- Email: [email protected]
Comparative study of Birge–Massart strategy and unimodal thresholding for image compression using wavelet transform
Accepted Manuscript Title: Comparitive Study Of Birge Massart Strategy And Unimodal Thresholding For Image Compression Using Wavelet Transform Author: Siraj Sidhik PII: DOI: Reference:
S0030-4026(15)00888-8 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.127 IJLEO 56051
To appear in: Received date: Accepted date:
13-8-2014 23-8-2015
Please cite this article as: S. Sidhik, Comparitive Study Of Birge Massart Strategy And Unimodal Thresholding For Image Compression Using Wavelet Transform, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.127 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Comparitive Study Of Birge Massart Strategy And Unimodal Thresholding For Image Compression Using Wavelet Transform
INTRODUCTION
d
I.
us
This paper has been organized as follows. Section II entails the proposed algorithm. Section III presents the Analysis and Design. Section IV covers the Results and Discussion. Section V provides us with Conclusion. II.
PROPOSED ALGORITHM
A. Wavelet Base Image Coding Discrete wavelet transform [3, 4] is considered as a powerful tool for image analysis and it overcomes the disadvantages of Discrete Fourier Transform and Discrete Cosine Transform .We know that the Fourier transform only provides the frequency resolution and not the time resolution while if we are using the wavelet transform we are able to represent a signal in time domain as well as frequency domain simultaneously. This property is used for analysis and compression of images
M
Keywords— wavelet transform; haar wavelet; peak signal to noise ratio; unimodal; compression;
in order to remove certain wavelet coefficients, thus compressing the image.
an
Abstract— This paper is mainly aimed at comparing Birge Massart Thresholding strategy which is the inbuilt thresholding method in the MATLAB to that of the Unimodal thresholding strategy for the compression of an image in the transform domain. It usually discusses the important features of the Wavelet transform in compression of still images, including the extent to which the quality of the image is degraded during compression and decompression. Image quality is measured objectively using Peak Signal to noise Ratio and Weighted Peak Signal to noise ratio. The effects of different wavelet functions, image contents and compression ratio are also assessed.
cr
Department of Optoelectronics, University of Kerala, Thiruvananthapuram, India 695581 [email protected]
ip t
Siraj Sidhik
te
Uncompressed multimedia which includes audio, video and images requires very much high storage capacity and transmission bandwidth. Despite the rapid enhancement in the storage density, processors speed, data rate and transmission bandwidth the demand for data storage capacity and data transmission bandwidth continues to outstrip the present available technologies. One approach to remove this problem is to reduce the volume of multimedia data transmitted over the communication channel which is done by adopting certain compression techniques such as JPEG, JPEG2000 and MPEG. This compression [1] technique aims at obtaining high compression ratio without affecting the quality of an image. But these techniques ignore the energy consumption during the compression and RF transmission. Another important factor which is not considered is the processing power requirement at both the ends that is Transmitter and the Receiver. Thus in this work we have considered all these parameters like the processing power required in the mobile handset which is limited and also the processing time consideration.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Since images constitute the larger part of the transmission data we focus in the work on developing energy efficient, computing efficient and adaptive image compression technique. We are therefore considering the filter bank implementation by using regular tree structure and also based on popular compression algorithm called as Wavelet compression. Wavelet compression [2] uses thresholding method
The Wavelet transform is usually a sub band decomposition process and an image can be decomposed using a high pass and low pass filter in horizontal and vertical directions. In this process first level decomposition produces four bands given by low-low (LL), low-high (LH), high-low (HL) and highhigh (HH) bands. The LL band is obtained by applying low pass filter to the rows and columns of an image; the LH band is obtained by applying a low pass filter to the rows and a high pass filter to the column; the HL band is obtained by passing a low pass filter to column and a high pass filter to rows; the HH band is obtained by applying a high pass filter to rows and the columns. This decomposition process produces Wavelet coefficients which are further utilized for compression purpose. In 2 levels decomposition the LL band from the first level is decomposed and replaced with four new bands, while the other bands are left without any change or decomposition. The new sub band is half the width and half the height of the LL sub band from the previous level. This decomposition results in generation of wavelet coefficients. .
