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An Efficient Parameter Selection Criterion for Image Denoising

Hamed Pirsiavash1, Shohreh Kasaei2, Farrokh Marvasti1

h_pirsiavash@ee.sharif.edu, skasaei@sharif.edu, marvasti@sharif.edu

1Multimedia Research Lab., Sharif University of Technology, Tehran, Iran

2Department of Computer Engineering, Sharif University of Technology, Tehran, Iran

Abstract - The performance of most image denoising systems

depends on some parameters which should be set carefully based

on noise distribution and its variance. As in some applications

noise characteristics are unknown, in this research, a criterion

which its minimization leads to the best parameter set up is

introduced. The proposed criterion is evaluated for the wavelet

shrinkage image denoising algorithm using the cross validation

procedure. The criterion is tested for some different values of

thresholds, and the output leading to the minimum criterion value

is selected as the final denoised output. The resulting outputs of our

method and the previous threshold selection scheme for the wavelet

shrinkage, i.e. the median absolute difference (MAD), are

compared. The objective and subjective test results show the

improved efficiency of the proposed denoising algorithm.

Keywords – image denoising, wavelet shrinkage, noise estimation,

parameter selection.

1. INTRODUCTION

Nowadays, by improving image acquisition systems,

many types of cameras are available. Some of these cameras

use very simple hardware in order to have low cost and to be

embedded in other devices like mobile phones. Hence, the

output images of these devices are noisy and poor. In

addition, in most image processing systems, the taken image

should be fed to some processing stages like compression

and recognition. The parasitic noise in the input image could

suffer the other processes and make them inefficient.

To overcome these shortcomings, many image denoising

algorithms have been developed during recent years. For

instance, Gaussian smoothing, neighborhood filtering, and

wavelet shrinkage can be mentioned [1].

In general, all denoising methods have some parameters

and thresholds which should be adjusted to gain the best

performance. Generally, these parameters depend on the

noise distribution and its variance. Most algorithms suppose

the noise to have a white Gaussian distribution with a known

variance. However, in practical situations, we have no

information about the noise variance. Hence, another

problem rises which is the parameter and threshold selection

algorithm. During recent years, some researchers considered

this problem and made some solutions [2, 3, and 4]. The

generalized cross validation method is proposed by Jansen

et al. for multiple wavelet threshold selection [2, 3]. They

defined a criterion which its minimum roughly minimizes

the mean square error (MSE), but their method works in

some special conditions and as proved in [2], it works only

for wavelet shrinkage with orthogonal transforms. In

addition, as they mentioned in their paper, its output has low

MSE, but it is not guaranteed to yield a good visual quality.

In this paper, assuming the additive noise to have an

arbitrary distribution, a novel criterion for image denoising

is introduced. The minimization of this criterion leads to

near optimum parameter set for denoising purposes. In order

to evaluate the performance of this criterion, it is applied for

optimum parameter selection in a popular image denoising

algorithm, the wavelet thresholding.

The layout of this paper is as follows: Section 2

introduces the proposed criterion and its efficiency in

parameter selection. In section 3, wavelet shrinkage

algorithm is described briefly. The experimental results and

the performance comparison are presented in Section 4.

Finally, Section 5 concludes the paper.

2. PROPOSED CRITERION FOR PARAMETER

SELECTION

In image denoising algorithms with additive noise, the

input image is assumed to be the summation of original

image and an additive random noise. An important

knowledge which is used in the proposed criterion is the

independency of these two signals (the original image and

the additive noise). Here, the aim of denoising algorithms is

to remove the parasitic noise. In fact, the difference between

the input and the output of the denoising stage is the

estimated noise which has been removed (see Figure 1).

Therefore, the distribution of the estimated noise should

approach that of the additive noise.

The estimated noise for image denoising with two

distinct parameter sets is shown in Figure 2. In this figure,

the estimated noise is exaggerated to be shown clearly. It

could be seen that there is a large similarity between the

estimated noise and the original image, but this similarity in

Figure 2(f) is less than that in Figure 2(e). It means that for

an optimum image denoising algorithm, the correlation

between the estimated noise and the output image which is

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an expectation of the original image should be minimized.

This result is in agreement with the assumption of

independency between the original image and the additive

noise. Now, the correlation for each parameter set can be

computed. Consequently, by minimizing that, the best

parameter set for the denoising algorithm can be found.

Fig. 1. Noise estimation

Fig. 2. Form left to right and top to bottom: (a) Original image(Lena), (b) noisy image (AWGN, sigma=15), (c) blurred denoised image, (d) denoised

image using proper parameters, (e) estimated noise of (c), (f) estimated noise of (d). (The estimated noise is exaggerated to be seen clearly).

According to our knowledge, most denoising

algorithms assume the original image to have more energy

in lower frequency components compared to the noise. In

addition, the additive noise usually has near flat spectrum

(i.e. white noise). Hence, a huge energy of noise can be

removed by removing the higher frequency components.

