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Volumetric medical image compression using 3D listless embedded block partitioning
SpringerPlus volume 5, Article number: 2100 (2016)
Abstract
This paper presents a listless variant of a modified threedimensional (3D)block coding algorithm suitable for medical image compression. A higher degree of correlation is achieved by using a 3D hybrid transform. The 3D hybrid transform is performed by a wavelet transform in the spatial dimension and a Karhunen–Loueve transform in the spectral dimension. The 3D transformed coefficients are arranged in a onedimensional (1D) fashion, as in the hierarchical nature of the waveletcoefficient distribution strategy. A novel listless block coding algorithm is applied to the mapped 1D coefficients which encode in an orderedbitplane fashion. The algorithm originates from the most significant bit plane and terminates at the least significant bit plane to generate an embedded bit stream, as in 3DSPIHT. The proposed algorithm is called 3D hierarchical listless block (3DHLCK), which exhibits better compression performance than that exhibited by 3DSPIHT. Further, it is highly competitive with some of the stateoftheart 3D wavelet coders for a wide range of bit rates for magnetic resonance, digital imaging and communication in medicine and angiogram images. 3DHLCK provides rate and resolution scalability similar to those provided by 3DSPIHT and 3DSPECK. In addition, a significant memory reduction is achieved owing to the listless nature of 3DHLCK.
Background
As the amount of patient data increases, compression techniques for the digital storage and transmission of medical images become mandatory. Imaging modalities such as ultrasonography (US), computer tomography (CT), magnetic resonance imaging (MRI) and Xrays provide flexible means of viewing anatomical cross sections for diagnosis. Three dimensional (3D) medical images can be viewed as a time sequence of radiographic images, the tomographic slices (images) of a dynamic object, or a volume of a tomographic slice images of a static object (Udupa and Herman 2000). In this paper, a 3D medical image corresponds to a volume of tomographic slices, which is a rectangular array of voxels with certain intensity values. Progressive lossy to lossless compression from a unified bit string is highly desirable in medical imaging. Lossy compression is tolerated as long as the required diagnostic quality is preserved. Lossless to lossy compression techniques are primarily used in telemedicine, teleradiology and the wireless monitoring of capsule endoscopy.
A compression technique using wavelets provides better image quality compared to joint photographic experts group compression (JPEG) (Pennebaker and Mitchell 1993; Santacruz et al. 2000). It also provides a rich set of features such as a progressive in quality and resolution, the region of interest (ROI) and optimal ratedistortion performance with a modest increase in computational complexity. The JPEG standard uses an 8 × 8 discrete cosine transform (DCT) and the JPEG2000 standard uses two dimensional discrete wavelet transform (2DDWT). The Karhunen–Loueve transform (KLT) is an optimal method for encoding images in the mean squared error (MSE) sense. The compression performance of 2D cosine, Fourier, and Hartley transforms was compared using positron emission tomography (PET) and magnetic resonance (MR) images in Shyam Sunder et al. (2006). The authors claimed that the discrete Hartley transform (DHT) and the discrete Fourier transform (DFT) perform better than the DCT. Several techniques based on the threedimensional discrete cosine transform (3DDCT) have been proposed for volumetric data coding (Tai et al. 2000). Nevertheless, these techniques fail to provide lossless coding coupled with quality and resolution scalability, which is a significant drawback for teleradiology and telemedicine applications.
Several works on waveletbased 3D medical image compression have been reported in the literature (Schelkens et al. 2003; Xiong et al. 2003; Chao et al. 2003; Gibson et al. 2004; Xiaolin and Tang 2005; Sriram and Shyamsunder 2011; Ramakrishnan and Sriram 2006; Srikanth and Ramakrishnan 2005; He et al. 2003). A method based on blockbased quadtree compression, layered zerocoding, and contextbased arithmetic coding was proposed by Schelkens et al. (2003). They claimed that the method gives an excellent result for lossless compression and a comparable result for lossy compression. Modified 3DSPIHT and 3DEBCOT schemes for the compression of medical data were proposed by Xiong et al. (2003). Their method gives a comparable result for lossy and lossless compression. An optimal 3D coefficient tree structure for 3D zerotree coding was proposed by Chao et al. (2003). They suggested that an asymmetrical tree can produce a higher compression ratio than a symmetrical one. Gibson et al. (2004) incorporated an ROI and texture modelling stage into the 3DSPIHT coder for the compression of angiogram video sequences based on bit allocation criteria. Xiaolin and Tang (2005) presented a 3D scalable coding scheme which aimed to improve the productivity of a radiologist by providing a high decoder throughput, random access to the coded data volume, progressive transmission, and coding gain in a balanced design approach. Sriram and Shyamsunder (2011) proposed an optimal coder by making use of wavelets db4, db6, cdf9/7, and cdf5/3 with 3DSPIHT, 3DSPECK, and 3DBISK. They found that cdf 9/7 with 3DSPIHT yields the best compression performance. Ramakrishnan and Sriram (2006) proposed a waveletbased SPIHT coder for DICOM images for teleradiology applications. Similarly, many works based on 3DSPECK, 3DBISK, and 3DSPIHT used for the compression of hyperspectral images have been reported (Tang et al. 2003; Fowler and Rucker 2007; Lu and Pearlman 2001).
