A noise-ranking switching filter for images with general fixed-value impulse noises

Signal Processing 106 (2015) 198–208

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A noise-ranking switching filter for images with general fixed-value impulse noises Hsien-Hsin Chou a, Ling-Yuan Hsu b,n,1 a b

Department of Electronic Engineering, National Ilan University, Yilan 26047, Taiwan Department of Information Management, St. Mary's Junior College of Medicine, Nursing and Management, Yilan 26644, Taiwan

a r t i c l e in f o

abstract

Article history: Received 26 March 2014 Received in revised form 22 July 2014 Accepted 22 July 2014 Available online 2 August 2014

Impulse noise is generally classified into random-value and fixed-value types. However, most of the previous literatures recognized salt and pepper (S&P) noise as the fixed-value impulse noise because it is easily detected and recovered. This article studies a general fixed-value impulse noise in which the corrupted pixels are set to not only the minimum– maximum values but also any fixed intensities. A novel noise-ranking switching filter (NRSF) is proposed for suppressing these tough noise types. In order to clearly distinguish the noise-free pixels from noisy pixels having the same gray value, the detection stage of NRSF consists of three modified methodologies, named global–local statistics analysis, sectional boundary discriminative noise detection, and a directional test. Once a corrupted pixel has been identified, the same matrix convolution technique used in the last test is employed again in the second NRSF stage to recover the noise. Additionally, it is suggested that the corrupted pixels are processed via the rank of noise density for improving the suppression capability at very high noise ratio. Although the NRSF is driven for a new impulse noise model, it outperforms several state-of-the-art algorithms in terms of noise suppression and detail preservation even for images with the traditional S&P noise. & 2014 Elsevier B.V. All rights reserved.

Keywords: Switching filter General fixed-value impulsive noise Noise ranking

1. Introduction During the acquisition, transmission and storage processes, images are often corrupted by impulse noises that occur primarily due to hardware imperfections. Corrupted pixel recovery is an essential part of any image processing job with numerous proposed denoising methods [1]. Nonlinear filters have been developed for noise removal due to the nonlinear characteristics of impulse noises. One of the most popular non-linear filters is the median filter with good denoising capability, simple implementation and n

Corresponding author. Tel.: þ886 3 9897396; fax: þ 886 3 9897396. E-mail address: [email protected] (L.-Y. Hsu). 1 Postal address: No.100, Ln. 265, Section 2, Sanxing Road, Sanxing Township, Yilan County 266, Taiwan, ROC. http://dx.doi.org/10.1016/j.sigpro.2014.07.015 0165-1684/& 2014 Elsevier B.V. All rights reserved.

high computational efficiency. However, the median filter will not perform well enough in high noise density cases. Various median-based filter modifications have been proposed to improve its performance. Nevertheless, those methods still process the whole image in spite of whether the current pixel may be noise-free. As a result, it would inevitably change undamaged pixels and cause image quality degradation. To overcome this drawback one solution is to introduce a noise-detection mechanism for determining damaged pixels first, then replace them with estimated values while leaving the remaining pixels unchanged [2]. It is obvious that the denoising capability of the switching median filter depends on the noisedetection performance [3]. A variety of filters with different noise detectors have been proposed in recent years. The more sophisticated median filters include the vector