But, we know that using these methods, the edge points

which have higher frequency components will be blurred.

As we know, because the human visualize system is more

sensitive to the edges, the blurring effect will be perceived

obviously. In this research, in order to adapt to the human

visualize system, the criterion is altered. As can be seen in

Figure 2, the edge points can be found in the estimated

noise. Therefore, the correlation between the estimated

noise and the edge map of the output image is used.

Because the output image for some parameter sets has

high amount of noise, the edge map should be extracted

using a robust edge detection method; thus, here, Canny

edge detector is used [5]. Then, the following value should

be minimized.

(C Correlation Estimated noise output image edge

=

With weak denoising parameters, the estimated noise

approaches to a zero field and makes the correlation to

have a low value. Consequently, the minimum correlation

,)

(1)

Image

Denoising

Differentiator

Output Image

Estimated

Noise

Input Image

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will be found for the weak denoising parameters. In order

to suppress this defect, the estimated noise energy is used

in the proposed criterion. The final criterion can be written

as follows:

(

log(

Input image energy

+

The minimum of this criterion is found for the

optimum parameters set. Some optimization methods like

the genetic algorithms can be used to achieve the

minimum, yet in this paper, showing the performance of

the proposed criterion is the final goal; consequently,

finding the minimum is performed using a simple method,

i.e. cross validation. This criterion is computed for some

parameter sets which are predefined multiplications of the

parameter set of the MAD method, described in the next

section, and the minimum value is chosen.

3. WAVELET SHRINKAGE

As the defined criterion should be evaluated and

compared with the other parameter selection methods, in

this research, a usual image denoising algorithm, i.e. the

wavelet shrinkage is implemented. In this section, a brief

description of this method

implementation results and details are discussed in Section

4.

Wavelet shrinkage is an efficient signal denoising

algorithm introduced by Donoho et al. in [1, 6, and 7].

That method is based on the idea that the original image

has large wavelet coefficients and the noise is distributed

over all coefficients. Thus, by thresholding the small

coefficients, the image will not be damaged although a

large amount of noise energy will be removed. The hard

thresholding is applied using:

>

Txx

where T is a predetermined threshold value. This basic

idea causes some oscillations near the edges. As a result,

they proposed soft thresholding method in which small

wavelet coefficients are cancelled and the others are

changed in order not to destroy the continuity in wavelet

coefficients.

−⋅

Txx Sign)()(

where Sign(x) denotes the signum function. Using this

method, oscillations are suppressed [7].

In wavelet thresholding methods, the selection of

thresholds for each resolution level is very important

because according to the other denoising algorithms,

wrong selection can make the output image blurred or

,)

)

l

Correlation Estimated noise output image edge

C

estimated noise energy

=

(2)

is presented. The

≤

=

Tx0

HWT(x)

(3)

>

≤

=

Tx

Tx0

SWT(x)

(4)

noisy. Some threshold selection methods are introduced

for Gaussian noise distributions with known variances.

Three commonly used methods are the universal, SURE,

and MiniMax. The mathematical details can be found in

[6, 7]. For instance, universal method is as follows:

N

T

n

⋅=

σ

N

log2

∧

(5)

where N is the number of data points and

noise variance defined below. In most denoising

algorithms, the median of absolute difference (MAD) is

used for noise variance estimation [8].

(

6745 . 0

This estimation yields to good results for Gaussian

distributed noise. A typical wavelet shrinkage algorithm is

shown in Figure 3.

Wavelet

Transform

n

σ

∧

is the

6745 . 0

) )x( Medianx Median

MAD

n

−

==

∧

σ

(6)

Fig. 3. Block diagram of a simple wavelet shrinkage denoising system.

4. EXPERIMENTAL RESULTS

In this section, the implementation results and the

performance of the proposed algorithm when compared to

other available approaches are presented. An efficient

criterion is computed for several parameter sets in a

denoising algorithm and the parameters leading to the

minimum of the criterion are chosen as the best

parameters for the input image and the related noise

statistics. The proposed algorithm is implemented using

Matlab package for wavelet shrinkage image denoising

process.

As briefly discussed in Section 3, wavelet shrinkage is

a powerful image denoising algorithm, and thus many

researchers have proposed different modified versions of

that algorithm. In this research, wavelet shrinkage is

implemented in two resolution levels. Here, Daubechies

wavelet with 6 tabs is used. The initial threshold for each

subspace is chosen independently based on the MAD

variance estimation and MiniMax threshold selection

methods. Next, the criterion is computed for 11 different

Input Image

MAD

Estimator

Threshold

Selection

Soft

Thresholding

Inverse

Wavelet

Transform

Output Image

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multiplications of this initial threshold set. The

multiplications are chosen uniformly in the logarithmic

scale in the range of

10−

to

is taken as the best solution. The MSE of this method and

the MAD method for some standard images are listed in

Table 1. The last column contains the best MSE among 11

tested threshold sets. As can be seen, the obtained

denoising system (with selected variance) approaches the

minimum MSE. The resulting PSNR and the calculated

criterion for a sample image are plotted in Figure 4. As

seen in this figure, choosing the minimum value of the

proposed criterion matches the maximum PSNR that leads

to the best parameter selection for denoising purposes. A

sample output result is shown in Figure 5.

