BLOCK ARTIFACT REDUCTION BY OPTIMIZATION UTILIZING SUBBAND DECOMPOSITION Hwayong Joung, Uipil Chong, and S. P. Kim
Department of Electrical Engineering Polytechnic University Brooklyn, New York, 11201
[email protected] [email protected] [email protected] ABSTRACT
The block discrete cosine transform (BDCT) is extensively used for compression of image and videos. One of its disadvantages, however, is the blocking artifacts at the block boundaries at low bit-rates. Most previous techniques for removing block artifacts utilize lowpass ltering or projection-on-convex-set (POCS) concept that gives an iterative restoration. These techniques produce unnecessary blurring in the image and require high computational power. In this paper, we introduce a new method for removing blocking e ects using an optimization technique and localized ltering through subband decomposition. We use transform domain ltering (TDF) technique in order to reduce the computational complexity. The subband decomposition allows localized ltering around the block boundaries where block artifacts would occur. The DCT coecient quantization error correction utilizes Kuhn-Tucker theorem which is a Lagrangian method with inequalty constraints.
1. INTRODUCTION In the compression of image and video coding standards, the block discrete cosine transform (BDCT) is the most popular transform. Each of transform coecients from transform blocks (usually 88) are quantized before further (lossless) compression. Quantization step sizes vary among transform coecients as well as frames in order to satisfy a given bit-rate constraint. When large quantization step sizes are used for quantization of BDCT coecients, the well known \block artifacts" would occur at the block boundaries. Our proposed approach has two stages of processing. The rst stage is based on the direct minimization of a cost function that represents discontinuity of block boundaries. Hence, no iteration is required. Note that This work was supported by Samsung Electronics
the block artifacts are essentially due to quantization errors of DCT coecients. The variables in the restoration process are DCT coecient correction factors. Since the the quantization errors are bounded by the quantization step sizes, the minimization is constrainted by the quantization step sizes of DCT coecients. In practice, it is dicult to correct all the quantization errors due to a large size system of equations. In the proposed approach only a few low frequency components that have most signi cant e ect are corrected. Further reduction of block artifact is obtained through localized ltering. This is based on the subband decomposition [5],[6] using FIR lters. These optimization and subband decomposition are directly processed in the DCT domain. The algorithm also does not require iterative computation as POCS based approaches do. As a result, the restoration can be implemented directly on the incoming DCT coecients. Since a large portion of DCT coecients are zero in practice, the amount of computations are signi cantly smaller than the equivalent spatial domain processing. The rest of this paper is organized as follows. In section 2, we formulate the restoration as optimization problems based on two di erent cost functions; the rst based on the continuity of two adjacent block boundary pixels and the second on the continuity of slopes across the boundary. Section 3 describes how the subband decomposition can be applied to the block e ect reduction. In Section 4 computer simulation and experimental results are provided. Finally, in Section 5 we present our conclusion and furture directions.
2. PROBLEM FORMULATION BASED ON OPTIMIZATION METHOD 2.1. Jump Discontinuity of Block Boundary
The block artifact is mainly due to jump discontinuities around the block boundaries. It is natural to try to minimize such discontinuities in order to achieve an
Block Boundary
North w(n1 ; N
? 1)
n(N
Neighbor Block
? 1; n2 )
Current Block
f (n1 ; n2 )
West
Current
N-3N-2N-1 0 1 2
Figure 1: Jump Discontinuity improved image quality. The discontinuity constraint around the causal block boundaries (i.e., north and west) of the current block gives the following cost function JfFg to be minimized.
JfFg =
NX ?1
jf (n1 ; 0) ? w(n1 ; N ? 1)j2
n1 =0 NX ?1
+
n2 =0
jf (0; n2 ) ? n(N ? 1; n2 )j2 (1)
where w(i; j ) and n(i; j ) represent the pixels along the west and north boundaries of the neighbor blocks as depicted in Fig. 1. To minimize JfFg by restoring the quantized DCT coecients, it is necessary to represent the cost function in the DCT domain. It is straightforward to express the above jump discontinuities in the DCT domain as follows.
JfFg = (Dt FDUt ? w )t(Dt FDUt ? w ) +(UDt FD ? n )(UDt FD ? n)t : (2) The variable F, the argument of the cost function JfFg, represent a correction vector of DCT coecients. Using the relation F = Fq + F, JfFg = (DtFDUt + Cw )t (Dt FDUt + Cw ) +(UDt FD + Cn )(UDt FD + Cn )t = At1 A1 + A2 At2 (3) where
A1 = DtFDUt + Cw ; Column Matrix A2 = UDtFD + Cn ; Row Matrix: Notice that the cost functional JfFg is a quadratic function of F. If the optimization is unconstrainted, nding the optimal solution will be straightforward for minimizing JfFg. Since, however, F should satisfy a
set of given inequality constraints set by the quantizer in the encoder, the solution can be obtained by the KuhnTucker theorem [4]. If F is Kuhn-Tucker point, the Kuhn-Tucker theorem gives necessary condition but it is also sucient if the cost function JfFg is a convex functional as in this case. Suppose we want to adjust the three lowest frequency components such that the block e ect can be reduced. Then F has only three components as shown below. Also de ne the elements of the DCT base matrix as follows. 2
x1 6 x 3 F = 664 .. . 0
0 3 0 77
x2 0 .. . 0
2
a00 6 a 6 10 . . . .. 75 ; D = 64 .. . . 0 a70
a01 a11 .. . a71
a07 3 a17 77
. . . .. 75 :(4) . a77
Then the terms that appear in the cost function are expanded as 2
DtFDUt =
6 6 6 4
a00 a00 x1 + a00 a10 x2 + a10 a00 x3 a01 a00 x1 + a01 a10 x2 + a11 a00 x3 .. . a07 a00 x1 + a07 a10 x2 + a17 a00 x3
3 7 7 7 5
and the cost function is expanded as
JfFg = At1 A1 + A2 At2
= c0 x21 + c1 x22 + c2 x23 + c3 x1 x2 + c4 x1 x3 +c5 x2 x3 + c6 x1 + c7 x2 + c8 x3 + c9 :
JfFg is a quadratic function of x1; x2 and x3 and obviously is a convex function of the variables. Hence the sucient condition is always satis ed if F is KuhnTucker point. The restoration problem can now be stated as a constrainted optimization problem as follows. Minimize JfFg = c0 x21 + c1x22 + c2 x23 + c3 x1 x2 + c4 x1 x3 +c5 x2 x3 + c6 x1 + c7 x2 + c8 x3 + c9
g1 (F) = ?x1 ? q1 0 g ?q1 x1 q1 g32 ((FF)) == ?x1x?2 ?q1q2 0 0 subject to : ?q2 x2 q2 )> g (F) = x ? q 0 4 2 2 ?q3 x3 q3 > > > g ( F ) = ? x ? q3 0 > 5 3 > : g6 (F) = x3 ? q3 0 8
> > > > >