H.-H. Chou, L.-Y. Hsu / Signal Processing 106 (2015) 198–208

median filter [4,5], directional weighted filter [6,7], decision-based algorithm [8,9], fuzzy switching median filter [10,11], and so on. Generally, impulse noise is classified into two types: random-value impulse noise and fixed-value impulse noise. The gray level of random-value impulse noises are uniformly distributed in the [0–255] interval. Filters that can remove random-value impulse noise or any mixture thereof have also been proposed. For example, the signaldependent rank ordered mean (SDROM) [12], directional weighted median (DWM) [6,13,14], bilateral filter [15,16], minimum–maximum exclusive mean (MMEM) [17,18] filters, the fuzzy impulse noise detection and reduction method (FIDRM) [19,20] and the improved decision-based method [21,22]. Basically, researchers can use the cluster test to separate a noise and its uncorrupted neighbors based on the assumption that the probability of noises with the same gray level is small. Nevertheless, it is known that random-value impulse noise removal is more difficult because the intensity value of the corrupted pixel and that of its uncorrupted neighbors can be quite close. This statement works on the assumption that in most of the literature the fixed-value impulse noise is just assumed to be the maximum or minimum gray level (0 or 255) which is also known as “salt and pepper” (S&P) noise. This is certainly not the only case while equipment is partially imperfect or corrupted by some fixed disturbances [19,23]. For example, if a data bus has a few central bits been flipped over, say, “xx0000xx”, where “x” may be “1” or “0”. The value of pixels will fixedly be corrupted as {0–3, 64–67, 128–131, 192–195}. In order to extend the scope of application, Ng and Ma proposed four general fixedvalue impulse noise models [24]. However, these models still limit the intensity value of corrupted pixels close to the extreme values to facilitate the detection and removal of noise. A more general noise model is also studied in FIDRM [19], but it can only deal with the single beam type and low-density noises. This article studies a novel and more general fixedvalue impulse noise model. The impulse noise discussed here may fixedly occur at any range in [0–255]. Thus the difference between the noise and its neighbor may not be as significant as that of S&P noise. Different from the random-value impulse noise, noise repetition with the same gray-level is quite usual and the random-value impulse noise detection method will be not applicable for this new noise model. Therefore, it is clear that the object of how to interleave the noise and the true image is tougher than both the classical impulse noise. In this paper we present a novel method to solve the above obstacle. Based on the switching- filter idea the NRSF consists of two stages. However, as mentioned, the key issue in NRSF is good ability of general fixed-value impulse noise detection and these two stages will all serve the detection object more or less. Several novel and/or modified techniques are proposed to improve the detection capability. The first NRSF stage considers both the global histogram and local statistics for discriminating the candidate noise. Then a sectional boundary discriminative noise detection (SBDND) technique is applied again. The second NRSF stage employs a multiple matrix convolution

199

to final confirm the true noise and then yield a directional mean to recover it. For improving the suppression capability at very high noise values, the sparse matrix transformation is used to determine the processing sequence of the corrupted pixels via the noise-density rank in the working window. Extensive simulations show that the NRSF outperforms several state-of-the-art algorithms, in terms of noise suppression and detail preservation, no matter for images with the new impulse noise or the traditional S&P noise. The rest of this paper is organized as follows: Section 2 defines the novel general fixed-value impulse noise. Section 3 introduces the proposed NRSF algorithm for impulse noise. Section 4 presents some simulation results. Conclusions are presented in Section 5. 2. The general noise model The novel fixed-value impulse noise model is introduced in (1) for examining the performance of our proposed filter. The intensity is stored as an 8-bit integer giving 256 possible different shades of gray going from black to white, which can be represented as a [0,255] integer interval. In this interval we consider several graylevel values g1, g2, …, gn with (gb  1 agb) and n are a small number for fixed-value impulse noises. The reasonable assumption on n is without loss of generality since the noise can be thought of as a random-value if n is too large. The impulse noises could be more realistically modeled using some ranges that appear at both ends with corresponding lengths of l and r. That is, for each image pixel at location (i,j) with intensity value oi,j, the corresponding pixel in the noisy image is given by xi,j, in which the probability density function of xi,j is 8 1  p; f or xi;j ¼ oi;j > > > p1 > > ; for xi;j A ½g 1  l1 ; g 1 þ r 1  > > < l1 þ r 1 þ 1 p2 ð1Þ f ðxi;j Þ ¼ l2 þ r2 þ 1; for xi;j A ½g 2  l2 ; g 2 þ r 2  > > > ⋮ ⋮ > > > > : pn ; for xi;j A ½g n  ln ; g n þ r n  ln þ r n þ 1

where ½g t  lt ; g t þ r t ; t A ½1; n denote the gray-level interval of fixed-value impulse noises. The value pt indicates the probability that an original pixel becomes a value within the t'th interval and p ¼ Σ nt ¼ 1 pt . In the special case of S&P noise there are only two extreme values g1 and g2 (with l1 ¼ r1 ¼0 and l2 ¼r2 ¼0), which are the minimum (i.e., 0) and maximum (i.e., 255) pixel value of the considered integer interval. The four models proposed by [24] with two extreme bands are easily derived from (1). However, here the gt is not limited to the near extreme values and the model can represent all general fixed-value impulse noises. 3. The NRSF design We focus on gray-scale images that are corrupted by fixed-value impulse noises. Our filtering scheme is divided into two stages but both stages performing more or less detection because the switching filter performance is greatly affected by the detector efficiency. Since the

