Invertible update-then-predict integer lifting wavelet for lossless image compression
- Dong Chen^{1}Email author,
- Yanjuan Li^{2},
- Haiying Zhang^{3} and
- Wenpeng Gao^{1}
DOI: 10.1186/s13634-016-0443-y
© The Author(s) 2017
Received: 6 July 2016
Accepted: 30 December 2016
Published: 14 January 2017
Abstract
This paper presents a new wavelet family for lossless image compression by re-factoring the channel representation of the update-then-predict lifting wavelet, introduced by Claypoole, Davis, Sweldens and Baraniuk, into lifting steps. We name the new wavelet family as invertible update-then-predict integer lifting wavelets (IUPILWs for short). To build IUPILWs, we investigate some central issues such as normalization, invertibility, integer structure, and scaling lifting. The channel representation of the previous update-then-predict lifting wavelet with normalization is given and the invertibility is discussed firstly. To guarantee the invertibility, we re-factor the channel representation into lifting steps. Then the integer structure and scaling lifting of the invertible update-then-predict wavelet are given and the IUPILWs are built. Experiments show that comparing with the integer lifting structure of 5/3 wavelet, 9/7 wavelet, and iDTT, IUPILW results in the lower bit-rates for lossless image compression.
Keywords
Integer lifting Invertibility Lossless image compression Update-then-predict Wavelet1 Introduction
Discrete wavelet transforms and perfect reconstruction filter banks have become one of the dominant technologies in numerous areas such as signal and image processing [1–3]. The second-generation wavelets based on lifting scheme have achieved substantial recognition [4–6], which are used in the fields of signal analysis [7], image coding [8–11], palmprint identification [12], moving object detection [13], especially since their integration in the JPEG2000 standard [14–18]. The lifting scheme is an efficient and powerful tool to compute the wavelet transform. It can improve the key properties of the first-generation wavelet step by step. Moreover, it has many advantages compared to the first-generation wavelet such as in-place computation, integer-to-integer transforms, and speed.
Update-first structure is useful to build the adaptive lifting wavelet [19, 20]. G. Piella and B. Pesque-Popescu present some adaptive wavelet decompositions that can capture the directional nature of images [20]. Claypoole, Davis, Sweldens, and Baraniuk introduce a kind of nonlinear wavelet transform for image coding via lifting [21]. To keep the stability and eliminate the propagation of error, they constructed the update-then-predict lifting wavelet using Donoho’s average-interpolation [22], and they apply it to construct the nonlinear wavelet transforms. However, unfortunately it is not perfect invertible for lossless image compression using integer-to-integer structure because there is a fractional factor 1/2 in its low-pass channel (see Fig. 3), which will be discussed in detail in Section 2.2 in this paper.
Our contributions can be summarized as follows. (1) The update-then-predict lifting structure is reviewed and its limitation is given in Section 2.2. Our analysis shows that the fractional factor 1/2 destroys the perfect reconstruction property of the integer structure of update-then-predict wavelet and makes the structure is not invertible. (2) The solution method is given. To perfect the update-then-predict lifting structure, we consider some central issues such as normalization, invertibility, integer structure, and scaling lifting. We re-factor the channel representation of the previous update-then-predict lifting wavelet with normalization into lifting steps, and then the invertible update-then-predict integer lifting wavelets (IUPILWs) for lossless image compression is obtained and named in Sections 3.1 to 3.3. (3) The computational complexity analysis and comparison between IUPILWs and other methods are given in Section 3.4. Furthermore, the experimental comparison and analysis for lossless image compression are given and the advantages of our IUPILWs are introduced in Section 4.
The remainder of the paper is organized as follows. Section 2 gives a brief description of the background of integer-to-integer lifting wavelet and update-then-predict lifting wavelet. Section 3 introduces the invertible update-then-predict integer lifting wavelets with scaling lifting. According to reference [21], the channel representation of the update-then-predict lifting wavelet with normalization is given firstly. Furthermore, we re-factor the channel representation into lifting steps and then the invertibility is guaranteed. Then the integer structure and scaling lifting of the invertible update-then-predict wavelet filter banks are investigated. Finally, the computational complexity is analyzed. Sections 4 and 5 give the experiments and conclusion, respectively.
2 Integer-to-integer lifting wavelet and update-then-predict lifting wavelet
2.1 Integer-to-integer lifting wavelet
Figure 1 denotes the analysis part and synthesis part of integer lifting structure. In Fig. 1 a, the “Round()” operations are given following the steps prediction p(z) and update u(z), respectively. However, the scaling factors K and K ^{−1} (K≠1) make the approximate coefficients a(z) or detail coefficients d(z) are not the integer point numbers, then make the structure is not integer-to-integer.
