to image restoration and enhancement. Authorea (2022). https://doi.org/10.22541/au.165717367.72990650/v1
[2] Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 845–866 (1992)
[3] Ambrosio, L., Caselles, V., Masnou, S., Morel, J.M.: The connected components of sets of finite perimeter. Eur. J. Math. 3, 39–92 (2001)
[4] Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147. Springer, New York (2006)
[5] Bertalmío, M., Caselles, V., Provenzi, E., Rizzi, A.: Perceptual color correction through variational techniques. IEEE Trans. Image Process. 16(4), 1058–1072 (2007)
[6] Black, M.J., Sapiro, G., Marimont, D.H., Heger, D.: Robust anisotropic diffusion. IEEE Trans. Image Process. 7(3), 421–432 (1998)
[7] Blomgren, P., Chan, T.F., Mulet, P., Wong, C.: Total variation image restoration: numerical methods and extensions. In: Proceedings of the 1997 IEEE International Conference on Image Processing, Santa Barbara, CA, USA, 1997, vol. 42, pp. 384–387 (1997)
[8] Bungert, L., Coomes, D.A., Ehrhardt, M.J., Rasch, J., Reisenhofer, R., Schönlieb, C.B.: Blind image fusion for hyperspectral imaging with the directional total variation. Inverse Probl. 34(4), 044003 (2018)
[9] Bungert, L., Ehrhardt, M.J.: Robust image reconstruction with misaligned structural information. IEEE Access 8, 222944–222955 (2020)
[10] 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)
[11] Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)
[12] Chen, Y., Levine, S., Stanich, J.: Image restoration via nonstandard diffusion. Figshare (2014). https://www.mathcs.duq.edu/tech-reports/tr04-01.pdf
[13] Chipot, M., de Oliveira, H.B.: Some results on the $ p(u) $-Laplacian problem. Math. Annal. 375, 283–306 (2019)
[14] D’Apice, C., De Maio, U., Kogut, P.I.: Gap phenomenon in homogenization of parabolic optimal control problem. IMA J. Math. Control Inf. 25, 461–480 (2008)
[15] D’Apice, C., De Maio, U., Kogut, P.I.: Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain. Discrete Contin. Dyn. Syst. B 11(2), 283–314 (2009)
[16] D’Apice, C., De Maio, U., Kogut, P.I.: An indirect approach to the existence of quasi-optimal controls in coefficients for multi-dimensional thermistor problem. In: Sadovnichiy, V.A., Zgurovsky, M. (eds.) Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, pp. 489–522. Springer, New York (2020)
[17] D’Apice, C., Kogut, P.I., Kupenko, O., Manzo, R.: On variational problem with nonstandard growth functional and its applications to image processing. J. Math. Imaging Vis. 65(3), 472–491 (2023)
[18] D’Apice, C., Kogut, P.I., Manzo, R., Uvarov, M.: Variational model with nonstandard growth conditions for restoration of satellite optical images using synthetic aperture radar. Eur. J. Appl. Math. 34(1), 77–105 (2023)
[19] D’Apice, C., Kogut, P.I., Manzo, R.: On coupled two-level variational problem in Sobolev-Orlicz space. Differ. Integral Equ. 36(7/8), 621–660 (2023)
[20] D’Apice, C., Kogut, P.I., Manzo, R.: A two-level variational algorithm in the Sobolev-Orlicz space to predict daily surface reflectance at LANDSAT high spatial resolution and MODIS temporal frequency. J. Comput. Appl. Math. 434, 1–23 (2023)
[21] Dautray, R., Lion, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Springer, Berlin (1985)
[22] Diening, L., Harjulehto, P., Hästö, P., Ru??iĉka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, New York (2011)
[23] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
[24] Jia, Z., Ng, M.K., Wang, W.: Color image restoration by saturation-value total variation. SIAM J. Imaging Sci. 12(2), 972–1000 (2019)