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The NRSF Algorithm Step 1: Histogram analysis Step 1.1) Obtain the histogram H of the corrupted image X. Step 1.2) Starting from t=1, find peak gt and the corresponding lt, rt from H (Refer to (3) to (5)). Step 1.3) Count the global noise ratio pt of group t and check whether the noise group is not significant: if then goes to Step 1.4. The and denote the limit of probability and group amount respectively. Otherwise register noise group in record G and excludes this group from H, then increase t and repeat Step 1.2. Step 1.4) If no noise group is found the candidate noise map C and the refine-flag matrix F are set as the default one matrix and the process jumps to Step 4. Otherwise the noise group record is sorted in ascending gray level order and a candidate noise map C is created with corresponding group value t for each pixel in the noisy image. Here ‘0’ denotes noise-free pixels. Step 2: Statistic testing for noise-map refinement Step 2.1) Reset the refine-counter matrix F. Step 2.2) For each candidate noise, calculate its local statistic in the elements in F if the local statistic is greater than

window and set the corresponding .

Step 3: The sectional boundary discriminative noise detection (SBDND) Step 3.1) For each candidate noise , set two boundaries as Step 3.2) Exclude the pixels in the

windows but outside the working section , and

a vector S. Step 3.3) Find the maximum value

Step 3.4) Check if the refine-flag Else check if

, where

. Then sort the remainder as

and minimum value

of S, where

at the maximum intensity difference between Increase the refine-flag

.

if

and

are determined

and med(S), med(S) and

, respectively.

. , reset noise map Ci,j as noise free then go to Step 5. and repeat Step 3.2.

, then set

Step 4: Multiple convolution I – directional test Step 4.1) Perform the directional testing

as (8) and check if the refine-flag

, reset noise map

ci,j as noise free. Step 5: Noise-Ranking Stage Step 5.1) Check if , generate the processing rank threshold of trigger probability.

from equations (9) and (11,12), here

is a presetting

Step 6: Multiple convolution II - restoring Step 6.1) Following the rank of , for each corresponding (i, j) repeat Calculate yi,j according to the multiple convolution from (10). Fig. 1. The NRSF algorithm.

intensity value of a general fixed-value noise and that of its uncorrupted neighbors can be quite close to each other, the core object is that the detector is expected to have good ability to discriminate noise-free pixels from corrupted pixels having same intensity value. In order to achieve this goal, three testing procedures are employed and any candidates that pass more than two tests will be reset as noise-free. The NRSF algorithm details are shown in Fig. 1. The algorithm first performs a histogram analysis of the image with size M  N pixels to determine each of the fixed-value noise regions. All of the pixels whose gray-scale values are inside these noise regions are selected and registered into a noisy candidate map, such that G ¼ fGt jGt ¼ ½g t ; lt ; r t ; pt ;

t ¼ 1 – ng

ð2Þ

lt, rt are determined at the first hit of histogram intensities less than α percent of the peak as either two side searches from gt or the end of the histogram, i.e., ( lt ¼ ( rt ¼

0; lt ;

if g t ¼ 0 if ðg t  lt Þ ¼ 0 or Hðg t  lt 1Þ r αHðg t Þ;

0; rt ;

if gt ¼ 255 if ðgt þ rt Þ ¼ 255 or Hðgt þ rt þ 1Þ rαHðgt Þ;

1r lt r H w :

ð3Þ

1r rt rH w

ð4Þ where H(  ) is the histogram function and Hw is the limit of width. The algorithm first seeks the peak value of histogram at gt, then search from it to left and right until finding the rapid decrement of histogram intensities in both sides. However, while gt is close to the minimum or

