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A combined scheme of pixel and block level splitting for medical image compression and reconstruction
Alexandria Engineering Journal (2017) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
A combined scheme of pixel and block level splitting for medical image compression and reconstruction H. Sunil a,*, Sharanabasaweshwar G. Hiremath b a b
Department of Electronics and Communication, Vemana IT, Bengaluru, India Department of Electronics and Communication, EWIT, Bengaluru, India
Received 7 December 2016; accepted 1 March 2017
KEYWORDS DICOM; Pixel; Compression
Abstract Medical images are very important source of information of human body diagnosis or therapy. These images contain huge amount of information which needs to be compressed to reduce the storage capacity issues. These images require lossless compression techniques to save the information. Another crucial issue in medical imaging is to reconstruct image efficiently without losing information. In order to achieve the reconstruction of medical image we present a new approach by introducing convex-smoothing problems. This approach is implemented by dividing the input image into sub-problems. According to this approach we divide the input image into sub-problems which are solved by using compressed sensing method. After achieving the output of compressed sensed image, this problem is passed to the iterative process of problem solver to reduce the time, computational complexity and reconstruction error. Proposed model is implemented using MATLAB tool and tested with medical images. Results are carried in terms of the PSNR, reconstruction error and time. Ó 2017 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction 1.1. Background of medical imaging Medical imaging system is having great impact on the diagnosis, recognition and surgical planning of the disease. Medical images contain huge amount of data which help doctors to analyze the data efficiently and planning the diagnosis for the patient. Storage of these medical images is the crucial task * Corresponding author. E-mail address: [email protected] (H. Sunil). Peer review under responsibility of Faculty of Engineering, Alexandria University.
for the hospitals because of storage requirements. These images are stored into the digital form which is easier to analyze. DICOM images are one of the examples of medical captured digital image. In order to overcome the issue of storage of images, compression is required. Due to the compression images lose their information which causes risk during treatment or diagnosis, so it motivates to design an efficient algorithm for compression and reconstruction to preserve the image quality and lesser computational time for transmission. Based on the loss of the data, image compression techniques are classified into two major categories: (i) Lossless image compression, (ii) Lossy Compression and (iii) Fractal Coding [2,3]. Lossless method is divided into two phases – image modeling and coding. During the modeling phase, a model is designed
http://dx.doi.org/10.1016/j.aej.2017.03.001 1110-0168 Ó 2017 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
2 from the given image, used for encoding and during the coding phase, statistical analysis is performed to obtain the symbols of the given image. Data loss is very less in lossless compression scheme when compared to the lossy compression due to the quantization nature of the scheme. Fractal coding based image compression can be lossless or lossy compression; it is used for the redundancy removal from the original data after compression. For medical imaging system, lossless compression scheme is a promising technique to store the huge medical data. Many researchers have proposed various architectures for image compression by keeping the information preserved in mind for medical imaging system. In [4] fractals coding based compression schemes are discussed for MRI data compression. These schemes are categorized as (i) standard fractal coding, (ii) quasi-lossless fractal coding and (iii) improved quasilossless fractal coding. This work proposes quasi-lossless fractal coding scheme to preserve the rich feature of an image. These features include domain, blocks, and by using fractal transformation the image is reconstructed. This approach reduces the computation time for encoding and improves the performance of the compression scheme. Miaou et al. [5] proposed a lossless compression scheme for medical image compression using JPEG-LS and Interframe Coding scheme. 1.2. Problem definition As discussed before that lossless image compression is the efficient technique to compress the data without losing the original information during reconstruction. Lossless image compression contains three sub-categories of image compression which are (i) Entropy-coding, (ii) Dictionary-based and (iii) Prediction methods. These schemes have some disadvantages which motivate researchers to improve the approach of image compression by introducing different techniques. These schemes generate extra overhead for the user during the encoding and decoding process on the image. Sometimes these schemes are complex enough for implementation which decreases the reliability. Finally, during the transmission of the image through a channel, noise affects the quality of compressed image which results in the lower reconstruction quality. Quality improvement of the reconstructed image is still a challenging task for the researchers. In order to achieve the efficiency in compression and reconstruction, a new approach has to be introduced. 1.3. Contribution of the paper By considering all the challenges and problems related to the medical image compression, we propose an efficient algorithm for medical image compression and reconstruction. In this work our main aim was to improve the performance of image compression and reconstruction. The performance of the system is evaluated in terms of PSNR and time taken. Proposed work contains two sub-sections, and in the first sub-section input image is considered as a convex problem, in order to optimize that the problem is decomposed into two sub problems. In next stage, these problems are solved by using an iterative optimization process. This approach is applied for the medical images to reconstruct the images by using lowerrank tensor construction. Experimental results show the efficiency of the proposed approach with lower computation time.
