# American Institute of Mathematical Sciences

October  2019, 13(5): 903-930. doi: 10.3934/ipi.2019041

## Piecewise constant signal and image denoising using a selective averaging method with multiple neighbors

 1 Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China 2 School of Mathematical Sciences, Nankai University, Tianjin 300071, China 3 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

* Corresponding author: Chunlin Wu

Received  March 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by HDU grant KYS075618129. The second author is supported by NSFC grant 11871035 and 11531013 and Recruitment Program of Global Young Expert. The third author is supported by NSFC grant 11771420.

Piecewise constant signals and images are an important kind of data. Typical examples include bar code signals, logos, cartoons, QR codes (Quick Response codes), and text images, which are widely used in both general commercial and automotive industry use. One previous work called a general selective averaging method (GSAM) was introduced to remove noise from them. It chooses homogeneous neighbors from the two closest pixels (one pixel at each side) to update the current pixel. One limitation is that it suffered from appearing sparse noisy pixels in the denoised result when the noise level is high. In this paper, we try to solve this problem by proposing a selective averaging method with multiple neighbors. To update the intensity value at each pixel, the proposed algorithm averages more homogeneous neighbors selected from a large domain, which is based on the property of the local geometry of signals and images. This greatly reduces sparse noisy pixels left in the final result by GSAM. Similarly, our method adopts the Neumann boundary condition at edges, and thus preserves edges well. In 1D case, some theoretical results are given to guarantee the convergence of our algorithm. In 2D case, except eliminating additive Gaussian noise, this algorithm can be used for restoring noisy images corrupted by speckle noise. Intensive experiments on both gray and color image denoising demonstrate that the proposed method is quite effective for piecewise constant image denoising and achieves superior performance visually and quantitatively.

Citation: Weina Wang, Chunlin Wu, Jiansong Deng. Piecewise constant signal and image denoising using a selective averaging method with multiple neighbors. Inverse Problems & Imaging, 2019, 13 (5) : 903-930. doi: 10.3934/ipi.2019041
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An illustration of the updating order from $i = 1$ to $i = m$