[25] Karami, F., Meskine, D., Sadik, K.: A new nonlocal model for the restoration of textured images. J. Appl. Anal. Comput. 9(6), 2070–2095 (2019)
[26] Kogut, P.I.: Variational S-convergence of minimization problems. Part I. Definitions and basic properties. Problemy Upravleniya i Informatiki (Avtomatika) 5, 29–42 (1996)
[27] Kogut, P.I.: $ S $-convergence of the conditional optimization problems and its variational properties. Problemy Upravleniya i Informatiki (Avtomatika) 4, 64–79 (1997)
[28] Kogut, P.I.: On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discret. Contin. Dyn. Syst. A 34(5), 2105–2133 (2014)
[29] Kogut, P.I.: On optimal and quasi-optimal controls in coefficients for multi-dimensional thermistor problem with mixed Dirichlet-Neumann boundary conditions. Control Cybern. 48(1), 31–68 (2019)
[30] Kogut, P.I., Kohut, Y., Manzo, R.: Existence result and approximation of an optimal control problem for the Perona-Malik equation. Ric. Mat. (2022). https://doi.org/10.1007/11587-022-00730-4
[31] Kogut, P.I., Kupenko, O.P.: Approximation Methods in Optimization of Nonlinear Systems. De Gruyter Series in Nonlinear Analysis and Applications, vol. 32. Walter de Gruyter GmbH, Berlin (2019)
[32] Kogut, P.I., Leugering, L.: On S-homogenization of an optimal control problem with control and state constraint. Z. fur Anal. ihre Anwend. 20(2), 395–429 (2001)
[33] Kohr, H.: Total variation regularization with variable Lebesgue priors. arXiv:1702.08807 (2017)
[34] Lieu, L.H., Vese, L.A.: Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces. Appl. Math. Optim. 58, 167–193 (2008)
[35] Manzo, R.: On Neumann boundary control problem for ill-posed strongly nonlinear elliptic equation with $ p $-Laplace operator and $ {L}^1 $-type of nonlinearity. Ric. Mat. 68(2), 769–802 (2019)
[36] Manzo, R.: On Tikhonov regularization of optimal Neumann boundary control problem for an ill-posed strongly nonlinear elliptic equation with an exponential type of non-linearity. Differ. Integral Equ. 33(3/4), 139–162 (2020)
[37] Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 2. AMS, Providence (2002)
[38] Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27(4), 458–474 (1990)
[39] Piella, G.: Image fusion for enhanced visualization: a variational approach. Int. J. Comput. Vis. 83(1), 1–11 (2009)
[40] Pierre, F., Aujol, J.-F., Bugeau, A., Steidl, G., Ta, V.-T.: Variational contrast enhancement of gray-scale and RGB images. J. Math. Imaging Vis. 57, 99–116 (2017)
[41] Prasath, V.B.S., Urbano, J.M., Vorotnikov, D.: Analysis of adaptive forward-backward diffusion flows with applications in image processing. Inverse Probl. 31, 1–30 (2015)
[42] Ring, W.: Structural properties of solutions to total variation regularization problems. ESAIM: Math. Model. Numer. Anal. 34(4), 799–810 (2000)
[43] Schönlieb, C.B.: Total Variation Minimization with an $ H^{-1} $ Constraint. Research Gate Publication (2009)
[44] Sugimura, D., Mikami, T., Yamashita, H., Hamamoto, T.: Enhancing color images of extremely low light scenes based on RGB/NIR images acquisition with different exposure times. IEEE Trans. Image Process. 24(11), 3586–3597 (2015)
[45] Wunderli, T.: On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudosolutions. J. Math. Anal. Appl. 364(2), 591–598 (2010)
[46] Zhikov, V.V.: Solvability of the three-dimensional thermistor problem. Proc. Steklov Inst. Math. 281, 98–111 (2008)
[47] Zhikov, V.V.: On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. 175(5), 463–570 (2011)
Options
Outlines

/