JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 3, ISSUE 2, OCTOBER 2010 23
A method of suppressing impulse noise using PIE-LT filter Aditya Goel, Anamika Jain Abstract— This paper presents a concept of developing one novel nonlinear and adaptive digital image filter to efficiently suppress SPN, RVIN and mixed noise, especially, for low level noise conditions. This filter suppresses noise through primary implicant elimination applied to logical system representation (minimized Boolean form) of the data. The filters are used to identify and remove noisy pixels from the noise affected areas of image while preserving the fine details such as edges, lines etc. The filtering algorithm used PIE-LT in addition to conventional adaptive median based switching filter. Simulation results are compared with the conventional methods. It has been observed that the filter improves peak signal to noise ratio (PSNR), mean square error (MSE) as well as the structural similarity (SSIM) index in both binary and grayscale images. Index Terms— Primary implicant elimination (PIE), Logical transform (LT), Peak signal to noise ratio, Mean square structural similarity index, Random-valued impulse noise (RVIN).
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1 INTRODUCTION
D
ifferent types of noise frequently contaminate images. Impulsive noise is one such noise, which may affect images at the time of acquisition due to noisy sensors, at the time of transmission due to channel errors or in storage media due to faulty hardware. The present day state-of-art technology offers very high quality photo sensors, high quality electronic circuitry, e.g., system on chip (SOC), and high quality channel as well. Therefore, the noise level has drastically reduced. But the filters that are quite efficient at high noise levels don’t perform so well at low noise levels. Therefore, it is very important to design and develop highly efficient image filters that suppress low power noise quite effectively. The most common filters used for eliminating noise are median-based filters [3]. But it blurs the image and edges are not preserved. When eliminating impulse noise, median filters tend to modify both the noisy pixels as well as the undisturbed good pixels due to the uniform application of the filter across the signal [2], leading to unnecessary errors. The present PIE-LT filter algorithm filter impulse noise by deletion of terms from the minimized Boolean form of the image data. The algorithm improve the results generated by the median filter by combining the general form PIE-LT filter with iterative and switching schemes variations. Both quantitative and qualitative evaluation are performed on a single 2-D image with different noise percentage in terms of: peak signal to noise ratio (PSNR), mean square error (MSE) and the structural similarity (SSIM) index. SSIM is based on examinations of the human visual system and provide structural information within the image, as well as structural distortion throughout. In the literature [14], [12], it has been shown that this method can provide numerical distinctions in images which may produce identical MSE outputs. Some ideas presented here were initially developed in [5]. This paper is organized as follows. Section 2 provides the
———————————————— Aditya Goel is with the Maulana Azad National Institute of Technology, Bhopal,INDIA 462051. Anamika Jain is with the Department of Electronics and Communication Engineering, Maulana Azad National Institute of Technology, Bhopal,INDIA 462051.
Mathematical definition and properties of the logical transform, and the process by which the sum of primary implicants is generated. Section 3 details the algorithm based on this PIE-LT denoising procedure. Section 4 present result and discussion of qualitative and quantitative evaluation of the 2-D-binary and 2-D-grayscale image compared to the median filter. A conclusion found in Section 6 ends the paper.
2 LOGICAL SYSTEM REPRESENTATION The logical system transform was introduced in as a method for converting binary data into a sum of Primary implicants. The logical transform, which is useful for quickly determining the minimized Boolean form (vector y) from an input signal f ,
{
}
y = δ1 Bnδ 0 ATn δ 0 ( f )
( 1) where δp is the Kronecker delta function which accepts and returns a 1-D vector, where p-valued elements are set to one and all others to zero, and where the matrices A and B are defined as
1 0 AnT = 0 1 1 1
⊗n
1 01 Bn = 011 0 01
⊗n
(2)
Unlike other common transforms, the inverse operation of the logical transform is not a mirror image of the forward transform Instead, the original binary data are generated from the sum of primary implicants through a process called implicant expansion. Equation (1.1) of the binary (base-2) representation of m, and since the maximum of m is N − 1, i goes from zero to n − 1. y is the 1-D binary vector output of the logical transform, representing the terms within the sum of primary implicants. Each one in the y-vector represents a term (all other locations are zero).