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maximum gray-level value, (gt  lt)¼ 0 and (gt þrt)¼255 will be checked first as the boundary conditions. The global probabilities of each noise group can be yielded directly from the histogram analysis and are used to perform a statistical comparison with the local density of candidate noise to distinguish the first-layer noise-free pixels. pt ¼

gt þ rt 1 ∑ p M  N b ¼ g t  lt b

ð5Þ

This process will be repeated until no more significant peaks can be found. Once the histogram analysis is complete the candidate noise map C can be yielded immediately as ( ci;j ¼

t;

if g t  lt rxi;j r g t þr t

0;

otherwise

ð6Þ

To achieve a good ability to discriminate noise-free pixels from corrupted pixels having same intensity value, the first NRSF stage consists of three tests. The concept behind the first statistical testing is quite straightforward: for a t-group candidate under testing, if the ratio of t-group candidate pixels in the local area is significantly larger than the global noise density pt, then it would quite possibly be the true image but have the same gray-level value as the noise. A 5  5 window is used here to retain enough statistical samples and a refine counter fi,j is utilized to denote that xi,j has passed how many tests. A xi,j with corresponding ci,j 4 0 but fi,j Z2 will be refined as noise-free. The second test is inspirited by the BDND method [24– 26]. The BDND algorithm first sorts and classifies the pixels in a localized window into three groups. The pixel within the center group will then be considered as “uncorrupted”. However, because the noise is no longer only 0 or 255, the boundaries are changed to those of the two noise groups nearest the processing pixel. Each candidate noise with different group values is checked into a different section. This modified version is called “Sectional BDND”. The last test will accompany the restoration process using the multiple matrix convolution skill. It is wellknown that the object of a filter is to remove the noise and also preserve the fine details in an image. In order to achieve this goal special filters are used that take the local features into account to recover the edges and textures [13,14,22]. These special filters utilize the correlation information between the current pixel and its neighbors aligned with four main or more directions in the window. A simple directional detection and the removal method is introduced based on a similar stratagem. However, because the intensity value of the corrupted pixel and that of its uncorrupted neighbors may be quite close, one has to concern the difference of noise type before comparing the gray level of two pixels. Instead of just using a single set of convolution operators on the image, three sets of operators are employed on both the image and candidate noise map simultaneously to achieve multiple goals. These four sets of modified matrix convolution operators are shown in Fig. 2. Here a 3  3 window is used to emphasize the local correlation. An extension matrix is

201

first yielded from C C 3 ¼ fc3i;j jci;j A Cg

ð7Þ

The third test checks if the candidate is aligned with two neighbors in one line. It will then be confident of refining the candidate as noise-free. 8 < f i;j þ 1; if C 3  K 0;v ¼ 0 i;j ð8Þ f i;j ¼ : f i;j ; otherwise where Ku,v is the v'th kernel of set Ku, C 3i;j is the 3  3 submatrix central at c3i;j ,  denotes a convolution operation for v ¼1–4 and each performs an element-wise inner product of two matrices. Certainly, the convolution will immediately stop at first hit. The cube of group value t is needed to avoid a miss cancelation. For example, if Diag (Ci,j)¼[123], Ci,j  K0,1 ¼0 but is not a wanted result. This can be correctly detected by (8) since C 3i;j  K 0;1 ¼ 12 a0: The noise map is affirmed after completing the directional test. The next stage of NRSF utilizes a simple directional mean to recover noisy pixels. However, because the intensity value of the corrupted pixel and that of its uncorrupted neighbors may be quite close, the directional mean need to take both of the gray level and the noise type into account. A noise-free map can be yielded as ( 1; for ci;j ¼ 0 di;j ¼ ð9Þ 0; for ci;j 40 Each noisy pixel is then located at di,j ¼0 will be restored via a linked-convolution procedures again. ( yi;j ¼

meanðX i;j  K 1;v Þ medianðDi;j  X i;j Þ

if ðDi;j  K 1;v ¼ 2Þ⇉ðX i;j  K 2;v r τÞ; v ¼ 1–4 otherwise

ð10Þ where Di,j and Xi,j are the sub-matrices central at di,j and xi,j, Kr,s is the s'th convolution kernel of operator set Kr, ⇉ denotes a linked operator with which the result of left operand switches whether the right operand is needed to proceed at the same operation direction s. I.e. Di,j  K1,1 ¼ 2 is checked first, if it is true then Xi,j  K2,1 will also be proceeded. Otherwise the linked convolution will directly pass to next kernel of direction. τ is a similarity threshold determined via experimental simulation. The first-line condition implies that there exists at least one directional pair of pixels which are noise-free and have a similar gray level.  in the second line is a mask operator. Once the directional detection is failed, NRSF throws away the noise pixels in the window and take the median as a default value. Two numerical examples are giving for illustrating the usage of (10). Here we suggest that the noises appear at 0, 125 and 255 for simplicity. Consider the 3  3 window X1 shown in Fig. 3(a) which contains an edge along the direction of 451. The corresponding noise map C1 and noise-free map D1 are obtained as Fig. 3(b) and (c). The linked convolution starts with the kernel K1,1 and fails in the D1  K1,v ¼2 test. The process directly passes to check the next directions and finds D1  K1,2 ¼2 is met. The righthand convolution X1  K2,2 has to be calculated now but get a difference that is large than the similarity tolerance.