Page 1 of 6
1↓2
LL
Hp_2
1↓2
HL
Lp_3
1↓2
LH
Hp_3
1↓2
HH
2↓1
2↓1
Hp_1
1. Birge Massart Strategy
This strategy works on the following wavelet selection rule which is given below:
+1 -1 0,
if 0 ≤ t ≤ ½ if ½ ≤ t ≤ 1 otherwise
(1)
cr
(a) At level J0+1 (coarser level), everything is kept.
us
Of all the Wavelets used HAAR Wavelet [5] is considered to be the simplest to implement the filter bank algorithm used for separating different frequency components which is given by,
Let J0 denotes the decomposition level, m be the length of the coarsest approximation over 2and α be a real value greater than 1, hence:
(b) For level J from 1 to J 0 the KJ larger coefficients are kept using the given formula.
an
Fig.1. Filter Bank Algorithm
Φ (t) =
ip t
Lp_1
Lp_2
using entropy coding compression. The use of wavelets and thresholding is to process the signal and to remove the wavelet coefficients having value less than the found out threshold values. This explains how the wavelet analysis compresses a signal with a given thresholding method. Hence we can say that more the number of zeros more will be the compression rate. Hence thresholding plays an important role in this wavelet compression approach .Two types of thresholding used in this work are given below:
(2)
The suggested value of α is 1 and is suggested in [6, 7].
M
2. Unimodal Thresholding
te
d
By using HAAR Wavelet, low frequency components are obtained by taking the average of the pixel values in the image provided whereas the high frequency coefficients are obtained by taking half of the difference of the pixel values of the image. Researchers have shown that in human perception, the retina of the eye splits the image into number of frequency channels having equal bandwidth which is similar to that of the multilevel decomposition and it is usually sensitive to only the low frequency components and not to the high frequency components. Because of this reason only wavelet transforms are used in this case for further operations. Other Wavelets that are also considered here for comparison are:
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table.1. Wavelet Families in MATLAB Wavelet Families Daubechies [6,7] Coiflets Symlets Discrete Meyer Biorthogonal Reverse Biorthogonal
Wavelets(MATLAB Notation) db1 or haar,db2,...,db10,…,db45 coif1,……coif5
sym2,..,sym8,…,sym45 Dmey bior1.1,bior1.5,bior2.2,bior2.4,bior2.6 rbio1.1,rbio1.3,rbio1.5,rbio2.2,rbio2.4
B. Thresholding of Wavelet coefficient For most of the signals the wavelet coefficients are having value close to or equal to zero. Thresholding can modify the wavelet coefficient to produce more number of zeros. In Hard thresholding any coefficient below a threshold λ, is set to zero. This should then produce many consecutive zeros which can be stored in much less space, and transmitted more quickly by
Most of the algorithms used for automatic image threshold selection assume that the intensity histogram is multimodal i.e. bimodal in nature. However some of the images are usually unimodal [8, 9] since much larger proportion one type of pixels are present in the image and it usually dominates the histogram. In such cases many of the threshold selection algorithms fails .However few algorithms are has been provided to cope up with these images. Maximum Deviation Algorithm/ Rosin Threshold 1. It assumes that there is one dominant population of pixel value in the image which produces one main peak that is located at the lower end of the histogram relative to the secondary population. 2. The latter class may or may not produce a discernible peak but needs to be very much separated from large peak to avoid being swamped off from it. 3. A straight line is drawn from the peak to the high end of the histogram. 4. More precisely speaking the line starts from the largest bin and finishes at the first empty bin of the histogram following the last filled bin. 5. If the i th entry of the histogram is written as Hi 6. The threshold point is selected as the histogram index I that maximizes the perpendicular distance between the line and the point (i, Hi).
Page 2 of 6
The peak signal-to-noise ratio [10, 11], (PSNR) usually depends on the Mean Square Error (MSE) which is given by:
(3)
Fig. 2 Procedure for determining threshold
The PSNR is defined as:
ANALYSIS AND DESIGN
d
III.
A. Retained Signal Energy
te
For finding out the quality of the Stego image generated we are:
It indicates the amount of energy in the compressed signal to that of the original signal. When compressing using orthogonal wavelets, the retained energy in percentage is given by:
(3)
B. Signal to Noise Ratio (SNR) This value gives the quality of reconstructed signal. Higher the value, better is the stego image. It is given by:
(4) Where and represents the mean square value of the input image and the means square difference between the cover and stego image.
