×

Existence and uniqueness of weak solutions for a new class of convex optimization problems related to image analysis. (English) Zbl 1477.49025

Summary: This paper proposes a new anisotropic diffusion model in image restoration that is understood from a variational optimization of an energy functional. Initially, a family of new diffusion functions based on cubic Hermite spline is provided for optimal image denoising. After that, the existence and uniqueness of weak solutions for the corresponding Euler-Lagrange equation are proven in an appropriate functional space, and a consistent and stable numerical model is also shown. We complement this work by illustrating some experiments on different actual brain Magnetic Resonance Imaging (MRI) scans, showing the proposed model’s impressive results.

MSC:

49K10 Optimality conditions for free problems in two or more independent variables
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
35K55 Nonlinear parabolic equations
90C25 Convex programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jourhmane, M., Méthodes numériques de résolution d’un problème d’électro-encéphalographie (1993), Rennes, France: University of Rennes 1, Rennes, France, Ph.D. thesis
[2] Aubert, G.; Kornprobst, P., Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (2006), New York, NY, USA: Springer, New York, NY, USA · Zbl 1110.35001
[3] Pinoli, J.-C., Mathematical Foundations of Image Processing and Analysis, Volume 1 (2014), Hoboken, NJ, USA: Wiley-ISTE, Hoboken, NJ, USA · Zbl 1398.94002
[4] Pinoli, J.-C., Mathematical Foundations of Image Processing and Analysis, Volume 2 (2014), Hoboken, NJ, USA: Wiley-ISTE, Hoboken, NJ, USA · Zbl 1398.94002
[5] Rudin, L. I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60, 1-4, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-f
[6] You, Y. L.; Xu, W.; Tannenbaum, A.; Kaveh, M., Behavioral analysis of anisotropic diffusion in image processing, IEEE Transactions on Image Processing : A Publication of the IEEE Signal Processing Society, 5, 11, 1539-1553 (1996) · doi:10.1109/83.541424
[7] Aubert, G.; Vese, L., A variational method in image recovery, SIAM Journal on Numerical Analysis, 34, 5, 1948-1979 (1997) · Zbl 0890.35033 · doi:10.1137/s003614299529230x
[8] Charbonnier, P.; Blanc-Feraud, L.; Aubert, G.; Barlaud, M., Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, 6, 2, 298-311 (1997) · doi:10.1109/83.551699
[9] Barcelos, C. A. Z.; Chen, Y., Heat flows and related minimization problem in image restoration, Computers & Mathematics with Applications, 39, 5-6, 81-97 (2000) · Zbl 0957.65069 · doi:10.1016/s0898-1221(00)00048-1
[10] Wang, L.; Zhou, S., Existence and uniqueness of weak solutions for a nonlinear parabolic equation related to image analysis, Journal of Partial Differential Equations, 19, 2, 97-112 (2006) · Zbl 1122.35061
[11] Barbu, T.; Barbu, V.; Biga, V.; Coca, D., A pde variational approach to image denoising and restoration, Nonlinear Analysis: Real World Applications, 10, 3, 1351-1361 (2009) · Zbl 1169.35341 · doi:10.1016/j.nonrwa.2008.01.017
[12] Wu, B.; Ogada, E. A.; Sun, J.; Guo, Z., A total variation model based on the strictly convex modification for image denoising, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.94032 · doi:10.1155/2014/948392
[13] Barbu, T.; Moroşanu, C., Image restoration using a nonlinear second-order parabolic pde-based scheme, Analele Universitatii “Ovidius” Constanta - Seria Matematica, 25, 1, 33-48 (2017) · Zbl 1389.35021 · doi:10.1515/auom-2017-0003
[14] Li, P.; Li, S., Weak solutions for a class of generalised image restoration models, International Journal of Dynamical Systems and Differential Equations, 8, 3, 190-203 (2018) · Zbl 1442.35210 · doi:10.1504/ijdsde.2018.10009557
[15] Maiseli, B. J., On the convexification of the perona-malik diffusion model, Signal, Image and Video Processing, 14, 6, 1283-1291 (2020) · doi:10.1007/s11760-020-01663-x
[16] Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 7, 629-639 (1990) · doi:10.1109/34.56205
[17] Whitaker, R. T.; Pizer, S. M., A multi-scale approach to nonuniform diffusion, CVGIP: Image Understanding, 57, 1, 99-110 (1993) · doi:10.1006/ciun.1993.1006
[18] Chen, P., Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations, Electronic Research Announcements in Mathematical Sciences, 24, 38-52 (2017) · Zbl 1404.35236 · doi:10.3934/era.2017.24.005
[19] Weickert, J., Anisotropic Diffusion in Image Processing (1998), Stuttgart, Germany: Teubner Verlag, Stuttgart, Germany · Zbl 0886.68131
[20] Tiarimti Alaoui, A.; Jourhmane, M., Existence and uniqueness of weak solutions for novel anisotropic nonlinear diffusion equations related to image analysis, Journal of Mathematics, 2021 (2021) · Zbl 1477.94022 · doi:10.1155/2021/5553126
[21] Stoer, J.; Bulirsch, R., Interpolation (2002), New York, NY, USA: Springer, New York, NY, USA
[22] Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions (2015), Abingdon, UK: Taylor & Francis Group, Abingdon, UK · Zbl 1310.28001
[23] Gaillard, F., Normal brain mr radiopaedia.org, rID: 42777 (2016)
[24] Gaillard, F., Normal brain mri (tle protocol) radiopaedia.org, rID: 40748 (2015)
[25] Gaillard, F., Normal Mri Brain Including Mr venogram radiopaedia.Org, rID: 51158 (2017)
[26] Gonzalez, R. C.; Woods, R. E., Digital Image Processing (2006), Upper Saddle River,, NJ, USA: Prentice-Hall, Upper Saddle River,, NJ, USA
[27] Wang, Z.; Bovik, A. C., Mean squared error: love it or leave it? a new look at signal fidelity measures, IEEE Signal Processing Magazine, 26, 1, 98-117 (2009)
[28] Wang, Z.; Bovik, A. C.; Sheikh, H. R.; Simoncelli, E. P., Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13, 4, 600-612 (2004) · doi:10.1109/tip.2003.819861
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.