International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
1
2
Department. of Electronics and Communication Engineering
Jabalpur Engineering College, Jabalpur,M.P. - India
AbstractThis article presents a novel method for the image compression based on the Fuzzy transform.The Approach for
The proposed method is based on the Fuzzy transform. In this method, the image is partitioned in to sub-
Image Compression, based on Fuzzy Transform, is attempted by normalizing the values of its pixels. Any image can be considered images of square size (16x16, 8x8, 4x4) called blocks, then any compressed sub-image is obtained using fuzzy transform as a fuzzy matrix (relation), which is subdivided in sub-matrices
(possibly square) called blocks. Each block is compressed with and uniform quantization according to the standard deviation of sub-image. These sub-images are decompressed with the the discrete fuzzy transform of a function in two variables and successively it is decompressed via the related inverse fuzzy related inverse fuzzy transform and recomposed for the transform. The decompressed blocks are recomposed for the reconstruction of the image. Further the quality of reconstructed reconstruction of the image. This method allows the user to select regions of interest and to compress each one with image is measured,(and compared with the some popular preexisting methods of image compression,) on the basis of PSNR different quality factor [3-11].
(Peak Signal to Noise Ratio), MSE (Mean Square Error).
II.
FUZZY TRANSFORM
I.
INTRODUCTION
Image compression has become one of the most important disciplines in digital electronics because of the ever growing popularity and usage of the internet and multimedia systems combined with the high requirements of the bandwidth and storage space. Compression of images is useful, where a tolerable degradation is required. With the wide use of computers and consequently need for large scale storage and transmission of data, efficient ways of storing of data have become necessary. With the growth of technology and entrance into the Digital Age,the world has found itself amid a vast amount of information. Dealing with such enormous information can often present difficulties. Image compression is minimizing the size in bytes of a graphics file without degrading the quality of the image to an unacceptable level.
The reduction in file size allows more images to be stored in a given amount of disk or memory space. It also reduces the time required for images to be sent over the Internet or downloaded from Web pages [1-2].
There are many methods exists for the image compression, JPEG and JPEG 2000 are main of them. The
Joint Photographic Experts Group (JPEG) standard is one of the most widely used for lossy image compression. The approach recommended by JPEG is a transform coding approach using the DCT. The JPEG specification consists of several parts, including a specification for both lossless and lossy encoding [1-2].
In classical mathematics, various kinds of transforms (Fourier,
Laplace, integral, wavelet) are used so commonly and frequently. A transformation method on fuzzy domain called fuzzy transform (or, shortly, F-transform) that encompasses both classical transforms as well as approximation methods
[3-5].
The F-transform establishes a correspondence between a set of continuous functions on an interval of real numbers and the set of n-dimensional (real) vectors. The formula to which we will refer as an inverse F-transform
(inversion formula) converts an n-dimensional vector into another continuous function which approximates the original one. The advantage of the inversion formula of the Ftransform is a simple approximate representation of the original function. Thus, in complex computations we can use the inversion formula instead of precise representation of the original function [6-9].
The concept of F-transform of a function establishes a correspondence between the set of continuous functions on the interval [a,b] and the set of n-dimensional vectors.
Conversely, the concept of inverse F-transform converts an ndimensional vector into a continuous function which approximates the original function up to a small quantity.
In many problems involving complex computations, then it is possible to operate with an image of the original function obtained by using the F-transform and hence by translating the functional problem into the respective problem of linear algebra, which is more convenient to manipulate
ISSN: 2231-5381 http://www.ijettjournal.org
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014 since one deals with vectors.After that the computations are made, the result (which is an n-dimensional vector) is converted, via the inverse F-transform, to a continuous function which approximates the original function[10-11].
The idea of fuzzy transform turned out to be very general and powerful. A function obtained by the inverse Ftransform has nice filtering properties which can be used to removing noise from images or from any other kind of data.
A.
Fuzzy Transform In 2- Dimension
We may define the matrix [ ] to be the discrete F-transform
[3,9]off with respect to { } and { } if we have for each and :
∑ ∑ (
∑ ∑
)
(
(
)
)
( )
( )
( )
Where is the rectangle showing the universe of discourse and let , and
be assigned points, called , such that
. and be a fuzzy partition of
be a of
and
( )
be a continuous function on .
We may give the discrete inverse F-transform of f with respect to { } and { } to be the following function defined in the same points ( )
with and , as
∑ ∑ ( ) ( ) ( )
III.
PROPOSED METHODOLOGY
3.
