# American Institute of Mathematical Sciences

April  2022, 16(2): 367-396. doi: 10.3934/ipi.2021054

## A fuzzy edge detector driven telegraph total variation model for image despeckling

 1 School of Basic Sciences, Indian Institute of Technology Mandi, PIN 175005, INDIA 2 Department of Mathematics & Scientific Computing, National Institute of Technology Hamirpur, PIN 177005, INDIA 3 Department of Mathematics, Indian Institute of Technology Delhi, PIN 110016, INDIA

* Corresponding author: Rajendra K. Ray

Received  October 2020 Revised  May 2020 Published  April 2022 Early access  September 2021

Speckle noise suppression is a challenging and crucial pre-processing stage for higher-level image analysis. In this work, a new attempt has been made using telegraph total variation equation and fuzzy set theory for image despeckling. The intuitionistic fuzzy divergence function has been used to distinguish between edges and noise. To the best of the authors' knowledge, most of the studies on the multiplicative speckle noise removal process focus only on diffusion-based filters, and little attention has been paid to the study of fuzzy set theory. The proposed approach enjoys the benefits of both telegraph total variation equation and fuzzy edge detector, which is robust to noise and preserves image structural details. Moreover, we establish the existence and uniqueness of weak solutions of a regularized version of the present system using the Schauder fixed point theorem. With the proposed technique, despeckling is carried out on natural, real synthetic aperture radar, and real ultrasound images. The experimental results computed by the suggested method are reported, which are found better in terms of noise elimination and detail/edge preservation, concerning the existing approaches.

Citation: Sudeb Majee, Subit K. Jain, Rajendra K. Ray, Ananta K. Majee. A fuzzy edge detector driven telegraph total variation model for image despeckling. Inverse Problems and Imaging, 2022, 16 (2) : 367-396. doi: 10.3934/ipi.2021054
##### References:
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Shanmugan and J. C. Holtzman, A model for radar images and its application to adaptive digital filtering of multiplicative noise, IEEE Trans. Pattern Anal. Mach. Intell., (1982), 157-166. doi: 10.1109/TPAMI.1982.4767223. [24] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2002. [25] J. W. Goodman, Some fundamental properties of speckle, JOSA, 66 (1976), 1145-1150.  doi: 10.1364/JOSA.66.001145. [26] Y. Hao, J. Xu, S. Li and X. Zhang, A variational model based on split Bregman method for multiplicative noise removal, Int. J. Electron. Commun., 69 (2015), 1291-1296.  doi: 10.1016/j.aeue.2015.05.009. [27] K. H. Ho and N. Ohnishi, Fedge fuzzy edge detection by fuzzy categorization and classification of edges, In International Workshop on Fuzzy Logic in Artificial Intelligence, Springer, 1188 (1995), 182-196. doi: 10.1007/3-540-62474-0_14. [28] C. Hua and T. 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Zeevi, Image enhancement using elastic manifolds, In Proceedings of the 14th International Conference on Image Analysis and Processing, ICIAP 2007, IEEE, (2007), 769-774. doi: 10.1109/ICIAP.2007.4362869. [48] V. Ratner and Y. Y. Zeevi, Denoising-enhancing images on elastic manifolds, IEEE Trans. Image Process., 20 (2011), 2099-2109.  doi: 10.1109/TIP.2011.2118221. [49] L. Rudin, P.-L. Lions and S. Osher, Multiplicative denoising and deblurring: Theory and algorithms, In Geometric Level Set Methods in Imaging, Vision, and Graphics, Springer, 2003,103-119. doi: 10.1007/0-387-21810-6_6. [50] X. Shan, J. Sun and Z. Guo, Multiplicative noise removal based on the smooth diffusion equation, J. Math. Imag. Vis., 61 (2019), 763-779.  doi: 10.1007/s10851-018-00870-z. [51] J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM J. Imaging Sci., 1 (2008), 294-321.  doi: 10.1137/070689954. [52] J. Song and H. 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##### References:
 [1] A. Achim, A. Bezerianos and P. Tsakalides, Novel Bayesian multiscale method for speckle removal in medical ultrasound images, IEEE Trans. Med. Imag., 20 (2001), 772-783. doi: 10.1109/42.938245. [2] R. Adams, Sobolev Spaces. Pure and Applied Mathematics, vol. 65, Academic Press, New York, London, 1975. [3] E. S. Agency, Esa earth online, https://earth.esa.int/handbooks/asar/CNTR1-4.html. [4] S. Aja, C. Alberola and A. Ruiz, Fuzzy anisotropic diffusion for speckle filtering, In 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No. 01CH37221), IEEE, 2 (2001), 1261-1264. [5] F. Argenti, A. Lapini, T. Bianchi and L. Alparone, A tutorial on speckle reduction in synthetic aperture radar images, IEEE Geosci. Remote Sens. Mag., 1 (2013), 6-35.  doi: 10.1109/MGRS.2013.2277512. [6] K. T. Atanassov, Intuitionistic fuzzy sets: Past, present and future, In EUSFLAT Conf., (2003), 12-19. [7] G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925-946.  doi: 10.1137/060671814. [8] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, vol. 147, Appl. Math. Sci. Springer, New York, 2006. [9] A. Averbuch, B. Epstein, N. Rabin and E. Turkel, Edge-enhancement postprocessing using artificial dissipation, IEEE Trans. Image Process., 15 (2006), 1486-1498.  doi: 10.1109/TIP.2006.875734. [10] J. J. J. Babu and G. F. Sudha, Adaptive speckle reduction in ultrasound images using fuzzy logic on coefficient of variation, Biomed. Signal. Process. Control., 23 (2016), 93-103. [11] G. Baravdish, O. Svensson, M. Gulliksson and Y. Zhang, Damped second order flow applied to image denoising, IMA J. Appl. Math., 84 (2019), 1082-1111.  doi: 10.1093/imamat/hxz027. [12] Y. Becerikli and T. M. Karan, A new fuzzy approach for edge detection, In International Work-Conference on Artificial Neural Networks, Springer, (2005), 943-951. doi: 10.1007/11494669_116. [13] K. Binaee and R. P. Hasanzadeh, An ultrasound image enhancement method using local gradient based fuzzy similarity, Biomed. Signal. Process. Control., 13 (2014), 89-101.  doi: 10.1016/j.bspc.2014.03.013. [14] C. B. Burckhardt, Speckle in ultrasound B-mode scans, IEEE Trans. Sonics Ultrason., 25 (1978), 1-6.  doi: 10.1109/T-SU.1978.30978. [15] Y. Cao, J. Yin, Q. Liu and M. Li, A class of nonlinear parabolic-hyperbolic equations applied to image restoration, Nonlinear Anal. Real World Appl., 11 (2010), 253-261.  doi: 10.1016/j.nonrwa.2008.11.004. [16] T. Chaira and A. K. Ray, Segmentation using fuzzy divergence, Pattern Recognit. Lett., 24 (2003), 1837-1844.  doi: 10.1016/S0167-8655(03)00007-2. [17] T. Chaira and A. Ray, A new measure using intuitionistic fuzzy set theory and its application to edge detection, Appl. Soft Comput., 8 (2008), 919-927.  doi: 10.1016/j.asoc.2007.07.004. [18] P. Dewaele, P. Wambacq, A. Oosterlinck and J.-L. Marchand, Comparison of some speckle reduction techniques for sar images, In Geoscience and Remote Sensing Symposium, 1990. IGARSS'90.'Remote Sensing Science for the Nineties'., 10th Annual International, IEEE, (1990), 2417-2422. doi: 10.1109/IGARSS.1990.689028. [19] G. Dong, Z. Guo and B. Wu, A convex adaptive total variation model based on the gray level indicator for multiplicative noise removal, In Abstr. Appl. Anal., 2013, (2013). doi: 10.1155/2013/912373. [20] eoPortal: Sharing Earth Observation Resources, Kompsat-5, https://directory.eoportal.org/web/eoportal/satellite-missions/k/kompsat-5. [21] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC press, 2015. [22] L. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, American Mathematical Society, Providence, RI, 1998. [23] V. S. Frost, J. A. Stiles, K. S. Shanmugan and J. C. Holtzman, A model for radar images and its application to adaptive digital filtering of multiplicative noise, IEEE Trans. Pattern Anal. Mach. Intell., (1982), 157-166. doi: 10.1109/TPAMI.1982.4767223. [24] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2002. [25] J. W. Goodman, Some fundamental properties of speckle, JOSA, 66 (1976), 1145-1150.  doi: 10.1364/JOSA.66.001145. [26] Y. Hao, J. Xu, S. Li and X. Zhang, A variational model based on split Bregman method for multiplicative noise removal, Int. J. Electron. Commun., 69 (2015), 1291-1296.  doi: 10.1016/j.aeue.2015.05.009. [27] K. H. Ho and N. Ohnishi, Fedge fuzzy edge detection by fuzzy categorization and classification of edges, In International Workshop on Fuzzy Logic in Artificial Intelligence, Springer, 1188 (1995), 182-196. doi: 10.1007/3-540-62474-0_14. [28] C. Hua and T. Jinwen, Speckle reduction of synthetic aperture radar images based on fuzzy logic, In First International Workshop on Education Technology and Computer Science, 1, IEEE, (2009), 933-937. doi: 10.1109/ETCS.2009.212. [29] Y.-M. Huang, M. K. Ng and Y.-W. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593. [30] S. K. Jain and R. K. Ray, Edge detectors based telegraph total variational model for image filtering, In Information Systems Design and Intelligent Applications, Springer, 433 (2016), 119-126. doi: 10.1007/978-81-322-2755-7_13. [31] S. K. Jain and R. K. Ray, Non-linear diffusion models for despeckling of images: Achievements and future challenges, IETE Technical Review, 37 (2020), 66-82.  doi: 10.1080/02564602.2019.1565960. [32] S. K. Jain, R. K. Ray and A. Bhavsar, Iterative solvers for image denoising with diffusion models: A comparative study, Comput. Math. Appl., 70 (2015), 191-211.  doi: 10.1016/j.camwa.2015.04.009. [33] S. K. Jain, R. K. Ray and A. Bhavsar, A nonlinear coupled diffusion system for image despeckling and application to ultrasound images, Circ. Syst. Signal Pr., 38 (2019), 1654-1683.  doi: 10.1007/s00034-018-0913-6. [34] J. S. Jin, Y. Wang and J. Hiller, An adaptive nonlinear diffusion algorithm for filtering medical images, IEEE Trans. Inf. Technol. Biomed., 4 (2000), 298-305.  doi: 10.1109/4233.897062. [35] Z. Jin and X. Yang, Analysis of a new variational model for multiplicative noise removal, J. Math. Anal. Appl., 362 (2010), 415-426.  doi: 10.1016/j.jmaa.2009.08.036. [36] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall PTR, New Jersey, 1995. [37] D. T. Kuan, A. A. Sawchuk, T. C. Strand and P. Chavel, Adaptive noise smoothing filter for images with signal-dependent noise, IEEE Trans. Pattern Anal. Mach. Intell., 7 (1985), 165-177.  doi: 10.1109/TPAMI.1985.4767641. [38] J.-S. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE Trans. Pattern Anal. Mach. Intell., 2 (1980), 165-168.  doi: 10.1109/TPAMI.1980.4766994. [39] M. Liu and Q. Fan, A modified convex variational model for multiplicative noise removal, J. Vis. Commun. Image Represent., 36 (2016), 187-198.  doi: 10.1016/j.jvcir.2016.01.014. [40] Q. Liu, X. Li and T. Gao, A nondivergence p-Laplace equation in a removing multiplicative noise model, Nonlinear Anal. Real World Appl., 14 (2013), 2046-2058.  