(4)
us
Here, MAXI indicates the maximum of the pixel value of the image and MSE represents the Mean Square Error. Typical values for the PSNR in lossy image and video compression are between 30 and 50 dB, where higher is better. C. Weighted Peak Signal-To-Noise Ratio (WPSNR)
an
The formula for WPSNR is shown below: (5)
The formula to calculate this factor as a simplified function is:
M
1. First of all we are reading an image, if it is a color image it is converted into gray image. 2. After that we are applying discrete wavelet transform to the input image to certain levels of decomposition. A level is chosen such that best the performance for the reconstructed image should be high at that level. 3. Then apply the thresholding strategy (Birge Massart Strategy and Rosin Threshold) to these wavelet coefficients in order to remove certain coefficients above the generated threshold. 4. Keep the retained coefficients and their position for reconstructing the image from them. 5. Then reconstruct the compressed image from the nonzero coefficients and replacing the missing ones by zeros.
cr
C. Algorithm
ip t
Where I indicate the input image and K represents the Stego image, m and n indicates the number of pixels in the image.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(6)
Where δ represents the luminance variance for the 8×8 block of the image and NORM represents the normalization function value. Typical values for the WPSNR in lossy image and video compression should be greater than 40 which indicate high quality. IV.
RESULTS AND DISCUSSION
Here we are using MATLAB 7.0.First of all the test image is taken, if it is a color image it is converted into grey level. Then we are performing various levels of decomposition. Here we are performing up to four levels of decomposition to obtain the coefficient matrix which is very much necessary for the compression purpose. We are performing four level decomposition, to obtain 16 sub band images which includes the Approximation details, Horizontal details, Vertical details and Diagonal details. After decomposition of the image , we are obtaining the coefficient matrix.next we are perfoming hard thresholding with the help of a threshold that is found by Birge Massart Strategy which is a default threshold in the matlab.with the help of hard thresholding we are discarding all the coeffecients value less than the threshold and keeping all those values equal to or greater than the threshold,and with help these thresholded coefficients we are constructing the image.
B. Peak Signal-To-Noise Ratio (PSNR) .
Page 3 of 6
Table.2 Perfomance Parameteers Family
Threshold
Compression
PSNR
WPSNR
Haar
1.0000
44.3542
58.0586
74.4997
Table.3 Performance Parameters
(b)
Fig. 3 (a) Test Image (b) Compressed Image
Threshold
Compression
PSNR
WPSNR
Haar
11.0000
75.2960
47.0908
63.9512
us
The compressed image is obtained with the Birge Massart Strategy and is analysed using various measurement parameters :
Family
cr
(a)
ip t
Next we are finding the threshold with the help of Unimodal Threshoding for the test image and with that we are compressing the test image and the performance measuerements are carried out which is given below
Quality Parameters
Compression
41.0034
43.5654
PSNR
58.9837
WPSNR
67.8124
Compression COIFLITS (coif1)
SYMLET (sym2)
BIORTHOG ONAL (bior 2.2)
Baboon
Brandy rose
Cameraman
Peppers
40.9439
41.2613
43.8339
45.3781
46.3442
55.6743
52.0690
53.0312
68.3709
65.6979
66.0812
61.5546
65.7005
45.3882
47.2850
41.3453
39.7999
50.1792
M
Water fall
60.6655
d
HAAR (haar)
Lena
te
Family
33.7698
an
Table.3 Study of Birge Massart Strategy
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
PSNR
57.3864
58.5417
46.3932
53.1506
53.5686
52.7670
WPSNR
66.0392
65.9147
66.1902
64.6366
62.6265
65.3194
Compression
34.3182
44.6470
47.3557
41.3500
39.7999
50.1792
PSNR
57.4420
58.5141
46.3500
53.1089
53.5686
52.7670
WPSNR
66.4560
65.8689
66.1446
64.6745
62.6265
65.3194
Compression
37.7826
48.0348
51.0793
45.5270
43.1266
55.4649
PSNR
55.9537
57.9311
44.8304
52.0164
52.0891
51.4115
WPSNR
65.6075
65.6460
65.9378
64.4640
62.0815
64.6380
Page 4 of 6
Now we are using unimodal thresholding strategy for finding out the threshold for compressing an image .Here also we are measuring various image quality parameters like PSNR
,WPSNR ,Threshold and Compression percentage for different images using different wavelet filters and is noted down in a table.
