- Open Access
Two-dimensional integer wavelet transform with reduced influence of rounding operations
© Strutz and Rennert; licensee Springer 2012
- Received: 26 May 2011
- Accepted: 11 March 2012
- Published: 4 April 2012
If a system for lossless compression of images applies a decorrelation step, this step must map integer input values to integer output values. This can be achieved, for example, using the integer wavelet transform (IWT). The non-linearity, introduced by the obligatory rounding steps, is the main drawback of the IWT, since it deteriorates the desired filter characteristic.
This paper discusses different methods for reducing the influence of rounding in 5/3 and 9/7 filter banks. A novel combination of two-dimensional implementations of the JPEG2000 9/7 filter bank with new filter coefficients is proposed and the effects of the methods on lossless image compression are investigated. In addition, these filter banks are compared to the 9/7 Deslauriers-Dubuc filter bank (97DD).
The analysed two-dimensional implementations generally perform better than their one-dimensional counterparts in terms of compression ratio for natural images. On average, the 2D 97DD filter bank performs best. In addition, it has been found that the compression results cannot be improved by simply reducing the number of lifting steps via 2D implementations of the JPEG2000 9/7 filter bank. Only the 2D implementation with a minimum number of lifting steps, in combination with modified lifting coefficients, leads to fewer bits per pixel than the separable implementation on average for a selected set of images.
- integer wavelet transform
- filter bank
- 2D lifting
- image compression
Efficient systems for the lossless compression of image data require a decorrelation step which maps the integer input samples to integer output values. In wavelet-based compression systems (see  for an overview), this is achieved by using the lifting implementation of a discrete wavelet transform (DWT)  in combination with the rounding of intermediate computation results, which is called integer wavelet transform (IWT) . Beginning with initial investigations on the IWT, which were also motivated by the standardization of the new image compression system JPEG2000 , and its application in the JPEG2000 framework , this topic has received growing attention. The idea of integer transforms relates back to the so-called S transform , the improved version called S + P transform , and the reversible TS transform . Since then, several integer wavelet transforms have been analysed in terms of their performance in image compression systems .
Wavelet filter banks are typically designed without taking the integer-to-integer mapping into account. The conversion into an integer wavelet transform requires rounding steps which introduce non-linear effects, which deteriorate the desired filter properties. That is one of the major reasons why JPEG2000 (Part 1) uses the simple 5/3 filter bank, which can be realised with a minimum of lifting steps, i.e. a minimum of rounding operators. In addition, this filter bank has favourable coefficients of −0.5 in its first lifting step, leading to rounding errors in only 50% of all cases. The 9/7 filter bank, in contrast, requires irrational filter coefficients, making it per se less suitable for integer-to-integer mapping.
Besides the degradation of the decorrelation performance, the non-linearity of rounding makes the two-dimensional application pseudo non-separable, because the result of a 2D transform is dependent on the order of row and column decomposition.
The effects of rounding have been investigated in  with the assumption that rounding errors are similar to quantization errors and can be modelled by additive noise. In , the optimizations of filters for a lossy-to-lossless framework and its implementation in hardware are discussed. Another issue of interest is to make the IWT adaptive to the statistics of the image to be processed. This can be achieved by either switching between filters of different length [12–14] or by optimising the lifting steps [15–19]. Many papers are dedicated to the approximation of the standard 9/7 filter coefficients by rational values for low-complexity hardware or software implementations [20–22]. More recent research has also addressed the problem of reducing the adverse effect of rounding operations in terms of designing new filters with more favourable lifting coefficients  or modifications of the signal flow, reducing the number of lifting steps [24, 25]. The construction of three-channel filter banks based on the lifting scheme has also been considered for lossless image compression .
This paper investigates methods for reducing the influence of rounding in terms of reduced number of lifting steps and new filter coefficients for the two-dimensional JPEG2000 9/7 filter bank. The derived processing structures are evaluated with respect to compression efficiency and complexity. In addition, the results are compared to the one- and two-dimensional implementation of the LeGall 5/3 filter bank  and the 9/7 Deslauries-Dubuc filter bank [3, 28–30], which both can be implemented with only two lifting steps.
The paper is organised as follows: first, Section 2 reviews the reversible 5/3 filter bank used in JPEG2000, discusses the handling of signal boundaries, and explains how the 2D implementation reduces the number of rounding steps. Section 3 is dedicated to the JPEG2000 9/7 filter bank. It explains the separable 2D signal flow and afterwards describes how the number of lifting steps can be decreased by rearranging the processing steps. Section 4 discusses a Deslauriers-Dubuc filter bank (97DD), which can be implemented with the same 2D structure as the 5/3 filter bank, while having filter lengths of nine and seven. In Section 5, the design of ‘rounding-friendly’ filters is addressed. Section 6 investigates all processing structures by means of sub-band entropy, compression performance, and complexity. Section 7 concludes the paper.
