Non-separable 3D integer wavelet transform for lossless data compression

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1 Science Journal of Circuits, Systems and Signal Processing 04; (6: 5-46 Published online January 6, 05 ( doi: 0.648/j.cssp ISSN: (Print; ISSN: (Online Non-separable D integer wavelet transform for lossless data compression Teerapong Orachon, *, Suvit Poomrittigul, Taichi Yoshida, Masahiro Iwahashi, Somchart Chokchaitam Department of Electrical Engineering, Nagaoka University of Technology, Nagaoka, Japan Department of Computer Technology, Pathumwan Institute of Technology, Bangkok, Thailand Department of Electrical Engineering, Thammasat University, Bangkok, Thailand address: oteerapong@gmail.com (T. Orachon, suvitp@gmail.com (S. Poomrittigul To cite this article: Teerapong Orachon, Suvit Poomrittigul, Taichi Yoshida, Masahiro Iwahashi, Somchart Chokchaitam. Non-Separable D Integer Wavelet Transform for Lossless Data Compression. Science Journal of Circuits, Systems and Signal Processing. Vol., No. 6, 04, pp doi: 0.648/j.cssp Abstract: This paper proposes a three-dimensional (D integer wavelet transform with reduced amount of rounding noise. Non-separable multi-dimensional lifting structures are introduced to decrease the total number of lifting s. Since the lifting contains a rounding operation, variance of the rounding noise generated due to the rounding operation inside the transform is reduced. This paper also investigates performance of the transform from various aspects such as variance of the noise in frequency domain and those in piel domain, the rate distortion curve in lossy coding mode and the entropy rate in lossless coding mode, computational time of the transforms, and 4 feature comparison with other methods. The proposed wavelet transform has a merit that its output signal, apart from the rounding noise, is eactly the same as the conventional separable structure which is a cascade of D structure. Due to this compatibility, it becomes possible to utilize legacy of previously designed D wavelet transforms with preferable properties such as the regularity. Furthermore, total amount of the rounding noise which is generated due to integer epression of signal values inside the transform is significantly reduced. This is because the total number of rounding operations is decreased by introducing the non-separable multi-dimensional lifting structure which includes multi-dimensional memory accessing. It contributes to increase coding performance of a system based on the D wavelet transform. As a result of eperiments, it was observed that the proposed method increases performance of data compression of various D input signals. Keywords: Wavelet, Transform, Lifting, D, Rounding, Coding. Introduction Over the past few decades, quality of image signals have been dramatically increased in respect of piel resolution, frame rate and dynamic range of piel values. The more the quality increases, the more data volume becomes. Therefore image compression techniques based on a transform have been playing an important role for storage and communications of digital media data. For still images, a two-dimensional (D transform such as the discrete cosine transform (DCT and the wavelet transform has been utilized. Recently, various types of three-dimensional (D transforms have been investigated for video, hyper spectral images, integral images and medical volumetric data [-4]. This paper deals with a D wavelet transform with reduced amount of rounding noise inside the transform for high performance lossless data compression. So far, a class of separable D wavelet transform has been widely developed for various applications. The JPEG 0 international standard also adopts this class of transform [5, 6]. It has been applied for digital cinema [7, 8]. Since its transfer function is composed of a product of the horizontal D transfer function and the vertical one, it can inherit legacy of previously designed D structure suitable for hardware implementations [9-]. It can also have the regularity and low sensitivity to various noises [, ]. However, most of them are designed for lossy coding of images for practically reasonable data compression rate.

