![ACEEE Int. J. on Signal & Image Processing, Vol. 02, No. 01, Jan 2011
Image Compression
Using Wavelet Packet Tree
Jagannath Sethi1, Sibaram Mishra2 ,Prajna Parimita Dash3, Sudhansu Kumar Mishra4,Sukadev Meher5
1,3
Department of Applied Electronics & Instrumentation ,ITER Bhubaneswar, India
4,5
Department of Electronics & Communication, NIT Rourkela, India
5
Panchayat College Dharamgarh, Orissa,India
1
jagannathsethi@gmail.com, sibaram.mishra@gmail.com,
3
pdash6@gmail.com,4sudhansu.nit@gmail.com, 5sukadevmeher@gmail.com
Abstract—Methods of compressing data prior to storage and 4. Finally conclusion and further research directions are
transmission are of significant practical and commercial discussed in the section 5.
interest. The necessity in image compression continuously II. FUNDAMENTAL OF DIGITAL IMAGE COMPRESSION
grows during the last decade. The image compression includes
transform of image, quantization and encoding. One of the A common characteristic of most of images is that the
most powerful and perspective approaches in this area is neighboring pixels are correlated. Therefore most
image compression using discrete wavelet transform. This important task is to find a less correlated representation of
paper describes a new approach called as wavelet packet tree image called as compression. The fundamental components
for image compression. It constructs the best tree on the basis of compression are reduction of redundancy and
of Shannon entropy. This new approach checks the entropy of
decomposed nodes (child nodes) with entropy of node, which
irrelevancy. Redundancy reduction aims at removing
has been decomposed (parent node) and takes the decision of duplication from the image. Redundancies can be spatial
decomposition of a node. In addition, authors have proposed redundancy, spectral redundancy and temporal redundancy.
an adaptive thresholding for quantization, which is based on In still image, the compression is achieved by removing
type of wavelet used and nature of image. Performance of the spatial redundancy and Spectral redundancy. Irrelevancy
proposed algorithm is compared with existing wavelet reduction omits parts of the signal that could not be noticed
transform algorithm in terms of percentage of zeros and by human visual system (HVS).
percentage of energy retained and signals to noise ratio.
III.WAVELET PACKETS TREE
Index Term-Discrete wavelet transform, wavelet packet
tree, percentage zero, percentage of energy retained, entropy A. Wavelet Packets
The wavelet packet method is a generalization of
I.INTRODUCTION wavelet decomposition that offers a richer signal analysis.
Visual communication is becoming increasingly Wavelet packet atoms are waveforms indexed by three
important with applications in several areas such as naturally interpreted parameters i.e. position, scale (as in
multimedia, communication, transmission, storage of wavelet decomposition), and frequency. For a given
remote sensing images, education, business documents and orthogonal wavelet function, we generate a library of bases
medical images etc. Since digital images are inherently called wavelet packet bases. Each of these bases offers a
voluminous, efficient data compression techniques are particular way of coding signals, preserving global energy,
essential for their archival and transmission. Discrete and reconstructing exact features. The wavelet packets can
Wavelet Transform (DWT) has emerged as a popular be used for numerous expansions of a given signal.
technique for image coding applications [1]. DWT has high B. Wavelets to wavelet packets decomposing.
decorrelation and energy compaction efficiency. The
blocking artifacts and mosquito noise are absent in a The orthogonal wavelet decomposition procedure
wavelet-based coder due to the overlapping basis functions splits the approximation coefficients into two parts.
[1]. The JPEG 2000 standard employs a discrete wavelet After splitting we obtain a vector of approximation
transform for image compression due to its merits in terms coefficients and a vector of detail coefficients both at a
of scalability, localization and energy concentration [2]. coarser scale. The information lost between two
JEPG 2000 suffers from blurring artifacts and ringing successive approximations is captured in the detail
artifacts [3]. coefficients. Then the new approximation coefficient
This paper is organized as follows. Section 2 outlines vector is split again. In the wavelet packet approach,
brief review of image compression. Wavelet Packet Tree is each detail coefficient vector also decomposed into two
presented and described in section 3. Simulation studies parts as in approximation vector splitting.
based on several numerical experiments are dealt in section
41
© 2011 ACEEE
DOI: 01.IJSIP.02.01.533](https://image.slidesharecdn.com/533-120926234746-phpapp01/85/Image-Compression-Using-Wavelet-Packet-Tree-1-320.jpg)
![ACEEE Int. J. on Signal & Image Processing, Vol. 02, No. 01, Jan 2011
TABLE III:PSNR After WT And WPT studied and generalization of this algorithm is to be done
Name of the PSNR in db PSNR in db for developing more effective algorithm for image
Image after WT after WPT compression.