2.1 One-dimensional decomposition
and are the impulse responses of the analysis high-pass and analysis low-pass filters, respectively. This filter bank can be easily implemented via the lifting scheme .
based on the detail signal and the original signal values . The property of integer-to-integer mapping, which is essential for lossless compression, is imposed simply by properly rounding the intermediate values to integer values .
Setting the lifting coefficients equal to and , the lifting scheme (without rounding) performs the same operations as the filters in equation (1) of the conventional filter bank.
2.2 Two-dimensional decomposition
The actual number of rounding steps can be determined based on Figure 1. Rounding is required at each lifting step before the summation, i.e. eight times in total. The rounding steps are interdependent, however. The horizontal dual steps, for instance, process values from DD and DA positions, which have already been affected by the rounding in previous primal steps. This possible accumulation of rounding errors continues in the second dimension of the image signal.
Section 6 investigates the influence of the reduced number of rounding steps on the performance of an IWT-based compression system.
3.1 One-dimensional decomposition
The standard 9/7 filter bank can be implemented based on the lifting scheme in the same manner as the 5/3 filter bank shown in the previous section. The only difference is that, in total, four lifting steps are needed in each direction (, Figure 2).
In literature, the factorization of a scaling factor is also discussed. We would like to remark that in the case of lossless compression, this scaling is not necessary and would merely introduce more rounding operations degrading the performance of the wavelet transform. In lossy compression, the scaling can be shifted into the quantization step, thus, there is no reason to treat the scaling within the transformation stage.
Further, it must be pointed out that for the practical implementation and in order to avoid an exception handling at the signal boundaries, it is sufficient to extend the original or intermediate signals by a single value at both boundaries.
3.2 Two-dimensional decomposition
This makes the structure in Figure 5 applicable for the 97DD filter bank as well. The only difference concerns following lifting polynomials: and both in combination with . The other lifting steps remain and with .
The superiority of the separable 5/3 filters over the separable standard 9/7 IWT filter bank, documented in literature [3, 9, 23], has its reason in the minimal accumulated influence of rounding at each lifting step, leading to a minimal change in the filter characteristics. Not only is the number of steps lower (only two instead of four in each direction), but the rounding error in its first lifting step is also smaller, since the factor of only results in errors when the sum of and is odd (see eq. (3)). The degradation of the magnitude response of the standard 9/7 filter is distinctly higher.
With this degradation in mind, we investigated whether some of the filter-design constraints leading to a maximum number of vanishing moments can be released, to obtain lifting coefficients which introduce fewer rounding errors, while keeping the essential characteristic of the magnitude responses.
In general, the relaxation of these filter constraints leads to poorer decorrelation of the input image. If, however, the adverse effects of rounding decrease due to the modification of the lifting coefficients, then there might be an optimal compromise. Reducing the number of vanishing moments from four to two increases the degree of freedom in selecting lifting coefficients with suitable properties.
6.1 Effects of rounding in 1D
The effects of rounding on the impulse responses of the high and low-pass filters have been investigated in the following manner.
The smaller the input value v, the higher the effect of rounding.
6.2 Effects of rounding in 2D
In order to illustrate the rounding effects in two dimensions, the same procedure as in the previous section was performed, but with the distinction that 2D filters were used.
6.2.1 Effects in the 5/3 filter bank
Figures 13(c) and (d) compare the original with the deformed magnitude responses for the filter .
in case of the 2D implementation.
6.2.2 Effects in the 9/7 filter bank
For the case of the standard 9/7 filter bank, we have to compare three different implementations, which are called ‘97v1’, ‘97v2’, and ‘97v3’ in the following, according to the flow charts in Figures 6, 7, and 9, respectively.
Similar alterations of the filter coefficients can be observed for the , , and filters. Another effect of non-linearity is that the filters are no longer the transposes of the filters.
Comparing the similarities of the two-dimensional impulse responses between the different implementations, the deviation is higher for the and filters than for the filter. This continues for the filter, which shows the highest variation from one implementation to another.
Note that the degradation is dependent on the magnitude of the impulse functioning as input signal. The shown magnitude responses merely give an impression of the effects, but do not allow a general conclusion or even a theoretical analysis.
6.3 Efficiency of decorrelation
The efficiency of the different filter banks was first analysed based on the zero-order entropies of single sub-bands.
Table 1-Table 4 contain the zero-order entropies of the sub-bands DD, AD, DA and AA after one 2D decomposition step is applied to different grey-scale images. Barbara-Zelda are taken from , cats_g-educ from . kodim07-kodim09 are the green components of true colour images found at . The meaning of the column titles is
– 53v1: 5/3 filter bank, implementation of Figure 1,
– 53v2: 5/3 filter bank, implementation of Figure 5,
– 97D1: 9/7 Deslauriers-Dubuc, implementation of Figure 1,
– 97D2: 9/7 Deslauriers-Dubuc, implementation of Figure 5,
– 97v1: 9/7 filter bank, implementation of Figure 6,
– 97v2: 9/7 filter bank, implementation of Figure 7,
– 97v3: 9/7 filter bank, implementation of Figure 9,
– 97v1a - 97v3a: same as before, but with lifting coefficients from equation (39).
