Robust Image Compression Algorithm using Discrete
Fractional Cosine Transform
VIVEK ARYA
Department of Electronics & Communication Engineering (FET)
Gurukula Kangri (Deemed to be University) Haridwar, INDIA
Abstract:- The discrete fractional Fourier transform become paradigm in signal processing. This transform process
the signal in joint time-frequency domain. The attractive and very important feature of DFrCT is an availability of
extra degree of one free parameter that is provided by fractional orders and due to which optimization is possible.
Less execution time and easy implementation are main advantages of proposed algorithm. The merit of
effectiveness of proposed technique over existing technique is superior due to application of discrete fractional
cosine transform by which higher compression ratio and PSNR are obtained without any artifacts in compressed
images. The novelty of the proposed algorithm is no artifacts in compressed image along with good CR and PSNR.
Compression ratio (CR) and peak signal to noise ratio (PSNR) are quality parameters for image compression with
optimum fractional order.
Key-words:- Image compression, fractional transform, discrete fractional cosine transform, redundancy.
Received: March 5, 2021. Revised: October 10, 2021. Accepted: December 1, 2021. Published: January 5, 2022.
1 Introduction
The need for robust techniques that can store visual
information and transmit has been increased in the
advancement of multimedia applications [1]. For
downloading the images from internet is a time
consuming process and because of this, nowadays
image compression become an attractive tool.
Therefore, the requirement of efficient and effective
algorithms that can provide high compression ratio
with less information loss has increased [2]. Image
data is the major portion of multimedia
communication and it consumes more bandwidth [3]
during transmission. Hence, development of
optimum and novel techniques for image
compression has become important paradigm [4].
Various lossy and lossless image compression
algorithms have been developed for the increasing
need of medical images, virtual conferencing and
multimedia. The existing technique’s model based on
the analyzing two dimensional singularities and
getting the captivating characteristics such as high
peak-signal-to-noise ratio (PSNR). Discrete cosine
transform (DCT) [5-8] is a very useful and important
transform for compression and it is an adaptation of
Fourier series. DCT provides fewer transform
coefficients [9] after approximation of a signal. The
Discrete sine transform (DST) is a factitive transform
of DCT. Nowadays, DST is used in compression
applications of audio coding and low rate image [10-
11]. Discrete Walsh Hadamard transform (DWHT) is
the simplest transform among all, although its energy
compaction capability very inferior than DCT.
Therefore, it does not used for image or data
compression [12-13]. DST, KLT and DCT are the
linear orthogonal blocked transforms, which remove
the correlating information or pixels inside the block.
These transforms do not work well with interrelating
over the block boundaries [14]. In last few years,
hybrid fractal image compression techniques [15] are
more in demand due to high compression rate while
it needs more execution time because of high
complexity.
In 1807, Jean Baptise Joseph Fourier introduced
Fourier transform, while he was working with a heat
conduction dilemma. However, with the
advancement of research areas and theme, in 1929,
the fractional power of Fourier transform noticed in
the mathematical literature [16-18]. The Fractional
Fourier Transform (FrFT) is commonly called as
rotational or angular Fourier transform in different
research papers [19-20]. Quantum mechanics [21],
signal processing [22-24], pattern recognition [25],
and optical, video and audio processing [26-27] are
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the significant applications of FrFT. In optical
domain, the continuous FrFT is applied [28]. It is a
scientifically and systematically proved fact that
difference gives a richer solution set than their
continuous limit differential equations [29]. In FrFT
domain the decomposition of continuous discrete
signals and systems has been developed [30].
Nowadays, FrFT have various authentic application
areas [31-36,45]. In last few years, two dimensional
discrete fractional cosine transform (DFrCT) is
applied for pattern recognition, encryption and image
compression [46] due to its robustness and reliability.
Some other image compression algorithms also
proposed by experts [47] where they have used
discrete cosine transform. The important research
gaps in the field of image compression are removal
of correlating information from the image and
obtained less PSNR and more blocking artifacts in
compressed images. The main motivation for the
research is to provide the robust image compression
technique which work efficiently on all type of
images and can give good compression ratio along
with good PSNR.
2 Proposed Image Compression Model
Two dimensional discrete fractional cosine transform
(DFrCT) is utilized at optimum value of fractional
order. The quantization of all those coefficients is
carried out to arbitrarily remove the coefficients that
consist of very less information. This is achieved by
adjusting the threshold value or cutoff value of the
quantizer also called the coarseness. After this zigzag
and Huffman coding is applied for further
compression. For decompression of image, Huffman
decoding and zigzag decoding was performed
respectively. Now, inverse DFrCT (IDFrCT) is
applied after quantization to the quantized image to
obtain the original image as shown in Figure 1.
