Integer fast lapped transforms based on direct-lifting of DCTs for lossy-to-lossless image coding
- Taizo Suzuki^{1}Email author and
- Masaaki Ikehara^{2}
https://doi.org/10.1186/1687-5281-2013-65
© Suzuki and Ikehara; licensee Springer. 2013
Received: 24 July 2012
Accepted: 3 December 2013
Published: 27 December 2013
Abstract
The discrete cosine transforms (DCTs) have found wide applications in image/video compression (image coding). DCT-based lapped transforms (LTs), called fast LTs (FLTs), overcome blocking artifacts generated at low bit rate image coding by DCT while keeping fast implementation. This paper presents a realization of more effective integer FLT (IntFLT) for lossy-to-lossless image coding, which is unified lossless and lossy image coding, than the conventional IntFLTs. It is composed of few operations and direct application of DCTs to lifting blocks, called direct-lifting of DCTs. Since the direct-lifting can reuse any existing software/hardware for DCTs, the proposed IntFLTs have a great potential for fast implementation which is dependent on the architecture design and DCT algorithms. Furthermore, the proposed IntFLTs do not need any side information unlike integer DCT (IntDCT) based on direct-lifting as our previous work. Moreover, they can be easily extended to larger size which is recently required as in DCT for the standard H.26x series. As a result, the proposed method shows better lossy-to-lossless image coding than the conventional IntFLTs.
1 Introduction
The most popular image/video compression (image coding) standards, JPEG [1, 2] and H.26x series [3, 4], employ discrete cosine transform (DCT) [5] at their transformation stages. DCT can be basically classified into types I to IV (DCT-I to IV) and has numerous fast implementations [6–10] and applications for signal processing. In them, DCT-II, so-called DCT, has excellent energy compaction capability and DCT-III is its inverse transform, so-called inverse DCT (IDCT). However, DCT generates annoying blocking artifacts at low bit rates because the DCT bases are short and create discontinuities at block boundaries due to non-overlapping. To overcome this drawback, lapped transforms (LTs), which are classified into lapped orthogonal transform (LOT) and lapped biorthogonal transform (LBT), have received much attention. DCT-based fast LTs (FLTs), which are classified into fast LOT (FLOT) and fast LBT (FLBT), are well-known as fast and effective transform for image coding [11]. FLTs are constructed by cascading DCT-II, DCT-III, DCT-IV, rotation matrices with π/4 angles, ±1 operations, scaling factors, a delay matrix, and permutation matrices. To improve the coding performance and reduce the complexity more, LiftLT with VLSI-friendly implementation has been proposed by Tran [12] ^{a}. However, the LTs cannot be applied to the lossless mode.
On the other hand, JPEG achieves the lossless mode by using differential pulse code modulation (DPCM) in place of DCT. JPEG 2000 [13] employs 9/7-tap and 5/3-tap discrete wavelet transforms (9/7-DWT and 5/3-DWT) for lossy and lossless modes, respectively [14]. They mean that JPEG and JPEG 2000 do not have compatibility between the lossy and lossless mode. Of course, lossless transform such as 5/3-DWT is applicable to lossy-to-lossless image coding. However, its lossy performance is not good compared with 9/7-DWT because each transform is suitable only in each mode. The next standard JPEG XR [15] has solved this problem by achieving lossy-to-lossless image coding which is unified lossy and lossless image coding. JPEG XR employs only hierarchical lapped transform (HLT) for both of lossy and lossless modes [16]. The HLT is composed of lifting structures [17–19] with rounding operations and achieves integer-to-integer transform, whereas it does not have enough coding performance, especially for images with many high frequency components. Various lifting-based filter banks (L-FBs) [20–28], which contain integer DCTs (IntDCTs) [29–35], have been researched to improve coding performance. However, these except for IntDCTs are not practical due to the complexity.
This paper presents a realization of integer FLT (IntFLT), which is constructed by lifting structures with rounding operations, for lossy-to-lossless image coding. Although FLT can be easily applied to lossy-to-lossless image coding by simple lifting factorizations of rotation matrices and scaling factors, the obtained integer transform is unsuitable due to large rounding error because of many rounding operations. The conventional IntFLTs also have many operations, whereas the proposed IntFLTs have simple implementations with few operations and direct application of DCTs to lifting blocks, called direct-lifting of DCTs. The direct-lifting can reuse any existing software/hardware for DCTs^{b}. As a result, although the proposed IntFLTs are apparently sacrificing the complexity to achieve the lossless mode compared with LiftLT, they have a great potential for fast implementation which is dependent on the architecture design and DCT algorithms. Furthermore, the proposed IntFLTs do not need any side information unlike IntDCT based on direct-lifting as our previous work [35]. Moreover, they can be easily extended to larger size which is recently required as in DCT for H.26x series. Such IntFLT already proposed in [36] cannot achieve enough coding performance due to the orthogonality. This paper introduces IntFLT without such a restriction. Finally, the proposed method shows better lossy-to-lossless image coding than the conventional IntFLTs.