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Fig. 2. Three sets of four 3  3 convolution kernels.

0

86

85

1

0

0

0

1

1

85

125

255

0

2

3

1

0

0

86

182

182

0

0

0

1

1

1

86

86

85

0

0

0

1

1

1

85

125

255

0

2

3

1

0

0

0

182

182

1

0

0

0

1

1

Fig. 3. Two hypothetical noisy sub-images and their corresponding noise and noise-free maps. (a) X1 (b) C1 for X1 (c) D1 for X1 (d) X2 (e) C2 for X2 (f) D2 for X2.

Therefore the linked-convolution goes on to the third direction and still fails on the left-hand test. Finally, both conditions of D1  K1,4 ¼2 and X1  K2,4 rτ are met and the central pixel is replaced by the mean of (85, 86). The next example given in Fig. 3(d) is modified from Fig. 3(a). The pixels x1,1 and x3,1 are interchanged in this case and all the results of linked-convolution failed at the first-line tests. The default value for restoration is obtained from the trimmed median process, i.e., the central pixel is replaced by median{85, 85, 56, 86, 182, 182}¼86. At last, the fundamental pixel-wise filter performance depends on enough noise-free pixels existing to restore the reference image. As can be seen, this requirement is difficult to reach in very high noise environments. Several effective algorithms have been proposed to remove image

noise with high noise density. For example, the adaptive method [24,27] enlarges the working window until it yields enough uncorrupted pixels. The recursive approach repeats the filtering process using the result from last iteration as in Refs. [28,29]. However, these algorithms take the risk of losing the local details and increasing the computational cost. A simple technique is suggested here to overcome these obstacles. Based on the uncertain property the local noise density will not be unique over the whole image, i.e. some working windows may still exist that have lower noise density in a heavily corrupted image. The core idea of this approach is to restore the corrupted pixels within a lightly corrupted window first, then take the restored pixel as a reference for its neighbors to improve the noisy condition of those neighbors. Once

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203

the noise-free map D is detected the binary complement matrix D is compressed by the sparse transform and used as a key to sort the filter process ranking. The confidence matrix Q can be given as

performances of the proposed algorithm were measured quantitatively using the Peak Signal to Noise Ratio (PSNR) and Mean Structural SIMilarity (MSSIM) [30] as defined in (15)–(18).

qi;j ¼ ∑ ∑ SPðDi;j Þ

MSE ¼

i

ð11Þ

j

where SPð U Þ denotes the sparse matrix transformation which throws away the zero components of D (i.e., the noise-free pixels) and generates a compressed index list of each corrupted pixel xi,j. qi,j and then accumulates the noise in a working window centered at xi,j. This means that there are more useful references for recovering the corrupted pixel xi,j while qi,j is small, and such a pixel should have a higher processing priority. We then have the processing rank as ξ ¼ SortðQ Þ

ð12Þ

where Sort(  ) stands for the sorting operation in ascending order. This completes the whole filter design for a highly corrupted image. 4. Simulation results Simulations were carried out on well-known Matlab R2010A environment images to verify the efficiencies of the proposed algorithm. The NRSF thresholds were determined using experiments as: percentage of maximum intensity α¼0.6, minimum noise ratio pmin ¼ 0.03, maximum width Hw ¼25, maximum group count nmax ¼ 10, local density bound β¼2.5, ranking trigger δ¼0.4 and similarity τ ¼11. The noise densities in these images varied widely from 10% to 90% with increments of 10%. A comparison of the noise removal capabilities of the proposed model against the filters motion above, i.e., bilateral filtering [15], directional weighted median (DWM) filter [13], fuzzy impulse noise detection and reduction method (FIDRM) [19,20], improvement BDND (IBDND) [25] filtering algorithm, median filter (MF) [1], minimum–maximum exclusive mean (MMEM) [17,18] filters and signaldependent rank ordered mean (SDROM) [12] filter. The

M N 1 ∑ ∑ ðoi;j  yi;j Þ2 M  Ni¼1y¼1

PSNR ¼ 10 log10

SSIM ¼

2552 MSE

!