2.1 Discrete Fractional Cosine Transform
The advancement of DCT is DFrCT. Sequence of
DCT {
[]xn
,
01nN
} is defined [37-41] with
equation (1) and (2);
1
0
21
cos 2
N
n
n
x k k x n N
,for
01kN
(1)
where,
10
() 211
for k
N
k
for k N
N
(2)
Kernel matrix of 1D-DCT [42] is given by equation
(3)
1, 0 0 1
( , ) 2 (2 1)
cos 1 1;0 1
2
DCT
k n N
N
kn nk Nn
N
E
kN
N
(3)
The inverse DCT is
1
0
(2 1)
[ ] ( ) ( )cos ;0 1
2
N
k
nk
x n k X k n N
N
The kernel matrix of N point DFrCT [42-44] is
(2 / )
,
T
N
N
NN
C V DV
2
2( 1)
10
j
N
jN
Ve
e
where
10 22
[]
N N
V V V V
, k the order DFT
Hermite eigenvector is
k
V
.
Fig. 1: Proposed Compression Model
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Unitarity, additivity of rotations, periodicity and
reality [36] are the impoertant mathematical
properties of DFrCT. 2D-DFrCT is used for
encryption, decryption and image compression. In 2-
D, angle of rotation
and
are used separately.
2.2 Zigzag Coding
In zigzag coding data access from the low frequency
components to high frequency components. Zigzag
coding lead to long run of 0’s and it converts the two
dimensional array of an image into one dimensional
array. In other words it converts the M x N matrix
into 1xN matrix as shown in Figure 2, therefore
zigzag coding play very vital role in image
compression.
2.3 Huffman Coding and Decoding
Generally, the Huffman coding is used to eliminate
the coding redundancy from the input image. It is a
one form of statistical coding that minimize number
of bits needs to represent the string of symbol. The
main advantage of Huffman coding is to utilized the
optimal code word that have least
average length. Its inverse process of coding is called
decoding, by which original input matrix can be
obtained easily.
2.4 Zigzag Decoding
In zigzag decoding reconstruction of original matrix
from the one dimensional array. In other words, it
convert one dimensional matrix into two dimensional
matrix of an image.
2.5 Quantization
Quantization works efficiently to quantize all the
coefficients obtained after applying DFrCT. The
small coefficients obtained after applying DFrCT
are quantized coarsely and the coefficients which
are having large values are quantized to the nearest
integer. After quantization, the coefficients are
written on the compressed stream.
3 Results and Discussion
The most common used parameters to evaluate the
image compression algorithms are PSNR and CR.
From Equation (6), the PSNR value (in dB) is used
to compare the difference between the compressed
image ‘
o
’ and the original input image ‘
i
’. In
general, if obtained PSNR after compression is less
means poor visual quality of image and for large
PNSR means image visual quality is good, therefore
for better image quality, the researcher aim is to get
the larger PSNR value.
11 2
00
1,,
255
MN
ij
MSE o i j i i j
(5)
10
10log MN
PSNR MSE
(6)
where
MN
is the size of the image.
In order to evaluate the effectiveness and efficiency
of proposed algorithm, four different test images
have been used i.e,: “Lena”, “Peppers”, “Barbara”
and “Baboon” as shown in Figure 3. It directly
worked on the entire image without blocking or
partitioning. Hence, this proposed algorithm
performed well and provides compressed images
which are free from blocking artifacts. This proposed
technique provide good PSNR for all the images as
compare to existing techniques due to which image
quality has been increased after the compression
without any blocking artifacts.
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Fig. 2: An Example of DFrCT-Based Coding Technique. (A) An 8x8 Matrix(Subimage). (B) The Quantized
DFrCT Coefficient Matrix of (A). (C) The Zig–Zag Scanning Pattern. (D) The Zig–Zag Scanned One
Dimensional Matrix of (B)
As there is no possibility of optimization in other
transforms based existing compression techniques.
So, one more important advantage of this technique
is that it uses DFrCT due to which optimization is
possible by varying the fractional order ‘
a
’.