As seen in Table 1, for most tested images the MSE

obtained from the proposed algorithm is less than that of

the MAD method. Moreover, for some cases the obtained

MSE by our algorithm is close to the minimum available

MSE.

As the MSE is not the best measurement for

performance analysis in image processing systems, the

outputs should be examined in a subjective test as well.

The results of the subjective test among 20 boys and girls

are presented in Table 2. Some particular cases in which

the MSEs of our method are high are examined in this

test. For instance, in the 21st row of Table 1, the resulting

MSE is higher than that of the MAD method. The outputs

of the 21st case are shown in Figure 6. It is obvious that

the output of our method is subjectively better than the

MAD output, which proves that both objective and

subjective tests should be run. In fact, because the

criterion uses the edge map, our algorithm leads to less

defects in the edge areas and thus results in higher

subjective performance; although it may have a higher

MSE. Another result obtained from these implementations

is that for lower input noise variances, our method

performs much better than the MAD method. Because for

images with a low level of noise, after denoising the edge

map can be extracted more efficiently, and thus the

proposed algorithm can better calculate the minimum that

matches the maximum of the PSNR. As another result, for

images with small size (about 256x256), our method

performs better than the MAD and MiniMax methods,

because our method is less directly dependent to the

statistics of the images. This fact motivated us to examine

this method in spatially adaptive wavelet shrinkage

algorithms introduced in [9].

5. CONCLUSION

In this paper, an efficient criterion for performance

analysis of denoising systems is introduced. It is shown

that using a cross validation procedure, we can adjust the

system parameters to achieve a better performance. This

criterion is examined for wavelet shrinkage as a common

8 . 02 . 1

10

. Finally, the minimum

denoising algorithm. According to the results, the obtained

subjective tests show the superiority of the proposed

algorithm when compared to the MAD approach. Another

important advantage of this method is its independency on

the noise distribution and its variance. As mentioned

above, the algorithm performs even better for images with

lower noise variances.

Table 1. MSE of our method in comparison with MAD and minimum

available MSE.

Image

name standard

deviation

1 Barbara512 5

2 Barbara512 10

3 Barbara512 15

4 Barbara512 20

5 Barbara512 25

6 Barbara256 5

7 Barbara256 10

8 Barbara256 15

9 Barbara256 20

10 Lena512 5

11 Lena512 10

12 Lena512 15

13 Lena512 20

14 Lena512 25

15 Lena256 5

16 Lena256 15

17 Boat 5

18 Boat 10

19 Boat 15

20 Boat 15

21 Boat 20

22 Peppers 5

23 Peppers 10

Average 13.04

Noise

MSE of

proposed

method

18.79

56.22

100.89

201.52

228.89

22.66

70.74

131.45

190.61

12.84

32.87

58.44

84.00

98.74

17.35

95.65

18.45

54.47

113.38

81.59

192.18

15.18

49.97

84.65

MSE of

MAD

method

112.18

173.19

222.49

260.27

292.18

162.87

237.11

294.70

345.20

35.57

55.13

73.91

92.65

110.67

79.24

153.05

80.26

107.66

133.26

133.68

157.22

38.55

55.90

148.13

Min

available

MSE

18.79

56.22

100.89

152.77

197.45

22.66

65.52

121.27

190.61

12.84

32.87

55.79

78.10

98.74

17.35

88.24

18.45

48.90

82.43

81.59

114.07

15.18

35.56

74.19

Table 2. Subjective test results among 20 boys and girls (score 5 is

assigned to the original image).

No. Image

name standard

deviation

1 Barbara512 25

2 Barbara256 20

3 Lena256 15

4 Boat 15

Average 18.75

Noise Proposed

method

MAD

method

3.30

3.47

3.49

4.12

3.59

2.62

2.44

2.99

3.19

2.81

ACKNOWLEDGEMENT

The authors would like to acknowledge the support of

Iran Telecommunication Research Center (ITRC) for this

project.

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Fig. 4. Results for Lena256 (with Gaussian noise, standard deviation=15) (a) PSNR vs. threshold ratio, (b) proposed criterion vs. threshold ratio.

(Threshold is the multiplication of the threshold ratio and the MAD threshold, i.e. setting threshold ratio equal to one leads to the MAD method).

(a) (b)

Fig. 5. From left to right and top to bottom: (a) Original image (Barbara256), (b) noisy image (Gaussian, standard deviation=15), (c) MAD output

(MSE=295), (d) output of our method (MSE=131).

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Fig. 6. From left to right and top to bottom: (a) Original image (fishing boat), (b) noisy image (AWGN, standard deviation=20), (c) MAD output

(MSE=157), (d) output of our method (MSE=192).