The compression rate
( )
is given by:
( )
( ) ( )
( ) ( )
( )
TABLE I
COMPRESSION RATES USED IN THE EXPERIMENTS
( ) M(B) N(B) m(B) n(B)
0.035156
0.062500
16
8
16
8
3
2
3
2
0.140625
0.250000
8
8
8
8
3
4
3
4
4.
Equation (2) is used to obtain a decompressed image of size N
( ) ( )
.
5.
Each block is then de-normalized to get reconstructed image.
IV.
PARAMETERS
1) MSE and PSNR: Techniques that commonly employed for image compression result in some degradation of the reconstructed image. A widely used measure of reconstructed image fidelity for an N x M size image is the mean square error (MSE) and is given by -:
∑ ∑ | ( ) ̂( )| ( )
( ) ( )
2)Execution Time: The different algorithm of image compression follows the different methodology. Due to which a measurable difference is obtained in there execution time. For a method to be suitable for the run time processing, its execution time should be small. The execution time is measured in seconds.
3)Compression Ratio: For the proposed method the
Compression Ratio is the inverse of the Compression
Rate.
The methodology of the proposed method is simple enough to understand, which can be explained in the following steps:
1.
Original image is divided into blocks of N
( ) ( )
.
2.
Pixel values of a gray scale image range from 0-255 are normalized by dividing the each pixel value with 255.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
V.
RESULT AND DISCUSSION
Input Image Input Image
Input Image
Input Image
Reconstructed Image Reconstructed Image
Reconstructed Image
Reconstructed Image
Reconstructed Image Reconstructed Image
Reconstructed Image Reconstructed Image
Reconstructed Image Reconstructed Image
Reconstructed Image Reconstructed Image
Reconstructed Image Reconstructed Image
Reconstructed Image
Reconstructed Image
Fig 1: (first row) Original images (second, third, fourth, fifth row) Compressed using Fuzzy transform for the value of compression rate
( )
equal to 0.035156, 0.062500, 0.140625, 0.250000 respectively.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
( )
0.035156
0.062500
0.140625
0.250000
TABLE II
PSNR VALUES FOR DIFFERENT IMAGE COMPRESSED USING FUZZY TRANSFORM.
Cameraman(256x256)
20.6266
21.5374
23.5360
25.2599
Lena(256x256)
22.5606
23.5344
25.7949
27.6914
Circuit(256x256)
24.7417
26.4248
29.8921
32.6822
Peppers(256x256)
25.09
26.19
28.89
30.73
( )
0.035156
0.062500
0.140625
0.250000
TABLE III
MSE VALUES FOR DIFFERENT IMAGE COMPRESSED USING FUZZY TRANSFORM.
Cameraman(256x256)
562.8914
456.3957
288.0555
193.6805
Lena(256x256)
360.5979
288.1667
171.2337
110.6475
Circuit(256x256)
218.2269
148.1163
66.6602
35.0642
Peppers(256x256)
201.35
156.32
83.80
54.92
TABLE IV
EXECUTION TIME OF COMPRESSION IN SECOND FOR DIFFERENT IMAGE COMPRESSED USING FUZZY TRANSFORM.
( )
0.035156
0.062500
0.140625
0.250000
Cameraman(256x256)
1.560100
0.868413
1.638462
2.715788
Lena(256x256)
1.574090
0.867797
1.682732
2.748999
We have seen from the results that with the increment in the value of compression rate the PSNR is increasing and MSE is decreasing. The time of compression is inversely proportional to M(B).N(B) i.e. block size before compression and directly proportional to the m(b).n(b) i.e. corresponding block size after compression.
VI.
CONCLUSSION
We have seen the requirements and the significance of the image compression. We seen that for any compression scheme the quality of the image after reconstruction and the time of compression and reconstruction are the two important issues. In the JPEG the Quality of compression is not a big problem but its execution time is very high in the 100 seconds, where as for the proposed technique it’s near about
1second only.
Circuit(256x256)
0.969000
0.656000
1.016000
1.594000
Peppers(256x256)
0.938000
0.641000
1.031000
1.594000
In the proposed method the compression rate (1/ compression ratio) is also sufficiently lower with the good
PSNR value. Thus we can say that the proposed method is an ideal method for the run time image compression i.e. image uploading and downloading,high speed recording cameras etc.
Fuzzy transform is a useful transform for the purpose of the data compression. In the future many other uses of Fuzzy transform can be found. The image compression using the Fuzzy transform is found good. In the nearest future, furthermore researches can be done for the improvement in the PSNR and Compression ratio with the high speed performance.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
VII.
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