2 6 Teerapong Orachon et al.: Non-Separable D Integer Wavelet Transform for Lossless Data Compression For lossless coding of images, it is necessary to have transformed values as integers for entropy coding, and also inversely transform them to reconstruct the original integer piel values without any loss. It has been implemented in the lifting structure with rounding operations as the integer wavelet transform and etended based on the international standard [4, 5]. The radar network has also been utilized for the reversible color transform and the integer DCT [6-]. Those are indeed lossless. A set of forward and backward transform guarantees lossless reconstruction of the original integer piel values. However, there eist rounding noise just after the forward transform. To reduce those noise due to the rounding operations applied at each of the lifting s, a non-separable structure was introduced to decrease the total number of lifting s as well as the rounding operations []. Most of the non-separable structures have been initially introduced to increase precision of the prediction by adapting to local contet of neighboring piels [4-6]. More general designing problem based on a sparse criteria has been investigated [7]. Recently, directionality has been utilized in a generalized poly-phase representation [8, 9]. Those have been focusing on designing adaptive high pass filters of the wavelet transform. A new class of non-separable D structure has been reported in [0-]. Its transfer function can be epressed as a product of D function. In this sense, the transform based on this structure is compatible to the separable transform. However, in its implementation, it is not a cascade of D signal processing in the D structure. It requires multi-dimensional memory accessing. In addition, those are not adaptive to local contet of piels. Even though it has some drawbacks yet, it has advantage that it decreases the total number of lifting s as well as the rounding operations. Unlike those previous studies on the D case, this paper proposes a D integer wavelet transform for lossless coding of D signals with reduced amount of the rounding noise. In a conference paper [4], a non-separable D lifting structure was introduced to decrease the total number of lifting s. However its computational compleity becomes huge since it requires D memory accessing. In case of video signal processing, it requires inter-frame memory accessing. Therefore it costs huge memory space. Therefore this paper newly introduces non-separable D structures to construct the D integer wavelet transform. In addition, investigation in [4] on performance of the D integer wavelet transform was limited to noise variance in frequency domain only. Its input signal was also limited to only one MRI data set. Therefore this paper clarifies not only variance of the noise in frequency domain and those in piel domain, but also the rate distortion curve in lossy coding mode and the entropy rate in lossless coding mode, computational time of the transforms, and 4 feature comparison with other methods. In addition to the MRI data set, various kinds of input data are tested in this paper. The proposed wavelet transform has a merit that its output signals, apart from the rounding noise, are eactly the same as a conventional transform whose transfer function is epressed as a product of D transfer functions. Due to this compatibility, it becomes possible to utilize legacy of previously designed D wavelet transforms. Furthermore, total amount of the rounding noise is significantly reduced. It contributes to increase coding performance of data compression system based on the D wavelet transform. This paper is organized as follows. The reason why the non-separable structure reduces the lifting s is eplained in. Discussion is etended to D case in. Non-separable D structures for the D wavelet transform is newly introduced in 4. The non-separable multi-dimensional structures are compared in various aspects for various input signals in 5. This paper is concluded in 6.. Wavelet Transform and Lifting Structure It is eplained how the rounding noise is reduced introducing a non-separable D structure... One-Dimensional (D Wavelet Transform Fig. illustrates a one-dimensional (D integer wavelet transform for lossless coding of discrete D signal. In encoding side, the input signal is fed into the forward transform which outputs low frequency band signal Y L and high frequency band signal Y H. Each of them are coded with an entropy encoder to generate a bit stream for storage and communications. In decoding side, the band signals are decoded and inversely transformed. S Figure. One-dimensional (D integer wavelet transform. This system is referred to as the integer wavelet transform for lossless coding since it reconstructs the original signal without any loss. This is because all the signal values inside the transform are rounded to integers, and the rounding noise generated by the rounding operation is canceled at the output of the backward transform. In detail, the original signal given in integer values with length Nis split into two half-length sequences 0 and as for m=0,,,m- where 0 forward 0 = [ = [ (0 = [ (0 0( m = (m ( m = (m (0 0 V V Y L = rounding ( 0 ( ( Y H V - ( N ] ( M ] 0 ( M ] V - backward 0 S - ( (

