Available Online atwww.ijcsmc.com
International Journal of Computer Science and Mobile Computing
A Monthly Journal of Computer Science and Information Technology
ISSN 2320–088X
IMPACT FACTOR: 5.258
IJCSMC, Vol. 5, Issue. 4, April 2016, pg.518 – 524
Speeding- Up Fractal Image
Compression Using Entropy Technique
Haider A. Muhsin
1, Dr. Hazeem B. Taher
2¹
Computer Science & Thi-Qar University/ Faculty of Education for Pure Science, Iraq²
Computer Science & Thi-Qar University/ Faculty of Education for Pure Science, Iraq1
allhaider@yahoo.com; 2 haze_comp792004@yahoo.com
Abstract— Fractal image encoding is attractive due to its potential high compression ratio, fast decompression and multi-resolution properties [1] . Fractal image compression explores the self-similarity property of a natural image and utilizes the partitioned iterated function system (PIFS) to encode it. It is time-consuming in the encoding process and such drawback renders it impractical for real time applications [2]. In this paper, a new method is proposed to reduce the encoding time based on using entropy on domain blocks The entropy have been used to speed up the Iterated Function System (IFS) matching stage. During the match, every range block is checking all domain blocks and choose the least error and this process needs more time, and this is the main disadvantage in Fractal Image Compression. The proposed method, compute the entropy for each domain block then compares each entropy of domain block with threshold (selected threshold through experience), if the entropy of the domain block less than threshold, matching process conducted, in this way, each range block no match with all domain blocks. The results of the tests conducted on Lenna (512x512) pixel, resolution 8 bits/pixel, where the image is partitioning into blocks of size (4x4, 8 x8and 16x16) pixel, Where was the time before adding the proposed method(4x4 pixel encoding time 1680.586 sec with appropriate PSNR (24.9963dB) and encoding time after adding the proposed method (91.473 sec with appropriate PSNR (27.0815dB) ) ,8x8 pixel encoding time 1159.539 sec with appropriate PSNR (23.2658dB) and encoding time after adding the proposed method(40.564 sec ) with appropriate PSNR (23.831 dB), 16x16 pixel 935.682 sec with appropriate PSNR (20.2326dB) and encoding time after adding the proposed method(35.205 sec ) with appropriate PSNR (20.1544).
Keywords— Compression, Image Compression, Fractal Image Compression, entropy, Speed up Fractal
1. INTRODUCTION
method is that the image blocks are compute entropy for domain block and check it with threshold before matching, so as to speed up the encoding process[6].
2. Traditional PIFS Encoder
The basic idea of PIFS is partitioning the image into non overlapping range blocks by fixed, quadtree and HV(Horizontal- Vertical) partitioning [7] .Domain pool generation module is responsible for generating domain array (Hd x Wd), which is quarter the size of the original image array (H xW),(i.e.,Hd=
H/2 and Wd= W/2).the data of domain array is taken from the range array ; there are many ways to
choose the data from the range array to fill the domain array.For each four neighboring pixels in range array ,only one value of these pixels is stored in domain array. In this work the average of every four neighboring pixels in the range is used to fill the corresponding position in the domain [8].
After generation the range and domain pools, the matching process implies for each range block, with all domain blocks (D) are listed in domain pool , the best matching between domain and range blocks which are satisfied the minimum the distortion error R [9 ].
n
1
i
n
1
i
)
n
1
i
b
i
2
o(no
)
i
a
o
2
n
1
i
a
i
b
i
2
2
i
a
n
1
i
s(s
2
i
b
n
1
R
…. (1)The IFS coefficients of the equation (1) ,(i.e. scale "S' and the offset "O" ) ,is determined by using :
(3)
Where:
ai : is the pixels values of the domain block.
bi : is the pixels values of the range block.
n : is the number of pixels in each block ( i.e. the block size ).
If
)
2
0
n
1
i
a
i
(
n
1
i
2
i
a
n
then S = 0 , and
n
1
i
b
i
n
1
o
There is a simple formula for R, butit is best to use this one as we’ll see later. The rms error is equal to
R
[ 10].The quantization of IFS coefficients is done by assigning number of bits for both , scale and offset ,to store their quantization indices . The quantized scale and offset values have been computed by using the following equations[11].
Qs= {
………….(4) ) 2 ...( ... ... ... 2 2 1 1 1 .
n i i ai ia n
n
i bi n
i ai n
i aibi n
is= round ( ) ……… ………..……….………..(5)
Sq= is Qs ………..…………(6)
Where S max is the highest permissible value of scale coefficients.
S min is the lowest permissible value of scale coefficients.
bs is the number of allocated bits to represent the quantization index of the scale coefficients.
Qs is the quantization step of the scale coefficients.
Is is the quantization index of the scale coefficients.
Sq is the quantized value of the scale coefficients.
Qo= {
……….(7)
Io= round ( ) ……….…………..(8)
Oq= io Qo ………..………..………..(9)
Where O max is the highest permissible value of offset coefficients.
Omin is the lowest permissible value of offset coefficients.
bo is the number of allocated bits to represent the quantization index of the offset coefficients.
Qo is the quantization step of the offset coefficients.
Io is the quantization index of the offset coefficients.
Oq is the quantization value of the offset coefficients.
The quantized values of scale (S) and offset (O) parameters should be used calculate the sum of square error R by using equation (1).
After computation of the IFS parameters and the sum of error R for any matching instance between the range and each domain block listed in the domain pool ,then the value of R is compared with the minimum registered error (Rmin).