One solution method we can imagine is omitting the scaling factors K and K ^{−1} in Fig. 1 a, b. If the scaling factors are omitted, the approximate coefficients a(z) and detail coefficients d(z) are all integer point numbers via lifting wavelet transform. Therefore, it seems that the integer-to-integer lifting is achieved. However, the problem is whether the structure obtained by omitting the scaling factors is a kind of wavelet filter with normalization. Obviously the answer is no. The reason is that the lifting wavelets are usually obtained by factoring the traditional wavelets, and the scaling factors are the important parts of the factoring. If we omit the scaling factors, the structure of the traditional wavelet is also destroyed. The function of the scaling factors is to keep the same energy for the coefficients in different scale. “Keeping the same energy” is important to image compression, it can make the encoding algorithm using less bits to encode the wavelet coefficients. Therefore, the method by omitting the scaling factors is not a good choice.
Another solution method is to lift the scaling factors, which is introduced in [4]. We will review the lifting of scaling factors and build our invertible update-then-predict integer lifting wavelet filter bank with scaling lifting in Section 3.3.
2.2 Update-then-predict lifting wavelet with normalization
The update-then-predict lifting wavelets are introduced in reference [21] by Claypoole, Davis, Sweldens, and Baraniuk. To ensure the stability of the wavelet transform for the image coding, the authors introduced the update-then-predict lifting structure and applied them to design the nonlinear wavelet. In [21], the authors discussed the advantages of the update-then-predict lifting structure. That is, comparing with the predict-then-update lifting structure, it has more stability and synchronization.
Prediction filters of update-then-predict lifting wavelets (UPLWs)
N | z ^{−k } | ||||||
---|---|---|---|---|---|---|---|
z ^{−3} | z ^{−2} | z ^{−1} | z ^{0} | z ^{1} | z ^{2} | z ^{3} | |
1 | −1 | ||||||
3 | \(1 \over 8\) | −1 | \(-1 \over 8 \) | ||||
5 | \(-3 \over {128}\) | \({11} \over {64}\) | −1 | \(-{11} \over {64}\) | \(3 \over {128}\) | ||
7 | \(5 \over {1024}\) | \(-{11} \over {256}\) | \({201} \over {1024}\) | −1 | \(-{201} \over {1024}\) | \({11} \over {256}\) | \(-5 \over {1024}\) |
To build the integer-to-integer lifting structure of Fig. 3, we can replace the analysis part and synthesis part using the structure in Fig. 1 a, b. However, the factor 1/2 in Fig. 3 must be remained and it is an obstacle for the implementation of integer-to-integer. For example, considering the situation there is a “Round()” operation after fractional factor 1/2, then after multiplying by the fractional factor 1/2, the integer values 7 and 8 have the same “Round()” value 4, but we cannot reconstruct the original integer values 7 and 8 using the same value 4 in the synthesis part. That is, the factor 1/2 destroys the perfect reconstruction property of the integer structure of update-then-predict wavelet and makes the structure is not invertible. Therefore, we will discuss how to preserve the perfect reconstruction property of the update-then-predict lifting wavelet and then give the design of the invertible update-then-predict lifting wavelet in Section 3.
3 Invertible update-then-predict integer lifting wavelets with scaling lifting
In this section, the polyphase representation and channel representation of the update-then-predict wavelet in Fig. 3 are given firstly. Secondly, the invertible update-then-predict lifting wavelet is obtained by re-factoring the channel representation into lifting steps. Then the integer structure of the invertible update-then-predict lifting wavelet with scaling lifting are constructed. Finally, the computational complexity is analyzed.
3.1 Channel representation of the update-then-predict wavelet filter bank
\(h_{e} (z)=\sum \limits _{k} {h_{2k} z^{-k}} \) and \(h_{o} (z)=\sum \limits _{k} {h_{2k+1} z^{-k}}\)
Therefore, the channel representation of the update-then-predict wavelet filter bank is obtained. In the next section, we will construct the invertible update-then-predict lifting wavelet filter bank by re-factoring the channel representation into lifting steps.