1.1 Notations
Several special matrices with reserved symbols are as follows: I, J, 0, and D are an identity matrix, a reversal identity matrix, a null matrix, and a diagonal matrix with alternating ±1 entries (i.e., diag{1,-1,1,-1,⋯ }), respectively. Also, ·^{ T } and ·^{-1} are transpose and inverse of a matrix, respectively.
2 Review
2.1 Fast lapped transform (FLT)
2.2 Direct-lifting structure
Thus, the parallel block system of T and T^{-1} can be efficiently implemented by the block-liftings as shown at the right side of Figure 2. This is a breakthrough structure because any block T and its inverse one T^{-1} can be directly applied to the block-lifting coefficients without breaking their forms. Although any existing software/hardware for DCT cannot be directly reused for the conventional IntDCTs, we can admit any of them as the lifting blocks when T = C _{ II }.
3 IntFLTs based on direct-lifting of DCTs
This section presents a realization of IntFLT for lossy-to-lossless image coding. The IntFLTs have simple implementations with few operations and direct-lifting of DCTs.
3.1 Direct-lifting of DCTs
where C _{ III } C _{ II } = I and $\stackrel{~}{\mathbf{E}}\left(z\right)$ are used to distinguish from the original E(z) in Equation 3. Of course, $\stackrel{~}{\mathbf{E}}\left(z\right)$ is the same transfer function as E(z). The FLT with this polyphase matrix $\stackrel{~}{\mathbf{E}}\left(z\right)$ is implemented as shown at the bottom half in Figure 1.
by substituting it into Equation 4.
3.2 Lifting structure of rotation matrix with π/4 angle
Similarly, the i th column signals and the j th column signals, i.e., the red and blue areas in Figure 3, are processed by FLTs in Equations 9 and 10, respectively. Consequently, the scales are changed temporarily for fast implementation and restored after two-dimensional transform.
3.3 Lifting structure of scaling part
where ${s}_{1}={s}_{0}^{-1}$. The lifting coefficients s _{0} and s _{1} in the scaling part are empirically determined.
4 Results
4.1 Coding gain
This paper designed 8 × 16 and 16 × 32 IntFLTs. First, the comparison of coding gain of the ideal FLTs and the proposed IntFLTs is shown.
Comparisons of coding gain of the ideal FLTs and the proposed IntFLTs (dB)
Ideal FLTs | Proposed FLTs | |
---|---|---|
8 × 16 FLOT | 9.2189 | 9.2189 |
8 × 16 FLBT | 9.4475 | 9.4475 |
16 × 32 FLOT | 9.7593 | 9.7593 |
16 × 32 FLBT | 9.8455 | 9.8455 |
For comparison, the coding gain of LiftLT [12] is 9.5378 (dB) which is higher than the proposed 8 × 16 IntFLTs because this is optimized for lossy coding.
4.2 Lossy-to-lossless image coding
Lossy-to-lossless image coding results by the designed IntFLTs are shown in this subsection. As targets for comparison, LiftLT [12], 5/3-DWT and 9/7-DWT for JPEG 2000 [14], HLT for JPEG XR [16], and the conventional 8 × 16 and 16 × 32 IntFLTs were applied. The conventional 8 × 16 and 16 × 32 IntFLTs are based on simple three-step lifting factorizations of rotation matrices and scaling factors [14]. The periodic extension was used for image boundaries except for DWTs and HLT. To evaluate transform performance fairly, a very common wavelet-based zerotree coder SPIHT [39] was adopted for all^{c}. Moreover, we used 8-bit gray scale test images with 512 × 512 size such as Barbara.