Quality Parameters
Lena
Waterfall
Baboon
Brandy rose
186
16
228
133
75
68.2236
75
74.9969
74.6017
74.9725
29.1872
33.6240
29.2347
29.7280
26.1436
60.0479
59.3920
63.5869
60.2550
58.9389
32.9509 5 9.4833
186
16
228
133
97
93
75
68.8639
75
75
74.8047
75
31.2341
33.8664
30.8048
31.8796
26.8162
36.2010
57.3397
53.5943
62.2919
55.7750
53.0068
57.8289
186
16
228
133
97
93
75
75
74.8092
74.9940
Threshold
HAAR (haar)
us
PSNR WPSNR
Compression
WPSNR Threshold
SYMLET (sym2)
Compression
WPSNR
68.7068
te
75 PSNR
M
PSNR
an
Threshold
COIFLITS (coif1)
97
cr
Compression
Cameraman
Peppers
ip t
Family
d
Table. 4 Study of Unimodal Thresholding (Rosin Threshold)
93
31.0076
33.7481
30.8371
31.9200
27.0630
35.9063
56.6573
52.5005
62.4541
56.1533
53.1564
55.6891
186
16
228
133
97
93
75
69.9941
75
75
74.9024
75
31.5671
33.4247
31.2366
32.2816
23.9368
36.8769
58.3057
54.9685
63.2930
57.0910
53.7661
58.3741
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Threshold
BIORTHOGO NAL (bior 2.2)
Compression PSNR
WPSNR
It is evident from the Comparative study performed for the Birge Massart thresholding Strategy and the Unimodal Thresholding strategy, that the image quality parameters like PSNR, WPSNR measured using MATLAB 7.0 is poor for Unimodal thresholding but the compression ratio achievable with the Unimodal thresholding is much higher. V.
CONCLUSION
A MATLAB simulation of the work was implemented successfully for compression of an image using thresholding method involving Wavelet transform. Two of the thresholding strategies were used namely Birge Massart Strategy and
Unimodal Thresholding strategy which includes Rosin Threshold Method otherwise called as Maximum Deviation Algorithm. Various Image quality measurement parameters like PSNR, WPSNR and Compression percentage were evaluated for both the strategies and it is concluded that the Birge Massart Strategy is the found to be the best method among the two methods discussed, in terms of the quality of the compressed image generated but high compression ratio is achieved by Unimodal Thresholding. There are many possible extensions to this paper. These include finding the best Thresholding strategy, finding the best wavelet for a given image, finding new wavelet Families
Page 5 of 6
ip t
[6]
us
[5]
an
[4]
M
[3]
d
[2]
S.G Chang. B.Yu and M.Vetterli.,” Adaptive Wavelet Thresholding for Image Denoising and Compression,”IEEE Transaction on Image Processing, Vol. 9,No. 9, September 2000. Mulcahy, colm.,” Image Compression using the Haar Wavelet Tranform,”Spelman Science and Math Journal. Deand . L. Wavelet Transformation and their Application.Birkhauser Boston 2002 N.Bi,Q. Sun,D. Huang,Z. Yang and J.Huang based on multiband wavelets and empirical mode decomposition”,IEEE Transaction on image processing ,August 2007 P. Jay and J. Havlicek, “ Image Watermarking Using Wavelets,” in Proceedings of the 2002 IEEE ,pp.II. 258-II.261, 2002. Karam, j.,A Global Threshold Wavelet-Based Scheme for Speech Recognition, Third International Coference on Computer Science,
te
[1]
Software Engineering Information Technology, E-Bussiness and Applications, Cairo,Egypt,Dec.27-29 2004. [7] Karam, J., Saad, R., The Effect of Different Compression Schemes on Speech signals, International Journal of Biomedical Sciences, Vol. 1 No. 4, pp: 230 234,2006 [8] A.D. Brink and N.E. Pendock. Minimum cross-entropy thresholdselection. Patt. Recog.,29(1):179-188, 1996. [9] J.A. Gong, L.Y. Li, and W.N. Chen, Fast Recursive algorithms for 2Dimensional threholding. Patt.,Recog., 31(3):295-300, 1998 [10] Welstead,Stephen T.(1991),”Fractal and Wavelet Image Compression Techniques,”SPIE Publication, pp. 155-156. [11] Huynh-Thu. Q,Ghanbari.M. (2008) ,”Scope of Validity of PSNR in image/video quality assessment,” Electronics Letter 44(13):800-801.