H. Sunil, S.G. Hiremath The rest of the paper is organized as follows: Section 2 describes the literature review of this approach which includes most recent works in the field of medical imaging system. Proposed model is discussed in Section 3, Section 4 describes the results achieve by using proposed model and finally the manuscript is concluded in Section 5. 2. Literature survey This section describes about the related works proposed in this field of medical imaging during last decade. Various approaches have been proposed for medical image compression and reconstruction. Bhavani and Thanushkodi [4], presented a comparative study of fractal-coding schemes for MRI images. In this work standard fractal coding, quasi-lossless fractal coding and improved quasi-lossless fractal coding are presented and their efficiency is evaluated. For MRI image (size is 512X512) fractal coding scheme is applied which helps to preserve the rich-feature of the image. Zhang and Wang [6] proposed a new approach for image compression by combining wavelet transform technique and neural network approach. In [7], authors proposed region based compression scheme. According to this approach, in medical images small region might be useful for diagnostic but due to the wrong interpolation cost of the diagnostic increases. To overcome these issues, Region Based Coding is proposed for compression and image transmission. In [8] a new approach for compression, to save the image information is discussed. In order to improve the compression ratio, diagonal pixel values are considered for computation along with horizontal and vertical pixels of the image. Modified Hierarchical Prediction and Context Adaptive (MHPCA) coding is introduced to overcome the issue of enormous prediction error rate near edges and preserves the sharpness of images. In the process of image compression, image coding plays an important role. CT images are acquired from the series of X-Ray exposures resulting in 3D-images. In [9] authors proposed a new approach to achieve the similarity in the medical image by dividing it into slices. This work presents a novel approach of correlation computation based on the r factor. Based on the fractal approach of image compression, Barnsely introduced iterated function system for fractal image compression [10]. Based on the iterated function system Nadira Banu Kamal et al. [11] introduced new approach which removes the iterative process for fractal image compression. Xu et al. [12] proposed sparsity based image compression for lower bit-rate image compression. Conventional approaches for the image compression based on sparsity utilize matching pursuit algorithms which are not stable in terms of sparsity of the image which reduces the image compression quality performance. In [13], Bo Wang used neural network based method for image compression and [14] describes a comparative study of fractal image compression using DWT. Lustig et al. has proposed a CG (conjugate gradient) methods [15] for compression and reconstruction, which is our existing approach. 3. Proposed system In this section we describe the proposed approach for medical image reconstruction. For this purpose various methodologies that are based on the total-variation and linear combination are used in the literature. In this work we use compressive sens-
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
Medical image compression and reconstruction
3
ing based approach, according to this approach forms a energy model which is represented by using a smooth convex function CðxÞ (see Table 1). This is used to formulate solutions according to the given problem with various properties. This is achieved by the linear combination of the non-smooth convex function Cx . In this work, the problem formulation is given as minp F ðxÞ ¼ CðxÞ þ xR
m X P i ðOi xÞ
ð1Þ
3.2. Building blocks of proposed model
i1
where Rp denotes the optimal solution for the given input medical image, P i denotes the prior modeling of each pixel and orthogonal pixel matrix of the pixels is denoted by Oi . In order to solve this we create two subproblems that perform the splitting of the given data, the splitted data contain pixel splitting and block splitting. Block splitting stage finds the value of x which is finally assigned to the maximalmonotone operator equal to zero. Splitting of pixel is another way to achieve the optimal solution by solving above given Eq. (1). This process uses alternate direction methods with the Langragian framework. According to this method pixels are splitted into m þ 1 pixels with the help of new pixel m which belongs to the image model P, then for each pixel Langragian method is applied and finally the decomposed Langragian is minimized to achieve the alternate directional solution of the image. In the later section we combine the splitting approaches to improve the performance. Original input problem is divided into subproblems by dividing the convex function into m function, dividing pixels into m pixels and minimization of the splitted image blocks and then finally the linear combination of this function gives the optimal solution of the image. Moreover the proposed approach is able to decompose the hard convex problems into subproblems to achieve the optimal solution with lesser computational complexity. 3.1. Proposed model for combined splitting method This section discusses about the proposed combined modeling approach for image reconstruction with completion of the lower rank tensor. In order to design this, we consider Eq.