Comparison of lossless image coding (LBR (bpp))
Test | 5/3-DWT | HLT | Conventional FLTs | Proposed FLTs | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Images | [14] | [16] | (A) | (B) | (C) | (D) | (E) | (F) | (G) | (H) |
Baboon | 6.25 | 6.23 | 6.24 | 6.24 | 6.23 | 6.24 | 6.23 | 6.22 | 6.22 | 6.22 |
Barbara | 4.97 | 4.96 | 5.00 | 4.95 | 4.95 | 4.93 | 4.95 | 4.85 | 4.90 | 4.83 |
Boat | 5.19 | 5.20 | 5.22 | 5.22 | 5.19 | 5.21 | 5.19 | 5.16 | 5.16 | 5.15 |
Elaine | 5.26 | 5.27 | 5.30 | 5.26 | 5.27 | 5.25 | 5.28 | 5.21 | 5.25 | 5.20 |
Finger | 5.88 | 5.89 | 5.91 | 5.79 | 5.85 | 5.78 | 5.89 | 5.75 | 5.84 | 5.75 |
Finger2 | 5.64 | 5.62 | 5.65 | 5.57 | 5.57 | 5.56 | 5.63 | 5.51 | 5.55 | 5.51 |
Goldhill | 5.08 | 5.12 | 5.21 | 5.20 | 5.17 | 5.19 | 5.18 | 5.15 | 5.15 | 5.14 |
Grass | 6.09 | 6.09 | 6.11 | 6.09 | 6.08 | 6.08 | 6.10 | 6.07 | 6.08 | 6.07 |
Lena | 4.58 | 4.64 | 4.74 | 4.77 | 4.71 | 4.75 | 4.71 | 4.69 | 4.66 | 4.67 |
Pepper | 4.96 | 5.00 | 5.03 | 5.06 | 4.99 | 5.04 | 4.99 | 5.00 | 4.96 | 4.98 |
Avg. | 5.39 | 5.40 | 5.44 | 5.42 | 5.40 | 5.40 | 5.42 | 5.36 | 5.38 | 5.35 |
Comparison of lossy image coding (PSNR (dB))
Comp. | LiftLT | 9/7-DWT | HLT | Conventional FLTs | Proposed FLTs | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Ratio | [12] | [14] | [16] | (A) | (B) | (C) | (D) | (E) | (F) | (G) | (H) | |
Baboon | 1:32 | 22.72 | 22.90 | 22.38 | 22.86 | 22.88 | 22.88 | 22.89 | 22.87 | 22.91 | 22.89 | 22.91 |
1:16 | 24.98 | 25.11 | 24.67 | 25.11 | 25.17 | 25.14 | 25.15 | 25.13 | 25.19 | 25.16 | 25.19 | |
1:8 | 28.31 | 28.55 | 28.10 | 28.49 | 28.45 | 28.48 | 28.42 | 28.55 | 28.53 | 28.52 | 28.51 | |
Barbara | 1:32 | 28.01 | 27.56 | 27.01 | 27.80 | 28.72 | 28.02 | 28.84 | 27.83 | 28.77 | 28.03 | 28.90 |
1:16 | 32.02 | 31.49 | 30.85 | 31.70 | 32.55 | 32.02 | 32.65 | 31.76 | 32.67 | 32.08 | 32.80 | |
1:8 | 37.07 | 36.28 | 36.00 | 36.33 | 36.65 | 36.60 | 36.66 | 36.59 | 37.13 | 36.84 | 37.19 | |
Boat | 1:32 | 29.20 | 29.43 | 28.80 | 28.93 | 28.98 | 29.21 | 29.08 | 28.97 | 29.04 | 29.23 | 29.13 |
1:16 | 32.36 | 32.45 | 32.02 | 32.05 | 32.00 | 32.26 | 32.08 | 32.13 | 32.13 | 32.33 | 32.22 | |
1:8 | 35.70 | 35.50 | 35.21 | 35.20 | 35.07 | 35.23 | 35.04 | 35.44 | 35.43 | 35.46 | 35.40 | |
Elaine | 1:32 | 31.83 | 31.99 | 31.54 | 31.28 | 31.29 | 31.55 | 31.42 | 31.32 | 31.35 | 31.61 | 31.50 |
1:16 | 32.86 | 32.98 | 32.20 | 32.39 | 32.63 | 32.53 | 32.74 | 32.49 | 32.78 | 32.62 | 32.87 | |
1:8 | 35.00 | 34.77 | 34.24 | 34.46 | 34.97 | 34.44 | 34.94 | 34.67 | 35.32 | 34.69 | 35.26 | |
Finger | 1:32 | 23.42 | 23.69 | 22.95 | 23.56 | 23.94 | 23.71 | 23.99 | 23.57 | 23.97 | 23.72 | 24.02 |
1:16 | 26.62 | 26.92 | 26.31 | 26.76 | 27.27 | 26.92 | 27.29 | 26.79 | 27.32 | 26.94 | 27.34 | |
1:8 | 30.67 | 30.50 | 30.12 | 30.57 | 31.29 | 30.76 | 31.30 | 30.64 | 31.44 | 30.82 | 31.44 | |
Finger2 | 1:32 | 24.36 | 24.63 | 23.27 | 24.40 | 24.72 | 24.62 | 24.79 | 24.42 | 24.76 | 24.63 | 24.82 |
1:16 | 27.73 | 28.04 | 27.09 | 27.84 | 28.19 | 28.09 | 28.26 | 27.86 | 28.25 | 28.12 | 28.32 | |
1:8 | 31.83 | 31.85 | 31.39 | 31.80 | 32.24 | 32.10 | 32.23 | 31.89 | 32.42 | 32.19 | 32.39 | |
Goldhill | 1:32 | 29.72 | 30.06 | 29.62 | 29.39 | 29.53 | 29.66 | 29.62 | 29.43 | 29.59 | 29.69 | 29.68 |