3DSPIHT and 3DSPECK use auxiliary lists [e.g., a list of insignificant pixels (LIP), a list of insignificant sets (LIS), and a list of significant pixels (LSP)] for tree/block partitioning. The auxiliary lists demand an efficient memory management technique, as the coefficients in the list are shuffled out during bitplane partitioning. This feature is not favorable for hardware realisation. Therefore, 2D variants of listless coders called no list SPIHT (NLS) (Latte et al. 2006), listless SPECK (Wheeler and Pearlman 2000), LEBP (Senapati et al. 2013), and HLDTT (Senapati et al. 2014a) use a state table to keep track of set partitions. These listless coders can be efficiently realised in hardware. Recently, a listless implementation of 3DSPECK for the compression of hyperspectral images was proposed by Ngadiran et al. (2010).
To the best of the authors’ knowledge, there have been few works on 3D listless implementations for medical images in the literature. This motivates us to develop a novel technique for encoding medical images using a modified 3D listless technique. The 3D listless algorithm uses static and dynamic marker state tables for encoding large clusters of insignificant blocks, which results in a rate reduction at earlier passes. From a unified bit string, the algorithm provides rate and resolution scalability for the compression of volumetric data. This set of features is a potential requirement in telemedicine and teleradiology applications.
The organization of the paper is as follows: “The proposed 3DHLCK embedded coder” section presents the proposed 3DHLCK algorithm and its memory allocation for 3D medical images. Simulation result and analysis with respect to coding performances and computational complexity using bigO notation are presented in “Results and discussion” section. Conclusions and further research directions are provided in “Conclusion” section.
The proposed 3DHLCK embedded coder
The block diagram of the proposed 3DHLCK algorithm is shown in Fig. 1. The 3D hybrid transformation is carried out in the 1st stage. Then, all 3D coefficients are mapped to one dimensional for processing by the proposed 3DHLCK algorithm. Figure 2 shows the coefficient arrangement algorithm. The arrangement is created by keeping in mind the hierarchical nature of a wavelet pyramid. Four image slices are shown here as an illustration. The experiment is carried out for eight slices in all images. The coefficients in each slice undergo Zscanning which maps two dimension to one dimension.
The coefficient is accessed using a linear indexing scheme (Wheeler and Pearlman 2000). Two types of marker state tables are used. They are a (1) dynamic marker table (Dm) and (2) static marker table (Sm). The onetoone correspondence between the coefficient values and the marker values are shown in Fig. 3. All marker values are initialised and loaded into memory along with the one dimensional (1D) arrangement of the image coefficient values. The dynamic markers in Dm update the values to indicate partitioning. The partitioning can be octal (8), tri (3) or quad (4). Octal partitioning takes place while there is a search for the significant coefficient in a composite wavelet level. Tri partitioning takes place while there is a search for the significant coefficient in a wavelet level. Quad partitioning takes place if a coefficient is found to be significant in a wavelet subband or a subblock inside a subband. The static marker table Sm is only used to skip a large cluster of areas, e.g. the entire composite level/wavelet level/wavelet subband. The length of the dynamic marker table is the same as that of the image array length. If each marker in the dynamic marker state table is 4 bits, then the memory consumes I/2 bytes for the state table. There are only three fixed markers per wavelet level. For five levels, there will be 15 × 3 = 45 markers in the constant marker table. The values of the markers depend on the image size (i.e. N × N) and the level of wavelet decompositions L. The initial marker value is \((log_2NL+1)\), and the final value is \((log_2N+1)\). For example, if the image dimension N = 128 and the level of decomposition L = 5, the marker values are 3, 4, 5, 6, 7, and 8 in each leading node of the wavelet level. Each bit plane undergoes three passes, as in conventional 3DSPIHT. They are (1) an insignificant coefficient pass, (2) an insignificant set pass, and (3) a refinement pass.
During the insignificant coefficient pass, a single coefficient will be tested for significance. During the insignificant set pass, a composite level/individual level/individual subband will be tested for significance. The refinement pass successively reduces the uncertainty interval between the reconstructed coefficient value and the actual coefficient value.
The symbol and meaning of each type of marker are specified below

