Document 13290182

Research Journal of Applied Sciences, Engineering and Technology 4(24): 5387-5390, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 18, 2012
Accepted: April 14, 2012
Published: December 15, 2012
Sub-band Based DCT for Image Compression
M. Marimuthu, R. Muthaiah and P. Swaminathan
School of Computing, SASTRA University Thanjavur, Tamilnadu, India
Abstract: Uncompressed multimedia such as data, image, audio and video requires transmission bandwidth
and storage capacity. Due to demand for data storage capacity and data transmission bandwidth, Image
compression is used which provides data security, reduces the storage requirements, provides rich quality
signals for visual data representation and reduces the image fidelity. The focus of this study is the Discrete
Cosine Transform (DCT) and Inverse Discrete Cosine Transform (IDCT) based on the sub-band decomposition
for image compression. The simulated results are shown that sub-band decomposition based DCT provides
good compression rate and better peak signal to noise ratio than conventional DCT.
Keywords: Discrete cosine transform, fast DCT, image compression, inverse discrete cosine transform, subband decomposition based DCT
In particular, lossless compression becomes vital
when loss of data is not tolerable. Hence, the techniques
pursuing for higher Compression Ratio (CR) with less
distortion and without data loss has been one of the
important research issues in image compression. DCT and
IDCT are extensively used as image compression
standards for the development of aviation, space
techniques, internet and communications applications.
Like other transforms, DCT efforts to decorrelate the data
and after decorrelation each transform coefficient can be
encoded separately without losing compression efficiency.
The Karhunen-Loeve Transform (KLT) is a linear
transform and can be adaptive. It is the basic function
derived from the statistical properties of the image data.
The transforms are not separable. KLT is data dependent
and these without a fast pre-computation transform.
Derivation of the respective basis for each image subblock requires unreasonable computational resources. The
Discrete Fourier Transform (DFT) is separable, linear and
symmetric. It is a complex transform and consequently
demands that, both image magnitude and phase
information can be encoded.
DCT offers better energy compaction than DFT for
most natural images. The transformed coefficients is
spread over low and high frequencies. The special
properties of the DCT are energy compaction for highly
correlated images, separability, symmetry, orthogonality,
highly computational intensive and optimal decorrelation.
These properties have been exploited to achieve high
Compression Ratio (CR) in Video and image compression
techniques such as H.26X, MPEG and JPEG (Kruti and
Rutvij, 2011; Sachin, 2011; Chaur, 1998; Dan and Peter,
2007; Vijayshri and Ajay, 2010). The vital concept behind
Sub-Band Coding (SBC) is to divide the frequency band
of a signal and then to code each sub-band using a coder
and bit rate precisely matched to the information of the
band. It is used in speech coding and in image coding
because of its advantages such as coding error
confinement within the sub-bands as well as variable bit
assignment among the subbands. At the decoder, the
subband signals are decoded, unsampled and passed
through a bank of synthesis filters and properly added to
yield the reconstructed image (Sindhu and Rajkamal,
Fast dct/idct based subband decomposition: Fast DCT
uses the orthonormal property for reducing complexity. It
is the key to note that DCT is not the real part of the DFT.
This can be easily established by inspecting the DCT and
DFT transformation matrices. The difficulty of a direct
DCT is as high as O(N3) but using fast DCT algorithms
it can be reduced to O(N2logN).
The N-point 1-D DCT is defined by:
 (2n  1) kx 
C[ k ]  [ k ] x[n]cos
n 0
N 1
where, k  0,1..., 7, N  1,
[0] 
, [ k ]  2 / N , for k  0
Corresponding Author: M. Marimuthu, School of Computing, SASTRA University Thanjavur, Tamilnadu, India
Res. J. Appl. Sci. Eng. Technol., 4(24): 5387-5390, 2012
Image Video
Image Video
Original image
Compressed image
Original image
Image from workspace
C8 (DCT)
C8 (DCT)
Subband based 8 point DCT
Subband based 8 point IDCT
Image Video
Reconstructed image
Psnr value in db for
PSNR reconstructed image
Fig. 1: Sub-band decomposition and composition for computing DCT/IDCT
The DCT of the signal C[k] equation can be rewritten
as follows:
C[ k ]  CL [ k ]  S H [ k ], where
 k 
CL [ k ]  2 cos 
 2N 
 k 
S H [ k ]  2 sin 
 2N 
( N / 2) 1
 (2n  1) kx 
 [ k ] x L [n]cos
 [ k ] x H [n]sin
n 0
( N / 2) 1
n 0
 (2n  1) kx 
CL[k] and SH[k] represents the Sub-Band Discrete
Cosine Transform (SB_DCT) and Sub-Band Discrete Sine
Transform (SB_DST), respectively. The DCT of a signal
is the summation of SB_DCT of the LF (low frequency)
sub signal and SB_DST of HF (high frequency) of the sub
The sub-band decomposition based DCT and DST
are used for computing DCT and its inverse (SB_IDCT
and SB_IDST) are used for signal composition and
computing IDCT as shown in the Fig. 1 (Yaw-Shih and
Yao-Te, 2008).