Table 1
(1) as the given problem. In this problem P i : Rp ! R is considered as a continuous non-smooth convex function, f : Rp ! R is the smooth convex function and ORpp is considered the orthonormal matrices. In this problem, if value of m ¼ 1, then it can be solved easily but for the higher values of m it is challenging (see Fig. 1).
From the above discussion it can be concluded that if a complex problem is given then it can be solved by using iterative method by achieving the optimal solution in the given number of iterations. This iterative process can be given as Algorithm 1 Input: d ¼ 1=Lc ; xo repeat for n = 1 to N do xn ¼ proxd ðgÞðxn1 drfðxn1 Þ End for Repeat until stopping criteria achieved
Lc is the constant, and d denotes the input scalar value. To make it faster by using iterative method we use output of this at each iteration which is given as fxn g and rn and the problem is formulated such as Fðxn ÞFðx Þ 6
2Lc jjxo x jj2 ðn þ 1Þ2
8x 2 X
ð2Þ
This problem can be solved by Algorithm 2 Input: d ¼ 1=Lc ; r1 ¼ xo ; t1 ¼ 1 Repeat For n ¼ 1 to N do xn ¼ proxd ðgÞðrn drfðrn ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tnþ1 ¼ ð1 þ 1 þ 4ðtk Þ Þ 12 1 rnþ1 ¼ xn þ ttnþ1 ðxn xn1 Þ End for Repeat until optimal criteria is achieved n
Symbol used in this work.
Symbol
Representation
CðxÞ F ðxÞ Pi Oi Rp m Rp ! R Lc d fxn g; rn jjxjjns1 jjhxjjns2 m Fp b; c
Denotes the convex problem Problem function Prior modeling Orthogonal matrix of the input image Optimal solution New pixel value for splitting Non-smooth convex function Input image optimal solution constant Scalar value Output achieved from the initial splitting Non-smooth problem Non-smooth problem Under sampled Fourier measurements Partial Fourier transform Positive constant parameter
3.3. Image reconstruction For image reconstruction we apply above discussed algorithm for image reconstruction by compressed sensing. This problem is formulated by using 1 minx FðxÞ ¼ jjFp x mjj2 þ bjjxjjns1 þ cjjhxjjns2 2
ð3Þ
jjxjjns1 and jjhxjjns2 are the non-smooth convex problems, m is the under sampled Fourier measurements, b and c are the two positive constant parameters and Fp is the Fourier transform of the given data. This model shows the efficiency for image recovery, used widely in the medical field. This process utilizes non-smooth
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
4
H. Sunil, S.G. Hiremath
Figure 1
(a) Brain image
(d) artery image Figure 2
Overall system flowchart.
(b) Kidney MR image
(c) Heart image
(e) Human skeleton image
Input images considered for compression using proposed approach.
convex problems which are given as jjxjjns1 and jjhxjjns2 . In order to solve this we use fast method of convex problem optimization using iterative method. The first step of this approach gives a closed form solution which can be computed using above discussed fast computation algorithm using iterative method. The achieved solution is given as xn ¼ proxd ðjjhxjj1 Þðrn drfðrn ÞÞ
ð4Þ
The computation time required for this type of problem is very less and can be solved easily by using iterative approach but if the problem is based on the joint L1 and total variation based which can be given as xn ¼ proxd ðbjjxjjns1 þ cjjhxjjns2 Þðrn drfðrk ÞÞ
ð5Þ
This problem requires more time to achieve the optimal solution due to the gradients of conjugate which makes it slower. In order to achieve the MR reconstruction we apply pixel splitting technique. The key feature of this approach is
to replace the linear solver with Fourier domain computation which converges faster and reduces the timing complexity issue. This algorithm is given as below mentioned algorithm 3 and applied for MR image reconstruction.
Algorithm 3 Input: d ¼ 1=Lc ; b; c; t1 ¼ 1; r1 ¼ xo For n ¼ 1 to N do xg ¼ rn drfðrn Þ x1 ¼ proxd ð2bjjxjjns1 Þðxg Þ x2 ¼ proxd ð2cjjhxjjns2 Þðxg Þ 2Þ xn ¼ ðx1 þx 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi2 kÞ Þ tnþ1 ¼ ð1þ 1þ4ðt 2 n1 rnþ1 ¼ xn þ ttnþ1 xn xn1 end for
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
Medical image compression and reconstruction
Figure 3
Table 2
5
Performance evaluation for brain image.
Performance evaluation.
Time taken
SNR (ES)
SNR (PS)
0.07020045 0.08580055 0.10140065 0.10140065 0.13260085 0.19500125 0.22620145 0.2340015 0.2496016
4.390606 6.691962 7.465152 7.766746 7.882058 7.925613 7.942536 7.949732 7.95339
6.004975 7.631901 8.527296 9.039236 9.439868 9.796934 10.11622 10.40066 10.65587
Figure 4
System performance.
