Chapter 4 Parameter Learning of Weighted Mean Based Noise Pixel
4.3 Operating Parameter Learning Mechanism
4.3.2 Proposed Training Method
In this section, we propose two training stages for the purpose to maintain image sharpness. In the first stage of our training method, we will choose four typical grayscale natural images with adding the same percentage of impulse noise rate to each natural images as the input training images, then we raster-scanningly train every pixel of training images. And at the end of each row, under the currently learned parameter values of s、t、T and s T, we test the noise and noise free pixel detection accuracy by summing of all four input training images. To enhance the noise detection accuracy, we weighted the noise pixel misclassification by a multiplier constant where value is larger than 1. From our experiment of obtaining good Peak Signal Noise Ratio (PSNR) , the multiplier constant is dependent on the impulse noise rate of adding on the image. We have experienced for a good PSNR that multiplier constant equal to 10 for impulse noise rate smaller than 10%, and equal to 100 otherwise. It is to be noted that our proposed learning scheme usually cannot obtain a
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perfect noise maps of training images. i.e., the number of wrongly classified noise or noise free pixels is zero for all the training images. Therefore, we will exploit the pocket algorithm to our learning phase. In the first epoch, we will select the best parameters with the smallest number of wrongly classified pixels. We store this number of smallest misclassified pixels and its associated parameter set as our best solution in this learning epoch. Then, we use the best parameters from the first learning epoch as the initial parameters of the second learning epoch. After a long enough learning epochs, we use the best parameters as the initial value of our second training stage.
Our learning algorithm will identify the central pixel of 3 3 window as a noisy pixel if there is a big enough difference with its eight neighbor pixels. Unfortunately, an edge point could be prone to be wrongly classified as a noise pixel for it could produce a big difference with neighbor pixels due to edge effect. To alleviate this shortage, we propose the second stage to retrain the images with edge pixel being identified differently. Similarly as the first stage, we add the edge detection process to our training method in the second stage. When we start training these four input images in the second stage, we first use the best parameter set from the first stage to repair the input training images. Then, we apply Canny edge detector to help us find the edge pixels from these repaired images. Finally, when the current training pixel is an edge pixel, we increase the threshold value by a factor of “1.1” to reduce the probability of edge pixel to be detected as a noisy pixel. By this way, we can achieved good performance by combining these two training stages.
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Chapter 5 Experimental Results
In our experiments, we focus on two common types of impulse noise, one is salt-and-pepper noise and the other is random-valued impulse noise. These two types of noise model are described in Section 1.2.1. For the measurement of the restoration quality, we employ the peak signal-to-noise ratio (PSNR) performance metric, which is based on the root-mean squared error (RMSE). The expression of RMSE and
respectively. For the evaluation of the detail preservation capabilities of the proposed filtering design, the mean absolute error (MAE) has been used as
43 Adaptive Peer Group (APG) [19], Center Weighted Median Filter (CWMF) [3], Peer Group Filter (PGF) [17], Fast Similarity-based impulsive noise removal Vector Filter (FSVF) [20-23], Switching Median Filter (SMF) [11]. Our first method applies s-norm and t-norm operators respectively to construct the interval-valued fuzzy relations by extensive combinatory trials, without s, t learning mechanism and saturation threshold process, we called it “ST” method. If “ST” method with saturation threshold process, we called it “ST with saturation” method. In order to make our experimental results more representative, we take average of 100 testing images with the same percentage of impulse noise rate to all four training images.
44
5.1 Results of Salt and Pepper Noise Correction
In this experiments, we have added the salt-and-pepper noise, as shown in Fig.
5.1, to four gray-scale images as our training input images. Based on our proposed weighted mean based interval-valued fuzzy relations for noise detection of noisy images, we have obtained good results by setting the initial values of
0.75, 0.25,
s t
learning constants 0.0035,
s t T Ts
noisy pixel
threshold T and saturation threshold 6 T s 18 in the first training stage. After 9 learning epochs in the first training stage, we select the best parameters as the initial values of our second training stage. In the second stage of our training method, we can obtain the best parameter set after 30 learning epochs. If the percentage of noise ratio is less than 20%, we will correct it by using alpha-trimmed mean filter using ranking central three pixels, when the pixel is regarded as a noisy pixel. Otherwise, we will correct it by using median filter when it is detected as a noisy pixel.
Fig. 5.1. Four 10% salt and pepper noise training images with size of 128 128.