fm
3 n−1
=
n −1
∑ yt ∏ [δ mi ( ti ) +δ 2 ( ti )] t =0 i =0
(3)
24
Noisy signal
Pre pro -cessing
Primary implicant Elimination
Logical Transform
Implicant Expansion
Post pro-cessing
Filtered signal
For each block in each bit plane
Fig1. General form of the PIE-LT filter where,
f
is the original 1-D binary bit-stream vector of n
data. It is of length N, or 2 , where n = log2N.
fm
th
is the m
binary value of the f -vector, where m goes from zero to th N−1 mi is the i value.
3 ANALYSIS OF LOGICAL TRANSFORM Analysis can be carried out by evaluating the following two parameters of this transform (i) Sum of Primary Implicants (ii) Histogram of primary Implicants.
3.1 Sum of Primary Implicants The logical transform converts binary data into a sum of primary implicants which includes don’t care conditions for the minimized terms. Thus, the sum of primary implicants is incorporated here as a representation of the block data that is both minimized and includes the don’t-care values [8]. The number of terms in the sum of primary implicants, MU, can be found n −1
MU =| y |= ∑ yt t =0
(4)
3.2 Histogram of Primary Implicants This paper applies the concept of a Histogram of Primary Implicants for analyzing the class of noisy signals. The graphical representation indicates the composition of the signal by calculating a running total of some aspect of the primary implicants required to represent each data block. One of these is MC count (number of don’t cares in each block) versus the number of terms found within each summation. Another is coined term distribution, which examines what terms are being used to represent each block. With a block size of 16 (4×4), there are four potential literals in each term of the sum of primary implicants (x1, x2, x3, and x4). Since each of these can have three values (zero, one, 4 or don’t-care), there are 81 (3 ) possible unique terms. The number of don’t-cares MCt from term t is calculated by n −1
MC t = yt ∑ δ 2 yt t =0
(5)
4 PIE-LT FILTER In the proposed method to achieve a reduction in noise found in the input signals, the process of PIE is introduced, where terms responsible implicants are identified and deleted. This process is based on the comparison of a threshold value to the number of don’t cares in each term of the sum of primary implicants. The threshold T is varied based
on not only the bit plane in which the block resides but also the number of terms found within the sum of primary implicants. An optimal set of threshold values for a given image can be customisable either by a genetic search [9] for best matches or through statistical analysis (obtained mathematically or experimentally) of a given class of signals. Fig.1 shows the general form of the procedure for removing noise.
4.1 PIE-LT Filtering Algorithm The steps of the algorithm are as follows: 1) The pre-processing step gray codes the image pixel’s values and decomposes them into bit planes. Each binary bit plane is further broken down into individual blocks. 2) Each block in each bit plane is passed through a logcal transform to produce a sum of primary implicants, the minimized Boolean representation of the data. 3) PIE deletes terms responsible for noise from this sum. 4) Implicant expansion, the inverse of the logical transform, returns the sum back to binary data. 5) The post processing step reassembles the blocks and bit planes, and inverse gray coding is performed. The gray coding [10],[11] and bit-plane decomposition [12] in step1 are only performed when the input data are grayscale (multi bit) in nature (similarly, step5 acts accordingly). Typically, when processing bit planes separately, any correlation between bit planes is ignored. However, by incorporating the gray coding of data values into the pre processing step before the decomposition (as in [11]), some association between the bit planes is kept throughout the process.
5
DESIGN EXAMPLES AND PERFORMANCE ANALYSIS
The performance analysis of the proposed PIE-LT(iterative and switching) filter has been done by testing it with standard 'Barbara' image instead of other available images like 'Lena' because 'Barbara' provides more information compared to 'Lena' for our application. The Barbara image distorted by introduction of RV and S&P noise can be easily distinguished as compared to other images. The 'Barbara' image is corrupted with varying noise density i.e. 1%, 3%, 5%, 7% with both Salt & Pepper noise and Random valued Impulse noise respectively.