Woman 26.5887 27.0241
Saturn 35.8079 35.8811 REFERENCES
Lena 30.9790 31.0530 [1]Howard L. Resnikoff, Raymond O. Wells, Jr., “Wavelet
Cameraman 31.6366 31.6503 Analysis” The scalable Structure of Information, Springer, ISBN-
0-387-98383-X, 1998.
The CPU time is the time taken for the completion of [2]Lokenath Debnath, “Wavelet Transforms and Time- Frequency
one algorithm. The program is run Signal Analysis”, Birkhauser, ISBN-0- 8176-4104-1, 2001.
on a computer with the specification of AMD 1.8 GHz [3]Anil K. Jain, “ Fundamental of Digital Image Processing”,
processor and 1024 MB of RAM. Prentice –Hall, 2000, ISBN-81-203- 0929-4.
[4]Subhasis Saha, “Image Compression – from DCT to Wavelets:
Table.IV. Comparison Of CPU Time Using Wt And Wpt
A review”, ACM Cross words students magazine, Vol.6, No.3,
CPU Time WT WPT Spring 2000.
[5]Sarah Betz, Nirav Bhagat, Paul Murhy & Maureen Stengler,
In second 554.6 311.54 “Wavelet Based Image Compression –Analysis of Results”,
From the table it is shown that the WPT requires about ELEC 301 Final Project.
311 second for training. But for implementing WT it needs [6]Rafael C. Gonzalez and Richard E. Woods, “Digital Image
about 554 second. Processing”, 2nd Edition,Pearson Education, ISBN-81-7808-629-
8, 2002.
V.CONCLUSION [7]B. Chanda and D. Dutrta Majumder, “Digital Image
Processing and Analysis” , Prentice-Hall of India, ISBN-81-203-
In this paper the results of discrete wavelet transform 1618-5, 2002.
and wavelet packet tree are compared. The results show [8]Agostino Abbte, Casimer M. DeCusatis, Pankaj K. Das,
that WPT is better than WT both subjectively as well as “Wavelets and Subbands” Fundamentals and Applications,
objectively. Future work includes introduction of different Birkhauser, 2002, ISBN-0-8176-4136- X, 2002.
operators in the proposed algorithm which allow better [9]L.Prasad, and S.S. Iyengar, “Wavelet Analysis with
applications to Image Processing”, CRC press, ISBN- 0-8493-
exploration and exploitation of the search space when
3169-2,1997.
applied to compression. Its adaptive capacity has to be
43
© 2011 ACEEE
DOI: 01.IJSIP.02.01.533](https://image.slidesharecdn.com/533-120926234746-phpapp01/85/Image-Compression-Using-Wavelet-Packet-Tree-3-320.jpg)
Image Compression Using Wavelet Packet Tree
- 1. ACEEE Int. J. on Signal & Image Processing, Vol. 02, No. 01, Jan 2011 Image Compression Using Wavelet Packet Tree Jagannath Sethi1, Sibaram Mishra2 ,Prajna Parimita Dash3, Sudhansu Kumar Mishra4,Sukadev Meher5 1,3 Department of Applied Electronics & Instrumentation ,ITER Bhubaneswar, India 4,5 Department of Electronics & Communication, NIT Rourkela, India 5 Panchayat College Dharamgarh, Orissa,India 1 [email protected], [email protected], 3 [email protected],[email protected], [email protected] Abstract—Methods of compressing data prior to storage and 4. Finally conclusion and further research directions are transmission are of significant practical and commercial discussed in the section 5. interest. The necessity in image compression continuously II. FUNDAMENTAL OF DIGITAL IMAGE COMPRESSION grows during the last decade. The image compression includes transform of image, quantization and encoding. One of the A common characteristic of most of images is that the most powerful and perspective approaches in this area is neighboring pixels are correlated. Therefore most image compression using discrete wavelet transform. This important task is to find a less correlated representation of paper describes a new approach called as wavelet packet tree image called as compression. The fundamental components for image compression. It constructs the best tree on the basis of compression are reduction of redundancy and of Shannon entropy. This new approach checks the entropy of decomposed nodes (child nodes) with entropy of node, which irrelevancy. Redundancy reduction aims at removing has been decomposed (parent node) and takes the decision of duplication from the image. Redundancies can be spatial decomposition of a node. In addition, authors have proposed redundancy, spectral redundancy and temporal redundancy. an adaptive thresholding for quantization, which is based on In still image, the compression is achieved by removing type of wavelet used and nature of image. Performance of the spatial redundancy and