Therefore, this proposed compression technique
fulfilling the all the research gaps as mentioned
earlier. Figure 4 gives the visual results of proposed
algorithm for different images. Comparative results
of various existing algorithms along with proposed
algorithm are shown in Figure 5 and Table 1. From
which we can conclude that proposed algorithm is
superior in performance. Table 2 gives PSNR values
for different fractional order ‘
a
’ for all compressed
images. Finally, it is found that the best image
compression is obtained by varying the fractional
order ‘
a
’ (0 to 1). For high PSNR values Table 2 and
Figure 7 shows that optimum value of ‘
a
’ is 0.6 for
Lena, Baboon and Barbara and 0.7 for Peppers. From
Table 2 and Figure 7 we can conclude that the
optimum value of fractional order ‘
a
’ is dependent
on image. Table 3 and Figure 6 shows that
comparative compression ratio for recently published
algorithm and proposed algorithm, which depicts that
proposed algorithm provide more compression ratio
as compare to others. Less execution time and easy
implementation are the attractive advantages of this
proposed technique. Image compression with DFrCT
works well in fractional domain and tries to save the
bandwidth by varying fractional order 0 to1.
(a) Lena (b) Peppers (c) Barbara (d) Baboon
Fig. 3: Original Images
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(a) Lena (b) Peppers (c) Barbara (d) Baboon
Fig. 4: Compressed Images By Proposed Technique
0
5
10
15
20
25
30
35
40
45
Lena Barbara Baboon Peppers
Proposed
JPEG
BTC
SVD
GP
C.S.Rawat
Fig. 5: Graph Shows the PSNR for Different Compression Techniques and Proposed Technique for Images
(a)Lena, (b)Barbara, (c)Baboon and (d)Peppers.
Table 1. Comparison of Proposed Method with Different Methods
Images
PSNR (in dB)
JPEG
Block Truncation
Coding (BTC)
Singular Value
Decomposition
(SVD)
Gaussian
Pyramid
(GP)
C.S.Rawat
[15]
Proposed
Method
Lena
29.8870
29.6116
22.6225
15.6656
31.5739
33.0649
Barbara
30.00
26.5894
20.4283
14.7322
32.7810
42.9752
Baboon
31.3421
25.1743
20.0996
16.1080
35.8160
36.0058
Peppers
34.2700
29.2346
22.3442
14.5351
39.2185
40.1281
Table 2. PSNR for Different Fractional Order ‘
a
’
Images
PSNR (in dB) for different fractional order ‘
a
’
0a
0.1a
0.2a
0.3a
0.4a
0.5a
0.6a
0.7a
0.8a
0.9a
1a
Lena
29.1933
30.1466
31.1514
31.4880
32.2296
32.3354
33.0649
32.7057
32.0428
31.5458
30.7834
Barbara
38.7292
40.1331
41.0359
41.5319
42.3117
42.3629
42.9752
40.9277
38.9353
37.2686
34.4562
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Baboon
32.1813
33.0001
34.0066
34.3771
35.1626
35.2664
36.0058
35.5677
34.8094
34.0582
32.2599
Peppers
34.0638
34.4808
35.5316
36.0736
36.9614
37.2041
38.1281
40.1281
37.7744
36.5518
33.8705
Table 3. Compression ratio for various images
Fig. 6: Graph Showing the Compression Ratio of
C.S. Rawat [15] and Proposed Technique
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
30
30.5
31
31.5
32
32.5
33
33.5
PSNR versus fractional order
Fractional Order 'a'
PSNR (in dB)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
37
38
39
40
41
42
43
PSNR versus fractional order
Fractional Order 'a'
PSNR (in dB)
(a) For Lena (b) For Barbara
Images
Compression Ratio
C.S. Rawat
[15]
Proposed
Method
Lena
11.1544
22.5113
Barbara
7.1266
22.4991
Baboon
4.5253
19.0918
Peppers
10.8559
16.0767
0
5
10
15
20
25
Lena Barbara Baboon Peppers
Proposed
C.S.Rawat
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
33
33.5
34
34.5
35
35.5
36
36.5
PSNR versus fractional order
Fractional Order 'a'
PSNR values (in dB)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
34
35
36
37
38
39
40
41
PSNR versus fractional order
Fractional Order 'a'
PSNR values (in dB)
(c) For Baboon (d) For Peppers
Fig. 7: Different values of PSNR with varying fractional order ‘
a
’ =0.1 to 0.9 for (a) Lena, (b) Barbara, (c) Baboon
and (d) Peppers
4 Conclusion
An efficient and novel technique for image
compression in fractional domain has been proposed.