ð2μo μy þ C 1 Þð2σ oy C 2 Þ ðμ2o þ μ2y þC 1 Þ þ ðσ 2o þ σ 2y þ C 2 Þ

MSSIM ¼

ð15Þ

1 W ∑ SSIMðok ; yk Þ Wk¼1

ð16Þ

ð17Þ

ð18Þ

where MSE stands for the Mean Square Error and SSIM stands for the Structural SIMilarity, M  N is size of the image, symbols o, x and y represent the original, corrupted and restored images, respectively. And μo and μy are the mean intensities of the original and restored images, σo and σy are the standard deviations of the original and restored images, σoy is the covariance of the original image and restored image, and C1 and C2 are set to 6.5025 and 58.5225, respectively (i.e. normal default), and W (¼1024) is the number of local windows in the image [30]. 4.1. Experimental results for S&P Experiments were carried out for S&P corruption cases using the test image Lena. We set g1 ¼0, g2 ¼255, l1 ¼r1 ¼ 0 and l2 ¼r2 ¼0. The pixels were corrupted randomly using two fixed extreme values, 0 (i.e., “pepper” noise) and 255 (i.e., “salt” noise), generated with the same probability. Fig. 4 shows visual results using the proposed filter over the various filters on the Lena image with different noise densities. These different filters can be divided into fixed-value noise (i.e., FIDRM, IBDND and MMEM), random-value noise (i.e., DWM and SDROM) and mixed-value noise (i.e., bilateral filter). We can clearly observe that the original image was not

Fig. 4. Comparison of different algorithms for the Lena image. (a) Corrupted image, (b) denoised by bilateral filter, (c) denoised by DWM (d) denoised by FIDRM, (e) denoised by IBDND, (f) denoised by MF, (g) denoised by MMEM , (h) denoised by SDROM, and (i) denoised by NRSF. Rows 1–3 show the processed results from various algorithms for the Lena image corrupted by 10%, 50% and 90% noise densities, respectively.

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50

1.2

45 1

40

DWM

30

FIDRM

25

IBDND

20

MMEM

15

Bilateral

DWM 0.6

FIDRM IBDND

0.4

SDROM

10

SDROM 0.2

NRSF

5 0

0.8

MSSIM

PSNR

35

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40

50

60

70

80

0

90

NRSF 10

20

30

noise density (%)

40

50

60

70

80

90

noise density (%)

Fig. 5. PSNR values of different filters for Lena in S&P noise model.

Fig. 8. MSSIM values of different filters for Lena in extreme sides noise model.

1.2

45 40

1

35

DWM

FIDRM 0.6

IBDND

PSNR

MSSIM

0.8

MMEM

0.4

SDROM 0.2

NRSF

30

Bilateral

25

DWM

20

FIDRM

15

IBDND

10

SDROM

NRSF

5 0

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90

noise density (%)

10

20

30

40

50

60

70

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90

noise density (%)

Fig. 6. MSSIM values of different filters for Lena in S&P noise model.

Fig. 9. PSNR values of different filters for Lena in one side noise model.

50

1.2

45

1

40

Bilateral

30

DWM

25

FIDRM

20

IBDND

15

Bilateral

DWM 0.6

FIDRM IBDND

0.4

SDROM

10

NRSF

5 0

0.8

MSSIM

PSNR

35

10

20

30

40

50

60

70

80

90

noise density (%)

SDROM 0.2 0

NRSF 10

20

30

40

50

60

70

80

90

noise density (%)

Fig. 7. PSNR values of different filters for Lena in extreme sides noise model.