Spectral redundancy. Irrelevancy proposed algorithm is compared with existing wavelet reduction omits parts of the signal that could not be noticed transform algorithm in terms of percentage of zeros and by human visual system (HVS). percentage of energy retained and signals to noise ratio. III.WAVELET PACKETS TREE Index Term-Discrete wavelet transform, wavelet packet tree, percentage zero, percentage of energy retained, entropy A. Wavelet Packets The wavelet packet method is a generalization of I.INTRODUCTION wavelet decomposition that offers a richer signal analysis. Visual communication is becoming increasingly Wavelet packet atoms are waveforms indexed by three important with applications in several areas such as naturally interpreted parameters i.e. position, scale (as in multimedia, communication, transmission, storage of wavelet decomposition), and frequency. For a given remote sensing images, education, business documents and orthogonal wavelet function, we generate a library of bases medical images etc. Since digital images are inherently called wavelet packet bases. Each of these bases offers a voluminous, efficient data compression techniques are particular way of coding signals, preserving global energy, essential for their archival and transmission. Discrete and reconstructing exact features. The wavelet packets can Wavelet Transform (DWT) has emerged as a popular be used for numerous expansions of a given signal. technique for image coding applications [1]. DWT has high B. Wavelets to wavelet packets decomposing. decorrelation and energy compaction efficiency. The blocking artifacts and mosquito noise are absent in a The orthogonal wavelet decomposition procedure wavelet-based coder due to the overlapping basis functions splits the approximation coefficients into two parts. [1]. The JPEG 2000 standard employs a discrete wavelet After splitting we obtain a vector of approximation transform for image compression due to its merits in terms coefficients and a vector of detail coefficients both at a of scalability, localization and energy concentration [2]. coarser scale. The information lost between two JEPG 2000 suffers from blurring artifacts and ringing successive approximations is captured in the detail artifacts [3]. coefficients. Then the new approximation coefficient This paper is organized as follows. Section 2 outlines vector is split again. In the wavelet packet approach, brief review of image compression. Wavelet Packet Tree is each detail coefficient vector also decomposed into two presented and described in section 3. Simulation studies parts as in approximation vector splitting. based on several numerical experiments are dealt in section 41 © 2011 ACEEE DOI: 01.IJSIP.02.01.533
- 2. ACEEE Int. J. on Signal & Image Processing, Vol. 02, No. 01, Jan 2011 C. Building Wavelet Packets IV. RESULTS The computation scheme for wavelet packets Wavelet packet tree using Shannon entropy has been generation is easy when using an orthogonal wavelet. implemented in the paper. The proposed algorithm is We start with the two filters of length 2N. tested on standard testing image of size 256×256. For the implementation of the algorithm, different intensity Wn(x), n=0,1,2,3) combination images are taken. Four different images like woman, lean, cameraman and saturn are used which are W ( ) = √2 h(k) Wn (2 − k) rich in different patterns. Results are observed in terms of percentage of zeros, percentage of energy retained, peak signal to noise ratio and CPU time. Figure 3 shows the W ( ) = √2 g(k) Wn (2x − k) decompressed images after applying wavelet compression and wavelet packet tree. MATLAB simulation tool has (1) been used for the simulation study. The performances of h(n) and g(n), corresponding to the wavelet. W0(x) = Ф wavelet transform and wavelet packet tree with these four (x) is the scaling function and W1(x) =.