doi: 10.1016/j.nonrwa.2013.02.008. [41] S. Majee, S. K. Jain, R. K. Ray and A. K. Majee, On the development of a coupled nonlinear telegraph-diffusion model for image restoration, Comput. Math. Appl., 80 (2020), 1745-1766.  doi: 10.1016/j.camwa.2020.08.010. [42] S. Majee, R. K. Ray and A. K. Majee, A gray level indicator-based regularized telegraph diffusion model: Application to image despeckling, SIAM J. Imaging Sci., 13 (2020), 844-870.  doi: 10.1137/19M1283033. [43] A. Mittal, A. K. Moorthy and A. C. Bovik, No-reference image quality assessment in the spatial domain, IEEE Trans. Image Process., 21 (2012), 4695-4708.  doi: 10.1109/TIP.2012.2214050. [44] M. Nadeem, A. Hussain and A. Munir, Fuzzy logic based computational model for speckle noise removal in ultrasound images, Multimed. Tools. Appl., 78 (2019), 18531-18548.  doi: 10.1007/s11042-019-7221-4. [45] R. Prager, A. Gee, G. Treece and L. Berman, Speckle detection in ultrasound images using first order statistics, University of Cambridge, Department of Engineering. [46] V. S. Prasath and R. Delhibabu, Image restoration with fuzzy coefficient driven anisotropic diffusion, In International Conference on Swarm, Evolutionary, and Memetic Computing, 8947, Springer, (2015), 145-155. doi: 10.1007/978-3-319-20294-5_13. [47] V. Ratner and Y. Y. 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Tizhoosh, Fuzzy anisotropic diffusion: A rule-based approach, In Proceeding of the 7th World Multiconference on Systemics, Cyebernetics and Informatics, (2003), 241-246. [53] E. Szmidt and J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets Syst., 114 (2000), 505-518.  doi: 10.1016/S0165-0114(98)00244-9. [54] D. N. Thanh, V. S. Prasath and S. Dvoenko et al., An adaptive method for image restoration based on high-order total variation and inverse gradient, Signal, Image and Video Process., 14 (2020), 1189-1197.  doi: 10.1007/s11760-020-01657-9. [55] M. Tur, K.-C. Chin and J. W. Goodman, When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159.  doi: 10.1364/AO.21.001157. [56] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861. [57] J. 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Set of sixteen $3 \times 3$ templates
Original images
Boat image ($512\times 512$). (a) Speckled image: $L = 3$. (b)-(f) Despeckled by various approaches. (g) Speckled image: $L = 10$. (h)-(l) Despeckled by various approaches
Brick image ($256\times 256$). (a) Speckled image: $L = 3$. (b)-(f) Despeckled by various approaches. (g) Speckled image: $L = 10$. (h)-(l) Despeckled by various approaches
(a) Ratio image for the original image 2b, (b)-(f) Ratio images for the despeckled images 4h-4l. (g) Indicate the one-dimensional slices. (h) Results for the Slice-1. (i) Results for the Slice-2. (j) Results for the Slice-3
Circle image ($299\times 299$). (a) Speckled image: $L = 3$. (b)-(f) Despeckled by various approaches. (g) Speckled image: $L = 10$. (h)-(l) Despeckled by various approaches
(a)-(f) 2D Contour map of the images 6g-6l. (g)-(l) 3D surface plot of the images 6g-6l
Results are plotted for the circle image when the image is degraded by $L = 10$. (a) Relative error vs. the iteration number. (b) Logarithmic Relative error vs. the iteration number. (c) PSNR value vs. the corresponding iteration number
(a) SAR Image-1: One look radar image [3]. (b)-(d) Restored by different models
(a) SAR Image-2: Image of KOMPSAT/Arirang-5 of a part of the Himalayan Arc [20]. (b)-(d) Restored by different models
A Ultrasound image of fetal foot and restored by different models
A Ultrasound image of liver cyst and restored by different models
Relative error vs. the iteration number for various models