4. Experimental results This section describes the obtained result by applying the proposed image compression methodology. The CPU used in experiment is an intel core i5 processor, which is clocked at 2.90 GHz and having 8.00 GB of RAM. Proposed approach is implemented using MATLAB tool on Windows 64-bit platform for different medical images. Proposed approach is implemented using MATLAB tool on Windows platform for different medical images. For experimental study we have considered MRI and Ct images. In the below given Fig. 2, input images are given. In our proposed approach we evaluate the performance in terms of time taken for compression and respective signal to noise ratio. Comparative analysis is shown in Fig. 3 (considering brain image). It can be seen from the figure that the proposed approach gives better SNR when compared to the existing approach. Time taken by the CPU increases, performance of the proposed work also increases in terms of SNR, and on other hand existing system [15] performance degrades, as the time taken increases. Key issue with the performance is the complexity. According to the existing system [15], input image is not
Figure 5
System performance CPU time vs iteration.
divided into sub-problems whereas we proposed to divide the image into smaller sub-problems which helps us to achieve the improved results. In other way for 10 iterations, performance is evaluated which is given in Table 2. Graphical representation of this is given in Fig. 4. It shows the performance of proposed system and existing system [15] in
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
6
H. Sunil, S.G. Hiremath stages: initially for a given image a problem is formulated on the problem we apply convergence theory to solve it, one the solution is achieved for this process and then later we apply the iterative process to reduce the computation complexity by introducing convergence for the given problem. Finally the image is splitted into pixel and blocks which is used for the reconstruction by applying hybrid approach of image reconstruction by using compressed sensing. Performance of the proposed model is achieved in terms of PSNR, Reconstruction error and time taken to perform the reconstruction. References
Figure 6
SNR vs CPU time performance for case II.
SNR Performance
SNR
Existing System
16 14 12 10 8 6 4 2 0
1
2
3
Proposed System
4
5
6
7
8
Iteration
Figure 7
Table 3
SNR performance for MRI image.
SNR performance for case II.
Existing system
Proposed system
6.269411134 9.914735097 11.45108472 12.12953799 12.40723566 12.51781347 12.56420924 12.58675518 12.60049576
8.634913284 11.36928701 12.82869845 13.37280002 13.676297 13.92724569 14.13313235 14.29648209 14.42867327
terms of SNR with respect to the iterations. Figs. 5 and 6 show the performance in terms of time and SNR. In the next stage of result evaluation of proposed approach, MRI scan of body image is considered for performance evaluation. In Fig. 7 we show the comparative study analysis of the proposed approach. Performance study evaluation is carried out in terms of SNR achieved vs iterations. For each iteration, SNR values are tabulated in the below given Table 3. 5. Conclusion In this work we propose a new approach for medical image reconstruction. The proposed approach is dived into the three
[2] A. Kumar Gupta, M. Dyer, A. Hirsch, S. Nooshabadi, D. Taubman, Design of a single chip block coder for the EBCOT engine in JPEG2000, in: Proceedings of the 48th midwest symposium on circuits and systems, 2005, pp. 63–66. [3] Isa Servan Uzun, Abbes Amira, Real-time 2-D wavelet transform implementation for HDTV compression, Real-Time Imag. 11 (2005) 151–165. [4] S. Bhavani, K.G. Thanushkodi, Comparison of fractal coding methods for medical image compression, IET Image Proc. 7 (7) (2013) 686–693. [5] S.G. Miaou, F.S. Ke, S.C. Chen, A lossless compression method for medical image sequences using JPEG-LS and interframe coding, IEEE Trans. Inf Technol. Biomed. 13 (5) (2009) 818– 821. [6] Suqing Zhang, Aiqiang Wang, An image compression method based on wavelet transform and neural network, TELKOMNIKA 13 (2) (2015) 587–596. [7] V.K. Bairagi, A.M. Sapkal, Automated region based hybrid compression for DICOM MRI images for telemedicine applications, IET Sci., Meas. Technol. 6 (4) (2012) 247–253. [8] P. Suresh Babu, S. Sathappan, Efficient lossless image compression using modified hierarchical prediction and context adaptive coding, Indian Journal of Science and Technology, December 2015, vol 8(34), http://dx.doi.org/10. 17485/ijst/2015/v8i34/74365. [9] J. Munoz-Gomez, J. Bartrina-Rapesta, M.W. Marcellin, J. Serra-Sagrist, Correlation modeling for compression of computed tomography images, IEEE J. Biomed. Health Informat. 17 (2013) 5. [10] M.F. Barnsley, Fractal image compression, Notices of the AMS, 1996. [11] A.R. Nadira Banu Kamal, S. Thamarai Selvi, Enhanced iteration free fractal image coding algorithm with efficient search and storage space, ICTACT J. Image Video Process. 2 (2010) 124–134. [12] M. Xu, S. Li, J. Lu, W. Zhu, Compressibility constrained sparse representation with learnt dictionary for low bit-rate image compression, IEEE Trans. Circuits Syst. Video Technol. 24 (10) (2014) 1743–1757. [13] Bo Wang, Yubin Gao, An image compression scheme based on fuzzy neural network, Vol. 13, No. 1, 2015, pp. 137–145. [14] Ansam Ennaciri, Mohammed Erritali, Mustapha Mabrouki, Jamaa Bengourram, Comparative study of wavelet image compression: JPEG2000 Standart, Vol. 16, No. 1, October 2015, pp. 83–90. [15] D.D.M. Lustig, J. Pauly, Sparse MRI: the application of compressed sensing for rapid MR imaging, Magn. Reson. Med. (2007).