cr
REFERENCES
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Page 6 of 6
S0030-4026(15)00888-8 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.127 IJLEO 56051
To appear in: Received date: Accepted date:
13-8-2014 23-8-2015
Please cite this article as: S. Sidhik, Comparitive Study Of Birge Massart Strategy And Unimodal Thresholding For Image Compression Using Wavelet Transform, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.127 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Comparitive Study Of Birge Massart Strategy And Unimodal Thresholding For Image Compression Using Wavelet Transform
INTRODUCTION
d
I.
us
This paper has been organized as follows. Section II entails the proposed algorithm. Section III presents the Analysis and Design. Section IV covers the Results and Discussion. Section V provides us with Conclusion. II.
PROPOSED ALGORITHM
A. Wavelet Base Image Coding Discrete wavelet transform [3, 4] is considered as a powerful tool for image analysis and it overcomes the disadvantages of Discrete Fourier Transform and Discrete Cosine Transform .We know that the Fourier transform only provides the frequency resolution and not the time resolution while if we are using the wavelet transform we are able to represent a signal in time domain as well as frequency domain simultaneously. This property is used for analysis and compression of images
M
Keywords— wavelet transform; haar wavelet; peak signal to noise ratio; unimodal; compression;
in order to remove certain wavelet coefficients, thus compressing the image.
an
Abstract— This paper is mainly aimed at comparing Birge Massart Thresholding strategy which is the inbuilt thresholding method in the MATLAB to that of the Unimodal thresholding strategy for the compression of an image in the transform domain. It usually discusses the important features of the Wavelet transform in compression of still images, including the extent to which the quality of the image is degraded during compression and decompression. Image quality is measured objectively using Peak Signal to noise Ratio and Weighted Peak Signal to noise ratio. The effects of different wavelet functions, image contents and compression ratio are also assessed.
cr
Department of Optoelectronics, University of Kerala, Thiruvananthapuram, India 695581 [email protected]
ip t
Siraj Sidhik
te
Uncompressed multimedia which includes audio, video and images requires very much high storage capacity and transmission bandwidth. Despite the rapid enhancement in the storage density, processors speed, data rate and transmission bandwidth the demand for data storage capacity and data transmission bandwidth continues to outstrip the present available technologies. One approach to remove this problem is to reduce the volume of multimedia data transmitted over the communication channel which is done by adopting certain compression techniques such as JPEG, JPEG2000 and MPEG. This compression [1] technique aims at obtaining high compression ratio without affecting the quality of an image. But these techniques ignore the energy consumption during the compression and RF transmission. Another important factor which is not considered is the processing power requirement at both the ends that is Transmitter and the Receiver. Thus in this work we have considered all these parameters like the processing power required in the mobile handset which is limited and also the processing time consideration.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Since images constitute the larger part of the transmission data we focus in the work on developing energy efficient, computing efficient and adaptive image compression technique. We are therefore considering the filter bank implementation by using regular tree structure and also based on popular compression algorithm called as Wavelet compression. Wavelet compression [2] uses thresholding method
The Wavelet transform is usually a sub band decomposition process and an image can be decomposed using a high pass and low pass filter in horizontal and vertical directions. In this process first level decomposition produces four bands given by low-low (LL), low-high (LH), high-low (HL) and highhigh (HH) bands. The LL band is obtained by applying low pass filter to the rows and columns of an image; the LH band is obtained by applying a low pass filter to the rows and a high pass filter to the column; the HL band is obtained by passing a low pass filter to column and a high pass filter to rows; the HH band is obtained by applying a high pass filter to rows and the columns. This decomposition process produces Wavelet coefficients which are further utilized for compression purpose. In 2 levels decomposition the LL band from the first level is decomposed and replaced with four new bands, while the other bands are left without any change or decomposition. The new sub band is half the width and half the height of the LL sub band from the previous level. This decomposition results in generation of wavelet coefficients. .