] ($T = 0.44, p = 0.49$), Algorithm 1 ($T = 0.44, k = 2, p = 0.25$), TV[37] ($\alpha = 1.3, r_p = 10$) and $L_{0}$[47] ($\lambda = 0.03, \kappa = 1.01$), when $\sigma = 0.12$. The last two rows show the results by GSAM [44] ($T = 0.43, p = 0.49$), Algorithm 1 ($T = 0.43, k = 3, p = 0.11$), TV[37] ($\alpha = 1.4, r_p = 10$) and $L_{0}$[47] ($\lambda = 0.05, \kappa = 1.01$), when $\sigma = 0.16$. The corresponding SNR values are listed in brackets">Figure 3.  The first two rows show the results by GSAM [44] ($T = 0.44, p = 0.49$), Algorithm 1 ($T = 0.44, k = 2, p = 0.25$), TV[37] ($\alpha = 1.3, r_p = 10$) and $L_{0}$[47] ($\lambda = 0.03, \kappa = 1.01$), when $\sigma = 0.12$. The last two rows show the results by GSAM [44] ($T = 0.43, p = 0.49$), Algorithm 1 ($T = 0.43, k = 3, p = 0.11$), TV[37] ($\alpha = 1.4, r_p = 10$) and $L_{0}$[47] ($\lambda = 0.05, \kappa = 1.01$), when $\sigma = 0.16$. The corresponding SNR values are listed in brackets
Test images. (a) Logo. (b) Cartoon1. (c) QRcode1. (d) Text. (e) QArtcode. (f) Blobs. (g) QRcode2. (h) QRcode3. (i) Cartoon2
], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets">Figure 5.  From top to down and left to right: the noisy Logo, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little line in each image are displayed for further comparison. The corresponding SNR values are listed in brackets">Figure 6.  From top to down and left to right: the noisy Cartoon1, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little line in each image are displayed for further comparison. The corresponding SNR values are listed in brackets
], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets">Figure 7.  From top to down and left to right: the noisy QRcode1, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets
], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets">Figure 8.  From top to down and left to right: the noisy Text, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets
], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets">Figure 9.  From top to down and left to right: the noisy QArtcode, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
], $L_{0}$[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the first image are displayed for further comparison. The corresponding SNR values are listed in brackets">Figure 10.  From left to right: the noisy QRcode2, the results by TV[1], $L_{0}$[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the first image are displayed for further comparison. The corresponding SNR values are listed in brackets
], $L_{0}$[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the second row-left image are displayed for further comparison. The corresponding SNR values are listed in brackets">Figure 11.  From up to down: Denoising results by different methods for two noisy QRcode3 and Cartoon2 images. From left to right: the noisy images, the results by TV[1], $L_{0}$[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the second row-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
