Volumetric medical image compression using 3D listless embedded block partitioning
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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.
Keywords
3D hierarchical listless embedded block Set partitioning in hierarchical trees Volumetric compression Embedded coder PeaksignaltonoiseratioBackground
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
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.

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.

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.

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.
\(k=129, 513, 2049, 8193, 32{,}719\) 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 \((\cap )\).

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.
Block partitioning of 3DHLCK algorithm
All the steps described above is presented below in the form of Pseudocode.
Pseudocode of 3DHLCK algorithm
Functions and parameters used in pseudocode
 1.
Significant test function \((\zeta _n(\gamma ))\): Significant test is obtained by logical AND \((\cap )\) operation.
Example
 2.
Function QuadSplit( ):
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)
 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.
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.
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
PSNR comparison of brain MRI image at 0.5 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  24.7942  24.8550  25.2461  24.9937  25.2052  25.3051  24.7844  24.9987 
Jyotheswar and Mahapatra (2007)  24.8210  24.9234  25.3012  25.1224  25.3045  25.3412  24.7724  25.0654 
3DHLCK  25.0136  25.4772  25.6954  25.1605  25.3025  25.5581  24.8334  25.0457 
PSNR comparison of brain MRI image at 1.0 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  28.7479  29.3962  29.6978  28.8805  28.7790  28.9684  28.6158  29.3115 
Jyotheswar and Mahapatra (2007)  29.1123  29.9129  29.8125  29.2224  29.4042  29.5512  28.9724  29.4654 
3DHLCK  28.9032  29.7086  29.7709  29.0900  29.3314  29.4190  28.8109  29.3783 
PSNR comparison of brain MRI image at 2.0 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  35.5649  35.8673  35.7901  35.5476  35.5843  35.6065  35.7257  35.6406 
Jyotheswar and Mahapatra (2007)  35.7210  35.9524  35.9612  35.8128  35.9245  35.8322  35.9724  35.9654 
3DHLCK  35.6260  35.9328  35.9546  35.7196  35.9179  35.7822  35.8127  35.8426 
PSNR comparison of DICOM knee image at 0.5 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  35.6328  35.2043  34.2868  34.9760  34.9612  34.9178  34.6008  34.3115 
Jyotheswar and Mahapatra (2007)  35.8276  35.3610  34.9234  35.3578  35.2997  35.2491  34.7612  34.4321 
3DHLCK  35.8646  35.3801  34.9428  35.3693  35.3564  35.2795  34.8423  34.4471 
PSNR comparison of DICOM knee image at 1.0 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  38.9826  38.7107  38.4028  38.5471  38.7088  38.5060  38.3513  37.8718 
Jyotheswar and Mahapatra (2007)  39.0214  38.8820  38.5221  38.7510  38.8997  38.6126  38.3901  37.9011 
3DHLCK  39.1471  38.8796  38.5239  38.7847  38.9222  38.6316  38.4140  37.9713 
PSNR comparison of DICOM knee image at 2.0 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  44.3862  44.2377  43.5063  43.9376  43.9620  43.7279  43.7797  43.4163 
Jyotheswar and Mahapatra (2007)  44.3901  44.3213  43.9112  44.2011  44.1930  43.9902  43.9128  43.5234 
3DHLCK  44.3887  44.3083  43.8962  44.1202  44.1539  43.9502  43.8313  43.5019 
Discussion
PSNR comparison of MRI angiogram image at 1.0 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  36.3810  36.5521  36.0900  36.4401  36.4912  36.3016  37.0721  36.5801 
Jyotheswar and Mahapatra (2007)  36.4891  36.4389  36.2523  36.5013  36.4434  36.4012  37.0121  36.6010 
3DHLCK  36.2908  36.4508  36.2815  36.5191  36.4380  36.3937  36.9916  36.0091 
PSNR comparison of MRI angiogram image at 2.0 bpp
Algorithm  Slice1  Slice2  Slice3  Slice4  Slice5  Slice6  Slice7  Slice8 

3DSPIHT (Sriram and Shyamsunder 2011)  45.1229  45.2415  45.0016  44.7210  44.6910  44.7890  44.8010  44.7892 
Jyotheswar and Mahapatra (2007)  45.1321  45.2312  45.1200  44.8310  44.7610  44.8231  44.8012  44.7891 
3DHLCK  44.8976  45.1531  45.1106  44.8234  44.7794  44.8434  44.8144  44.8627 
 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 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.
Notes
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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