Fig. 10. MSSIM values of different filters for Lena in one side noise model.

preserved by these filters, i.e., the bilateral filter, DWM, SDROM. This is because they are designed for uniform distributed random-value noise (or mixed-value noise) and the detection ability for the repeat value noise is not good enough. The S&P noise is part of a fixed-value noise and these fixed-value filters (such as FIDRM, IBDND and MMEM) will provide good performance. However, the serious distortion of border caused in FIDRM, IBDND and MMEM as they are unable to maintain relatively delicate image parts under highnoise density over 90%. The proposed NRSF overcomes the shortcomings and preserves the detail-protection advantages. Fig. 5 shows a comparison of the PSNR measures for Lena. It can be observed that all the fixed-value filters have

nice performance for the S&P noise. However, the PSNR value of the proposed NRSF algorithm is better than that achieved by the existing algorithms, especially at high noise densities above 50%. Also note, that the higher the noise level, the more significant improvement in the output from our filter. It is shown that the noise ranking process has good ability to remove high density S&P noises. The proposed algorithm was also quantitatively measured using MSSIM with the results given in Fig. 6. The figures indicate again that the performance of randomvalue filters is much worse than that of fixed-value filters. The directional-based approach FIDRM has very good

45

1

40

0.9

35

0.8

Bilateral

25

DWM

20

FIDRM

15

IBDND

10

SDROM

0.2

NRSF

0.1

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0.6

DWM

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FIDRM

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IBDND

0.3

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90

SDROM

NRSF 10

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noise density (%)

noise density (%) Fig. 11. PSNR values of different filters for Lena in uniform noise model.

Fig. 14. MSSIM values of different filters for Baboon in uniform noise model.

1.2

30

1

25

0.8

20

Bilateral DWM

0.6

FIDRM

PSNR

MSSIM

205

0.7

30

MSSIM

PSNR

H.-H. Chou, L.-Y. Hsu / Signal Processing 106 (2015) 198–208

IBDND

0.4

Bilateral

DWM 15

FIDRM IBDND

10

SDROM 0.2 0

SDROM 5

NRSF 10

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NRSF 10

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30

noise density (%)

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noise density (%)

Fig. 12. MSSIM values of different filters for Lena in uniform noise model.

Fig. 15. PSNR values of different filters for Barbara in uniform noise model.

30

1 0.9

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0.8

DWM 15

FIDRM IBDND

10

SDROM 5 0

0.7

Bilateral

NRSF 10

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80

90

noise density (%)

MSSIM

PSNR

20

Bilateral

0.6

DWM

0.5

FIDRM

0.4

IBDND

0.3

SDROM

0.2

NRSF

0.1 0

10

20

30

40

50

60

70

80

90

noise density (%)

Fig. 13. PSNR values of different filters for Baboon in uniform noise model.

Fig. 16. MSSIM values of different filters for Barbara in uniform noise model.

performance. However, the result from the proposed algorithm was better than that from the existing algorithm for the Lena image for all noise densities. This verifies the assumption that the NRSF can improve the detail-preservation performance in a wide-range of noise densities.

We set two noise groups near both sides with the center noise intensity values at g1 ¼2, g2 ¼253, and the width of each group is 5 (i.e., l1 ¼r1 ¼2 and l2 ¼r2 ¼2). The pixels on the extreme sides were randomly corrupted, i.e., 0–4, and 251– 255, generated with the same probability. Figs. 7 and 8 give comparisons of the PSNR and MSSIM measures for the Lena image with different filters. It can be observed that the IBDND outperformed all of the other filters because they have better detection capability on this type of noise. The proposed NRSF produced more significant improvement at the higher noise level. In the extreme noise rate up to 90%, our algorithm

4.2. Experimental results for noise corruption in extreme sides Experiments were carried out on extreme side corruption cases such as the noise model in [24] (using test image Lena).

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40 35 30

Bilateral

PSNR

25

DWM 20

FIDRM

15

IBDND

10

SDROM

NRSF

5 0

10

20

30

40

50

60

70

80

90

is better than the other algorithms by at least 3 db or more. Also, when the impulse noise density was lower than 10%, the performance was not so effective because the false detection may occur in the middle area of gray level. However, the PSNR value is still greater than 40 dB. This means it will provide a Table 1 The number of miss and false detections for different filters for Lena in uniform noise model. Noise density (%)