The proposed technique efficiently and effectively
compressed the images and optimum fractional order
‘
a
’ is computed using DFrCT. By working on blocks
many existing techniques produces blocking artifacts
but in this technique operation are not on non-
overlapped blocks or subimages. Therefore,
blocking artifacts were repudiated by working on the
entire image not by small blocks and this is the
novelty of proposed technique. This proposed image
compression model worked with single block (of size
256x256) which decreases the execution time. From
the experimental and simulation results, it can be
conclude that the proposed algorithm is robust and
effective for compressing different type of images
and provided very good compression and PSNR. The
future scope will be the application of other
fractional transform for image compression.
References:
[1] S. Annadurai and M. Sundaresan(2009). Wavelet
Based Color Image Compression using Vector
Quantization and Morphology. Proceedings of
the International Conference on Advances in
Computing, Communication and Control., pp.
391-396.
[2] L. Krikor, S. Baba, T. Arif, and Z. Shaaban
(2009). Image encryption using DCT and stream
cipher. European Journal of Scientific Research.
vol. 32, no. 1,pp. 48-58.
[3] A. Loussert, A. Alfalou, R. El-Sawda and A.
Alkholidi (2008). Enhanced system for image's
compression and encryption by addition of
biometric characteristics. International Journal
of Software Engineering and its Applications.
vol. 2, no. 2,pp. 111-118.
[4] G. Ames. Image compression. International
Journal of Advancements in Computing
Technology. vol. 1, no. 2, (2002),pp. 205-220.
[5] N. Ahmed, T. Natarajan and K.R. Rao (1974).
Discrete cosine transform. IEEE Trans. Comm.,
pp.90-93.
[6] Vivek Arya, P. Singh and K. Sekhon (2013).
RGB image compression using two dimensional
discrete cosine transform. International Journal
of Engineering Trends and Technology. pp. 828-
832.
[7] Vivek Arya, P. Singh and K. Sekhon (2015).
Medical image compression using two
dimensional discrete cosine transform.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.3
Vivek Arya
E-ISSN: 2224-2856
31
Volume 17, 2022
International Journal of Electrical and
Electronics Research. vol.3, issue 1, pp 156-164.
[8] Vivek Arya and J. Singh (2016). Robust image
compression using two dimensional discrete
cosine transform. International Journal of
Electrical and Electronics Research. vol.4, issue
2, pp.187-192.
[9] K.R. Rao and P. Yip (1990). Discrete cosine
transform: Algorithms advantages and
applications. Academic Press, New York.
[10] P.M. Fanelle and A.K. Jainn (1986). Recursive
block coding: A new approach to transform
coding. IEEE Trans. Communication. pp. 161-
179.
[11] M. Bosi and G. Davidson (1992). High quality
low bit rate audio transform coding for
transmission and multimedia applications.
Journal of Audio Eng. Soc. pp. 43-50.
[12] D. Solomaon, (2004). Data Compression: The
complete Reference. Springer Verlag, Newyork.
[13] F. J. Macwilliams and N.J.A. Slone (1977). The
Theory of Error Correcting Codes. Elsevier,
Amsterdam.
[14] Y. T. Chan, (1995). Wavelet Basics. Kluwer
Academic Publishers, Norwell, MA.
[15] C.S. Rawat and S. Meher (2013). A hybrid
image compression scheme using DCT and
fractal image compression. International Arab
Journal of Information Technology. vol.10, no.6.
[16] N. Wiener (1929). Hermitian polynomials and
Fourier analysis. J. Math. Phys., pp.70–73.
[17] E. U. Condon (1937). Immersion of the Fourier
transform in a continuous group of functional
transformations. Proc. Nat. Acad. Sci. USA,
pp.158–164.
[18] H. Kober and Wurzeln aus der Hankel (1939).
Fourier- und aus an-deren stetigen
transformationen. Q. J. Math. Oxford Ser., pp.
45–49.
[19] B. Santhanam, J. H. McClellan (1996). The
discrete rotational Fourier transform. IEEE
Trans. Signal Processing, pp. 94–998.
[20] A.W. Lohmann (1993). Image rotation, Wigner
rotation and the fractional Fourier transform. J.
Opt. Soc. Am, pp. 2181–2186.
[21] V. Namias (1980). The fractional order Fourier
transform and its application to quantum
mechanics. J. Inst. Maths. Appl., pp. 241–265.
[22] L.B. Almeida (1994). The fractional Fourier
transform and time frequency representations.
IEEE Trans. Signal Process, pp. 3084–3091.
[23] C. Vijaya, J.S. Bhat (2006). Signal compression
using discrete fractional Fourier transform and
set partitioning in hierarchical tree. Signal
Process, pp. 1976–1983.