Further reading [1] G.K. Wallace, The JPEG still picture compression standard, Commun. ACM 34 (4) (1991) 607–609.
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
A combined scheme of pixel and block level splitting for medical image compression and reconstruction H. Sunil a,*, Sharanabasaweshwar G. Hiremath b a b
Department of Electronics and Communication, Vemana IT, Bengaluru, India Department of Electronics and Communication, EWIT, Bengaluru, India
Received 7 December 2016; accepted 1 March 2017
KEYWORDS DICOM; Pixel; Compression
Abstract Medical images are very important source of information of human body diagnosis or therapy. These images contain huge amount of information which needs to be compressed to reduce the storage capacity issues. These images require lossless compression techniques to save the information. Another crucial issue in medical imaging is to reconstruct image efficiently without losing information. In order to achieve the reconstruction of medical image we present a new approach by introducing convex-smoothing problems. This approach is implemented by dividing the input image into sub-problems. According to this approach we divide the input image into sub-problems which are solved by using compressed sensing method. After achieving the output of compressed sensed image, this problem is passed to the iterative process of problem solver to reduce the time, computational complexity and reconstruction error. Proposed model is implemented using MATLAB tool and tested with medical images. Results are carried in terms of the PSNR, reconstruction error and time. Ó 2017 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction 1.1. Background of medical imaging Medical imaging system is having great impact on the diagnosis, recognition and surgical planning of the disease. Medical images contain huge amount of data which help doctors to analyze the data efficiently and planning the diagnosis for the patient. Storage of these medical images is the crucial task * Corresponding author. E-mail address: [email protected] (H. Sunil). Peer review under responsibility of Faculty of Engineering, Alexandria University.
for the hospitals because of storage requirements. These images are stored into the digital form which is easier to analyze. DICOM images are one of the examples of medical captured digital image. In order to overcome the issue of storage of images, compression is required. Due to the compression images lose their information which causes risk during treatment or diagnosis, so it motivates to design an efficient algorithm for compression and reconstruction to preserve the image quality and lesser computational time for transmission. Based on the loss of the data, image compression techniques are classified into two major categories: (i) Lossless image compression, (ii) Lossy Compression and (iii) Fractal Coding [2,3]. Lossless method is divided into two phases – image modeling and coding. During the modeling phase, a model is designed
http://dx.doi.org/10.1016/j.aej.2017.03.001 1110-0168 Ó 2017 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
2 from the given image, used for encoding and during the coding phase, statistical analysis is performed to obtain the symbols of the given image. Data loss is very less in lossless compression scheme when compared to the lossy compression due to the quantization nature of the scheme. Fractal coding based image compression can be lossless or lossy compression; it is used for the redundancy removal from the original data after compression. For medical imaging system, lossless compression scheme is a promising technique to store the huge medical data. Many researchers have proposed various architectures for image compression by keeping the information preserved in mind for medical imaging system. In [4] fractals coding based compression schemes are discussed for MRI data compression. These schemes are categorized as (i) standard fractal coding, (ii) quasi-lossless fractal coding and (iii) improved quasilossless fractal coding. This work proposes quasi-lossless fractal coding scheme to preserve the rich feature of an image. These features include domain, blocks, and by using fractal transformation the image is reconstructed. This approach reduces the computation time for encoding and improves the performance of the compression scheme. Miaou et al. [5] proposed a lossless compression scheme for medical image compression using JPEG-LS and Interframe Coding scheme. 1.2. Problem definition As discussed before that lossless image compression is the efficient technique to compress the data without losing the original information during reconstruction. Lossless image compression contains three sub-categories of image compression which are (i) Entropy-coding, (ii) Dictionary-based and (iii) Prediction methods. These schemes have some disadvantages which motivate researchers to improve the approach of image compression by introducing different techniques. These schemes generate extra overhead for the user during the encoding and decoding process on the image. Sometimes these schemes are complex enough for implementation which decreases the reliability. Finally, during the transmission of the image through a channel, noise affects the quality of compressed image which results in the lower reconstruction quality. Quality improvement of the reconstructed image is still a challenging task for the researchers. In order to achieve the efficiency in compression and reconstruction, a new approach has to be introduced. 1.3. Contribution of the paper By considering all the challenges and problems related to the medical image compression, we propose an efficient algorithm for medical image compression and reconstruction. In this work our main aim was to improve the performance of image compression and reconstruction. The performance of the system is evaluated in terms of PSNR and time taken. Proposed work contains two sub-sections, and in the first sub-section input image is considered as a convex problem, in order to optimize that the problem is decomposed into two sub problems. In next stage, these problems are solved by using an iterative optimization process. This approach is applied for the medical images to reconstruct the images by using lowerrank tensor construction. Experimental results show the efficiency of the proposed approach with lower computation time.