45
According to TABLE Ι, although our method is not the best method, the performance of our method is still above average. When the concentration of the salt and pepper noise is increased in an image, our method is better than the other methods gradually. Fig. 5.2, Fig. 5.3 and Fig. 5.4 show the correction results by different filters of noisy Boat image, with 20%, 40% and 60% salt and pepper noise, respectively.
(a) (b)
(c) (d)
(e) (f)
46
(g) (h)
(i) (j)
(k)
Fig. 5.2. Noisy pixel correction results of Boat image filtered by different filters. (a) Original image.
(b) Corrupted image with 20% salt and pepper noise. (c)(i) are filtering results. Image filtering results filtered by (c) our proposed filter with two stages. (d) our proposed filter with one stage. (e) Adaptive Peer Group (APG). (f) Center Weighted Median Filter (CWMF). (g) Peer Group Filter (PGF).
(h) Fast Similarity-based impulsive noise removal Vector Filter (FSVF). (i) Switching Median Filter (SMF). (j) “ST” method. (k) “ST with saturation” method.
47
(a) (b)
(c) (d)
(e) (f)
(g) (h)
48
(i) (j)
(k)
Fig. 5.3. Noisy pixel correction results of Boat image filtered by different filters. (a) Original image. (b) Corrupted image with 40% salt and pepper noise. (c) (i) are filtering results. Image filtering results filtered by (c) our proposed filter with two stages. (d) our proposed filter with one stage. (e) Adaptive Peer Group (APG). (f) Center Weighted Median Filter (CWMF). (g) Peer Group Filter (PGF). (h) Fast Similarity-based impulsive noise removal Vector Filter (FSVF). (i) Switching Median Filter (SMF). (j) “ST” method. (k) “ST with saturation” method.
49
(a) (b)
(c) (d)
(e) (f)
(g) (h)
50
(i) (j)
(k)
Fig. 5.4. Noisy pixel correction results of Boat image filtered by different filters. (a) Original image. (b) Corrupted image with 60% salt and pepper noise. (c) (i) are filtering results. Image filtering results filtered by (c) our proposed filter with two stages. (d) our proposed filter with one stage. (e) Adaptive Peer Group (APG). (f) Center Weighted Median Filter (CWMF). (g) Peer Group Filter (PGF). (h) Fast Similarity-based impulsive noise removal Vector Filter (FSVF). (i) Switching Median Filter (SMF). (j) “ST” method. (k) “ST with saturation” method.
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TABLE Ι
THE NOISE REMOVAL RESULTS BY DIFFERENT FILTERS
(a) Corrupted Boat image with 10% salt and pepper noise
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 7.746 1.729 30.3499 0.000 0.000 0.0009
CWMF 4.995 0.825 34.1612 0.620 0.820 1.4402
PGF 6.005 1.014 32.5615 0.393 0.649 1.0424 FSVF 5.267 0.882 33.7003 0.559 0.768 1.3273
SMF 3.313 0.626 37.7291 1.000 1.000 2.0001
STTAE 6.642 1.417 31.6858 0.249 0.283 0.5328
ST with
saturation 6.201 1.107 32.2827 0.349 0.564 0.9137
Our learning ST method (two stage)
5.911 1.065 32.6994 0.414 0.602 1.0165
Our learning ST method (one stage)
6.166 1.098 32.3326 0.357 0.572 0.9296
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(b) Corrupted Boat image with 20% salt and pepper noise
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 10.236 3.147 27.9299 0.000 0.000 0.0009
CWMF 8.023 1.795 30.0463 0.442 0.752 1.1943
PGF 8.200 2.098 29.8545 0.407 0.584 0.9914 FSVF 7.835 1.835 30.2522 0.480 0.730 1.2102
SMF 5.230 1.349 33.7621 1.000 1.000 2.0001
STTAE 9.059 2.820 28.9908 0.235 0.182 0.4178
ST with
saturation 8.353 2.192 29.6956 0.376 0.532 0.9086
Our learning ST method (two stage)
8.129 2.250 29.9304 0.421 0.499 0.9205
Our learning ST method (one stage)
8.399 2.223 29.6477 0.367 0.514 0.8817
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(c) Corrupted Boat image with 40% salt and pepper noise
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 15.335 6.518 24.4187 0.546 0.151 0.6976
CWMF 21.269 6.070 21.5779 0.000 0.272 0.2729
PGF 13.724 4.747 25.3822 0.694 0.628 1.3222 FSVF 18.866 5.788 22.6188 0.221 0.348 0.5697
SMF 10.399 3.367 27.7921 1.000 1.000 2.0001
STTAE 14.935 6.611 24.6485 0.583 0.126 0.7095
ST with
saturation 14.441 5.222 24.9394 0.628 0.501 1.1293
Our learning ST method (two stage)