5.1 PIE-LT Iterative Filter As in [2], it is possible to apply the PIE-LT filter iterative version multiple times (via a shift in the window locations). The filter proved producing better results. Similar to the iterative median filter, each time the process is performed, the overall computational requirements are increased.
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5.2 PIE-LT Switching Filter
tails better.
Due to the PIE-LT filter’s ability to accurately detect impulses, the filtering procedure can be incorporated into a switching filter configuration. The mathematical formulation [14] for the generalized form of a switching filter Equation can be given by
{
x , bi = 0 si = xb ( xi − yi ) = yi , bi = 1 i i i
(6)
The performance, of this proposed algorithm is evaluated based on three mathematical formulations: MSE, the SSIM index, and PSNR. The MSE is expressed [15], as shown in equation below, between the original noise free image and filtered image M
(b)
N
∑ ∑ (X MSE =
(a)
©©
i, j
− Yi , j ) 2
i =1 j =1
MN
(7)
where, (M×N) is the size of both the original Xi, j, and filtered image Yi j. SSIM provide structural information within the image throughout Ns ∑ j =1W j ( xj , yj ).SSIM ( xj , yj ) SSIM ( X , Y ) = Ns ∑ W j ( xj , y j ) j =1
(2η xη y + C1)(2σ xy + C 2) SSIM ( x, y ) = 2 (η x + η y2 + C1)(σ x2 + σ y2 + C 2)
(8)
2552 PSNR = 10 log10 M N 1 ∑ i=1∑ j=1( Xi , j −Yi , j )2 MN
(9)
© (d) Fig 2. (a)Barbara' image (binary) original, (b) with 1% S&P noise (c)median filter output,(d) PIE-LT filter output
5.3 Visual Comparison The PIE-LT filter provides better filter output as compared to median filter for the Barbara image, both for binary as well as grayscale. Fig.2 (a) shows Barbara binary image corrupted with salt & pepper impulse noise. The ability of the PIE-LT filter to accurately remove the noise with least effect on the image quality as compared to other median filter can be clearly visible from Fig.2 (b-d). In a close-up region, Fig.3, demonstrates the effectiveness of the proposed algorithm as compared to traditional median filter to accurately remove the salt & pepper noise of very low noise density i.e. 3% during processing. Fig.3(a) shows Barbara grayscale (close-up) original image, Fig.3(b) shows Barbara grayscale (close-up) corrupted with salt & pepper impulse noise. The noise is applied to the 8bit grayscale images by corrupting a subset of the pixels by randomly assigning values of either 0 or 255 to the data points with equal probability. In Fig.3(c-d) the Barbara(close-up) grayscale image corrupted with salt & pepper impulse noise is filtered via median filter and proposed PIELT filter. The median filtered image is blurred as the output modifies the value of processed block by the median value. Thus the pixel value is modified irrespective of the noise density. The PIE-LT filter, on the other hand, works on the logical transform thus the pixel value is modified as the other pixel values in the neighbourhood. Thus the blurring of the image is avoided. Similarly, Fig.4(b&c) shows filtered output of random valued corrupted Barbara grayscale image via median filter and PIE-LT filter. It is clearly visible that the median filter output blurs out the details of the image and additionally destroys the patterns within. On the other hand, the PIE-LT filter is able to removing the noise maintaining the fine de-
(a)
(b)
© (d) Fig 3. (a)Barbara' image close up (grayscale) original, (b) with 3% S&P noise, (c) median filter output, (d) PIE-LT filter output.
