A New Detection Statistic for Random-Valued Impulse

A New Detection Statistic for Random-Valued Impulse Noise Removal Yiqiu Dong∗

Raymond H. Chan†

Shufang Xu‡

Abstract This paper proposes a new image statistic for detecting random-valued impulse noise. Combining it with detail-preserving regularization, we obtain a powerful two-phase method for denoising even for noise level as high as 60%. Simulation results show that our method is significantly better than a number of existing techniques in terms of image restoration and noise detection.

Key words. random-valued impulse noise, noise detector, detail-preserving regularization, image denoising.

1

Introduction

Images are often corrupted by impulse noise due to noisy sensors or communication channels [1]. There are two models of impulse noise: the easier-to-restore salt-and-pepper noise and the more difficult random-valued impulse noise. This paper deals with the detection and denoising of random-valued impulse noise. Recently, a two-phase iterative method for removing random-valued impulse noise was proposed in [2]. In the first phase, they use ACWM filter [3] to identify noisy pixels. In the second phase, these noise candidates are restored by a detail-preserving regularization. The capability of this method is mainly limited by the accuracy of the noise detector. In [4], Garnett et al. introduced a local image statistic ROAD to identify impulse. For detecting random-valued impulse noise, one drawback of ROAD is that some noise values may be very close to those of their neighbors, in which case, the ROAD values may not be large enough to distinguish them. In this paper, we define a new local image statistic based on ROAD. By this new statistic, the differences between noisy pixels and noise-free pixels will be amplified so that the noise detection will be more accurate. We use this new statistic in phase one of the two-phase method [2]. We have compared our method with a number of methods. It outperforms the others in both image restoration and noise detection. In particular, when the noise ratio is as high as 60%, it still can remove most of the noise while preserving image details. The outline of this paper is as follows. In Section II, we define the new statistic. Section III describes our method in detail. Section IV includes simulation results to demonstrate the performance of the new method. Finally conclusions are drawn in Section V.

2

Definition of ROLD

Suppose the gray-level value yi,j is in [0, 1]. Let ΩN denotes the set of coordinates in a (2N + 1) × (2N + 1) window centered at (0, 0), i.e., ΩN = {(s, t)| − N ≤ s, t ≤ N }, ∗ LMAM,

School of Mathematical Sciences, Peking University, Beijing, 100871, China([email protected]). of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong([email protected]). ‡ LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China([email protected]). † Department

1

Mathematical Model for Multi-Channel Image Processing (MultIm’2006)

2

and let Ω0N = ΩN \(0, 0). Define dst as the absolute difference between gray-level values of yi+s,j+t and yi,j , i.e., dst (yi,j ) = |yi+s,j+t − yi,j |, ∀(s, t) ∈ Ω0N . Then we use a logarithmic function to amplify the middle range of its values. In order to keep it in [0, 1], we use a simple truncation and a linear transformation: Dst (yi,j ) ≡ 1 + max{loga |yi+s,j+t − yi,j |, −b}/b,

∀(s, t) ∈ Ω0N ,

where a, b are positive parameters to be chosen. Note that the value of b decides the truncation position and the value of a controls the shape of the curve of the logarithmic function. The selections of the parameters a and b have great effects on the accuracy of our detection. Based on [5], for an 8-bit graylevel image, we choose a = 2 and b = 5. Arrange all Dst in an increasing order, and let Rk be the kth smallest Dst for all (s, t) ∈ Ω0N . We define our local image statistic as ROLDm (yi,j ) =

m X

Rk (yi,j ).

k=1

We name this statistic as “Rank-Ordered Logarithmic Difference”, and ROLD for short.