14.347 5.277 24.9963 0.637 0.486 1.1234
Our learning ST method (one stage)
15.161 7.080 24.5176 0.562 0.000 0.5628
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(d) Corrupted Boat image with 60% salt and pepper noise
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 26.985 14.667 19.5087 0.816 0.493 1.3097
CWMF 48.974 21.161 14.3329 0.000 0.011 0.0119
PGF 26.259 10.449 19.7462 0.843 0.805 1.6482 FSVF 45.988 21.310 14.8788 0.111 0.000 0.1118
SMF 22.023 7.823 21.2741 1.000 1.000 2.0001
STTAE 26.895 12.443 19.5386 0.819 0.657 1.4766
ST with
saturation 26.865 11.292 19.5475 0.820 0.743 1.5633
Our learning ST method (two stage)
26.840 12.294 19.5563 0.821 0.669 1.4904
Our learning ST method (one stage)
26.843 12.309 19.5554 0.821 0.667 1.4895
55
5.2 Results of Random-Valued Impulse Noise Correction
In this experiments, we have added the random-valued impulse noise, as shown in Fig. 5.3, to four gray-scale images as our training input images. Similarly, we have obtained the good performances by setting the initial values of s 0.75, t 0.25,
learning constants 0.0035,
s t T Ts
noisy pixel threshold T 6 and
saturation threshold T s 18 in the first training stage. After 9 learning epochs in the first training stage, we select the best parameters as the initial values of our second training stage. In the second stage of our training method, we can obtain the best parameter set after 30 learning epochs. If the percentage of noise ratio is less than 20%, we will correct it by using alpha-trimmed mean filter using ranking central three pixels, when the pixel is regarded as a noisy pixel. Otherwise, we will correct it by using median filter when it is detected as a noisy pixel.
Fig. 5.5. Four 10% random-valued impulse noise training images with size of 128 128.
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According to TABLE ΙI, the performance of our proposed method is above average no matter what the sample corruption probability is. Fig. 5.5 and Fig. 5.6 show the correction results by different filters of noisy Pepper image, with 20% and 40% random-valued impulse noise, respectively. Fig. 5.7 shows the correction results by different filters of noisy Boat image with 60% random-valued impulse noise.
(a) (b)
(c) (d)
(e) (f)
57
(g) (h)
(i) (j)
(k)
Fig. 5.6. Noisy pixel correction results of Pepper image filtered by different filters. (a) Original image.
(b) Corrupted image with 20% random-valued impulse noise. (c)(i) are filtering results. Image filtering results filtered by (c) our proposed filter with two stages. (d) our proposed filter with one stage.
(e) Adaptive Peer Group (APG). (f) Center Weighted Median Filter (CWMF). (g) Peer Group Filter (PGF). (h) Fast Similarity-based impulsive noise removal Vector Filter (FSVF). (i) Switching Median Filter (SMF). (j) “ST” method. (k) “ST with saturation” method.
58
(a) (b)
(c) (d)
(e) (f)
(g) (h)
59
(i) (j)
(k)
Fig. 5.7. Noisy pixel correction results of Pepper image filtered by different filters.
(a) Original image. (b) Corrupted image with 40% random-valued impulse noise. (c)
(i) are filtering results. Image filtering results filtered by (c) our proposed filter with two stages. (d) our proposed filter with one stage. (e) Adaptive Peer Group (APG). (f) Center Weighted Median Filter (CWMF). (g) Peer Group Filter (PGF). (h) Fast Similarity-based impulsive noise removal Vector Filter (FSVF). (i) Switching Median Filter (SMF). (j) “ST” method. (k) “ST with saturation” method.
60
(a) (b)
(c) (d)
(e) (f)
(g) (h)
61
(i) (j)
(k)
Fig. 5.8. Noisy pixel correction results of Boat image filtered by different filters. (a) Original image. (b) Corrupted image with 60% random-valued impulse noise. (c) (i) are filtering results. Image filtering results filtered by (c) our proposed filter with two stages. (d) our proposed filter with one stage. (e) Adaptive Peer Group (APG). (f) Center Weighted Median Filter (CWMF). (g) Peer Group Filter (PGF). (h) Fast Similarity-based impulsive noise removal Vector Filter (FSVF). (i) Switching Median Filter (SMF). (j) “ST” method. (k) “ST with saturation” method.
