Abstract
This paper introduces an adaptive diffusion partial differential equation (PDE) for noise removal, which combines a total variation (TV) term and a p-Laplacian (1 < p ≤ 2) term. Utilizing the edge indicator, we can adaptively control the diffusion model, which alternates between the TV and the p-Laplacian(1 < p ≤ 2) in accordance with the image feature. The main advantage of the proposed model is able to alleviate the staircase effect in smooth regions and preserve edges while removing the noise. The existence of a weak solution of the proposed model is proved. Experimental results confirm the performance of the proposed method with regard to peak signal-to-noise ratio (PSNR), mean structural similarity (MSSIM) and visual quality.
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References
Andreu F, Mazn JM, Caselles V (2001) The Dirichelt Peroblem for the total varitaion flow. J Funct Anal 180(2):347–403. https://doi.org/10.1007/978-3-0348-7928-6_5
Andreu F, Caselles V, Diaz JI, Marzon JM (2002) Some qualitative properties for the total variation flow. J Funct Anal 188(2):516–547. https://doi.org/10.1006/jfan.2001.3829
Aubert G, Kornprobst P (2006) Mathematical problems in image processing: partial difierential equations and the calculus of variations, 2nd. Springer, New York
Blomgren P, Chan TF, Mulet P, Wong CK (1997) Total variation image restoration: numerical methods and extensions. Int Conf Image Process 3:384–387. https://doi.org/10.1109/ICIP.1997.632128
Canny J (1986) A Computational Approach to Edge Detection. IEEE Trans Pattern Anal Mach Intell 8(6):679–698. https://doi.org/10.1109/TPAMI.1986.4767851
Chambolle A, Lions PL (1997) Image recovery via total varition minimization and related problems. Numerische Mathematik 76(2):167–188. https://doi.org/10.1007/s002110050258
Chan TF, Esedoglu S, Park FE (2007) Image decomposition combining staircase reduction and texture extraction. J Vis Commun Image Represent 18(6):464–486. https://doi.org/10.1016/j.jvcir.2006.12.004
Chen Y, Levine S, Rao M (2006) Variable exponent, linear growth functionals in image restoration. Siam J Appl Math 66 (4):1383–1406. https://doi.org/10.1137/050624522
Gilboa G, Sochen N, Zeevi YY (2004) Image enhancement and denoising by complex diffusion processes. IEEE Trans Pattern Anal Mach Intell 26(8):1020–1036. https://doi.org/10.1109/TPAMI.2004.47
Guidotti P, Longo K (2011) Two enhanced fourth-order diffusion models for image denoising. J Math Imaging Vision 40(2):188–198. https://doi.org/10.1007/s10851-010-0256-9
Kuijper A (2007) P-Laplacian Driven Image Processing. IEEE Int Conf Image Process 5:257–260. https://doi.org/10.1109/ICIP.2007.4379814
Li F, Li Z, Pi L (2010) Variable exponent functionals in image restoration. Appl Math Comput 216(3):870–882. https://doi.org/10.1016/j.amc.2010.01.094
Lysaker M, Lundervold A, Tai X (2003) Noise removal using forth-order partial difffential equation with applications to medical magnetic resonance images in space and time. IEEE Trans Image Process 12(12):1579–1590. https://doi.org/10.1109/TIP.2003.819229
Perona P, Malik J (1990) Scale-space and edge detection using anisotropic diffucsion. IEEE Trans Pattern Anal Mach Intell 12(7):629–639. https://doi.org/10.1109/34.56205
Rafsanjani HK, Sedaaghi MH, Saryazdi S (2016) Efficient diffusion coefficient for image denoising. Comput Math Appl 72(4):893–903. https://doi.org/10.1016/j.camwa.2016.06.005
Rudin L, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60(1-4):259–268. https://doi.org/10.1016/0167-2789(92)90242-F
Simon J (1986) Compact sets in the space L p(0,T; B). Annali di Matematica 146(1):65–96. https://doi.org/10.1007/BF01762360
Suray Prasath VB, Urbano JM, Vorotnikov DA (2015) Analysis of adaptive forward-backward diffusion flows with applications in image processing. Inverse Prob 31 (10):105008. https://doi.org/10.1088/0266-5611/31/10/105008
Suray Prasath VB, Vorotnikov DA, Pelapur R, Jose S, Seetharaman G, Palaniappan K (2015) Multiscale Tikhonov-Total variation image restoration using spatially varying edge coherence exponent. IEEE Trans Image Process 24(12):5220–5235. https://doi.org/10.1109/TIP.2015.2479471
Temam R (1979) Navier-stokes equuations. Studies in Mathematics and its Applications, vol 2. North-Holland Publishing Co., Amsterdam
Vorotnikov D (2012) Global generalized solutions for Maxwell-alpha and Euler-alpha equations. Nonlinearity 25(2):309–327. https://doi.org/10.1088/0951-7715/25/2/309
Weickert J, Romeny BH, Viergever MA (1998) Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans Image Process 7(3):398–410. https://doi.org/10.1109/83.661190
You YL, Kaveh M (2000) Fourth-order partial differential equations for noise removal. IEEE Trans Image Process 9(10):1723–1730. https://doi.org/10.1109/83.869184
Zhou B, Mu CL, Feng J, Wei W (2012) Continuous level anisotropic diffusion for noise removal. Appl Math Model 36(8):3779–3786. https://doi.org/10.1016/j.apm.2011.11.026
Zvyagin VG, Vorontnikov DA (2008) Topological approximation methods for evoluctionary problems of nonlinear hydrodynamics. De Gruyter Series in Nonliear Analysis and Applications, vol 12, De Gruyter & Co., Berlin
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Zhang, X., Ye, W. An adaptive second-order partial differential equation based on TV equation and p-Laplacian equation for image denoising. Multimed Tools Appl 78, 18095–18112 (2019). https://doi.org/10.1007/s11042-019-7170-y
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DOI: https://doi.org/10.1007/s11042-019-7170-y