3

Our method

We combine the new noise detector ROLD with the detail-preserving regularization [6] to obtain a new two-phase method. To ensure high accuracy of detection, we apply our method iteratively with decreasing threshold. At every iteration, we decrease the threshold to include more noise candidates. Suppose the noisy image is y. Our algorithm is as follows. Algorithm 1: Step 1: Set k = 0 and u(0) = y. (k)

(k)

Step 2: (Noise detection) If ROLD(ui,j ) > Tk , then ui,j is noise, and (i, j) ∈ Nk , the noise candidate (k)

set; otherwise, ui,j is noise-free. Step 3: (Noise restoration) Restore all pixels in Nk by minimizing the following function: ¶ X µ X X (k) (k) F (u(k+1) ) = ϕ(ui,j − u(k) ) + 2 ϕ(u − y ) , m,n m,n i,j (i,j)∈Nk

(m,n)∈Vi,j ∩Nk

(m,n)∈Vi,j \Nk

where Vi,j is the set of the four closest neighbors of (i, j), and ϕ is an edge-preserving potential (k+1) (k) function [7]. For all (i, j) ∈ / N , take ui,j = ui,j . Step 4: Stop the iteration as soon as k is larger than Kmax , which is the maximum number of iterations. Otherwise, set k = k + 1, and go back to Step 2.

4

Simulations

In this section, we will compare the image restoration and noise detection capability of our method with a number of methods. For illustrations, the results for 512-by-512, 8-bit gray-level images “Lena” and “Bridge” are presented here.


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Table 1: Comparison of restoration results in PSNR (dB) Method SD-ROM Filter [8] MSM Filter [9] ACWM Filter [3] Trilateral Filter [4] Two-Phase Iterative Method [2] Our Method

“Lena” image 20% 40% 60% 35.29 28.59 21.64 35.44 29.26 22.14 36.07 28.79 21.19 36.70 31.12 26.08 36.57 32.21 24.62 37.46 33.01 27.32

“Bridge” image 20% 40% 60% 27.04 23.33 19.43 27.27 23.55 20.07 27.08 23.23 19.27 27.31 24.01 20.84 27.66 24.60 20.89 27.84 24.86 21.95

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1: Results of different methods in restoring 60% corrupted images: (a) and (e) original image, (b) and (f) the old two-phase iterative method, (c) and (g) the trilateral filter, (d) and (h) our method, .

4.1

Comparison of image restoration

In Table 1, we list the best results in PSNR for the two images with different noise ratios. From Table 1, it can be seen that in all cases our method provides the best results. For 60% noise ratio, our method outperforms all the others by more than 1 dB. To compare the results subjectively, we give in Figures 1, enlarged areas of the images restored by different methods. In the results of the old two-phase iterative method, there are still many noticeable noise patches. Although no noticeable noise is obtained by the trilateral filter, the details, such as edges and lines, are not restored well. In contrast, our method performs better, and can suppress the noise successfully while preserving more details. Considering the abundance of image details and the high noise level, the visual qualities of our restored images are quite good.

4.2

Comparison of noise detection

For good performance, the capability of noise detection is very important. Here, we compare our method with all the methods in Table 1 that have noise detectors. Table 2 lists the number of undetected noisy pixels (“miss” term) and the number of noise-free pixels which are identified as noise (“false-hit” term). Because some of the random-valued impulse noise values are not so different from their neighbors as in salt-and-pepper noise, there may be much more noise-free pixels detected as noise when detecting


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Table 2: Comparison of noise detection results for image “Lena” Method SD-ROM Filter [8] MSM Filter [9] ACWM Filter [3] Two-Phase Iterative Method [2] Our Method

miss 22842 16582 16052 13657 11459

40% false-hit 411 7258 1759 6192 6531

miss 32566 20857 23683 13868 11690

50% false-hit 998 10288 2895 12693 10349

miss 45365 26169 32712 23793 12424

60% false-hit 2651 15778 7644 17573 14789

random-valued impulse noise. Therefore, for a good random-valued impulse noise detector, on one hand it should be able to identify most of the noisy pixels, but on the other hand, its “false-hit” value should be as small as possible. Comparing with others, our method can distinguish more noise pixels with fewer mistakes. Even when the noise level is as high as 60%, our method can still identify most of the noisy pixels.

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Conclusions

In this paper, we propose a new local image statistic ROLD, by which we can identify more noisy pixels with less false-hit pixels. We combine it with the detail-preserving regularization to get a powerful method for removing random-valued impulse noise. Simulation results show that our method outperforms a number of existing methods both visually and quantitatively.

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