3.2 Re-factoring channel representation into lifting steps
The channel representation of wavelet filter bank can be factored into lifting steps using Euclidean algorithm [4]. In this section, we factor the synthesis low-pass filter h(z)(see Eq. (11)) into lifting steps and then the synthesis polyphase matrix P ^{new}(z) is obtained. Furthermore, the conjugate transpose matrix \(\tilde {P}^{\text {new}}(z^{-1})^{t}\) of analysis polyphase matrix can be given. Therefore, the factor 1/2 in Fig. 3 is gone and the invertible update-then-predict lifting wavelet filter bank is built.
Prediction filters of invertible update-then-predict lifting wavelet filter bank
N | z ^{−k } | ||||||
---|---|---|---|---|---|---|---|
z ^{−3} | z ^{−2} | z ^{−1} | z ^{0} | z ^{1} | z ^{2} | z ^{3} | |
1 | \(-{1 \over 2}\) | ||||||
3 | \({1 \over {16}}\) | \(-{1 \over 2}\) | \(-{1 \over {16}}\) | ||||
5 | \(-{3 \over {256}}\) | \({{11} \over {128}}\) | \(-{1 \over 2}\) | \(-{{11} \over {128}}\) | \({3 \over {256}}\) | ||
7 | \({5 \over {2048}}\) | \(-{{11} \over {512}}\) | \({{201} \over {2048}}\) | \(-{1 \over 2}\) | \(-{{201} \over {2048}}\) | \({{11} \over {512}}\) | \(-{5 \over {2048}}\) |
Comparing Table 2 with Table 1, we know that the slight difference is each value in Table 2 is the half of the corresponding value in Table 1. That is, \(p^{\text {new}}(z)={1 \over 2}p(z)\), where p(z) is the prediction filter in original update-then-predict lifting wavelets, and p ^{new}(z) is the prediction filter in our invertible update-then-predict lifting wavelets. Another difference between these two lifting structure are u ^{new}(z)=2×u(z) and the factor 1/2 is omitted in our invertible update-then-predict lifting structure.
Comparing Fig. 6 with Fig. 3, we note that there are some differences between them. First, the factor 1/2 in Fig. 3 is omitted in Fig. 6. This means the invertibility of the update-then-predict integer lifting wavelet can be guaranteed. Second, the update filter and prediction filter are different. Finally, the scaling factor K is different, \(K=\sqrt 2 \) in Fig. 3, but \(K=1 \left /\right. {\sqrt 2 }\) in Fig. 6.
Comparing Figs. 7 and 6, we know that just the operations “Round()” are added and followed prediction filter p(z) and update filter u(z). The operations “Round()” ensure the invertible of the prediction step and update step. Note that the above structure is not completely invertible because the scaling factors K and K ^{−1} are included in it. Therefore, we will discuss the scaling lifting (focus to K and K ^{−1}) in the next section.
3.3 Invertible update-then-predict integer lifting wavelet filter bank with scaling lifting
In Fig. 9, the update filter u(z) is given in Eq. (21), that is, u(z)=1. The prediction filters p(z) are given in Table 2. The integer lifting and scaling lifting are achieved by using the matrix factoring (see Eq. (24)) and rounding-off operations. The structure in Fig. 9 is perfect invertible, which means the processes from signal x(z) to a(z) and d(z), the process from a(z) and d(z) to the reconstruction signal \(\hat {x}(z)\) are all lossless, and the result \(\hat {x}(z)=x(z)\) can be obtained. We name the above new update-then-predict wavelet family as invertible update-then-predict integer lifting wavelets (IUPILWs), and we will do some experiment comparisons between IUPILWs and the integer lifting structure of 5/3 wavelet, 9/7 wavelet, and iDTT for lossless image compression in Section 4.
3.4 Computational complexity
In this section, we discuss the computational complexity of IUPILWs, integer lifting 5/3-wavelet, integer lifting 9/7-wavelet, and iDTT based on the lossless image compression. The unit we use to analyze the computation complexity is the cost, measured in number of multiplications, additions, and roundings. Besides, the scaling lifting (see Fig. 8) step can give four multiplications, four additions, and three rounds. For image compression, we suppose the size of image is m×n, where m is the height of the image and n is the width of the image.
Cost of analysis part (IUPILWs)
Item | No. of | No. of | No. of | Sum |
---|---|---|---|---|
multiplication | addition | rounding | ||
u(z) | 1 | 0 | 0 | 1 |
Round after u(z) | 0 | 0 | 1 | 1 |
+ after u(z) | 0 | 1 | 0 | 1 |
p(z) | N | N −1 | 0 | 2N −1 |
Round after p(z) | 0 | 0 | 1 | 1 |
+ after p(z) | 0 | 1 | 0 | 1 |
Scaling lifting | 4 | 4 | 3 | 11 |
Sum | N + 5 | N + 5 | 5 | 2N + 15 |
Also considering the synthesis part of IUPILWs, the size of image, the row and column lifting, we obtain the number of multiplications, additions, and rounding for IUPILWs is 2×(2N+15)×m×n.