INC: The coefficient is insignificant or untested for this bit plane.

NSC: The coefficient becomes significant so it shall not be refined for this bit plane.

SCR: The coefficient is significant and it shall be refined in this bit plane.
The markers listed below corresponds to the leading indices of each lower level of the pyramid. These markers shall be used to test the insignificance of a subband/block during each bitplane pass.
Static markers (Sm[k]):

Sm[1]: The coefficient is at the leading index of the combined wavelet level L. All the coefficients in the same wavelet level shall be skipped.

Sm[129]: The coefficient is at the leading index of the combined wavelet level L − 1. All coefficients in the same wavelet level shall be skipped.

Sm[513]: The coefficient is at the leading index of the combined wavelet level L − 2. All coefficients in the same wavelet level shall be skipped.
\(\vdots\)

Sm[32,719]: This coefficient is at the leading index of the finest pyramid level \(L5\). All coefficients in this level shall be skipped.
Dynamic markers (Dm[k]): The partitioning take place due to dynamic markers in a typical pyramid level (L1) is illustrated below. Similar illustration can be applied for other levels.

If Dm[129] = Sm[129], then the combined wavelet level L − 1 may be skipped.

If Dm[129] = Sm[129]1, then a wavelet level (for a single plane) L − 1 may be skipped.

If Dm[129] = Sm[129]2, then a single subband block in the wavelet level L − 1 may be skipped.

If Dm[129] = Sm[129]3, then \(\frac{1}{4}\)th of a subband block from a wavelet level may be skipped.
\(\vdots\)

If Dm[129] = 0, then a single coefficient is to be examined for significance.
Similar partitioning algorithm is applied to the other combined subbands as well as the composite coarsest subband.
\(k=129, 513, 2049, 8193, 32{,}719\) are the leading indices from resolution level (L − 1) to level 1(finest resolution level). There is a total of five combined level of arrangement in eight MRI slices, where each of 128 × 128 resolution.
The 3D coefficients are mapped to a 1D array of length I after hybrid transformation. The progressive encoder encodes the most significant bit plane and moves towards the lowest bit plane. It can be stopped whenever the bit budget matches the target rate. The significance level for each bit plane is s = \(2^n\), which is calculated with the bitwise logical AND operation \((\cap )\). The decoder performs reverse of encoding operation with some minor changes. The decoder generates the magnitude bits and sign bits of the coefficients with bitwise logical OR \((\cup )\) instead of bitwise logical AND \((\cap )\).
The 1D coefficient array \(\xi\) is bitplane coded and examined for significance in each bit plane pass. The initial threshold value can be computed as follows:
1. The initialization of Sm[k] and Dm[k] state table markers are illustrated below:

Sm[1, 17, 33, 49, 65, 81, 97, 113] = Dm[1, 17, 33, 49, 65, 81, 97, 113] = 3 for \(LL_5\) subband.

Sm[129, 177, 225, 273, 321, 369, 417, 465] =
Dm[129, 177, 225, 273, 321, 369, 417, 465] = 4 are the leading nodes of \(HL_5\), \(LH_5\) and \(HH_5\) subbands.

Sm[513, 705, 897, 1089, 1281, 1473, 1665, 1857] =
Dm[513, 705, 897, 1089, 1281, 1473, 1665, 1857] = 5 are the leading nodes of \(HL_4\), \(LH_4\) and \(HH_4\) subbands.
\(\vdots\)

Sm[2049, 2817, 3585, 4353, 5121, 5889, 6657, 7425] =
Dm[2049, 2817, 3585, 4353, 5121, 5889, 6657, 7425] = 6 are the leading nodes of \(HL_3\), \(LH_3\) and \(HH_3\) subbands.