MSSIM, PSNR, and SI
 Image $L$ AA [7] Dong[19] DDD[61] ZZDB[62] Proposed MSSIM PSNR SI MSSIM PSNR SI MSSIM PSNR SI MSSIM PSNR SI MSSIM PSNR SI Boat 1 0.5577 16.90 0.3695 0.5526 16.78 0.3368 0.5510 16.92 0.3417 0.5705 16.98 0.3289 0.5816 17.04 0.3162 3 0.6780 22.40 0.3759 0.6680 22.41 0.3712 0.6806 22.20 0.3720 0.6810 22.30 0.3569 0.6976 22.54 0.3472 5 0.7210 24.27 0.3783 0.7128 24.27 0.3755 0.7205 24.06 0.3662 0.7200 24.14 0.3637 0.7386 24.46 0.3558 10 0.7757 26.16 0.3796 0.7658 26.17 0.3782 0.7729 26.11 0.3794 0.7745 26.20 0.3709 0.7885 26.39 0.3658 Brick 1 0.2875 12.10 0.0816 0.2874 12.18 0.0805 0.2873 12.14 0.0779 0.2880 12.16 0.0728 0.2961 12.23 0.0719 3 0.3710 16.95 0.0933 0.3737 17.00 0.0901 0.3646 16.86 0.0879 0.3650 16.87 0.0865 0.3823 17.09 0.0854 5 0.4167 19.17 0.0998 0.4174 19.21 0.0978 0.4176 18.61 0.0955 0.4175 18.65 0.0923 0.4234 19.32 0.0908 10 0.4790 21.84 0.1063 0.4855 21.86 0.1051 0.4874 21.88 0.1043 0.4877 21.90 0.1005 0.4889 22.00 0.0996 Circle 1 0.9510 33.19 0.3106 0.9501 32.22 0.3165 0.9458 33.69 0.3219 0.9502 33.48 0.3098 0.9670 34.87 0.2982 3 0.9633 36.88 0.3270 0.9654 36.89 0.3245 0.9572 36.72 0.3271 0.9603 36.90 0.3215 0.9765 38.90 0.3163 5 0.9688 37.85 0.3285 0.9688 37.86 0.3271 0.9617 37.64 0.3289 0.9634 37.58 0.3249 0.9784 39.82 0.3198 10 0.9726 39.65 0.3291 0.9756 39.67 0.3279 0.9761 39.86 0.3290 0.9732 39.76 0.3253 0.9821 41.40 0.3241
 Image $L$ AA [7] Dong[19] DDD[61] ZZDB[62] Proposed MSSIM PSNR SI MSSIM PSNR SI MSSIM PSNR SI MSSIM PSNR SI MSSIM PSNR SI Boat 1 0.5577 16.90 0.3695 0.5526 16.78 0.3368 0.5510 16.92 0.3417 0.5705 16.98 0.3289 0.5816 17.04 0.3162 3 0.6780 22.40 0.3759 0.6680 22.41 0.3712 0.6806 22.20 0.3720 0.6810 22.30 0.3569 0.6976 22.54 0.3472 5 0.7210 24.27 0.3783 0.7128 24.27 0.3755 0.7205 24.06 0.3662 0.7200 24.14 0.3637 0.7386 24.46 0.3558 10 0.7757 26.16 0.3796 0.7658 26.17 0.3782 0.7729 26.11 0.3794 0.7745 26.20 0.3709 0.7885 26.39 0.3658 Brick 1 0.2875 12.10 0.0816 0.2874 12.18 0.0805 0.2873 12.14 0.0779 0.2880 12.16 0.0728 0.2961 12.23 0.0719 3 0.3710 16.95 0.0933 0.3737 17.00 0.0901 0.3646 16.86 0.0879 0.3650 16.87 0.0865 0.3823 17.09 0.0854 5 0.4167 19.17 0.0998 0.4174 19.21 0.0978 0.4176 18.61 0.0955 0.4175 18.65 0.0923 0.4234 19.32 0.0908 10 0.4790 21.84 0.1063 0.4855 21.86 0.1051 0.4874 21.88 0.1043 0.4877 21.90 0.1005 0.4889 22.00 0.0996 Circle 1 0.9510 33.19 0.3106 0.9501 32.22 0.3165 0.9458 33.69 0.3219 0.9502 33.48 0.3098 0.9670 34.87 0.2982 3 0.9633 36.88 0.3270 0.9654 36.89 0.3245 0.9572 36.72 0.3271 0.9603 36.90 0.3215 0.9765 38.90 0.3163 5 0.9688 37.85 0.3285 0.9688 37.86 0.3271 0.9617 37.64 0.3289 0.9634 37.58 0.3249 0.9784 39.82 0.3198 10 0.9726 39.65 0.3291 0.9756 39.67 0.3279 0.9761 39.86 0.3290 0.9732 39.76 0.3253 0.9821 41.40 0.3241
Comparison of SI and BRISQUE (BQ) values of despeckled images
 Image AA [7] Dong[19] DDD[61] ZZDB[62] Proposed SI BQ SI BQ SI BQ SI BQ SI BQ SAR Image-1 0.5076 43.21 0.5034 43.99 0.5283 42.83 0.4806 42.56 0.4398 42.45 SAR Image-2 0.6985 45.26 0.6874 45.58 0.6845 43.96 0.6563 40.75 0.6270 38.38 Fetal foot 1.052 45.40 1.055 42.80 1.0642 40.17 1.0507 40.94 1.024 39.09 Liver cyst 0.8480 39.28 0.8484 41.15 0.8580 40.39 0.8252 45.95 0.8101 38.18
 Image AA [7] Dong[19] DDD[61] ZZDB[62] Proposed SI BQ SI BQ SI BQ SI BQ SI BQ SAR Image-1 0.5076 43.21 0.5034 43.99 0.5283 42.83 0.4806 42.56 0.4398 42.45 SAR Image-2 0.6985 45.26 0.6874 45.58 0.6845 43.96 0.6563 40.75 0.6270 38.38 Fetal foot 1.052 45.40 1.055 42.80 1.0642 40.17 1.0507 40.94 1.024 39.09 Liver cyst 0.8480 39.28 0.8484 41.15 0.8580 40.39 0.8252 45.95 0.8101 38.18
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