Ψ(x) is the wavelet images are compared both subjectively as well as function. An idea of wavelet packet is the same as wavelet. objectively. Only difference is that wavelet packet offers a more complex and flexible analysis. In wavelet packet analysis the details as well as the approximation are split. The wavelet packet tree for 3-level decomposition is shown in Figure 2. Figure 2. Wavelet Packet Tree Decomposition In this paper Shannon entropy criteria is used to construct the best tree. Shannon entropy criteria find the information content of signal ‘S’. Figure 3. Decompressed Images using WT and WPT Entropy(S) = ∑ log (2) In this work computer simulations are carried out to The value of threshold is calculated based on nature of compare the proposed wavelet packet tree algorithm with image and type of wavelet used for decomposition. wavelet transform algorithm using percentage of zeros, Threshold =K–Sqrt(mean (w_energy(T)×100)) Percentage of Energy retained and PSNR value. From results D. The Proposed Algorithm shown in the table 1 it is clear that percentage of zero The algorithm is described as follows: obtained using WPT is more than WT. It means 1. Level counter = 1 performance of wavelet packet tree is better as compared to wavelet in terms of objective evaluation. 2. The current node = input image. 3. Decompose current node using wavelet packet tree. TABLE I: Percentage of zero after WT and WPT 4. Find the entropy of the current node. Name of the Percentage of Percentage of 5. Find the entropy of decomposed components, CA1, Image zero after WT zero after WPT CH1, CV1, CD1. Woman 80.5603 81.8253 6. Compare the entropy of parent node with the sum of Saturn 96.1829 96.2110 the entropy of child node. If the Lena 87.8540 87.8845 sum of the entropy of child nodes is less than that of Cameraman 87.8555 87.8387 parent node, then child node will be considered as a leaf node of a tree and repeat the steps 3, 4, 5, 6 for TABLE II: Percentage of Energy retained after WT and WPT each child nodes considering it as current node. Name of the Image Percentage of Energy Percentage of Energy Otherwise parent acts as a leaf node of a tree. retained after WT retained after WPT 7. Stop Woman 99.1948 99.2248 Saturn 99.7832 99.7864 Lena 99.5874 99.5944 Cameraman 99.7519 99.7527 42 © 2011 ACEEE DOI: 01.IJSIP.02.01.533
- 3. ACEEE Int. J. on Signal & Image Processing, Vol. 02, No. 01, Jan 2011 TABLE III:PSNR After WT And WPT studied and generalization of this algorithm is to be done Name of the PSNR in db PSNR in db for developing more effective algorithm for image Image after WT after WPT compression. Woman 26.5887 27.0241 Saturn 35.8079 35.8811 REFERENCES Lena 30.9790 31.0530 [1]Howard L. Resnikoff, Raymond O. Wells, Jr., “Wavelet Cameraman 31.6366 31.6503 Analysis” The scalable Structure of Information, Springer, ISBN- 0-387-98383-X, 1998. The CPU time is the time taken for the completion of [2]Lokenath Debnath, “Wavelet Transforms and Time- Frequency one algorithm. The program is run Signal Analysis”, Birkhauser, ISBN-0- 8176-4104-1, 2001. on a computer with the specification of AMD 1.8 GHz [3]Anil K. Jain, “ Fundamental of Digital Image Processing”, processor and 1024 MB of RAM. Prentice –Hall, 2000, ISBN-81-203- 0929-4. [4]Subhasis Saha, “Image Compression – from DCT to Wavelets: Table.IV. Comparison Of CPU Time Using Wt And Wpt A review”, ACM Cross words students magazine, Vol.6, No.3, CPU Time WT WPT Spring 2000. [5]Sarah Betz, Nirav Bhagat, Paul Murhy & Maureen Stengler, In second 554.6 311.54 “Wavelet Based Image Compression –Analysis of Results”, From the table it is shown that the WPT requires about ELEC 301 Final Project. 311 second for training. But for implementing WT it needs [6]Rafael C. Gonzalez and Richard E. Woods, “Digital Image about 554 second. Processing”, 2nd Edition,Pearson Education, ISBN-81-7808-629- 8, 2002. V.CONCLUSION [7]B. Chanda and D. Dutrta Majumder, “Digital Image Processing and Analysis” , Prentice-Hall of India, ISBN-81-203- In this paper the results of discrete wavelet transform 1618-5, 2002. and wavelet packet tree are compared. The results show [8]Agostino Abbte, Casimer M. DeCusatis, Pankaj K. Das, that WPT is better than WT both subjectively as well as “Wavelets and Subbands” Fundamentals and Applications, objectively. Future work includes introduction of different Birkhauser, 2002, ISBN-0-8176-4136- X, 2002. operators in the proposed algorithm which allow better [9]L.Prasad, and S.S. Iyengar, “Wavelet Analysis with applications to Image Processing”, CRC press, ISBN- 0-8493- exploration and exploitation of the search space when 3169-2,1997. applied to compression. Its adaptive capacity has to be 43 © 2011 ACEEE DOI: 01.IJSIP.02.01.533