Page 1 of 6
1↓2
LL
Hp_2
1↓2
HL
Lp_3
1↓2
LH
Hp_3
1↓2
HH
2↓1
2↓1
Hp_1
1. Birge Massart Strategy
This strategy works on the following wavelet selection rule which is given below:
+1 -1 0,
if 0 ≤ t ≤ ½ if ½ ≤ t ≤ 1 otherwise
(1)
cr
(a) At level J0+1 (coarser level), everything is kept.
us
Of all the Wavelets used HAAR Wavelet [5] is considered to be the simplest to implement the filter bank algorithm used for separating different frequency components which is given by,
Let J0 denotes the decomposition level, m be the length of the coarsest approximation over 2and α be a real value greater than 1, hence:
(b) For level J from 1 to J 0 the KJ larger coefficients are kept using the given formula.
an
Fig.1. Filter Bank Algorithm
Φ (t) =
ip t
Lp_1
Lp_2
using entropy coding compression. The use of wavelets and thresholding is to process the signal and to remove the wavelet coefficients having value less than the found out threshold values. This explains how the wavelet analysis compresses a signal with a given thresholding method. Hence we can say that more the number of zeros more will be the compression rate. Hence thresholding plays an important role in this wavelet compression approach .Two types of thresholding used in this work are given below:
(2)
The suggested value of α is 1 and is suggested in [6, 7].
M
2. Unimodal Thresholding
te
d
By using HAAR Wavelet, low frequency components are obtained by taking the average of the pixel values in the image provided whereas the high frequency coefficients are obtained by taking half of the difference of the pixel values of the image. Researchers have shown that in human perception, the retina of the eye splits the image into number of frequency channels having equal bandwidth which is similar to that of the multilevel decomposition and it is usually sensitive to only the low frequency components and not to the high frequency components. Because of this reason only wavelet transforms are used in this case for further operations. Other Wavelets that are also considered here for comparison are:
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table.1. Wavelet Families in MATLAB Wavelet Families Daubechies [6,7] Coiflets Symlets Discrete Meyer Biorthogonal Reverse Biorthogonal
Wavelets(MATLAB Notation) db1 or haar,db2,...,db10,…,db45 coif1,……coif5
sym2,..,sym8,…,sym45 Dmey bior1.1,bior1.5,bior2.2,bior2.4,bior2.6 rbio1.1,rbio1.3,rbio1.5,rbio2.2,rbio2.4
B. Thresholding of Wavelet coefficient For most of the signals the wavelet coefficients are having value close to or equal to zero. Thresholding can modify the wavelet coefficient to produce more number of zeros. In Hard thresholding any coefficient below a threshold λ, is set to zero. This should then produce many consecutive zeros which can be stored in much less space, and transmitted more quickly by
Most of the algorithms used for automatic image threshold selection assume that the intensity histogram is multimodal i.e. bimodal in nature. However some of the images are usually unimodal [8, 9] since much larger proportion one type of pixels are present in the image and it usually dominates the histogram. In such cases many of the threshold selection algorithms fails .However few algorithms are has been provided to cope up with these images. Maximum Deviation Algorithm/ Rosin Threshold 1. It assumes that there is one dominant population of pixel value in the image which produces one main peak that is located at the lower end of the histogram relative to the secondary population. 2. The latter class may or may not produce a discernible peak but needs to be very much separated from large peak to avoid being swamped off from it. 3. A straight line is drawn from the peak to the high end of the histogram. 4. More precisely speaking the line starts from the largest bin and finishes at the first empty bin of the histogram following the last filled bin. 5. If the i th entry of the histogram is written as Hi 6. The threshold point is selected as the histogram index I that maximizes the perpendicular distance between the line and the point (i, Hi).
Page 2 of 6
The peak signal-to-noise ratio [10, 11], (PSNR) usually depends on the Mean Square Error (MSE) which is given by:
(3)
Fig. 2 Procedure for determining threshold
The PSNR is defined as:
ANALYSIS AND DESIGN
d
III.
A. Retained Signal Energy
te
For finding out the quality of the Stego image generated we are:
It indicates the amount of energy in the compressed signal to that of the original signal. When compressing using orthogonal wavelets, the retained energy in percentage is given by:
(3)
B. Signal to Noise Ratio (SNR) This value gives the quality of reconstructed signal. Higher the value, better is the stego image. It is given by:
(4) Where and represents the mean square value of the input image and the means square difference between the cover and stego image.