H. Sunil, S.G. Hiremath The rest of the paper is organized as follows: Section 2 describes the literature review of this approach which includes most recent works in the field of medical imaging system. Proposed model is discussed in Section 3, Section 4 describes the results achieve by using proposed model and finally the manuscript is concluded in Section 5. 2. Literature survey This section describes about the related works proposed in this field of medical imaging during last decade. Various approaches have been proposed for medical image compression and reconstruction. Bhavani and Thanushkodi [4], presented a comparative study of fractal-coding schemes for MRI images. In this work standard fractal coding, quasi-lossless fractal coding and improved quasi-lossless fractal coding are presented and their efficiency is evaluated. For MRI image (size is 512X512) fractal coding scheme is applied which helps to preserve the rich-feature of the image. Zhang and Wang [6] proposed a new approach for image compression by combining wavelet transform technique and neural network approach. In [7], authors proposed region based compression scheme. According to this approach, in medical images small region might be useful for diagnostic but due to the wrong interpolation cost of the diagnostic increases. To overcome these issues, Region Based Coding is proposed for compression and image transmission. In [8] a new approach for compression, to save the image information is discussed. In order to improve the compression ratio, diagonal pixel values are considered for computation along with horizontal and vertical pixels of the image. Modified Hierarchical Prediction and Context Adaptive (MHPCA) coding is introduced to overcome the issue of enormous prediction error rate near edges and preserves the sharpness of images. In the process of image compression, image coding plays an important role. CT images are acquired from the series of X-Ray exposures resulting in 3D-images. In [9] authors proposed a new approach to achieve the similarity in the medical image by dividing it into slices. This work presents a novel approach of correlation computation based on the r factor. Based on the fractal approach of image compression, Barnsely introduced iterated function system for fractal image compression [10]. Based on the iterated function system Nadira Banu Kamal et al. [11] introduced new approach which removes the iterative process for fractal image compression. Xu et al. [12] proposed sparsity based image compression for lower bit-rate image compression. Conventional approaches for the image compression based on sparsity utilize matching pursuit algorithms which are not stable in terms of sparsity of the image which reduces the image compression quality performance. In [13], Bo Wang used neural network based method for image compression and [14] describes a comparative study of fractal image compression using DWT. Lustig et al. has proposed a CG (conjugate gradient) methods [15] for compression and reconstruction, which is our existing approach. 3. Proposed system In this section we describe the proposed approach for medical image reconstruction. For this purpose various methodologies that are based on the total-variation and linear combination are used in the literature. In this work we use compressive sens-
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
Medical image compression and reconstruction
3
ing based approach, according to this approach forms a energy model which is represented by using a smooth convex function CðxÞ (see Table 1). This is used to formulate solutions according to the given problem with various properties. This is achieved by the linear combination of the non-smooth convex function Cx . In this work, the problem formulation is given as minp F ðxÞ ¼ CðxÞ þ xR
m X P i ðOi xÞ
ð1Þ
3.2. Building blocks of proposed model
i1
where Rp denotes the optimal solution for the given input medical image, P i denotes the prior modeling of each pixel and orthogonal pixel matrix of the pixels is denoted by Oi . In order to solve this we create two subproblems that perform the splitting of the given data, the splitted data contain pixel splitting and block splitting. Block splitting stage finds the value of x which is finally assigned to the maximalmonotone operator equal to zero. Splitting of pixel is another way to achieve the optimal solution by solving above given Eq. (1). This process uses alternate direction methods with the Langragian framework. According to this method pixels are splitted into m þ 1 pixels with the help of new pixel m which belongs to the image model P, then for each pixel Langragian method is applied and finally the decomposed Langragian is minimized to achieve the alternate directional solution of the image. In the later section we combine the splitting approaches to improve the performance. Original input problem is divided into subproblems by dividing the convex function into m function, dividing pixels into m pixels and minimization of the splitted image blocks and then finally the linear combination of this function gives the optimal solution of the image. Moreover the proposed approach is able to decompose the hard convex problems into subproblems to achieve the optimal solution with lesser computational complexity. 3.1. Proposed model for combined splitting method This section discusses about the proposed combined modeling approach for image reconstruction with completion of the lower rank tensor. In order to design this, we consider Eq.