Sm[8193, 11,265, 14,337, 17,409, 20,481, 23,553, 26,625, 29,697] =
Dm[8193, 11,265, 14,337, 17,409, 20,481, 23,553, 26,625, 29,697] = 7 are the leading nodes of \(HL_2\), \(LH_2\) and \(HH_2\) subbands.

Sm[32,769, 45,057, 57,345, 69,633, 81,921, 94,209, 106,497, 118,785] =
Dm[32,769, 45,057, 57,345, 69,633, 81,921, 94,209, 106,497, 118,785] = 8 are the leading nodes of \(HL_1\), \(LH_1\) and \(HH_1\) subbands.
2. Dm[k] shall be initialize to an arbitrary value (i.e. \(Dm[k]\ge (log_2N+1)+1\)) and these are marked as INC.
Block partitioning of 3DHLCK algorithm
The block partitioning of composite/combined levels is demonstrated in Fig. 4. Figure 4a demonstrates how the partitioning takes place for the composite coarsest level for the 1D arrangement of coefficients, and Fig. 4b demonstrates the partitioning of the combined pyramid level. If a coefficient is found to be significant, the combined coarsest level is partitioned into eight levels, where each level corresponds to the coarsest level of individual slices. Further, recursive quad partitioning in each level takes place until a coefficient is found to be significant. Finally, the significance of the coefficient value along with the sign bit will be transmitted. No sign bit will be transmitted if the coefficient is found to be insignificant. Similarly, the combined pyramid level is first octal partitioned into individual pyramid levels which correspond to each image slice. Then, each pyramid level is tri partitioned to find the subbands. The subbands are further quad partitioned to find a significant coefficient. Then, the coefficient will be coded and transmitted.
All the steps described above is presented below in the form of Pseudocode.
Pseudocode of 3DHLCK algorithm
Bit plane pass1: Insignificant coefficient pass
Bit plane pass2: Insignificant set pass
Bit plane Pass3: Refinement pass
Functions and parameters used in pseudocode

1.
Significant test function \((\zeta _n(\gamma ))\): Significant test is obtained by logical AND \((\cap )\) operation.
Example
Let a (2 × 2) block \(\gamma =[127 \;109\; 19\; 24]\), and current threshold value \(n=6\). \(\zeta _{n}\) can be calculated as

2.
Function QuadSplit( ):
The function partition the subband into four equal block sizes. The algorithm for quard partitioning can be illustrated below:
Note that if \(Dm[k]=0\), quad partitioning stops. The corresponding coefficient in block ‘\(\gamma\)’ is an insignificant coefficient (INC), and it will be examined for significance in insignificant coefficients pass (Pass 1) of the algorithm.
The OctalSplit() and TriSplit() functions are similar to the algorithm for QuadSplit(). OctalSplit() produces eight equal partitioned blocks, whereas TriSplit() produces three equal partitioned blocks.
If block ‘\(\gamma\)’ is a composite coarsest subband, then ‘\(\gamma\)’ undergoes octal partitioning (shown in Fig. 4a). Each partitioned block belongs to the coarsest wavelet level of the extracted plane of 3D medical image.
If block ‘\(\gamma\)’ is a composite/combined wavelet level, then ‘\(\gamma\)’ also undergoes octal partitioning (shown in Fig. 4b). Each partitioned corresponds to a wavelet level having three subbands.

3.
If \(Dm[k] = Sm[k]\), then a combined wavelet level is to be tested for significance.

4.
If \(Dm[k] = Sm[k]1\), then a single wavelet level is to be tested for significance.

5.
If \(Dm[k] = Sm[k]2\), then a subband is to be tested for significance.
Comparison with listless embedded block partitioning (LEBP)
The main differences between our earlier work on LEBP algorithm (Senapati et al. 2014b) and 3DHLCK are:

1.
A 3D hybrid transform is used in 3DHLCK (Wavelet transform using CDF 9/7 filters (Daubechics and Sweldens 1998) along spatial dimension and KLT along spectral dimension), whereas 2D wavelet transform is used in LEBP algorithm.

2.
The 3D coefficient arrangement is mapped to an 1D arrangement in order to encode large clusters of insignificant coefficients in 3DHLCK. However, LEBP uses 2D to 1D mapping scheme.

3.
Rate reduction because of fixed state table (Sm[k] markers) at initial passes in 3DHLCK. For example, \(Sm[k]=Dm[k]\) indicates a composite wavelet level can be skipped instead of a single wavelet level as in LEBP.