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TABLE 3 MSE AND SSIM RESULTS FOR FILTERING OF (GRAYSCALE) BARBARA IMAGE WITH RANDOM VALUE IMPULSE NOISE RV Noise
Median filter
PIE-LT filter
MSE
SSIM
MSE
SSIM
187.1 189.6 192.1 194.4 197.2 201.1 205.2
0.8131 0.8110 0.8092 0.8075 0.8062 0.8040 0.8019
44.5 57.9 73.0 86.2 106.8 128.2 155.1
0.9365 0.9320 0.9085 0.9025 0.8780 0.8700 0.8520
(%)
(a) (b) (c) Fig 4. (a)Barbara image (gray-scale) with 1% RVI Noise (b) Median filter output (c) PIE-LT filter output
5.4 Quantitative Comparison Error measurements of the filtered images are presented here via the MSE (mean square error), PSNR (peak signal to noise ratio) and SSIM (structure similarity index) [13]. As discussed in BFO [16], MAE (mean absolute error) is considered as the quantitative measure, which in our view will not provide sufficient information about structural distortion throughout the image. As with both grayscale and binary data, visual quality can vary significantly even even when the MSE values are same [3],[4]. TABLE 1
MSE AND SSIM RESULTS FOR FILTERING OF BARBARA (BINARY) IMAGE WITH SALT & PEPPER IMPULSE NOISE Noise
Median filter
PIE-LT filter
(%)
MSE
SSIM
MSE
SSIM
1 2 3 4 5 6 7
0.0471 0.0475 0.0482 0.0485 0.0515 0.0700 0.0901
0.763 0.761 0.759 0.757 0.755 0.735 0.710
0.0041 0.0065 0.0091 0.0115 0.0150 0.0250 0.0350
0.951 0.918 0.884 0.873 0.860 0.841 0.820
TABLE 2
MSE AND SSIM RESULTS FOR FILTERING OF (GRAYSCALE) BARBARA IMAGE WITH SALT & PEPPER IMPULSE NOISE Noise
Median filter
PIE-LT filter
(%)
MSE
SSIM
MSE
SSIM
1 2 3 4 5 6 7
187.9 189.5 193.5 195.5 199.5 207.1 215.5
0.8133 0.8115 0.8099 0.8080 0.8060 0.8030 0.7999
55.1 80.3 105.5 128.7 156.6 180.2 204.3
0.9530 0.9340 0.9150 0.8950 0.8777 0.8500 0.8250
1 2 3 4 5 6 7
Table 1-3 provides the numerical results from median filter, SD-ROM implementation, LRC technique, and proposedPIE-LT filter. The MSE and SSIM index are pre sented in Table-1,where the filtering output is examined when the Barbara binary image is corrupted with salt & pepper noise. Supporting the visual inspection, Fig.5 shows the graph for Mean square error & SSIM salt & pepper noise. The noise density varies from 1-7%. From Fig.5 MSE for the Barbara binary image is constant at the value of (0.05) upto 5% of noise density after the median filter application on the other hand the value of MSE for PIE-LT filter is very low . Although MSE is varying linearly from (0.005) to (0.015) but at much lower rate and upto 5% of noise density it is about 5 times lower than that due to median filter. For the noise density greater than 5%, MSE for both filter output increases but the increase in PIE-LT filter output is still shows better performance as compared to median filter. Similarly, from the graph in Fig.5 SSIM for PIE-LT filter outputs is (0.95) as compared to (0.75) for median filter output. Thus the PIE-LT filter output is more close to the original one .Although SSIM index is decreasing almost linearly for the PIE-LT filter but the value of SSIM we got is still much higher as compared to give much less mean square error as compared to the quality for Barbara binary image provided by PIE-LT filter will be prove most close to median filter constant SSIM upto 5% of noise density. Thus the image quality for binary ‘Barbara ’image is provided by PIE-LT filter will be prove most close to the original one as compared to the median filter. The MSE and SSIM index in table-2 indicates the filtering output of Barbara grayscale image corrupted with salt & pepper impulse noise. As can be seen from Fig.6 that the MSE output of PIE-LT filter increase linearly from the value of 60, at 1% of noise density, to 180 ,at 7% of noise density, as compared to the almost constant median filter MSE output i.e185.It can be clearly seen that the MSE output for PIE-LT filter is much less compared to the median filter upto 5% of noise density for either type of images .Similarly Fig.6 indicates variation in SSIM output for PIE-LT and median filter of Barbara grayscale images.SSIM index of PIELT filter is decreasing almost linearly from 0.96 to 0.84 for S&P noise density varying from 1-7%. The value of SSIM we got from PIE-LT filter is still higher as compared to SSIM from