], $L_{0}$ [47], AGSAM [44] and our method. The zoom-in views of the region indicited by the little line in the each image are displayed for further comparison. The corresponding SNR values are listed in brackets">Figure 12.  From top to down and left to right: the clean and noisy "Blobs", the results by Wang et.al [43], $L_{0}$ [47], AGSAM [44] and our method. The zoom-in views of the region indicited by the little line in the each image are displayed for further comparison. The corresponding SNR values are listed in brackets
], $L_{0}$ [47], AGSAM [44] and our method. The corresponding SNR values are listed in brackets">Figure 13.  From up to down: Denoising results by different methods for Cartoon1 corrupted by two levels of noise. From left to right: the noisy images, the results by Wang et.al [43], $L_{0}$ [47], AGSAM [44] and our method. The corresponding SNR values are listed in brackets
Different types to compute $u^{(n+1)}_{i}$ based on five cases of $K^{l}_{i-1}$ and $K^{r}_{i-1}$ from (2) to (6). (a1-a2), (b1-b2), (c1-c2), (d) and (e) correspond to Case 1, Case 2, Case 3, Case 4 and Case 5, respectively
Iterative number and time comparison(in seconds) by GSAM [44] and Algorithm 1 in Fig. 3
 Signal $\sigma$ GSAM [44] Proposed iter t iter t Fig. 3 $0.12$ 6231 1.61 3695 1.07 $0.16$ 3322 0.98 2181 0.79
 Signal $\sigma$ GSAM [44] Proposed iter t iter t Fig. 3 $0.12$ 6231 1.61 3695 1.07 $0.16$ 3322 0.98 2181 0.79
The parameter settings of different methods for gray images
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed $\alpha$ $\lambda$ $h$/$T$ $T$ $T$/$p$ Logo $10$ 30 0.01 1.5/0.6 5 5/0.13 $15$ 20 0.02 0.9/1.12 4 4/0.13 $20$ 14 0.03 0.7/2 2.4 3.3/0.05 Cartoon1 $15$ 20 0.02 1.6/1.35 5 5/0.13 $20$ 15 0.03 1.1/2.8 4.5 4.5/0.13 $25$ 12 0.04 0.8/5 4 4/0.06 QRcode1 $15$ 20 0.02 1.8/1.12 5 5/0.13 $25$ 12 0.04 1.1/3.12 4.5 4.5/0.13 $35$ 8 0.1 0.9/11.03 3.2 3.2/0.07 Text $15$ 24 0.02 1.7/1.8 4.5 4.5/0.13 $25$ 14 0.05 1.1/3.75 4.5 4.5/0.13 $35$ 10 0.06 1/9.8 3.2 3.5/0.13 QArtcode $15$ 23 0.02 1.7/1.8 5 5/0.13 $25$ 14 0.04 0.9/4.38 4.5 4.5/0.13 $35$ 10 0.07 0.6/9.8 3.4 3.4/0.07
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed $\alpha$ $\lambda$ $h$/$T$ $T$ $T$/$p$ Logo $10$ 30 0.01 1.5/0.6 5 5/0.13 $15$ 20 0.02 0.9/1.12 4 4/0.13 $20$ 14 0.03 0.7/2 2.4 3.3/0.05 Cartoon1 $15$ 20 0.02 1.6/1.35 5 5/0.13 $20$ 15 0.03 1.1/2.8 4.5 4.5/0.13 $25$ 12 0.04 0.8/5 4 4/0.06 QRcode1 $15$ 20 0.02 1.8/1.12 5 5/0.13 $25$ 12 0.04 1.1/3.12 4.5 4.5/0.13 $35$ 8 0.1 0.9/11.03 3.2 3.2/0.07 Text $15$ 24 0.02 1.7/1.8 4.5 4.5/0.13 $25$ 14 0.05 1.1/3.75 4.5 4.5/0.13 $35$ 10 0.06 1/9.8 3.2 3.5/0.13 QArtcode $15$ 23 0.02 1.7/1.8 5 5/0.13 $25$ 14 0.04 0.9/4.38 4.5 4.5/0.13 $35$ 10 0.07 0.6/9.8 3.4 3.4/0.07
The parameter settings of different methods for color images
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed $\alpha$ $\lambda$ $T$ $T$/$p$ QRcode2 $10$ 16 0.01 5 5/0.13 $20$ 8 0.04 4 4/0.13 $30$ 5 0.08 3.5 3.5/0.13 $40$ 4 1.6 2.3 2.6/0.1 QRcode3 $10$ 17 0.02 5 5/0.13 $20$ 9 0.04 4 4/0.13 $30$ 6 0.12 2.5 2.6/0.11 $40$ 4 0.5 2 2/0.06 Cartoon2 $10$ 19 0.01 5 5/0.13 $20$ 10 0.05 4 4.2/0.13 $30$ 6 0.09 2.5 2.8/0.13 $40$ 5 1.16 1.9 2.3/0.1