1:16 | 32.35 | 32.37 | 32.02 | 31.97 | 32.06 | 32.15 | 32.10 | 32.05 | 32.20 | 32.22 | 32.25 | |
1:8 | 35.57 | 35.36 | 35.17 | 35.03 | 34.98 | 35.06 | 34.93 | 35.26 | 35.36 | 35.29 | 35.30 | |
Grass | 1:32 | 24.44 | 24.63 | 24.26 | 24.58 | 24.62 | 24.66 | 24.63 | 24.61 | 24.66 | 24.68 | 24.65 |
1:16 | 26.65 | 26.73 | 26.37 | 26.76 | 26.87 | 26.81 | 26.87 | 26.78 | 26.92 | 26.84 | 26.92 | |
1:8 | 29.67 | 29.62 | 29.39 | 29.74 | 29.90 | 29.77 | 29.86 | 29.80 | 30.00 | 29.81 | 29.98 | |
Lena | 1:32 | 32.80 | 33.46 | 32.76 | 32.27 | 32.44 | 32.83 | 32.69 | 32.37 | 32.55 | 32.89 | 32.83 |
1:16 | 36.04 | 36.32 | 35.90 | 35.42 | 35.42 | 35.92 | 35.65 | 35.59 | 35.71 | 36.08 | 35.96 | |
1:8 | 39.02 | 38.83 | 38.62 | 37.95 | 37.57 | 38.09 | 37.64 | 38.47 | 38.53 | 38.55 | 38.50 | |
Pepper | 1:32 | 32.35 | 32.89 | 32.52 | 31.62 | 31.73 | 32.33 | 31.99 | 31.68 | 31.84 | 32.38 | 32.13 |
1:16 | 34.69 | 35.13 | 34.71 | 34.13 | 33.78 | 34.58 | 34.09 | 34.29 | 34.02 | 34.73 | 34.34 | |
1:8 | 37.07 | 36.79 | 36.39 | 36.18 | 35.94 | 36.21 | 35.91 | 36.50 | 36.44 | 36.53 | 36.41 |
Comparison of number of rounding operations in each one-dimensional transform of M × 1 signals
Conventional FLTs | Proposed FLTs | |
---|---|---|
8 × 16 FLOT | 72 | 36 |
8 × 16 FLBT | 84 | 48 |
16 × 32 FLOT | 240 | 90 |
16 × 32 FLBT | 264 | 114 |
5 Conclusions
This paper presented integer fast lapped transforms (IntFLTs) for effective lossy-to-lossless image coding, which were constructed by few operations and direct-lifting of discrete cosine transforms (DCTs). Due to merging, many rounding operations and keeping small lifting coefficients by use of direct-lifting, the proposed IntFLTs performed better coding than the conventional IntFLTs in lossy-to-lossless image coding. Also, the proposed IntFLTs can preserve the high frequency components in the images. Since the direct-lifting can reuse any existing software/hardware for DCTs, the proposed IntFLTs have a great potential for fast implementation which is dependent on the architecture design and DCT algorithms. Furthermore, the proposed IntFLTs do not need any side information unlike IntDCT based on direct-lifting as our previous work.
Endnotes
^{a} “The conventional IntFLTs” do not include LiftLT in this paper.
^{b} Any other lifting-based DCTs cannot reuse all existing software/hardware for DCTs.
^{c} The block transform coefficients through 2^{ k }-channel ($k\in \mathbb{N}$) FLTs are rearranged to a k-level wavelet-like multi-resolution representation, and they are applied to the zerotree coder [40], e.g., 3-level wavelet-like multi-resolution representation when M = 8.
^{d} The IntFLT referred by [20] has less rounding operations. However, it performs undesirable coding due to large lifting coefficients. For example, although the 8 × 16 FLOT has only five rounding operations in each 4 × 4 DCT, its application of lossless image coding shows 5.06 (bpp) for Barbara.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for providing many constructive suggestions that significantly improve the presentation of this paper. This work was supported by JSPS Grant-in-Aid for Young Scientists (B) grant number 25820152.
Authors’ Affiliations
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