Table 1
(1) as the given problem. In this problem P i : Rp ! R is considered as a continuous non-smooth convex function, f : Rp ! R is the smooth convex function and ORpp is considered the orthonormal matrices. In this problem, if value of m ¼ 1, then it can be solved easily but for the higher values of m it is challenging (see Fig. 1).
From the above discussion it can be concluded that if a complex problem is given then it can be solved by using iterative method by achieving the optimal solution in the given number of iterations. This iterative process can be given as Algorithm 1 Input: d ¼ 1=Lc ; xo repeat for n = 1 to N do xn ¼ proxd ðgÞðxn1 drfðxn1 Þ End for Repeat until stopping criteria achieved
Lc is the constant, and d denotes the input scalar value. To make it faster by using iterative method we use output of this at each iteration which is given as fxn g and rn and the problem is formulated such as Fðxn ÞFðx Þ 6
2Lc jjxo x jj2 ðn þ 1Þ2
8x 2 X
ð2Þ
This problem can be solved by Algorithm 2 Input: d ¼ 1=Lc ; r1 ¼ xo ; t1 ¼ 1 Repeat For n ¼ 1 to N do xn ¼ proxd ðgÞðrn drfðrn ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tnþ1 ¼ ð1 þ 1 þ 4ðtk Þ Þ 12 1 rnþ1 ¼ xn þ ttnþ1 ðxn xn1 Þ End for Repeat until optimal criteria is achieved n
Symbol used in this work.
Symbol
Representation
CðxÞ F ðxÞ Pi Oi Rp m Rp ! R Lc d fxn g; rn jjxjjns1 jjhxjjns2 m Fp b; c
Denotes the convex problem Problem function Prior modeling Orthogonal matrix of the input image Optimal solution New pixel value for splitting Non-smooth convex function Input image optimal solution constant Scalar value Output achieved from the initial splitting Non-smooth problem Non-smooth problem Under sampled Fourier measurements Partial Fourier transform Positive constant parameter
3.3. Image reconstruction For image reconstruction we apply above discussed algorithm for image reconstruction by compressed sensing. This problem is formulated by using 1 minx FðxÞ ¼ jjFp x mjj2 þ bjjxjjns1 þ cjjhxjjns2 2
ð3Þ
jjxjjns1 and jjhxjjns2 are the non-smooth convex problems, m is the under sampled Fourier measurements, b and c are the two positive constant parameters and Fp is the Fourier transform of the given data. This model shows the efficiency for image recovery, used widely in the medical field. This process utilizes non-smooth
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
4
H. Sunil, S.G. Hiremath
Figure 1
(a) Brain image
(d) artery image Figure 2
Overall system flowchart.
(b) Kidney MR image
(c) Heart image
(e) Human skeleton image
Input images considered for compression using proposed approach.
convex problems which are given as jjxjjns1 and jjhxjjns2 . In order to solve this we use fast method of convex problem optimization using iterative method. The first step of this approach gives a closed form solution which can be computed using above discussed fast computation algorithm using iterative method. The achieved solution is given as xn ¼ proxd ðjjhxjj1 Þðrn drfðrn ÞÞ
ð4Þ
The computation time required for this type of problem is very less and can be solved easily by using iterative approach but if the problem is based on the joint L1 and total variation based which can be given as xn ¼ proxd ðbjjxjjns1 þ cjjhxjjns2 Þðrn drfðrk ÞÞ
ð5Þ
This problem requires more time to achieve the optimal solution due to the gradients of conjugate which makes it slower. In order to achieve the MR reconstruction we apply pixel splitting technique. The key feature of this approach is
to replace the linear solver with Fourier domain computation which converges faster and reduces the timing complexity issue. This algorithm is given as below mentioned algorithm 3 and applied for MR image reconstruction.