If R < (Rmin) then
Sopt = is ; Oopt = io ; Rmin = R
End if [7]
At decoding stage , the approximations could be performed several times ,starting with any random image ,till reaching the fixed point[6].
3. Usage of Entropy in Modified PIFS Encoder
In the improve PIFS coding stage , the range and domain pools generated first , The most computationally intensive part of the FIC process is the search step. One way to decrease encoding time is to decrease the number of comparisons between range and domain blocks. The proposed method based on entropy values to reduce the domain pool of matching process between the range and domain block by using the value of entropy of domain blocks. Each range block is compared only with domain blocks if the entropy of domain block less than threshold, then execution the matching process. The entropy of the domain block is given by [12]:
where Pi is the probability of domain block, n is the block size
The main steps of the proposed method can be summarized as: Step1: Load the image.
Step 2:choose minimum error €
.
Step3: Choose the threshold.
Step4: Partitioning the image into fixed blocks size with non-overlap (R1… Rn-1 ).
Step5: Generate the domain image from the original image by averaging method.
Step 6: Partitioning the domain into fixed blocks size with overlapping blocks (D1… Dn-1 ).
Step7: For each new domain block compute the entropyby using equation (10).
Step8: Compare entropy of the domain block with threshold. i. If the entropy of domain less than threshold. ii. Compute the coefficients S ,O using equation (2,3). iii. Quantization S, O using equations (4,5,6,7,8,9).
iv. Compute RMS by using equation (1), if RMS less than €, then store the IFS code; else go to the next domain block.
4. Test Results
The proposed method have been tested by using grayscale lenna image (512x512 pixels ) and Barbara image (512x512 pixels )TABLE(1) and TABLE(2) show the encoding time and PSNR before adding The proposed method ; the test results shown in TABLE (3) and TABLE (4) shows that the traditional system is the slower PIFS encoding and speed up with proposed method(entropy) ,the advantage of the proposed method is reduce encoding time but the Disadvantage of entropy method, is the choice of threshold for each partition of the image, as shown in the TABLES below :
Lenna (512x512) pixels
BLOCKSIZE PSNR ET
4x4 24.9961 1680.5862
8x8 23.1412 1159.5389
16x16 20.2326 935.6816
TABLE(1) test results Lenna (512x512) pixel before the proposed method (entropy ) (Traditional FIC)
Barbara (512x512) pixels
BLOCKSIZE PSNR ET
4x4 26.1362 1638.3150
8x8 23.3570 1158.1609
16x16 20.4295 924.7043
Lenna (512x512) pixels
THRESHOLD BLOCKSIZE PSNR ET
2.0000 4x4 27.0815 91.47359
2.5000 8x8 22.8317 40.56409
.3.5000 16x16 20.2926 35.20468
Barbara (512x512) pixels
THRESHOLD BLOCKSIZE PSNR ET
1.5000 4x4 26.7410 63.951308
2.5000 8x8 23.4143 54.391457
3.0000 16x16 20.2698 25.553411
(a)
Fig 1 : (a) show orginal Image (512x512) , (b) Lenna(512x512) with proposed method (entropy) PSNR=27.0815, ET= 91.47359, Blocksize= 4x4
(b)
(a) (b)
Fig 2 : (a) show original Image (512x512) , (b) Barbara image (512x512) with proposed method (entropy Technique ) PSNR=27.3805, ET= 38.408175, Blocksize= 4x4
5. Conclusions
The entropy is very useful to speed-up encoding ,first compute the entropy of domain blocks and then compare with threshold, this proposed method leads to decrease time of encoding from (1680.586232)sec to (91.47359)sec and increase PSNR from (24.5193 dB into 27.0815 dB) with partition image into 4x4 pixel.The disadvantage of The proposed method is for each partitioning must have threshold differs from another partitioning.
References
[1] Vahdati.G., Afarandeh.E., Yaghoubi.M.," Improvement Speed of Fractal Image Compression through Gray Level Difference and Normal Variance",International Conference of Soft Computing and Pattern Recognition,2009.
[2] Chih Teng.W, Jyh-Horng.J, Jer-Guang.H," Spatial correlation genetic algorithm for fractal image compression", Available online at www.sciencedirect.com, 2006.
[3] Salarian.M, Nadernejad.E, Naim. H," A new modified fast fractal image compression Algorithm", The Imaging Science Journal Vol 61.2013.
[4] Wei.W-Y,"An Introduction to Image Compression",Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC.
[5] Li Hsu. S., Chang. Y., Jeng. J.," A Study of Fractal Image Compression with Coefficient Quantization",Department of Information Engineering I-Shou University, Ta-Hsu Hsiang, Kaohsiung County, The 24th Workshop on Combinatorial Mathematics and Computation Theory, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.378.9750&rep=rep1&type=pdf [6] George. L. E, Minas. N. A.," Speeding Up Fractal Image Compression Using DCT Descriptors", Journal of Information and Computing Science,2011.
[7] Al-Hilo. E., "Speeding-up Fractal Colored Image Compression Using Moments Features", 978-0-7695-3456-5/08 $25.00 © 2008 IEEE.
[8] Taha M.H.,"Partitioning Development for Fractal Image Compression",M.Sc.Thesis ,Science college,AL-Mustansiriyah University, December 2005.
[9] Hussein A.A.,"Fractal Image Compression with Fasting Approaches",M.Sc.thesis,Collegs of Science,Saddam University,2003.
[10] Fisher Y.,"Fratal Image Compression :Theory and Application",Springier Verlage,New York,1994. [11] George ,L.,"IFS Coding for Zero-Mean Image Blocks",Iraqi Jornal of Science ,Vol.47,No.1,2006.