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed $\alpha$ $\lambda$ $T$ $T$/$p$ QRcode2 $10$ 16 0.01 5 5/0.13 $20$ 8 0.04 4 4/0.13 $30$ 5 0.08 3.5 3.5/0.13 $40$ 4 1.6 2.3 2.6/0.1 QRcode3 $10$ 17 0.02 5 5/0.13 $20$ 9 0.04 4 4/0.13 $30$ 6 0.12 2.5 2.6/0.11 $40$ 4 0.5 2 2/0.06 Cartoon2 $10$ 19 0.01 5 5/0.13 $20$ 10 0.05 4 4.2/0.13 $30$ 6 0.09 2.5 2.8/0.13 $40$ 5 1.16 1.9 2.3/0.1
The SNR values by different methods for gray images
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed Logo $10$ 30.07 46.80 46.41 48.05 $\bf{49.26}$ $20$ 24.41 $\bf{39.15}$ 34.96 29.05 34.77 Cartoon1 $15$ 26.86 44.60 43.22 44.95 $\bf{46.87}$ $25$ 22.64 40.17 36.01 41.20 $\bf{42.35}$ QRcode1 $15$ 27.20 41.98 42.00 43.78 $\bf{44.97}$ $25$ 22.93 37.54 35.77 39.85 $\bf{40.78}$ Text $15$ 23.55 39.19 39.75 42.57 $\bf{43.06}$ $35$ 16.48 $\bf{30.46}$ 25.00 29.79 29.97 QArtcode $15$ 24.45 36.14 $\bf{40.01}$ 39.37 39.51 $25$ 20.16 31.70 35.08 35.20 $\bf{35.25}$
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed Logo $10$ 30.07 46.80 46.41 48.05 $\bf{49.26}$ $20$ 24.41 $\bf{39.15}$ 34.96 29.05 34.77 Cartoon1 $15$ 26.86 44.60 43.22 44.95 $\bf{46.87}$ $25$ 22.64 40.17 36.01 41.20 $\bf{42.35}$ QRcode1 $15$ 27.20 41.98 42.00 43.78 $\bf{44.97}$ $25$ 22.93 37.54 35.77 39.85 $\bf{40.78}$ Text $15$ 23.55 39.19 39.75 42.57 $\bf{43.06}$ $35$ 16.48 $\bf{30.46}$ 25.00 29.79 29.97 QArtcode $15$ 24.45 36.14 $\bf{40.01}$ 39.37 39.51 $25$ 20.16 31.70 35.08 35.20 $\bf{35.25}$
The iterative number and time comparison(in seconds) by different methods for gray images
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed iter/t iter/t iter/t iter/t iter/t Logo $10$ 90/0.7 1551/5.5 1/141.4 139/0.5 $\bf{66}$/$\bf{0.2}$ $20$ $\bf{91}$/0.6 1481/5.0 1/145.7 182/0.6 138/$\bf{0.4}$ Cartoon1 $15$ $\bf{96}$/2.5 1481/19.6 1/698.3 170/3.5 102/$\bf{2.2}$ $25$ $\bf{98}$/$\bf{2.6}$ 1355/18.2 1/720.5 235/5.0 177/3.8 QRcode1 $15$ $\bf{83}$/$\bf{2.2}$ 1479/18.0 1/994.1 177/3.6 103/2.3 $25$ $\bf{84}$/$\bf{2.1}$ 1411/17.2 1/998.0 254/5.2 150/3.5 Text $15$ $\bf{112}$/$\bf{0.7}$ 1482/3.9 1/165.9 295/1.6 147/0.8 $35$ $\bf{109}$/$\bf{0.6}$ 1355/3.5 1/166.4 502/2.7 324/1.8 QArtcode $15$ $\bf{67}$/0.3 1478/2.9 1/77.6 175/0.3 109/$\bf{0.2}$ $25$ $\bf{69}$/0.3 1389/2.3 1/78.3 254/0.4 155/$\bf{0.2}$
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed iter/t iter/t iter/t iter/t iter/t Logo $10$ 90/0.7 1551/5.5 1/141.4 139/0.5 $\bf{66}$/$\bf{0.2}$ $20$ $\bf{91}$/0.6 1481/5.0 1/145.7 182/0.6 138/$\bf{0.4}$ Cartoon1 $15$ $\bf{96}$/2.5 1481/19.6 1/698.3 170/3.5 102/$\bf{2.2}$ $25$ $\bf{98}$/$\bf{2.6}$ 1355/18.2 1/720.5 235/5.0 177/3.8 QRcode1 $15$ $\bf{83}$/$\bf{2.2}$ 1479/18.0 1/994.1 177/3.6 103/2.3 $25$ $\bf{84}$/$\bf{2.1}$ 1411/17.2 1/998.0 254/5.2 150/3.5 Text $15$ $\bf{112}$/$\bf{0.7}$ 1482/3.9 1/165.9 295/1.6 147/0.8 $35$ $\bf{109}$/$\bf{0.6}$ 1355/3.5 1/166.4 502/2.7 324/1.8 QArtcode $15$ $\bf{67}$/0.3 1478/2.9 1/77.6 175/0.3 109/$\bf{0.2}$ $25$ $\bf{69}$/0.3 1389/2.3 1/78.3 254/0.4 155/$\bf{0.2}$