DD-band entropies in bit per pixel after first decomposition
The results for the DD band (high-pass filtering in both directions) show a clear trend with decreasing entropy from left to right (Table 1), when excluding 97D1 and 97D2. There is only one exception: the implementation 97v2 leads to lower entropy on average than 97v3 when the standard lifting coefficients (eq. (24)) are used. The Deslauriers-Dubuc filter bank (97D1) performs worst despite the four vanishing moments of its high-pass filter. One reason lies in the gain of , which is significantly higher than the gain of the JPEG2000 9/7 high pass. In comparison to the 5/3 filter banks (also having ), the filter characteristic seems to be more influenced by the rounding operations deteriorating the advantage of two additional vanishing moments.
The 2D implementations of the 5/3 filter bank (53v2) and of the Deslauriers-Dubuc filter bank (97D2) are superior to the separable implementations. All other 9/7 implementations yield better results, despite the higher number of lifting steps. It also can be seen that the modified lifting coefficients truly improve the decorrelation on average. However, the numbers in this table also reveal that the number of rounding steps is not a unique measure to estimate the influence of rounding steps. The DD values depend on three steps in 97v2 and only on two steps in 97v3; nevertheless, the result is better for 97v2.
AD-band entropies in bit per pixel after first decomposition
DA-band entropies in bit per pixel after first decomposition
AA-band entropies in bit per pixel after first decomposition
In addition, we would like to point out that the modified lifting coefficients again have a positive influence on the decorrelation.
6.4 Compression results
The overall performance of the different filter banks was tested in combination with a compression system, which applies the basic coding algorithm of JPEG2000 without using the header/marker structure. The number of decomposition steps was dependent on the image size, for example, five decompositions for images with pixels and seven decompositions for images with pixels. The focus was exclusively on lossless image compression. A rate-distortion analysis exploiting the scalability of the compression scheme is not considered in this paper.
Compression results in bits per pixel [bpp]
The 97v2(a) implementation does not result in any advantage over the separable 97v1(a) implementation. The new non-separable implementation 97v3a in combination with the lifting coefficients according to eq. (39) performs best within all 9/7 filter banks based on the JPEG2000 9/7 filter-bank structure.
6.5 Complexity of implementations
Complexity estimation for different filter-bank implementations
It can be noticed that the increase of complexity is rather moderate when switching from a one-dimensional to a two-dimensional implementation. This is largely due to the properties of the lifting scheme and the utilisation of the symmetry of filters. With respect to the memory access, the 2D implementation is even somewhat advantageous, as the algorithm runs only once through the data and not twice, separately in horizontal and vertical directions. Practical implementations, however, also have to consider the efforts of signal extension at the signal boundaries, which are highest for the 97Dx filter banks.
The paper has discussed and analysed different attempts to decrease the effects of rounding in implementations of the integer wavelet transform. The number of rounding steps could be reduced practically by special two-dimensional implementations of the 5/3 and 9/7 filter banks and virtually by using a first lifting coefficient of in the JPEG2000 9/7 filter bank. Dependent on the implementation, the rounding of intermediate values to integers has different effects on the impulse responses and, consequently, on the decomposition of the signal.
The set of test images contained twelve natural images. It can be shown that the various processing schemes affect the single sub-bands differently in terms of entropy. As soon as the low-pass filter comes into play, the 5/3 filter bank and the 97DD filter bank tend to decorrelate the image data better than the standard 9/7 filter bank and its derivatives. The gain of the low-pass filter at has a high impact on lossless compression performance.
The compression results of the 5/3 and 97DD filter bank have been improved on average by substituting the separable implementation with a special 2D processing, which would be compatible in the absence of rounding. The mere reduction of rounding steps by 2D implementations of the JPEG2000 9/7 filter bank does not lead to increased compression ratios. Only when the modified lifting coefficients enabling division-free integer arithmetic are combined with the new proposed 2D implementation does the average bitrate of the compressed images decrease from 4.230 bpp to 4.205 bpp. This is about the same bitrate as for the standard 5/3 filter bank while having distinctly higher complexity.
The results of the different implementations of the 9/7 filter bank reveal that the influence of rounding is not solely determined by the number of lifting steps, which means the rounding errors (strictly: their variances) do not simply sum up. The degradation of the filter characteristics under the presence of rounding also depends on the values of the lifting coefficients and on the structure of processing. The best compression result could be obtained with the new 2D implementation of the 9/7 Deslauriers-Dubuc filter bank (97D2). It requires only two sequential lifting steps as the 5/3 filter bank, while having more suitable filter characteristics. The improvement is 1.36% compared to the standard 5/3 implementation (53v1) for the set of images used.
The authors would like to thank the unknown reviewers who gave valuable comments on an earlier version of this manuscript.
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