(4)
us
Here, MAXI indicates the maximum of the pixel value of the image and MSE represents the Mean Square Error. Typical values for the PSNR in lossy image and video compression are between 30 and 50 dB, where higher is better. C. Weighted Peak Signal-To-Noise Ratio (WPSNR)
an
The formula for WPSNR is shown below: (5)
The formula to calculate this factor as a simplified function is:
M
1. First of all we are reading an image, if it is a color image it is converted into gray image. 2. After that we are applying discrete wavelet transform to the input image to certain levels of decomposition. A level is chosen such that best the performance for the reconstructed image should be high at that level. 3. Then apply the thresholding strategy (Birge Massart Strategy and Rosin Threshold) to these wavelet coefficients in order to remove certain coefficients above the generated threshold. 4. Keep the retained coefficients and their position for reconstructing the image from them. 5. Then reconstruct the compressed image from the nonzero coefficients and replacing the missing ones by zeros.
cr
C. Algorithm
ip t
Where I indicate the input image and K represents the Stego image, m and n indicates the number of pixels in the image.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(6)
Where δ represents the luminance variance for the 8×8 block of the image and NORM represents the normalization function value. Typical values for the WPSNR in lossy image and video compression should be greater than 40 which indicate high quality. IV.
RESULTS AND DISCUSSION
Here we are using MATLAB 7.0.First of all the test image is taken, if it is a color image it is converted into grey level. Then we are performing various levels of decomposition. Here we are performing up to four levels of decomposition to obtain the coefficient matrix which is very much necessary for the compression purpose. We are performing four level decomposition, to obtain 16 sub band images which includes the Approximation details, Horizontal details, Vertical details and Diagonal details. After decomposition of the image , we are obtaining the coefficient matrix.next we are perfoming hard thresholding with the help of a threshold that is found by Birge Massart Strategy which is a default threshold in the matlab.with the help of hard thresholding we are discarding all the coeffecients value less than the threshold and keeping all those values equal to or greater than the threshold,and with help these thresholded coefficients we are constructing the image.
B. Peak Signal-To-Noise Ratio (PSNR) .
Page 3 of 6
Table.2 Perfomance Parameteers Family
Threshold
Compression
PSNR
WPSNR
Haar
1.0000
44.3542
58.0586
74.4997
Table.3 Performance Parameters
(b)
Fig. 3 (a) Test Image (b) Compressed Image
Threshold
Compression
PSNR
WPSNR
Haar
11.0000
75.2960
47.0908
63.9512
us
The compressed image is obtained with the Birge Massart Strategy and is analysed using various measurement parameters :
Family
cr
(a)
ip t
Next we are finding the threshold with the help of Unimodal Threshoding for the test image and with that we are compressing the test image and the performance measuerements are carried out which is given below
Quality Parameters
Compression
41.0034
43.5654
PSNR
58.9837
WPSNR
67.8124
Compression COIFLITS (coif1)
SYMLET (sym2)
BIORTHOG ONAL (bior 2.2)
Baboon
Brandy rose
Cameraman
Peppers
40.9439
41.2613
43.8339
45.3781
46.3442
55.6743
52.0690
53.0312
68.3709
65.6979
66.0812
61.5546
65.7005
45.3882
47.2850
41.3453
39.7999
50.1792
M
Water fall
60.6655
d
HAAR (haar)
Lena
te
Family
33.7698
an
Table.3 Study of Birge Massart Strategy
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
PSNR
57.3864
58.5417
46.3932
53.1506
53.5686
52.7670
WPSNR
66.0392
65.9147
66.1902
64.6366
62.6265
65.3194
Compression
34.3182
44.6470
47.3557
41.3500
39.7999
50.1792
PSNR
57.4420
58.5141
46.3500
53.1089
53.5686
52.7670
WPSNR
66.4560
65.8689
66.1446
64.6745
62.6265
65.3194
Compression
37.7826
48.0348
51.0793
45.5270
43.1266
55.4649
PSNR
55.9537
57.9311
44.8304
52.0164
52.0891
51.4115
WPSNR
65.6075
65.6460
65.9378
64.4640
62.0815
64.6380
Page 4 of 6
Now we are using unimodal thresholding strategy for finding out the threshold for compressing an image .Here also we are measuring various image quality parameters like PSNR
,WPSNR ,Threshold and Compression percentage for different images using different wavelet filters and is noted down in a table.