3 Science Journal of Circuits, Systems and Signal Processing 04; (6: and M=N/. In the st lifting, is predicted from 0 as y H ( 0 m m = RV [ ( ] ( where R[] denotes an integer rounded from the original value, and V denotes convolution. The convolution is defined as ( ( m V 0 = h 0 0 (4 where h is a filter coefficient. In the nd lifting, 0 is updated from Y H as y L( 0 H m m = RV [ y ( ] (5 Fig. illustrates a two-dimensional (D integer wavelet transform. In this transform, the D transform is applied vertically to the D input image. Net, the same D transform is applied horizontally. As a result, 4 frequency band signals Y LL, Y LH, Y HL and Y HH are generated. Since this transform is a product of D transforms, it is referred to as the separable transform. Fig. illustrates another epression of Fig.. It outputs the same band signals to those in Fig.. It splits the D input signal (n,n into 4 groups as V H V V H H Y LL where the convolution V is defined as S V V H 0 Y LH ( y y ( m V yh = h H H (6 H 0 Y HL where h is a filter coefficient. As the rounding operations R[ ] are applied, all the signal values are epressed as integers inside this integer wavelet transform. In the lifting wavelet transforms, filter coefficients have been carefully determined in previous reports. For eample, a set (h, h = (-/,/4 is utilized in JPEG 0 international standard [5,6]. In this case, a low pass filter of the forward transform has unity gain for a constant signal, and zero gain for an alternating signal. Those properties are epressed as ( y = h y y ( m L 0 H H and a high pass filter has constant for n ( = constant = n 0 for n ( = ( ( h ( m m y = h ( m H 0 0 = (m ( ( = 0 for n ( = constant. It is desirable to utilize good properties like (7 and (8, and other properties such as the vanishing moment in designing a multi-dimensional transform... Two-Dimensional (D Wavelet Transform S V V V vertical V V horizontal H H H H H H H H Figure. Two-dimensional (D integer wavelet transform. Y LL Y LH Y HL Y HH (7 (8 Figure. Separable D structure for D wavelet transform. 0 0 = (m = (m = (m = (m, m, m, m, m where m =0,,,M -and m =0,,,M - for M =N / and M =N /. In the st lifting, 0 and are predicted from and 0 respectively as where ( 0 ( V V 0 = = 0 = h = h RV [ RV [ 0 ] ] ( ( m, m ( m, m ( ( m, m. 0 0 (9 (0 ( This convolution is performed vertically along with the variable m. Similarly, in the nd lifting, and 0 are updated from 0 and respectively as where ( ( 0 V V ( 0 ( = = 0 = h = h H RV [ RV [ ( 0 ( ] ] ( ( ( 0 ( m, m 0 ( m, m ( ( ( ( m, m. H H Y HH ( ( In the rd and 4th lifting s, the process is repeated V V 4

4 8 Teerapong Orachon et al.: Non-Separable D Integer Wavelet Transform for Lossless Data Compression horizontally along with the variable m. For eample, the horizontal convolutions are given as H H ( ( 0 = h = h ( ( ( ( m, m ( m, m ( ( (. 0 0 (4 Note that this separable D lifting structure has 8 rounding operations in total since each of 8 lifting has one rounding operation. Those are source of the rounding noise to be reduced by introducing a non-separable structure in this paper... Non-separable D Structure Fig. 4 illustrates a non-separable D lifting structure. Different from the separable stricture, it has multi-input single-output lifting s. In addition, it has less lifting s comparing to the separable one. Therefore it has less rounding operations as well as rounding noise. Comparing to Fig., the total number of lifting s is reduced from 4 to (75 %, and the total number of rounding operations is reduced from 8 to 4 (50 % in Fig.4. It also reduces the total amount of rounding noise []. Note that this transform has the properties of the D structure in (7 and (8 since its transfer function is the same as that of the separable transform in.. Figure 4. Non-separable D structure for D wavelet transform. In detail, the st lifting predicts from multi-inputs, 0 and 0 as y HH = R[ V H V 0 H 0 ] for (m=(m, m. In this prediction, a diagonal prediction V H ( = hh ( m, m ( m, m ( m, m is utilized. In the nd lifting, two predictions and y y S V V V H H H H 0 0 V H H V V LH( 0 HH m (5 (6 m = R[ H V y ( ] (7 HL( 0 HH m H m = R[ V H y ( ] (8 V V H H -V H Y LL Y LH Y HL Y HH are applied. Note that these two s can be done simultaneously with a parallel processor sine it is not necessary to wait for calculation results each other. Finally, the rd lifting completes the transform as y LL = V y HL R[ H V y LH H y HH ]. (9 As a result, this non-separable structure outputs the same band signals as the separable D transform apart from the rounding noise. It is referred to as compatibility. In the net section, this theory is etended to D case.. Three-Dimensional (D Wavelet Transform A non-separable D structure is summarized after describing the separable D structure... Separable D Structure Fig. 5 illustrates a separable D lifting structure. A D input signal is split into 8 groups and single-input single-output lifting s are applied to generate 8 frequency band signals. This structure has 6 lifting s and 4 rounding operations. We are going to reduce these numbers in this paper. In detail, the D input signal (n,n,n is split into 8 groups as 0 = ( m, m, m = ( m, m, m =, m 0 =, m = (m, m, m 0 = (m, m, m 0 = (m, m, m = (m, m, m (0 for (m=(m,m,m.in the st lifting, the vertical prediction is applied as ( ( 0 ( 0 ( = 0 0 RV [ RV [ RV [ RV [ 0 0 ] ] ] ] ( where V denotes the vertical convolution along with the variable m similar to (. After the updating ( 0 ( ( ( 0 = 0 0 RV [ RV [ RV [ RV [ ( ( 0 ( 0 ( ] ] ] ] (