27
(a) (a)
SSIM index in table-3 & Fig7(a-b) where the filtering output is examined for the Barbara grayscale image when it is corrupted with random-valued impulse noise. Variation in Fig.7(a) shows that the MSE output of PIE-LT filter is almost similar for both S&P and RV impulse noise. The difference lies in MSE output i.e. 40 at 1% and 160 at 7% of noise density against the value of 60 at 1% and 180 at 7% of S&P noise density. It can be clearly that the PIE-LT filter performs better as compared to the median filter up to 5% of random valued noise density .Fig.7(b) indicates variation in SSIM output for PIE-LT and median filter of Barbara grayscale images . SSIM index of PIE-LT filter is decreasing almost linearly from 0.95 to 0.86 for random valued noise in the same way as for S&P noise density varying from 1-7%. The value of SSIM from PIE-LT filter is higher as compared to the median filter or the quality of image from PIE-LT filter will be as close as to the original one as compared to the median filter for only lower level of random valued noise density .As the noise density increases the PIE-LT filter output also deteriorates.
(b)
Fig 5. MSE & SSIM graph for various levels ‘Barbara’ (binary) image for various levels of S&P impulse noise
(a) (a)
(a)
(b)
Fig 7. MSE & SSIM graph for various levels ‘Barbara’ (grayscale) closeup image for various levels of S&P impulse noise
(b) (b)
Fig 6. MSE & SSIM graph for various levels ‘Barbara’ (grayscale) closeup image for various levels of S&P impulse noise
the median filter .Thus the quality of image from PIE-LT filter is 88% as close as to the original one upto 5% of noise density or the output image quality for Barbara grayscale image provided by PIE-LT filter will also prove better as compared to the median filter. Similar error measurements are demonstrated via MSE and
Fig 8. PSNR graph for various levels ‘Barbara’ (binary) image for various levels of S&P impulse noise Supporting the graph Fig.(5-7), Fig.8 shows the graph for PSNR vs. Salt & pepper noise for different noise density varying from 1-7%. From Fig.8, PSNR of PIE-LT filter for the
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Barbara binary image is much higher (24) as compared to (13) of median filter at 1% noise density. The value of PSNR decreases continuously for the PIE-LT filter with the noise density upto typical value of 15 at 7% of noise density. The value of PSNR is still higher as compared to the median filter. Hence we conclude that the PIE-LT filter provide much better output as compared to the constant output of median filter upto 5% of noise density. Thus, the output image quality for Barbara binary image provided by PIE-LT filter will be prove most close to the original one as compared to the median filter. It can be seen from the image results that PIE-LT filter provide great improvement in terms of MSE, the SSIM over both iterative and switching median filter and other advanced SD-ROM, LRC and BFO optimized adaptive median filter performance. In addition PSNR improvement of PIE-LT filter can be easily visualized from the Fig.8.
6 CONCLUSION This paper presents an approach to remove low level impulse noise in images for real time applications. The concept of detecting and removing impulse noise through PIE was shown to be effective at correctly identifying pixels and accurately restoring their values. The effectiveness of the procedure is demonstrated through application of the proposed PIE-LT filter on the ‘Barbara’ binary and grayscale image. It was found that PIE-LT class of filters produced better results as compared to the median filter in terms of MSE, SSIM and PSNR for either salt & pepper or random valued noise for either type of images. Numerical and visual both results indicates that the output from PIE-LT filter will be as close to the original ones for low levels (upto 5% noise density ) of impulse noise.
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