1.2 1

MSSIM

0.8

DWM IBDND SDROM 0.2 0

NRSF 10

20

30

40

50

60

70

80

90

3346 11,865

5295 501

5261 516

20

Nmiss Nfalse

22,403 26,365

7229 11,334

12,911 602

2985 8194

30

Nmiss Nfalse

33,613 26,061

12,234 10,949

23,378 874

3704 8138

40

Nmiss Nfalse

43,952 26,686

18,717 11,813

36,162 1397

3355 8116

50

Nmiss Nfalse

53,695 29,349

27,422 14,792

51,841 2974

2483 7458

60

Nmiss Nfalse

61,499 31,458

39,305 20,394

70,864 6479

1361 6552

70

Nmiss Nfalse

67,418 28,510

54,484 27,178

89,336 11,615

471 5150

80

Nmiss Nfalse

73,529 20,403

71,788 25,958

107,720 13,731

89 3980

90

Nmiss Nfalse

76,542 11,804

91,452 17,710

124,329 10,348

6 2434

325,977 151,993

521,836 48,521

19,715 50,538

noise density (%) Fig. 18. MSSIM values of different filters for Peppers in uniform noise model.

NRSF

11,157 25,614

FIDRM

0.4

SDROM

Nmiss Nfalse

Bilateral

0.6

DWM

10

noise density (%) Fig. 17. PSNR values of different filters for Peppers in uniform noise model.

IBDND

Sum (Nmiss) Sum (Nfalse)

443,808 226,250

Fig. 19. Lena with (a) 10%, (b) 50%, and (c) 90% corrupted noise density shown in Row 1 and denoising results of NRSF shown in Row 2.

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Fig. 20. (a) Baboon, (b) Barbara, and (c) Peppers with 90% corrupted noise density shown in Row 1 and denoising results of NRSF shown in Row 2.

rather good visual quality. Certainly, this defect is easily solved via a simple adaptive method, exchanging a suitable tolerance or using an alternative filter at 10% density. 4.3. Experimental results for one-side noise In this experiment we set three noise groups in a halfside only. The center intensity values were g1 ¼40, g2 ¼70 and g3 ¼100 and the width of each group was 5 (i.e., l1 ¼ r1 ¼2, l2 ¼r2 ¼2 and l3 ¼r3 ¼2). The pixels were corrupted randomly in intensity ranges, i.e., 38–42, 68–72, and 98–102, generated with the same probability. Figs. 9 and 10 give comparisons of the comparisons of the PSNR and MSSIM measures for the Lena image with different filters. Because all of the other algorithms cannot catch this noise model, it could be observed that the proposed NRSF was significantly better than the other filters for different noise densities. However, when the impulse noise density was smaller than 10% the difference between the noise and other image values was not large enough for the filter to discriminate between them and the NRSF did not provide so good output. 4.4. Experimental results for noise corruption in uniform In this experiment the proposed algorithm was tested by using four typical 512  512 8-bit gray-scale images, namely, Lena, Baboon, Barbara and Peppers. We set three noise groups with different widths. The center intensity values were g1 ¼70, g2 ¼ 120 and g3 ¼185, and the width of each group was l1 ¼ r1 ¼1, l2 ¼r2 ¼2 and l3 ¼r3 ¼3, respectively. The pixels were corrupted randomly in intensity

ranges, i.e., 69–71, 118–122 and 182–188, generated with the same probability. Figs. 11–18 give a comparison of the PSNR and MSSIM measures for these four images. It can be observed that the filters for the random-value noise still showed no ability to detect such noises. It is observed that the performance of NRSF is the identical on these four images (i.e., Lena, Baboon, Barbara and Peppers). The proposed new noise model is clearly worthy of further study, and the NRSF gives a possible solution and works quite well. However, when the impulse noise density was lower than 10%, the performance of NRSF was not so effective because the low repetition rate of noise results in more miss detections. Certainly, this defect is easily solved via a simple adaptive method, for example, using an alternative filter under 10% noise densities. The number of miss (Nmiss) and false detections (Nfalse) for several different filters for Lena using uniform noise models are shown in Table 1. It is shown that the NRSF works well for these novel noises and outperforms all other filters in miss-detection count. Fig. 19 shows the visual results for the Lena image with different corrupted noise densities (i.e., 10%, 50% and 90%) and the NRSF denoising results. Fig. 20 shows different images with 90% corrupted noise density denoising results of NRSF. It can be observed that the proposed NRSF works quite well even with the noise density up to 90%. The NRSF can still maintain the image detail. 5. Conclusions This paper proposed a novel algorithm for fixed-value impulse noise removal from an image. The noise models

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