Cost of analysis part (integer lifting 5/3-wavelet)
Item | No. of | No. of | No. of | Sum |
---|---|---|---|---|
multiplication | addition | rounding | ||
p(z) | 2 | 1 | 0 | 3 |
Round after p(z) | 0 | 0 | 1 | 1 |
+ after p(z) | 0 | 1 | 0 | 1 |
u(z) | 2 | 1 | 0 | 3 |
Round after u(z) | 0 | 0 | 1 | 1 |
+ after u(z) | 0 | 1 | 0 | 1 |
Scaling lifting | 4 | 4 | 3 | 11 |
Sum | 8 | 8 | 5 | 21 |
Also considering the synthesis part of integer lifting 5/3-wavelet, the size of image, the row and column lifting, we obtain the number of multiplications, additions, and rounding for integer lifting 5/3-wavelet is 2×21×m×n.
Cost of analysis part (integer lifting 9/7-wavelet)
Item | No. of | No. of | No. of | Sum |
---|---|---|---|---|
multiplication | addition | rounding | ||
α(1 + z) | 2 | 1 | 0 | 3 |
Round after α(1 + z) | 0 | 0 | 1 | 1 |
+ after α(1 + z) | 0 | 1 | 0 | 1 |
β(1 + z) | 2 | 1 | 0 | 3 |
Round after β(1 + z) | 0 | 0 | 1 | 1 |
after β(1 + z) | 0 | 1 | 0 | 1 |
γ(1 + z) | 2 | 1 | 0 | 3 |
Round after γ(1 + z) | 0 | 0 | 1 | 1 |
+ after γ(1 + z) | 0 | 1 | 0 | 1 |
δ(1 + z) | 2 | 1 | 0 | 3 |
Round after δ(1 + z) | 0 | 0 | 1 | 1 |
+ after δ(1 + z) | 0 | 1 | 0 | 1 |
Scaling lifting | 4 | 4 | 3 | 11 |
Sum | 12 | 12 | 7 | 31 |
Also considering the synthesis part of lifting 9/7-wavelet, the size of image, the row and column lifting, we obtain the number of multiplications, additions, and rounding for integer lifting 9/7-wavelet is 2×31×m×n.
Cost of IUPILWs, integer lifting 5/3, integer lifting 9/7, and iDTT
Wavelets | Cost (multi., add., and roundings) |
---|---|
IUPILWs | 2×(2N+15)×m×n |
Integer lifting 5/3 | 2×21×m×n |
Integer lifting 9/7 | 2×31×m×n |
iDTT | 4×m×n |
Time cost of IUIPLWs, integer lifting 5/3, integer lifting 9/7, and iDTT
Wavelets | Time cost (unit: ms) |
---|---|
IUPILWs | 483 (N=5) |
Integer lifting 5/3 | 469 |
Integer lifting 9/7 | 516 |
iDTT | 47 |
4 Experiments
For example, for a 512 ×512 8-bit gray-scale image, letting the “final code file” equal to the “original image”, then the value of “bitRates” is “8”. It means that the encoding for each pixel of the original image consists of 8-bit. Obviously, the small value of “bitRates” means the less encode bits for each pixel of original image.