[24] R.Tao, B. Deng, Y. Wang (2006). Research
progress of the fractional Fourier transform in
signal processing. Sci. China Ser. F Inf. Sci., pp.
1–25.
[25] Bracewell, R. N (1986). The Fourier Transform
and Its Applications. 2nd Revised. McGraw Hill,
New York.
[26] Abomhara, M., Zakaria, O., Khalifa, O.O
(2010). An overview of video encryption
techniques. Int. J. Comput. Theory Engineering,
pp. 103–110.
[27] R. C. Gonzalez and R. E. Woods (2008).
Digital Image Processing, 3rd edition.
[28] D. Mendlovic and H.M. Ozaktas (1993).
Fractional Fourier transforms and their optical
implementation. I. J. Opt. Soc. Am. A, pp. 1875–
1881.
[29] N. M. Atakishiyev, L. E. Vicent and K.B. Wolf
(1999). Continuous vs. discrete fractional Fourier
transforms. J. Comput. Appl. Math. pp. 73–95.
[30] S.I. Yetik, M.A. Kutay and M.H. Ozaktas
(2000). Continuous and discrete fractional
Fourier domain decomposition. In IEEE Trans.
pp. 93–96.
[31] M.H. Ozaktas, Z. Zalevsky and M.A. Kutay
(2000). The Fractional Fourier Transform with
Applications in Optics and Signal Processing.
Wiley, New York.
[32] S.C. Pei, M.H. Yeh (2001). A novel method for
discrete fractional Fourier transform
computation. ISCAS, pp. 585–588.
[33] M.H. Yeh and S.C. Pei (2003). A method for the
discrete-time fractional Fourier transform
computation. Signal Process, pp. 1663–1669.
[34] S.C. Pei, M.H. Yeh, C.C. Tseng (1999). Discrete
fractional Fourier transform based on orthogonal
projections. IEEE Trans. Signal Processing., pp.
1335–1348.
[35] I. Djurovic, S. Stancovic and I. Pitas (2001).
Digital watermarking in the fractional Fourier
transformation domain. Journal of Netw.
Comput. Applications. pp. 167–173.
[36] Rubio, J.G.V. and B. Santhanam (2005). On the
multiangle centered discrete fractional Fourier
transform. IEEE Signal Process. Letters.
[37] Gerek, O.N., Erden, M.F. (2000) The discrete
fractional cosine transform. Proceedings of the
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.3
Vivek Arya
E-ISSN: 2224-2856
32
Volume 17, 2022
IEEE Balkan Conference on Signal Processing,
Communications, Circuits and Systems. Istanbul,
Turkey.
[38] Lohmann, A.W., Mendlovic, D., Zalevsky, Z.,
Dorch, R.G. (1996), Some important fractional
transformations for signal processing. Opt.
Commun., pp. 18–20.
[39] Yip, P., Sine and Cosine transforms (1996). In
Poularikas, A.D. (ed.) Transforms, The
Handbook Applications. CRC Press, Alabama.
[40] Prudnikov, A.P., Brychkov, Y.A., Marichev,
O.I. (1986), Integrals and Series VI: Elementary
Functions. Gordon and Breech Science
Publishers, New York.
[41] Ahmed, N., Natarajan, T., Rao, K.R. (1974),
Discrete cosine transform. In IEEE Trans.
Comp. C, pp.90–93.
[42] S.C. Pei, M.H Yeh, M.H. (2001). The discrete
fractional cosine and sine transforms. IEEE
Trans. Signal Process., pp.1198–1207.
[43] Narasimha, M.J., Peterson, A.M. (1978). On the
computation of the discrete cosine transform. In
IEEE Trans. Communication., pp.934–936.
[44] M.H. Yeh and S.C. Pei. (2003). A method for
the discrete fractional Fourier transform
computation. In IEEE Trans. Signal Processing.
pp. 889–891.
[45] T. Zhao and Q. Ran (2019). The weighted
fractional Fourier transform and its application in
image encryption. Mathematical Problems in
Engineering Hindawi. Vol. 2019, pp. 1-10.
[46] N. Kumar R., B.N. Jagadale and J.S. Bhat
(2018),. An improved image compression
algorithm using wavelet and fractional cosine
transform. International Journal of Image,
Graphics and Signal Processing. vol.11, pp. 19-
27.
[47] Vivek Arya, & Jugmendra Singh (2016). “Image
Compression Algorithm Using Two Dimensional
Discrete Cosine Transform” Published in
Imperial Journal Interdisciplinary Research
(IJIR), Vol-2, Issue-8.
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