5 Science Journal of Circuits, Systems and Signal Processing 04; (6: in the nd lifting, the horizontal prediction ( ( 0 (4 0 (4 = ( ( 0 ( 0 ( R[ H R[ H R[ H R[ H ( 0 ( ( ( 0 ] ] ] ] ( is applied where H and H denote the horizontal convolution along with the variable m similar to (4. In the 5th and 6th lifting s, the convolution D and D are performed along with the variable m. V H D 0 Y LLL D D V H H S D D Y LLH V H H H D Y LHL V D D S D D 0 Y LHH V V S V V H D Y HLL V H H D D S D D 0 Y HLH V H H H D 0 Y HHL V D D S D D Y HHH 4 Figure 5. Separable D structure SepD for D wavelet transform. 5 6 In this separable structure, the Nth lifting must wait for calculation results of its previous N-th lifting. It increases delay from the input to the output. These lifting s are reduced introducing non-separable structures composed of multi-dimensional memory accessing in the net subsection... Non-separable D Structure 0 Y LLL V H D V H V D H D D V H D Y LLH V H V H H Y LHL V D V D D H -H D 0 Y LHH V V Y HLL H D H D D V -V D 0 Y HLH H H V -V H 0 Y HHL D D H V -H D -V D -V H V H D Y HHH Figure 6. Non-separable D structure NsD for D wavelet transform. 4 Fig. 6 illustrates the non-separable D lifting structure in [4]. In this structure, multi-input single-output lifting s are introduced. It means D memory accessing. Comparing to the eisting separable structure, the total number of lifting s is reduced from 6 to 4 (66.7 %. Therefore it is epected to reduce the delay due to waiting for results of the previous

6 40 Teerapong Orachon et al.: Non-Separable D Integer Wavelet Transform for Lossless Data Compression lifting utilizing parallel signal processing. The total number of rounding operations is also decreased from 4 to 8 (. %. We eperimentally demonstrate that it contributes to reducing total amount of the rounding noise in 5. In detail, the st stage predicts from all of the remaining groups as = RV [ H D ( 0 V H V D V 0 H D H 0 D 0 ]. (4 It includes convolution with multi-dimensional memory accessing. For eample, the convolution V H D is defined as where V H D = h 0, m, m 0 0, m, m 0 0, m, m 0 0, m, m 0 0 { } (5 m = m, m = m, m = m. (6 In the nd lifting, three predictions = RV [ H ( V H D ( ( ( = RV [ D V D H ( ( ] = RH [ D H D V ] ] (7 (8 (9 can be done simultaneously on a parallel signal processing platform since it is not necessary to wait for calculation results each other. Similarly, updating in the rd and 4th lifting s are performed. Note that there is no difference between the separable structure in Fig.5 and the non-separable structure in Fig.6 in respect of signals. It means that both of them are epressed with the same transfer function which is a product of D function. In this sense, the transform to be implemented is separable. However, in respect of noise, those are different. The structure in Fig.6 is epected to have less rounding noise as it has less rounding operations. It is investigated in section Two-Dimensional (D Structures for D Wavelet Transform Two-dimensional (D structures for the D integer wavelet transform is newly introduced. 4.. Non-separable D Structures for D Transform 0 Y LLL V H D D H D Y LLH V H H Y LHL V D H D -D H 0 Y LHH V V Y HLL V H D D H D 0 Y HLH V H H 0 Y HHL V D H D -D H Y HHH 4 5 Figure 7. Non-separable D structure NsD( for D wavelet transform.