Bit-rates (bit/pixel) for lossless image compression (18 images)
Image | 5/3-wavelet | 9/7-wavelet | iDTT | IUPILW-(1, 1) | IUPILW-(1, 3) | IUPILW-(1, 5) |
---|---|---|---|---|---|---|
Baboon | 6.029224 | 6.065701 | 6.053941 | 6.217995 | 6.040749 | 5.986946 |
Barbara | 5.040646 | 5.079731 | 5.186513 | 5.468422 | 5.100750 | 5.037720 |
Bike | 5.690926 | 5.806850 | 5.622491 | 5.440147 | 5.614979 | 5.626175 |
Bridge | 5.922947 | 5.956379 | 5.951451 | 6.127350 | 5.930901 | 5.877102 |
Couple | 5.135845 | 5.166874 | 5.159809 | 5.358124 | 5.142929 | 5.056225 |
Crowd | 4.457260 | 4.482311 | 4.541467 | 5.007710 | 4.518784 | 4.421471 |
Elaine | 5.208218 | 5.285564 | 5.266576 | 5.374374 | 5.202930 | 5.139389 |
Goldhill | 5.104885 | 5.141258 | 5.133493 | 5.332794 | 5.114780 | 5.034573 |
Lake | 5.385849 | 5.409073 | 5.403192 | 5.592735 | 5.395832 | 5.316547 |
Lena | 4.532589 | 4.611752 | 4.567688 | 4.858097 | 4.558022 | 4.514534 |
Man | 4.909214 | 4.934772 | 4.942688 | 5.237564 | 4.936646 | 4.869335 |
Milkdrop | 4.106567 | 4.187325 | 4.148641 | 4.324787 | 4.112820 | 4.023701 |
Peppers | 4.872906 | 4.954685 | 4.902747 | 5.090134 | 4.877186 | 4.823788 |
Plane | 4.323769 | 4.370487 | 4.367616 | 4.650158 | 4.347626 | 4.250137 |
Portofino | 5.170605 | 5.233532 | 5.213912 | 5.324055 | 5.172428 | 5.094902 |
Woman1 | 5.010029 | 5.086510 | 5.045542 | 5.231766 | 5.014275 | 5.101509 |
Woman2 | 3.596012 | 3.690968 | 3.664954 | 4.023357 | 3.628143 | 3.543446 |
Zelda | 4.255531 | 4.357315 | 4.312983 | 4.630722 | 4.276470 | 4.230957 |
Table 8 shows the bit-rates using the integer lifting structure of 5/3 wavelet, 9/7 wavelet, iDTT, and IUPILWs, respectively. Compared with the integer lifting structure of 5/3 wavelet, 9/7 wavelet, and iDTT, and IUPILW-(1, 5) gets the lowest bit-rates, which means IUPILW-(1, 5) has the best performance for lossless image compression.
Bit-rates (bit/pixel) for lossless image compression (corpus ISO 12640-1)
Image | 5/3-wavelet | 9/7-wavelet | iDTT | IUPILW-(1, 1) | IUPILW-(1, 3) | IUPILW-(1, 5) |
---|---|---|---|---|---|---|
N1 | 4.424217 | 4.493907 | 4.462199 | 4.656300 | 4.444999 | 4.346868 |
N2 | 5.273573 | 5.308329 | 5.302399 | 5.537801 | 5.293032 | 5.188928 |
N3 | 4.291140 | 4.393228 | 4.339916 | 4.478541 | 4.298331 | 4.214809 |
N4 | 4.606598 | 4.703839 | 4.659298 | 4.734381 | 4.603954 | 4.519682 |
N5 | 4.591891 | 4.645296 | 4.624030 | 4.811606 | 4.603206 | 4.509288 |
N6 | 3.681239 | 3.807993 | 3.719131 | 3.817874 | 3.678956 | 3.586124 |
N7 | 5.473100 | 5.574008 | 5.504893 | 5.563710 | 5.465703 | 5.400206 |
N8 | 5.751989 | 5.808060 | 5.770165 | 5.924475 | 5.756146 | 5.665746 |
One of the reasons why the IUPILW-(1, 5) has the better performance than the predict-then-update lifting wavelet may be the update-then-predict structure can reduce the errors during the wavelet decomposition. Update-first means the approximate coefficients will be obtained firstly during each decomposition-level, and then the approximate coefficients of next decomposition-level will be obtained using the approximate coefficients of the current decomposition level. It means that the errors will not spread between the approximate coefficients. However, for the predict-first lifting structure, the detail coefficients must be get using the approximate coefficients of the upper level, then computing the approximate coefficients of the current level using these detail coefficients. Therefore, for the predict-first structure, the errors will spread between detail coefficients and approximate of the same decomposition level.
5 Conclusions
A new update-then-predict integer lifting wavelet family for lossless image compression is built and named in this paper. It is a perfect invertible update-then-predict structure and compared with the integer lifting structure of 5/3 wavelet, 9/7-wavelet, and iDTT, IUPILW-(1, 5) results in the lower bit-rates for lossless image compression.
Declarations
Acknowledgments
This work was supported in part by National Natural Science Foundation of China (Nos. 61300098, 61303080), the Fundamental Research Funds for the Central Universities (No. DL13BB02), and Self-Planned Task (No. SKLRS201407B) of State Key Laboratory of Robotics and System (HIT).
Authors’ contributions
DC built the theoretical framework of this paper. YL and HZ drew all the figures and provide funding support. DC, YL, and WG finished the experimental section in this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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