The original signal x[n] can be obtained from xL[n]
and xH[n] as follows:
x L [ n] 
x H [ n] 
N 1
 k 
 (2n  1) kx 
 C[ k ]cos
 2N 
 [ k ]cos
 [ k ]sin
n 0
N 1
 k 
 (2n  1) kx 
 C[ k ]sin
 2N 
n 0
The sub-band based 8-point 1-D DCT is defined by:
C8 
F R x , where x8   x[0]  x[7] in wich
4 8 8 8
x L [n]  x L [0]... x L [3]
, x H [n]  x H [0]... x H [3] T
xL[n] and xH[n] represents the low frequency sub-band
signals and high frequency sub-band signals of original
signal x[n] respectively. xL[n] and xH[n] can be obtained
from C[k] (Lu et al., 2011).
Subband decomposition Based dct/idct: Based on the
sub-band decomposition of a signal, the DCT coefficients
of a signal are approximated by the sub-band DCT
coefficients. The applications of sub-band decomposition
in image resizing were proposed by (Mukherjee and
Mitra, 2002; Dugad and Ahuja, 2001). The Image is
decomposed into two sub-bands namely low frequency
sub-band, which gives the average of the signal and high
frequency sub-band, which gives the difference of the
signal. It is otherwise called as multiple level resolutions
DCT. The sub-band decomposition based 8-point
DCT/IDCT is shown in the Fig. 2.
The input image is subjected to various stages of
decomposition. There are several stages for compression
and reconstruction of an image. The first stage contains
matrix vector multiplication of decomposition matrix R8
with input signal x8 for producing y8.The second stage
pre-shuffles the pre-transportation matrix T8 and z8 which
is produced by matrix vector multiplication of coefficient
matrix with y8. This is followed by the post-transportation
matrix T8.The third stage is the multiplication of 0.3536
with z8 producing C8 (sub-band DCT). The fourth stage is
the multiplication of 0.3536 with C8 producing u8.The
fifth stage pre-shuffles the inverse pre-transportation
matrix and w8 which is produced by matrix vector
multiplication of inverse coefficient matrix with u8.This is
followed by the inverse post transportation matrix T8.
The value of Decomposition matrix and its inverse,
coefficient matrix and its inverse, pre transportation
matrix and its inverse, post transportation matrix and its
inverse etc. are well defined (Lu et al., 2011).
The sub-band based 8-point 1-D IDCT is defined by:
x8  2 2 F81 R81 C8 where C8  C[0]  C[7]
The comparison of quality measure for various
reconstructed images is shown in the Table 1. The Peak
Res. J. Appl. Sci. Eng. Technol., 4(24): 5387-5390, 2012
Fig. 2: Fast DCT and IDCT based sub-band decomposition with various stages of decomposition
Fig. 3: Fast DCT and IDCT based sub-band decomposition with various stages of decomposition
Table 1: Sub-band decomposition based DCT/IDCT for various images
Image File size
Image File size
Lena.bmp 512x512
257. 0KB
Boat.png 512X512
House.png 256x256
Peppers.png 256x256
Signal to Noise Ratio (PSNR) is used to measure the
quality of a reconstruction image. The Mean Square Error
(MSE) is the average squared difference between
reference and distorted image and Compression Ratio
(CR) is equal to the file size of the original image to the
file size of the compressed image. The various stages of
sub-band decomposition based DCT to achieve the
compressed image are shown in the Fig. 3.
The principal aim of this research study is to
emphasize the uniqueness of the sub-band decomposition
based DCT for image compression. The sub-band
decomposition based DCT, which is simulated and tested
for various images are given in the Table 1. The results of
sub-band decomposition based DCT are superior to the
Res. J. Appl. Sci. Eng. Technol., 4(24): 5387-5390, 2012
conventional DCT in terms of Peak Signal to Noise Ratio
(PSNR), Mean Square Error (MSE) and computation
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