Article Contents
Article Contents

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

• * Corresponding author: Rajendra K. Ray
• 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.

Mathematics Subject Classification: 35L70, 65M06, 68U10.

 Citation:

• Figure 1.  Set of sixteen $3 \times 3$ templates

Figure 2.  Original images

Figure 3.  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

Figure 4.  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

Figure 5.  (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

Figure 6.  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

Figure 7.  (a)-(f) 2D Contour map of the images 6g-6l. (g)-(l) 3D surface plot of the images 6g-6l

Figure 8.  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

Figure 9.  (a) SAR Image-1: One look radar image [3]. (b)-(d) Restored by different models

Figure 10.  (a) SAR Image-2: Image of KOMPSAT/Arirang-5 of a part of the Himalayan Arc [20]. (b)-(d) Restored by different models

Figure 11.  A Ultrasound image of fetal foot and restored by different models

Figure 12.  A Ultrasound image of liver cyst and restored by different models

Figure 13.  Relative error vs. the iteration number for various models

Table 1.  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

Table 2.  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

Figures(13)

Tables(2)