Optimum word length allocation of integer DCT and its error analysis
Introduction
The discrete cosine transform (DCT) is a well-known transform used in many international standards of image compression such as JPEG [6] and MPEG [4]. The DCT-based systems have huge advantage to image applications because they provide a high compression ratio. However, their coding systems are limited to operating in only lossy coding because distortion of decoded image is unavoidable with these lossy algorithms.
On the other hand, the integer transform [1], which includes rounding operations in the lifting structure [9], is becoming popular as a key technique to lossless and lossy unified waveform coding [11]. Especially the integer DCT [7], [2], [3] is attractive as the unified coding with compatibility to the conventional DCT-based algorithms. In Fig. 1, encoder applied the conventional lossy DCT, whereas decoder applied the Int-DCT to illustrate its compatibility to the conventional DCT-based algorithms. Notice that the coding performance of the Int-DCT is similar to that of the conventional lossy DCT in a low bit-rate but it is slightly worse than that of the conventional lossy DCT in a high bit-rate because of rounding error discussed in Section 4.2.
So far, relevant to the integer DCT, previous reports focused on reducing the rounding operations with the non-separable 2D structuring [7] and reducing multipliers with the integer Hadamard transform [3]. Optimization of the basis function of the orthogonal transform (integer KLT) is also reported [10]. What seems to be lacking, however, is how to express multipliers’ word length as short as possible for the reduction of hardware complexity.
In this report, we define a new “SNR sensitivity” as an indicator of how the word length truncation of multiplier coefficients affects quality of a reconstructed image. Based on the newly defined sensitivity, we propose a new word length allocation method. We also theoretically analyze errors in a reconstructed signal to confirm an effectiveness of the proposed method. This report is organized as follows. Overview of the integer DCT is summarized in Section 2. An error generated from finite word length allocation is theoretically analyzed in Section 3 and errors in a reconstructed signal are theoretically analyzed in Section 4. The “SNR sensitivity” is newly defined and applied to an optimum word length allocation using the least square method in Section 5. An effectiveness of the proposed method is confirmed in Section 6.
Section snippets
The integer DCT (Int-DCT) [6–8]
Algorithm of the integer DCT (Int-DCT), illustrated in Fig. 2, is composed of the 4-point integer Hadamard transform (4-IHT) and integer rotation transform (IRT) described in 2.2 The 4-point integer Hadamard transform (4-IHT), 2.3 Integer rotation transform (IRT), respectively. The integer DCT transforms integer input vector x(n), (n=0,1,…,7) into integer output vector y(n), (n=0,1,…,7). Therefore, it is possible to achieve effective lossless coding by applying an entropy coding directly to the
Finite word length expression
The multiplier coefficient mj(i), (i=A, B, C, D, E and j=1,2,3), is expressed as hk, (k=0,1,…,14), bywhere Bj (j=0,1,…) is 0 or 1. Under the finite word length expression in this report, hk is truncated into Wk [bit] binary value hk′. Namely,
Value Δhk is defined as a difference between value hk and binary value hk′ as
An error generated from finite word length allocation
Considering errors generated from finite word length allocation, we can find an equivalent circuit of
Analysis on errors in a reconstructed signal
In this section, we analyze errors between an original signal and a reconstructed signal. A variance of the errors (σE2) is calculated fromwhere x(n) and x′(n) denote an original signal and a reconstructed signal, respectively. “n” denotes a sequence of input signal where “n”=0,1,2,…,N−1.
The SNR sensitivity
From , , , , we can rewrite errors generated from finite word length allocation aswhere the called “SNR sensitivity” is defined as an effect of the finite word length expression on a quality of the decoded image.where
Simulation results
In this section, we practically confirm an effectiveness of the optimum word length allocation by applying AR(1) model and standard images as input signals in 6.1 Simulation results based on AR(1) model, 6.2 Simulation results based on standard images, respectively. In this report, we emphasize on finite-word-length effect (the different coefficients are used between encoder and decoder), so we consider an effectiveness of the proposed method in two conditions: no quantization and a small
Conclusion
In this report, the “SNR sensitivity” was newly defined as an indicator of how the word length truncation of multiplier coefficients affects the quality of a reconstructed image. We proposed a new word length allocation method based on the SNR sensitivity. The optimum word length allocation depends on a frequency spectrum of an input signal. Both theoretical analysis and simulation results confirm an effectiveness of the proposed method.
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