Quality Parameters
Lena
Waterfall
Baboon
Brandy rose
186
16
228
133
75
68.2236
75
74.9969
74.6017
74.9725
29.1872
33.6240
29.2347
29.7280
26.1436
60.0479
59.3920
63.5869
60.2550
58.9389
32.9509 5 9.4833
186
16
228
133
97
93
75
68.8639
75
75
74.8047
75
31.2341
33.8664
30.8048
31.8796
26.8162
36.2010
57.3397
53.5943
62.2919
55.7750
53.0068
57.8289
186
16
228
133
97
93
75
75
74.8092
74.9940
Threshold
HAAR (haar)
us
PSNR WPSNR
Compression
WPSNR Threshold
SYMLET (sym2)
Compression
WPSNR
68.7068
te
75 PSNR
M
PSNR
an
Threshold
COIFLITS (coif1)
97
cr
Compression
Cameraman
Peppers
ip t
Family
d
Table. 4 Study of Unimodal Thresholding (Rosin Threshold)
93
31.0076
33.7481
30.8371
31.9200
27.0630
35.9063
56.6573
52.5005
62.4541
56.1533
53.1564
55.6891
186
16
228
133
97
93
75
69.9941
75
75
74.9024
75
31.5671
33.4247
31.2366
32.2816
23.9368
36.8769
58.3057
54.9685
63.2930
57.0910
53.7661
58.3741
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Threshold
BIORTHOGO NAL (bior 2.2)
Compression PSNR
WPSNR
It is evident from the Comparative study performed for the Birge Massart thresholding Strategy and the Unimodal Thresholding strategy, that the image quality parameters like PSNR, WPSNR measured using MATLAB 7.0 is poor for Unimodal thresholding but the compression ratio achievable with the Unimodal thresholding is much higher. V.
CONCLUSION
A MATLAB simulation of the work was implemented successfully for compression of an image using thresholding method involving Wavelet transform. Two of the thresholding strategies were used namely Birge Massart Strategy and
Unimodal Thresholding strategy which includes Rosin Threshold Method otherwise called as Maximum Deviation Algorithm. Various Image quality measurement parameters like PSNR, WPSNR and Compression percentage were evaluated for both the strategies and it is concluded that the Birge Massart Strategy is the found to be the best method among the two methods discussed, in terms of the quality of the compressed image generated but high compression ratio is achieved by Unimodal Thresholding. There are many possible extensions to this paper. These include finding the best Thresholding strategy, finding the best wavelet for a given image, finding new wavelet Families
Page 5 of 6
ip t
[6]
us
[5]
an
[4]
M
[3]
d
[2]
S.G Chang. B.Yu and M.Vetterli.,” Adaptive Wavelet Thresholding for Image Denoising and Compression,”IEEE Transaction on Image Processing, Vol. 9,No. 9, September 2000. Mulcahy, colm.,” Image Compression using the Haar Wavelet Tranform,”Spelman Science and Math Journal. Deand . L. Wavelet Transformation and their Application.Birkhauser Boston 2002 N.Bi,Q. Sun,D. Huang,Z. Yang and J.Huang based on multiband wavelets and empirical mode decomposition”,IEEE Transaction on image processing ,August 2007 P. Jay and J. Havlicek, “ Image Watermarking Using Wavelets,” in Proceedings of the 2002 IEEE ,pp.II. 258-II.261, 2002. Karam, j.,A Global Threshold Wavelet-Based Scheme for Speech Recognition, Third International Coference on Computer Science,
te
[1]
Software Engineering Information Technology, E-Bussiness and Applications, Cairo,Egypt,Dec.27-29 2004. [7] Karam, J., Saad, R., The Effect of Different Compression Schemes on Speech signals, International Journal of Biomedical Sciences, Vol. 1 No. 4, pp: 230 234,2006 [8] A.D. Brink and N.E. Pendock. Minimum cross-entropy thresholdselection. Patt. Recog.,29(1):179-188, 1996. [9] J.A. Gong, L.Y. Li, and W.N. Chen, Fast Recursive algorithms for 2Dimensional threholding. Patt.,Recog., 31(3):295-300, 1998 [10] Welstead,Stephen T.(1991),”Fractal and Wavelet Image Compression Techniques,”SPIE Publication, pp. 155-156. [11] Huynh-Thu. Q,Ghanbari.M. (2008) ,”Scope of Validity of PSNR in image/video quality assessment,” Electronics Letter 44(13):800-801.
cr
REFERENCES
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Page 6 of 6