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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 INTRODUCTION 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, 2009). METHODOLOGY 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 2N n 0 N 1 where, k 0,1..., 7, N 1, 1 [0] , [ k ] 2 / N , for k 0 N Corresponding Author: M. Marimuthu, School of Computing, SASTRA University Thanjavur, Tamilnadu, India 5387 Res. J. Appl. Sci. Eng. Technol., 4(24): 5387-5390, 2012 Image Video viewer Image Video viewer Original image Input Image Compressed image Original image Image from workspace C8 (DCT) C8 (DCT) Subband based 8 point DCT Reconstructed Subband based 8 point IDCT Image Video viewer Reconstructed image I1 30.83 PSNR I2 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 N [ k ] x L [n]cos [ k ] x H [n]sin n 0 ( N / 2) 1 n 0 (2n 1) kx N 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 signal. 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 N [ k ]cos [ k ]sin n 0 N 1 k (2n 1) kx C[ k ]sin 2N N n 0 The sub-band based 8-point 1-D DCT is defined by: C8 2 T F R x , where x8 x[0] x[7] in wich 4 8 8 8 x L [n] x L [0]... x L [3] T , 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). EXPERIMENTAL RESULTS The sub-band based 8-point 1-D IDCT is defined by: x8 2 2 F81 R81 C8 where C8 C[0] C[7] T The comparison of quality measure for various reconstructed images is shown in the Table 1. The Peak 5388 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 Uncompressed Compressed Images Image File size Image File size Lena.bmp 512x512 257. 0KB 34.0KB Boat.png 512X512 162.0KB 43.5KB House.png 256x256 34.1KB 8.26KB Peppers.png 256x256 39.2KB 11.6KB 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. Compression Ratio 8:1 3:1 8:2 5:2 MSE 56.0416 103.9767 116.7764 180.4113 PSNR (DB) 30.83 28.08 25.57 25.63 CONCLUSION 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 5389 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 complexity. REFERENCES Chaur, C.C., 1998. On the selection of image compression algorithms. 14th International Conference on Pattern Recognition (ICPR'98), Brisbane, Australia. Dan, L. and K.J. Peter, 2007. A survey of parallel algorithms for fractal image compression. J. Algorithms Comput. Technol., 1(2): 171-186, ISSN: 1748-3018. Dugad, R. and N. Ahuja, 2001. 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