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TABLE ΙI
THE NOISE REMOVAL RESULTS BY DIFFERENT FILTERS
(a) Corrupted Pepper image with 10% random-valued impulse noise.
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 7.900 1.687 30.1798 0.714 0.627 1.3418 CWMF 4.054 0.663 35.9761 1.000 1.000 2.0001
PGF 4.726 0.806 34.6446 0.950 0.948 1.8986
FSVF 4.329 0.726 35.4072 0.980 0.977 1.9572 SMF 17.490 3.408 23.2759 0.000 0.000 0.0009
STTAE 5.199 0.967 33.8147 0.915 0.889 1.8047 ST with
saturation 4.608 0.808 34.8625 0.959 0.947 1.9065 Our learning
ST method (two stage)
4.422 0.792 35.2213 0.973 0.953 1.9263 Our learning
ST method (one stage)
4.499 0.828 35.0724 0.967 0.940 1.9074
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(b) Corrupted Pepper image with 20% random-valued impulse noise.
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 13.040 4.107 25.8268 0.637 0.517 1.1548
CWMF 6.319 1.501 32.1191 1.000 1.000 2.0001
PGF 7.359 1.817 30.7965 0.944 0.941 1.8855 FSVF 6.859 1.570 31.4082 0.971 0.987 1.9582
SMF 24.826 6.894 20.2339 0.000 0.000 0.0009
STTAE 7.874 2.123 30.2077 0.916 0.885 1.8017
ST with
saturation 7.375 1.820 30.7776 0.943 0.941 1.8846
Our learning ST method (two stage)
7.056 1.729 31.1613 0.960 0.958 1.9183
Our learning ST method (one stage)
7.189 1.849 30.9994 0.953 0.936 1.8894
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(c) Corrupted Pepper image with 40% random-valued impulse noise.
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 28.480 13.748 19.0408 0.347 0.108 0.4558
CWMF 13.614 4.578 25.4521 1.000 0.976 1.9761
PGF 14.835 5.094 24.7076 0.946 0.927 1.8736 FSVF 15.138 4.327 24.5323 0.933 1.000 1.9332
SMF 36.392 14.894 16.9119 0.000 0.000 0.0009
STTAE 15.269 5.786 24.4567 0.927 0.862 1.7897
ST with
saturation 14.750 4.981 24.7564 0.950 0.938 1.8884
Our learning ST method (two stage)
14.695 4.910 24.7892 0.953 0.945 1.8983
Our learning ST method (one stage)
14.781 5.119 24.7385 0.949 0.925 1.8745
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(d) Corrupted Boat image with 60% random-valued impulse noise.
Filter RMSE MAE PSNR RMSE- normalized
MAE-
normalized SUM
APG 45.189 33.151 15.0308 0.259 0.000 0.2599
CWMF 37.469 18.380 16.6582 0.987 1.000 1.9871
PGF 37.329 20.461 16.6901 1.000 0.859 1.8592 FSVF 43.470 21.081 15.3167 0.397 0.817 1.2147
SMF 47.939 26.934 14.5179 0.000 0.421 0.4218
STTAE 38.733 23.599 16.3705 0.868 0.647 1.5155
ST with
saturation 37.978 21.201 16.5404 0.939 0.809 1.7484
Our learning ST method (two stage)
37.924 20.893 16.5533 0.944 0.830 1.7743
Our learning ST method (one stage)
38.933 24.292 16.3256 0.849 0.600 1.4496
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Chapter 6 Conclusions
In this thesis, we integrate the weighted mean aggregation and Interval-Valued Fuzzy Relation (IVFR) for detecting noise of an image. For each 3 3 sliding window, the upper and lower weighted mean aggregations of central pixel and its eight neighbor pixels can be calculated, which constitute the interval-valued fuzzy relations. To counter the over-weighting of a big difference term, we introduce a saturation threshold transfer function for pixel difference values. Moreover, the difference between the upper and lower aggregations reflects the degree of intensity variation between central pixel and its eight neighbor pixels. That is, we will identify the central pixel as a noise candidate if it is larger than threshold. Along this line of reasoning, the learning formula of the weighting parameters are derived to decrease the noise detection error of an image. However, there could be no solution for perfect noise map. Therefore, we have exploited the pocket algorithm to our learning algorithm.
The effectiveness of our noise detection scheme is verified by various impulse noise images. Finally, our designed model is applied to noise detection of natural image. Results indicate that the method we proposed is proven to be more superior than the other noise detection and correction algorithms.
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