Reduction of Gaussian Noise and Impulse Noise

Impulse Noise

and impulse noise. The goal of the algorithm is to eliminate these two types of noise for color images and videos. Practically, in order to reduce the hardware storage and computation time, we demand the following requirements in the development of the noise reduction algorithm.

1

and fine-detail regions.

2. The algorithm is simple, computations.

3.3.1

The bilateral filter, as described i

sian noise while remaining the edge information. Each pixel is replaced by a weighted average of the intensities in a (2N+1) × (2N+1) neighborhood. The weighting function is designed to smooth regions with similar intensity while keeping edges intact.

More precisely, let x be the location of the pixel under consideration, and let

( )

Ω = Ω_x

( )

{ ( )

}

Ω_x = x+ − ≤ ≤

b y∈Ω

with respect to x is the product of two components, one spatial weight an photometric weight:

2

wP(x,y) is further normalized and the smoothed pixel

2

S weighting function decreases as the spatial distance in the image between x and y

The parameters σS and σP are used to adjust the influence of the spatial and the

Bilateral filter has been proved to have excellent performance for Gaussian noise mo

increases; while the wP weighting function decreases as the intensity difference between the color vectors increases. The spatial component decreases the influence of the distant pixels and reduces the blurring effect; while the photometric component reduces the influence of those pixels that are perceptually different with respect to the one under processing. In this way, only perceptually similar areas of pixels are averaged and thus the sharpness of edges can be preserved.

photometric weightings, respectively. They can be considered as rough thresholds for identifying pixels sufficiently close or similar to the central pixel. Note that when σP

approaches infinity, the bilateral filter becomes a Gaussian filter. On the other hand, when both σP and σS approach infinity, the bilateral filter behaves like the mean filter.

re val. The algorithm is adequately simple and can be used for color images with low computation cost. However, for strong destructive noise, such as impulse noise, the bilateral filter performs rather poorly. In the following sections, we will make some modification over the bilateral filter to improve the reduction of impulse-like noise.

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Detection of Impulse Noise

mpulse noise. Let x be the

3.3.2

Here, we design a simple statistical measure to detect i

location of a pixel and Ωx(N) the neighborhood of x, as defined in Eq. 3-1. Considering the case N = 1, we can get

( ) { }

0 1 /

Ω = Ω_{x x} x Eq. 3-8

hich represents the neighbor pixels of x. For each point

w y∈Ω⁰_x, we define the

e., absolute difference in intensity of the pixels between x and y; i.

x,y x y

We define the total absolute difference (TAD) as

0

The TAD statistic provides us a measure of intensity similarity between the center pixel and its neighboring pixels. Figure 3-8 shows examples from the Lena image, in which we compare the neighborhood of an impulse noise pixel with the neighborhood of an edge pixel.

TAD=863

TAD=64

Figure 3-8 An impulse noise pixel (upper side) and a typical edge pixel (right side). TAD of impulse＝863; TAD of edge pixel = 88

( 2, 2) ( 2, 2) (0, 2) (1, 2) (2, 2) c the TAD value of the center pixel is obviously bigger than e TAD value of the pure edge pixel. To confirm that the TAD statistic is a good detec

Group I:

TAD influenced by central impulse noise

Group II:

pure and smooth pixels Gr TAnoi wi

oup III:

D influenced by impulse se of corresponding ndow

re 3-9 An 5×5 window with impulse noise on the el, together with the

Figure 3-9 shows examples from the Lena image, with an impulse noise enter pixel. We can find that

th

tor for impulse noise, we check the details of the TAD distribution in Figure 3-9.

The center pixel T(0,0) is the only impulse noise in this 5×5 block, and the TAD of the 8 neighboring pixels (group I in Figure 3-9) are correlated with the center noisy pixel.

The range of these values distribute over the range [120,165], while the range of outer pure pixels (group II in Figure 3-9) distribute in [15, 55]. Both groups have a tremendous difference with respect to the TAD value of impulse pixel. If we set a proper threshold value of TAD, the TAD difference between Group I and Group II would be negligible. Group III can be considered to have a similar distribution as Group I, because their TAD values are correlated with noisy pixels. Figure 3-10 shows the result of impulse noise detection in three color space.

Figure 3-10 Impulse noise detection by TAD

.3.3 Reduction of Impulse Noise

ine it with the original ilateral filter to develop our noise reduction algorithm. Here, we propose the third

3

In this section, we utilize the TAD statistic and comb b

weighting function to indicate how likely an image pixel possesses an impulse noise.

This weighting function can be defined as the “impulsive weighting function”:

TAD( )x 2

2 2

( )

w

x = e

The parameter σI determines the penalty of high TAD values for impulse noise.

ter to rm a new weighting function, which includes spatial, photometric and impulsive

sed in [13], we introduce an adaptive function to the how much to use the impulsive weighting and photometric weighting.

We combine the impulsive weighting function with the original bilateral fil fo

properties. However, the direct combination of impulsive weighting function and photometric weighting function is not appropriate. This is because the impulsive weighting function works well to remove strong impulse outliers, while the photometric weighting function is used to smooth weak impulses. These two weighting functions need to be combined in a proper way to handle not only strong impulses but also weak impulses.

Based on the method propo determine

Considering a central pixel x, and the neighbor pixels y∈Ω, we define the “adaptive weighting” T of y with respect to x as

22

The range of adaptive weighting is [0, 1]. The σT parameter controls the shape of the function. If x or y has a large TAD value with respect to σT, then . On the other hand, if neither pixel has large TAD value, then

( , ) 1x y ≈ T

0 ( , )

T x y ≈ . For the property of adaptive weighting function, we define the final weighting function as

1 T( , ) T

S P I

w ( , ) x y = w ( , ) x y w ( , ) x y

w ( , ) x y

The restored pixel can be computed as expressed below:

The modified bilateral filter can remove Gaussian noise and im

choosing proper parameters σS, σP, σI, and σT For the impulse level p<20%, we can get good performance by 3×3 window size and without iterative calculation. The selection of parameters and experiment results will be introduced in the following sections.

As indicated in [14], for Gaussian noise removal, the optimal σS value is relatively ately equal to 1.7 × σn, where σn

e. In [15], Immerkaer provides a st method to estimate the σn of Gaussian noise in an image. The estimator is given by

( , )

3.3.4 Selection of Parameters

insensitive to the optimal σP, and σP can be approxim represents the standard deviation of the Gaussian nois fa

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W and H represent the weight and height of the image, and L denotes the Laplacian filter. To suppress impulse noise, we take σS ＝ 5. Experiment results have shown that the proper value of σI is around 2400, and σJ ＝ 0.1 is a good choice for oth impulse noise and Gaussian noise. For compression artifact reduction , similar parameters are selected to smooth dusty regions. The experimental results will be showed in the following chapter.

removing b