Optimum word length allocation of integer DCT and its error analysis

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Abstract

Recently, the integer DCT (Int-DCT), which transforms an integer input to an integer output, is attracting many researchers’ attention as an effective method for DCT-based lossy/lossless unified coding. So far, there are many reports relevant to the Int-DCT, but they have been limited to a few topics such as how to reduce the number of multipliers with the four-point lossless Hadamard transform and the non-separable two-dimensional Int-DCT. However, none of them is focused on how to express the 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 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. As a result, the optimum word length allocation, which depends on a frequency spectrum of an input signal, significantly improves quality of a reconstructed image.

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), byhk=(−1)B0·∑j=1∞Bj2−j,k=0,1,…,14,where 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,hk′=(−1)B0·∑j=1WkBj′2−j,k=0,1,…,14.

Value Δhk is defined as a difference between value hk and binary value hk′ asΔhk=hk−hk′.

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 fromσE2=1N∑n=0N−1{x′(n)−x(n)}2,where 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 (NTF) asNTF=∑k=014(SHk·Δhk),where the SHk called “SNR sensitivity” is defined as an effect of the finite word length expression on a quality of the decoded image.SH(j)=F1−1·NHF(j)·F1·Xforj=0,1,2F2−1·NHF(j)·F2·Xforj=3,4,5,6,7,8F3−1·NHF(j)·F3·Xforj=9,10,11,12,13,14,whereNHF(j)=HFjZ4Z4Z4forj=0,1,2,…,8,Z4Z4Z4HFjforj=9,10,11,…,14,HF0=Z2Z2Z2GF1A,HF1=Z2Z2Z2GF2A,HF2=Z2Z2Z2GF3A,HF3=GF1BZ2Z2Z2,HF4=GF1BZ2Z2Z2,HF5=GF1BZ2Z2Z2,HF6

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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