Algorithm 3 Input: d ¼ 1=Lc ; b; c; t1 ¼ 1; r1 ¼ xo For n ¼ 1 to N do xg ¼ rn drfðrn Þ x1 ¼ proxd ð2bjjxjjns1 Þðxg Þ x2 ¼ proxd ð2cjjhxjjns2 Þðxg Þ 2Þ xn ¼ ðx1 þx 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi2 kÞ Þ tnþ1 ¼ ð1þ 1þ4ðt 2 n1 rnþ1 ¼ xn þ ttnþ1 xn xn1 end for
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
Medical image compression and reconstruction
Figure 3
Table 2
5
Performance evaluation for brain image.
Performance evaluation.
Time taken
SNR (ES)
SNR (PS)
0.07020045 0.08580055 0.10140065 0.10140065 0.13260085 0.19500125 0.22620145 0.2340015 0.2496016
4.390606 6.691962 7.465152 7.766746 7.882058 7.925613 7.942536 7.949732 7.95339
6.004975 7.631901 8.527296 9.039236 9.439868 9.796934 10.11622 10.40066 10.65587
Figure 4
System performance.
4. Experimental results This section describes the obtained result by applying the proposed image compression methodology. The CPU used in experiment is an intel core i5 processor, which is clocked at 2.90 GHz and having 8.00 GB of RAM. Proposed approach is implemented using MATLAB tool on Windows 64-bit platform for different medical images. Proposed approach is implemented using MATLAB tool on Windows platform for different medical images. For experimental study we have considered MRI and Ct images. In the below given Fig. 2, input images are given. In our proposed approach we evaluate the performance in terms of time taken for compression and respective signal to noise ratio. Comparative analysis is shown in Fig. 3 (considering brain image). It can be seen from the figure that the proposed approach gives better SNR when compared to the existing approach. Time taken by the CPU increases, performance of the proposed work also increases in terms of SNR, and on other hand existing system [15] performance degrades, as the time taken increases. Key issue with the performance is the complexity. According to the existing system [15], input image is not
Figure 5
System performance CPU time vs iteration.
divided into sub-problems whereas we proposed to divide the image into smaller sub-problems which helps us to achieve the improved results. In other way for 10 iterations, performance is evaluated which is given in Table 2. Graphical representation of this is given in Fig. 4. It shows the performance of proposed system and existing system [15] in
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001
6
H. Sunil, S.G. Hiremath stages: initially for a given image a problem is formulated on the problem we apply convergence theory to solve it, one the solution is achieved for this process and then later we apply the iterative process to reduce the computation complexity by introducing convergence for the given problem. Finally the image is splitted into pixel and blocks which is used for the reconstruction by applying hybrid approach of image reconstruction by using compressed sensing. Performance of the proposed model is achieved in terms of PSNR, Reconstruction error and time taken to perform the reconstruction. References
Figure 6
SNR vs CPU time performance for case II.
SNR Performance
SNR
Existing System
16 14 12 10 8 6 4 2 0
1
2
3
Proposed System
4
5
6
7
8
Iteration
Figure 7
Table 3
SNR performance for MRI image.
SNR performance for case II.
Existing system
Proposed system
6.269411134 9.914735097 11.45108472 12.12953799 12.40723566 12.51781347 12.56420924 12.58675518 12.60049576
8.634913284 11.36928701 12.82869845 13.37280002 13.676297 13.92724569 14.13313235 14.29648209 14.42867327
terms of SNR with respect to the iterations. Figs. 5 and 6 show the performance in terms of time and SNR. In the next stage of result evaluation of proposed approach, MRI scan of body image is considered for performance evaluation. In Fig. 7 we show the comparative study analysis of the proposed approach. Performance study evaluation is carried out in terms of SNR achieved vs iterations. For each iteration, SNR values are tabulated in the below given Table 3. 5. Conclusion In this work we propose a new approach for medical image reconstruction. The proposed approach is dived into the three
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Further reading [1] G.K. Wallace, The JPEG still picture compression standard, Commun. ACM 34 (4) (1991) 607–609.
Please cite this article in press as: H. Sunil, S.G. Hiremath, A combined scheme of pixel and block level splitting for medical image compression and reconstruction, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.001