7 Science Journal of Circuits, Systems and Signal Processing 04; (6: Fig. 7 illustrates a newly introduced structure NsD( for the D lifting wavelet transform. The st and the nd lifting s are the same as those of the separable structure in Fig. 5. However, the rd, 4th, 5th 6th lifting s are implemented as the non-separable D structure. It contributes to reduce hardware compleity comparing to NsD in Fig. 6 since it does not use the D memory accessing. 0 Fig. 8 illustrates another variation NsD( for the D transform. Unlike the structure in Fig. 7, the st, the nd and the rd lifting s are implemented as the non-separable D structure. Whereas the 4th and the 5th lifting s are the same as those of the separable structure in Fig. 5. In brief, the structures appeared in this paper are epressed as Y LLL V H H V D D Y LLH V H H V H Y LHL V H D D 0 Y LHH V V Y HLL H V D D 0 Y HLH H V H -H V 0 Y HHL V H -H V D D Y HHH 4 Figure 8. Non-separable D structure NsD( for D wavelet transform. 5 Sep D VV HH DD ( Fig.5 Ns D [ VV HH DD ] ( Fig.6 Ns D( VV ( HH DD ( Fig.7 Ns D( ( VV HH DD ( Fig.8 (0 where [ ] in NsD and ( in NsD denote the non-separable D structure and the non-separable D structure. 4.. Comparison of Structures Table summarizes the variations of the structure of the D integer wavelet transform. Unlike NsD, the newly introduced structures NsD( and NsD( do not use inter-frame memory accessing which requires huge memory space. Instead, the total number of lifting s is increased comparing to NsD. The number of rounding operations is in the middle between SepD and NsD. In this paper, we compare the variations in (0 for various input signals from various aspects in 5. Table. Comparison of the structures. SepD NsD NsD( NsD( Rounding operations Lifting s Memory accessing D D D D 5. Eperimental Results In the following eperiments, a set of MRI volumetric data in Fig. 9 provided by MATLAB is tested. Each frame has piels and each piel is epressed with an 8 bit depth integer. Fig. 0 illustrates results of applying the D wavelet transform to the 8 frames of the MRI data. Note that each frequency band signals are normalized to the range of [0,55] for display purpose in this figure. In this paper, not only variance of the noise in frequency domain and those in piel domain, but also the rate distortion curve in lossy coding mode and the entropy rate in lossless coding mode, computational time of the transforms, and 4 feature comparison with other methods are investigated. In this paper, the input signal is not limited to the MRI data like [4], but also a random D input signal is included. In addition, the D auto-regressive model epressed as ( n, n, n = ' ( n, n, n ρ ( n, n, n ' ( n, n, n = ''( n, n, n ρ ' ( n, n, n ''( n, n, n = w ( n, n, n ρ ''( n, n, n ( is included into our eperiments. Note that ρ is set to 0.9 in the eperiments in this paper.

8 4 Teerapong Orachon et al.: Non-Separable D Integer Wavelet Transform for Lossless Data Compression Figure 9. Tested data set MRI. Figure 0. Results of the D wavelet transform. 5.. Evaluation of Rounding Noise noise variance Sep NsD NsD( NsD( MRI LLL LHL HLL HHL LLH LHH HLH HHH frequency band (a MRI data noise variance AR Sep NsD NsD( NsD( LLL LHL HLL HHL LLH LHH HLH HHH frequency band (b AR model

9 Science Journal of Circuits, Systems and Signal Processing 04; (6: noise variance (c Random input Figure. Rounding noise in each frequency band. Firstly, the structures summarized in table are compared in respect of the rounding noise. Fig. (a, (b and (c indicate variance of the rounding noise in each frequency bands for MRI data, AR model, and random input, respectively. In all cases, variance of the noise in HHH band of NsD is This is because the noise is not amplified. Since value of the rounding noise is in the range of [-0.5, 0.5], its variance is /=0.08 just after each of the rounding operation. On the contrary, noise variance in LLL band of NsD is greater than This is because multiple noise amplified though convolutions are summed up in Y LLL. The highest noise variance is observed in LHH band and HHH band of Sep. Noise variance is magnified approimately 0.45/0.08=5.6 times in those frequency bands in the separable structure in Fig. 5. PSNR (db Sep NsD NsD( NsD( Rand LLL LHL HLL HHL LLH LHH HLH HHH frequency band Sep NsD NsD( NsD( method MRI AR Rand Figure. Rounding noise in frequency domain. Fig. illustrates averaged variance over all frequency bands in peak-signal to noise ratio (PSNR defined as 55 PSNR = 0log0 ( db ( σ where σ denotes variance of the rounding error. It was observed that NsD and NsD increase PSNR by 5 (db and (db respectively comparing to Sep. PSNR (db Sep NsD NsD( NsD( Figure. Rounding noise in piel domain. Fig. illustrates variance of the rounding noise in piel domain. The input signal is transformed in forward without the rounding operation, and transformed in backward with the rounding operations. Variance of the rounding noise in this reconstructed signal is measured in this figure. Similarly to Fig., NsD is the best and NsD is the second. There is no significant difference among input signals. ratio of entropy PSNR (db % 98% 96% 94% 9% 90% method MRI AR Rand NsD NsD( NsD( 8 bit 6 bit 4 bit bit depth of input Figure 4. Performance in lossless coding mode. Sep NsD bit rate (bpp Figure 5. Performance in lossy coding mode

