Frequency domain post-processing technique based on POCS
where KBij ¼ NZðAij Þ þ VBij
Yoon Kim, Chun-Su Park and Sung-Jea Ko A novel post-processing technique in the DCT domain, based on the theory of the projection onto convex sets (POCS), to reduce the blocking artifacts in the block discrete cosine transform (BDCT)coded images, is proposed. Computer simulation results indicate that the proposed scheme performs better than conventional post-processing algorithms. Furthermore, the proposed algorithm can be implemented without DCT=IDCT operations. Estimates of computation savings vary between 41 and 64% depending on the task.
and NZ() represents the order of the last nonzero DCT coefficient in the zig-zag scan. In (5), VBij is a factor representing how many DCT coefficients of Bij do contribute to the blocking artifacts. Thus, VBij should be chosen based on the local statistics of the image and the human perceptual properties. We define VBij by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u5 uP u jACi j t ð6Þ VBij ¼ Z i¼1 1 þ DC where Z is a scaling factor, and DC and ACi are DCT coefficients of Bij. By observing the smoothness constraint set in (4), the projection operator onto that set can be easily found. By discarding the nonzero coefficients of Bij at higher orders than the KBij in (5), we can obtain the projected element B¯ ij. N
aij bij
N
Introduction: Image compression techniques based on block discrete cosine transform (BDCT) have been the popular choice in both still and moving image coding standards, such as JPEG, H.26x, and MPEG. However, BDCT has a major drawback, which is usually referred to as blocking artifacts at low bit rates. The iterative approaches based upon the theory of projections onto convex sets (POCS) have been proposed for the reduction of blocking artifacts [1–3]. These methods, however, require iterative DCT=IDCT operations with heavy computational burden. Therefore, in spite of better performance than other post-processing techniques, the POCS-based post-processing techniques have not been widely used in real-time applications. In this Letter, we propose a new smoothness constraint set and its projection operator based on the theory of POCS in the DCT domain, to reduce blocking artifacts in the BDCT-coded images.
ð5Þ
Proposed SCS and its projection: Let us denote an mN nN input image and its pixel by f and f ( , ), respectively, where m and n are positive integers. In a block-based transform coding, f ( , ) is composed of non-overlapping N N blocks aijs, where aij ¼ {aij(0, 0), . . . , aij(N 1, N 1)} with aij(x, y) ¼ f (iN þ x, jN þ y), i ¼ 0, 1, . . . , (m 2) and j ¼ 0, 1, . . . , (n 2). Fig. 1 shows a block bij, that is diagonally shifted one pixel from the block aij where bij ¼ {bij(0, 0), . . . , bij(N 1, N 1)}, where bij(x, y) ¼ f (iN þ x þ 1, jN þ y þ 1). Note that the block bij includes both right vertical and bottom horizontal block boundaries of aij. Let the DCT of aij(x, y) be Aij(u, v). Then, for N ¼ 8, the DCT=IDCT transform is given by 7 P 7 P pð2x þ 1Þu Aij ðu; vÞ ¼ aðuÞaðuÞ aij ðx; yÞcos 16 x¼0 y¼0 cos
aij ðx; yÞ ¼
7 P 7 P
pð2y þ 1Þv 16
u¼0 v¼0
cos
The DCT coefficients of shifted blocks can be assembled directly from the existing DCT blocks [4]. Therefore, the proposed method is fully operated in the frequency domain without performing the DCT= IDCT operations. Estimates of computation savings vary between 41 and 64% depending on the task.