4.
Separate encoding techniques are used in 3DHLCK for combined coarsest and combined wavelet levels so as to reduce the number of zeros for insignificant coefficients in the coarsest subband.
Memory allocation
In 3DHLCK, the mapped 3D coefficient array, \(L_{max}\) has length 8I, where I is a 1D length of each slice/plane. If Y bytes are allocated for each subband coefficient, then the total storage memory required is 8IY for the subband coefficients and RC / 2 for the Dynamic state table \(\mathbf{Dm}\) as each marker is half a byte. In the case of L level of wavelet decomposition, \(\mathbf{Sm}\) needs \(\frac{(8L+1)}{2}\) bytes, as the number of fixed markers are \((8L+1)\) and each marker is half a byte.
Hence, the total memory needed by 3DHLCK is:
As said earlier Sm[k] markers are fixed markers. These are used in association with Dm[k] markers to check for insignificance (refer to pseudocode).
In 3DSPIHT coder, dynamic memory is determined by the auxiliary lists. The 3DSPIHT uses of LIP, LIS, and LSP as auxiliary lists. LIS has type ‘A’ or ‘B’ information to distinguish the coefficients.
Let, \(N_{LIP}\) be the number of coefficients in LIP, \(N_{LSP}\) be the number of coefficients in LSP, \(N_{LIS}\) be the number of coefficients in LIS, and Y be the number of bits to store the addressing information of a coefficient.
Then the total memory required (in bytes) due to auxiliary lists is given by Senapati et al. (2014a):
As the memory size increases in each bit plane pass, The worst case values are,
The memory required by Jyotheswar and Mahapatra (2007) is \(\left(\frac{37}{16}+\frac{5}{16}\times (Y+1)\right)\times M\times N\times\) (No. of planes).
For a 128 × 128 image using 3 bytes per coefficient and five levels of wavelet transform, and having the optional precomputed maximum length array (i.e, 8IY for 3DHLCK), the worst case memory (RAM) required is \(\frac{(128\times 128)}{2}\times (8\,{\rm bits})+\frac{(8L+1)}{2} \simeq 8\) kB for 3DHLCK, 204 kB for 3DSPIHT and 60 kB by Jyotheswar and Mahapatra (2007). Therefore 3DHLCK is a suitable candidate over 3DSPIHT and work in Jyotheswar and Mahapatra (2007) in terms of memory saving. This calculation is based using only memory consumption by the algorithms without regard to wavelet transform. Efficient wavelet transform techniques that take less memory have been reported recently in Mendlovic et al. (1997).
Results and discussion
Simulation was carried out on a Window XP platform having an Intel core i5 processor operating at a frequency of 2.6 GHz and 6 GB of internal RAM. The bit rate was varied from 0.5 to 2 bpp for compressing the images. Brain MRI, DICOM knee, and angiogram images were used in our experiment. Each image with a size 128 × 128 was used for the experiment. Tables 1, 2 and 3 summarise the PSNR comparison between 3DSPIHT (Sriram and Shyamsunder 2011), the algorithm by Jyotheswar and Mahapatra (2007) and the proposed 3DHLCK algorithm for brain MRI images. Tables 4, 5 and 6 summarise the PSNR comparison for DICOM knee images. Figures 5, 6, and 7 show compressed brain MRI images at a bit rate of 1.0 bpp using 3DHLCK, the algorithm in Jyotheswar and Mahapatra (2007), and 3DSPIHT respectively. It is apparent from the Figs. 5, 6, and 7 that the visual quality of the compressed images using 3DHLCK is better than that obtained by using 3DSPIHT and comparable with the algorithm in Jyotheswar and Mahapatra (2007). Figures 8 and 9 show the DICOM knee and angiogram images compressed at 2.0 bpp using proposed 3DHLCK algorithm.
Discussion
Comparisons of the coding performance (PSNR vs. Slice no.) for MRI brain image, DICOM knee image, and MRI angiogram images for a constant bit rate are summarised in Tables 1, 2, 3, 4, 5, 6, 7 and 8 respectively. It is observed that the proposed 3DHLCK algorithm exhibits a PSNR improvement between 0.05 and 0.5 dB for MRI brain images and 0.05–0.6 dB for DICOM knee images for a bit rate of 0.5–2.0 bpp compared to 3DSPIHT. 3DSPIHT shows higher a PSNR in the Slice1, Slice2, and Slice7 images in comparison to 3DHLCK in the MRI angiogram images at 1 bpp. A similar trend is also observed at 2 bpp. In comparison to the work by Jyotheswar and Mahapatra (2007), 3DHLCK shows an improvement of 0.05–0.5 dB for the given slices at 0.5 bpp. However, at higher rates, the work in Jyotheswar and Mahapatra (2007) shows a PSNR improvement of around 0.15 dB in the MRI images compared to 3DHLCK. In the DICOM and angiogram images, the algorithm in Jyotheswar and Mahapatra (2007) shows a slight PSNR improvement with respect to 3DHLCK.
The proposed algorithm exhibits a better PSNR improvement for other slices in 3DSPIHT because of the following reasons:

1.
3D SPIHT uses 3D DWT coefficients for encoding, whereas hybrid transformed (2D DWT+KLT) coefficients are encoded by 3DHLCK.

2.
Large clusters of zeros are efficiently coded (both inter and intra) by 3DHLCK.

3.
Coefficients are efficiently arranged among different subbands of slices to exploit inter and intrasubband correlations within and across slices.
The work in Jyotheswar and Mahapatra (2007) outperforms 3DHLCK at higher rates (above 1 bpp) for MRI images for the following reasons: (i) The execution of a refinement pass before the sorting pass. (ii) The ordering of the coefficient scanning process for simple hardware implementation. (iii) Optimisation for lossless encoding using 5/3 filters in the spatial and spectral dimensions.
The proposed 3DHLCK algorithm will occupy a fixed amount of memory, irrespective of the number of bitplane passes, owing to the fixed number of state table markers. Partitioning takes place by updating the marker values. Each marker holds a maximum 4 bits. The algorithm in Jyotheswar and Mahapatra (2007) requires a fixed memory size and exhibits simple hardware portability. However, in 3DSPIHT, the linked lists (LIP, LIS, and LSP) add/remove/move additional nodes for every bitplane pass. Therefore, the memory usage grows exponentially. Rate and resolution scalability on par with 3DSPIHT is achieved by 3DHLCK. Memory saving is trivial, as in most applications, the cost of memory is cheap. However, the proposed algorithm is potentially suitable for applications such as the progressive transmission of DICOM images, lossless archival, telemedicine, teleradiology, and capsule endoscopy. Therefore, 3DHLCK can be a preferred option over 3DSPIHT for the aforementioned applications. A further reduction in the overall complexity can be achieved by using fractional wavelet transforms (FrWTs) (Mendlovic et al. 1997) for such applications.
From the simulation, it is observed that the average encoding and decoding times for 3DHLCK are 12 times more than those for 3DSPIHT at 2 bpp. Further optimisation can be done for 3DHLCK to reduce the time complexity. However, it can be proved mathematically that the computational complexity of 3DHLCK will be O(N) operations compared to O(N log N) for 3DSPIHT (Senapati et al. 2014a).
Conclusion
A new 3D coder called 3DHLCK is proposed in this paper. Owing to the listless nature of 3DHLCK, significant memory reductions of over 96 and 86% are achieved compared to 3DSPIHT and the work by Jyotheswar and Mahapatra respectively. 3DHLCK has features such as rate and resolution scalability. In brain MRI, DICOM knee and angiogram images, a PSNR improvement of 0.05–0.5 dB is also achieved compared to 3DSPIHT. The proposed coder exhibits a comparable coding efficiency and easy hardware portability with the work by Jyotheswar and Mahapatra. Therefore, it can be used in applications such as telemedicine, teleradiology, wireless capsule endoscopy and the Internet transmission of DICOM images. Future work will incorporate additional features such as the ROI coding, random access coding, and video coding using 3DHLCK.
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Authors' contributions
RKS first conceived the idea, carried out the simulation work and drafted the manuscript. PMK carried out the literature review. GS assisted in MATLAB simulation during revision process. TNS participated in the sequence alignment and assisted in correcting the vocabulary. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Senapati, R.K., Prasad, P.M.K., Swain, G. et al. Volumetric medical image compression using 3D listless embedded block partitioning. SpringerPlus 5, 2100 (2016). https://doi.org/10.1186/s400640163784y
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DOI: https://doi.org/10.1186/s400640163784y
Keywords
 3D hierarchical listless embedded block
 Set partitioning in hierarchical trees
 Volumetric compression
 Embedded coder
 Peaksignaltonoiseratio