The SNR values by different methods for color images
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed QRcode2 $10$ 31.29 46.49 46.02 $\bf{47.62}$ $30$ 22.23 36.37 37.53 $\bf{37.81}$ $40$ 20.01 29.14 33.39 $\bf{33.85}$ QRcode3 $10$ 29.68 44.10 44.44 $\bf{46.11}$ $20$ 23.97 38.08 38.91 $\bf{40.77}$ $40$ 18.48 $\bf{25.57}$ 21.10 25.30 Cartoon2 $10$ 29.39 40.31 40.93 $\bf{41.05}$ $20$ 23.65 38.36 38.46 $\bf{39.53}$ $30$ 20.37 36.09 35.80 $\bf{36.99}$
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed QRcode2 $10$ 31.29 46.49 46.02 $\bf{47.62}$ $30$ 22.23 36.37 37.53 $\bf{37.81}$ $40$ 20.01 29.14 33.39 $\bf{33.85}$ QRcode3 $10$ 29.68 44.10 44.44 $\bf{46.11}$ $20$ 23.97 38.08 38.91 $\bf{40.77}$ $40$ 18.48 $\bf{25.57}$ 21.10 25.30 Cartoon2 $10$ 29.39 40.31 40.93 $\bf{41.05}$ $20$ 23.65 38.36 38.46 $\bf{39.53}$ $30$ 20.37 36.09 35.80 $\bf{36.99}$
The iterative number and time comparison(in seconds) by different methods for color images
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed iter/t iter/t iter/t iter/t QRcode2 $10$ 76/3.0 1551/33.3 144/2.7 $\bf{68}$/$\bf{1.4}$ $30$ $\bf{82}$/3.3 1342/29.6 253/3.4 172/$\bf{2.9}$ $40$ $\bf{75}$/$\bf{3.0}$ 1041/23.9 311/5.8 215/4.4 QRcode3 $10$ $\bf{66}$/2.6 1411/31.3 139/2.7 68/$\bf{1.5}$ $20$ $\bf{63}$/2.8 1340/32.3 183/3.5 118/$\bf{2.7}$ $40$ $\bf{72}$/$\bf{2.9}$ 1158/25.9 308/6.0 283/6.1 Cartoon2 $10$ $\bf{70}$/3.4 1551/46.1 146/3.4 79/$\bf{1.9}$ $20$ $\bf{54}$/3.2 1355/44.1 189/4.2 116/$\bf{2.8}$ $30$ $\bf{78}$/3.8 1240/39.5 245/5.4 152/$\bf{3.7}$
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed iter/t iter/t iter/t iter/t QRcode2 $10$ 76/3.0 1551/33.3 144/2.7 $\bf{68}$/$\bf{1.4}$ $30$ $\bf{82}$/3.3 1342/29.6 253/3.4 172/$\bf{2.9}$ $40$ $\bf{75}$/$\bf{3.0}$ 1041/23.9 311/5.8 215/4.4 QRcode3 $10$ $\bf{66}$/2.6 1411/31.3 139/2.7 68/$\bf{1.5}$ $20$ $\bf{63}$/2.8 1340/32.3 183/3.5 118/$\bf{2.7}$ $40$ $\bf{72}$/$\bf{2.9}$ 1158/25.9 308/6.0 283/6.1 Cartoon2 $10$ $\bf{70}$/3.4 1551/46.1 146/3.4 79/$\bf{1.9}$ $20$ $\bf{54}$/3.2 1355/44.1 189/4.2 116/$\bf{2.8}$ $30$ $\bf{78}$/3.8 1240/39.5 245/5.4 152/$\bf{3.7}$
The parameter settings of different methods for gray images with speckle noise
 Image $\sigma$ Wang et.al [43] $L_{0}$[47] AGSAM[44] Proposed $\lambda$/$\mu$ $\lambda$ $T$ $T$/$p$/$k$ Blobs $2$ 1.2/11 0.04 60 60/0.04/12 Cartoon1 $2$ 0.8/2 0.05 58 58/0.13/4 $2.5$ 0.6/2 0.06 45 45/0.03/5
 Image $\sigma$ Wang et.al [43] $L_{0}$[47] AGSAM[44] Proposed $\lambda$/$\mu$ $\lambda$ $T$ $T$/$p$/$k$ Blobs $2$ 1.2/11 0.04 60 60/0.04/12 Cartoon1 $2$ 0.8/2 0.05 58 58/0.13/4 $2.5$ 0.6/2 0.06 45 45/0.03/5
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