ð1Þ
aðuÞaðuÞAij ðu; vÞcos
Fig. 1 Graphical explanation of aij and bij
pð2x þ 1Þu 16
pð2y þ 1Þv 16
ð2Þ
Experimental results: In this Section, the performance of four postprocessing techniques, namely, Zakhor’s [1], Yang et al.’s [2], Paek et al.’s [3], and the proposed technique are evaluated on various still images. The decoded images, with visible blocking artifacts, are obtained by JPEG at 0.2 bit=pixel (bpp).
where 8 rffiffiffiffi 1 > > > < N; k ¼ 0 aðkÞ ¼ rffiffiffiffi ð3Þ > 2 > > : ; otherwise N In the same manner, bij(x, y) and Bij(u, v) can be formulated. To reduce the blocking artifacts in the decoded image, we first observe the characteristics of Bij, the DCT block of bij. The Bij has more nonzero AC coefficients when compared with Aij, the DCT block of aij, since Bij includes the high frequency block boundaries (blocking artifacts), i.e. nonzero coefficients of Bij occur at higher order in the zig-zag scan than highest nonzero valued location of Aij. Based on this observation, we establish an algorithm for reducing the blocking artifacts by proposing a smoothness constraint set and its projection operator. Let us define a closed convex set CBij as CBij ¼ f f jbij f ; NZðBij Þ KBij g
ð4Þ
Fig. 2 PSNR performance variation on Lena decoded image
ELECTRONICS LETTERS 30th October 2003 Vol. 39 No. 22
pictures of Lena in Figs. 3a–d. As shown in these Figures, although Zakhor’s technique alleviates the blocking artifacts, it yields an excessively smoothed image due to the use of a global lowpass filter. It is also observed that the blocking artifacts are alleviated more effectively and edges are better preserved by the proposed techniques than the other two techniques. Similar results are also observed on the other test images.
Table 1: PSNR for different post-processing methods Lena
a
Zelda Bridge Pepper Church
Zakhor [1] 27.85 30.33
b
24.98
29.27
28.77
Yang [2]
27.78 29.97
24.77
28.71
28.35
Paek [3]
27.80 29.99
24.85
29.01
28.22
Proposed
27.92 30.16
24.99
29.23
28.74
In Table 1, the PSNR performance of the proposed technique is compared with those of Zakhor, Yang et al. and Paek et al. The simulation results show that the performance of the proposed algorithm is better than those of other algorithms. # IEE 2003 Electronics Letters Online No: 20031025 DOI: 10.1049/el:20031025 c
d
Yoon Kim, Chun-Su Park and Sung-Jea Ko (School of Electrical Engineering, Korea University, 5-1 Anam-Dong, Sungbuk-Ku, Seoul 136-701, Korea)
Fig. 3 Comparison of subjective quality on Lena Post-processed images: a Zakhor’s algorithm c Paek et al.’s algorithm
1 September 2003
b Yang et al.’s algorithm d Proposed method
E-mail:
[email protected] References
In Fig. 2, the PSNR performance of the proposed technique is compared with those of Zakhor’s, Yang et al.’s and Paek et al.’s techniques, with increasing iteration. The Figure shows that the proposed technique converges within a few iterations, regardless of the input images, and can be easily implemented. We can also observe that Zakhor’s technique fails to converge, in the sense of the theory of POCS. In addition, the performance of the proposed technique is better than the techniques of Yang et al. and Paek et al. in terms of PSNR. Thus, it is believed that our technique is robust to the input images, in terms of convergence behaviour and performance improvement. To make a clear comparison of the subjective quality, we present
1 2 3 4
ZAKHOR, A.: ‘Iterative procedures for reduction of blocking effects in transform image coding’, IEEE Trans. Circuits Syst. Video Technol., 1992, 2, pp. 91–95 YANG, Y., GALATSANOS, N.P., and KATSAGGELOS, A.K.: ‘Projection-based spatially adaptive reconstruction of block-transform compressed images’, IEEE Trans. Circuits Syst. Video Technol., 1995, 5, pp. 298–304 PAEK, H., KIM, R.C., and LEE, S.U.: ‘On the pocs-based postprocessing technique to reduce the blocking artifacts in transform coded images’, IEEE Trans. Circuits Syst. Video Technol., 1998, 8, pp. 358–367 CHANG, S.-F., and MESSERSCHMITT, D.G.: ‘Manipulation and compositing of MC-DCT compressed video’, IEEE J. Sel. Areas Commun., 1995, 13, pp. 1–11
ELECTRONICS LETTERS 30th October 2003 Vol. 39 No. 22