10 44 Teerapong Orachon et al.: Non-Separable D Integer Wavelet Transform for Lossless Data Compression 5.. Evaluation of Coding Performance structure was observed. Secondly, we investigate the entropy rate to evaluate lossless coding performance. Fig. 4 compares the structures in respect of lossless data compression rate for various bit depth of the input signal MRI. The compression rate is defined as HStr η = (% ( H for Str {NsD, NsD(, NsD(} where H denotes the entropy rate in bit per piel (bpp averaged over all frequency band signals which approimates the average code length of the compressed data volume of the transformed signal. According to Fig. 4, no significant difference was observed between the three methods for the original 8 bit depth signal. However, when the bit depth of the input signal is reduced to 4 (bit, the ratio of the entropy rate is decreased to 9.9 (% and 96. (% by NsD and NsD( respectively. Note that the bit depth of piel values in the range of [0, B -] is defined as B (bit. In conclusion, performance in lossless coding mode is improved by 8. (% and.8 (% by NsD and NsD( respectively for 4 bit depth MRI volumetric data. Fig.5 indicates rate-distortion curve of Sep and that of NsD. The horizontal ais and the vertical ais indicate entropy rate and PSNR of the reconstructed signal, respectively. In this eperiment, quantization is introduced just after the forward transform. These curves indicate performance of the methods in lossy coding mode. It was observed that NsD is higher than Sep by approimately.5 (db at 6.5 (bpp. In conclusion, superiority of the non-separable structure over the conventional separable Sep Table. Comparison with other methods 5.. Evaluation of Computational Time Thirdly, computational time of the structures for the D transform was investigated. Table summarizes computational time of SepD, NsD, NsD( and NsD(, respectively. These algorithms were performed with MATLAB program on Dell Inspiron 580 PC, Intel Core i-550 Processor. GHz, 4GB RAM on Windows 7. According to the table, it was observed that NsD is the fastest. Both of NsD( and NsD( are almost the same, and the second in computational speed. Those are faster than SepD by 9 (ms. The result confirms that less lifting s mean faster computation of the transform. Table. Comparison of computational time. SepD NsD NsD( NsD( Computational time 0.7 ms 97.9 ms 0. ms 0.4 ms 5.4. Comparison with Other Methods Fourthly, the methods investigated in this paper are compared to other conventional methods. Table summarizes features of those methods. The well-known international standards JPEG, JPEG-LS and JPEG 0 performs lossless coding. However, those are limited to D signals. Other methods such as Huffyuv and its improved version Lagarith can be applied to D signals. However those do not support the scalability. This feature makes it possible to analyze the input signal in frequency domain. Therefore this paper focused on SepD, NsD and NsD. Among these methods, it was found that NsD has the minimum lifting s, the minimum rounding noise and the minimum computational time. JPEG JPEG Luffyuv Lagarith JPEG 0 SepD NsD NsD Input signal D D D D D D D D Huffman Entropy Arithmetic Golumb- Huffman Run Length coding coding Rice coding Arithmetic EBCOT EBCOT EBCOT EBCOT Transform or adaptive median (5, (5, (5, (5, DPCM DPCM prediction prediction prediction wavelet wavelet wavelet wavelet Scalability N.A. N.A. N.A. N.A. available available available available Frequency analysis N.A. N.A. N.A. N.A. available available available available Lifting s many minimum reduced Memory accessing D D D D D D D D 6. Conclusions A D integer lifting wavelet transform with reduced amount of rounding noise was proposed. Non-separable multi-dimensional lifting structures were introduced to decrease the total number of lifting s as well as the rounding operations. It was eperimentally observed that the non-separable D structure increases PSNR by 5 (db in frequency for MRI, AR model and random input signals. It was also observed that the non-separable D structure increases PSNR by (db without using inter-frame memory accessing. Those PSNR improvements were also observed in rate-distortion curves at high bit rates in lossy coding mode. In lossless coding mode, data compression performance is improved for low bit depth input images. Computational time was also improved by the proposed method. Acknowledgements This work was supported by JSPS KAKENHI Grant Number 6897.

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