Efficient Compression of 4D fMRI Images using Bandelet Transform and Fuzzy Thresholding R. Rajesh School of Computer Science and Engineering Bharathiar University Coimbatore-641046, India Email:
[email protected]
R. Rajeswari School of Computer Science and Engineering Bharathiar University Coimbatore-641046, India Email:
[email protected]
Abstract
of fMRI images is very crucial for their efficient storage. In [5] compression of fMRI images by taking the difference between adjacent volumes is proposed. In [6] authors propose 4D SPIHT (Set Partitioning in Hierarchical Trees) for compressing fMRI images. LOCO-I (Low Complexity Lossless Compression of Images) is used to compress fMRI volumetric images in [7]. In this paper we propose compression scheme, which uses bandlet transform, fuzzy thresholding of the bandlet coefficients and arithmetic coding to compress 4D fMRI images. The rest of the paper is organized as follows. Section II gives an overview of transform coding and bandelet transform. Section III provides a detailed description of the proposed compression scheme for compressing 4D fMRI images. Section IV provides the result of compressing various 4D fMRI images using the proposed compression scheme. Section V discusses the conclusion and the scope for future work.
Medical image compression techniques should give high compression ratios apart from preserving vital information in medical images. In this paper we propose a compression technique for four-dimensional functional Magnetic Resonance Images (fMRI). The proposed technique uses bandelet transform to capture the anisotropic regularity of edge structures apart from capturing regularity information from smooth regions. Then fuzzy thresholding is performed to retain the important bandelet coefficients. To further improve the compression we make use of variable length encoding technique viz., arithmetic coding. The proposed method is applied to 4D fMRI images and the results are compared with bandelet based ordinary thresholding. Test results show that the proposed method gives better results in terms of PSNR and compression ratio.
1. Introduction 2. Review of Bandelet Transform Medical images require enormous amount of memory for storage. Also transmission of these images through network may be very slow. Image compression techniques help in reducing the amount of data required to represent medical images. Medical image compression techniques must be lossless so that all the relevant and important image information present in medical images are retained and this helps in effective diagnosis of images. Four-dimensional (4D) medical images are sequences of three-dimensional (3D) volumetric images obtained over a period of time that require large amount of storage space. 4D medical images represent changes of 3D medical images over the fourth dimension, the time, so redundancies in the fourth dimension are higher than redundancies in the 3D images. Various image compression algorithms have been developed to exploit this redundancy in the fourth dimension [1]–[4]. Medical imaging techniques like functional Magnetic Resonance Imaging (fMRI) generate 4D image datasets, which determine regions of brain that are activated due to various cognitive and/or motor functions. Compression
c 978-1-4244-5612-3/09/$26.00 2009 IEEE
In transform based image compression methods, a digital image is represented by a set of coefficients, which are then coded. Transform methods like Discrete Cosine Transform (DCT) [8], Discrete Wavelet Transform (DWT) [9] can be used to represent the image as a set of coefficients. These transforms exploit correlation between pixels in the image data. They represent the image data as a set of less correlated coefficients and thus the coefficients are packed into a specific area of the transform domain. Image compression can be done by setting the coefficients below a given threshold to zero and then coding the non-zero coefficients. This property makes these transform techniques to be widely used in various image compression schemes [10]–[14]. The non-zero coefficients can be encoded using Huffman coding [15] and arithmetic coding [16], which are popular variable length entropy coding techniques. Standard Wavelet bases [9] are efficient to represent functions with pointwise singularities. But they do not capture the geometric regularity along the singularities of a
543
surface, due to their isotropic support. To exploit anisotropic regularity of a surface along edges, the basis must include elongated functions that are nearly parallel to the edges. Contourlets [17] are also bases constructed with elongated basis functions using the combination of multiscale and a directional filter bank. The bandelet decomposition [18]–[21] is computed with a geometric orthogonal transform that is applied on orthogonal wavelet coefficients. Wavelet transform, when applied to an pixels, computes the set of dot products image of
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
(1)
0.1
0
functions produces a coarse where the projection on represents the level at which we approximation at scale stop the wavelet transform. Those values can be conveniently pixels. A dyadic square is a square stored in an array of obtained by recursively splitting the original wavelet transformed image into four sub-squares of equal size. Let the . For width of the squares be pixels with and orientation s each dyadic square at a given scale of the wavelet transform 1D reordering of the grid points is performed. The possible number of 1D reordering may be equal to the number of directions joining pairs of points in square of width . 1D reordering is done by projecting the sampling location along d and sorting the resulting 1D points from left to right. To the resulting 1D discrete signal, , 1D wavelet discrete wavelet transform is performed. For a given threshold , the direction , which generated denote the the less approximation error, is selected. Let be the coefficients of 1D wavelet transform of , and number of bits needed to code the quantized coefficients . To select the best geometry, the direction d that minimizes the Lagrangian (2) where is the signal recovered from the quantized coefis the number of bits needed to code the ficients and geometric parameter with an entropy coder. is taken as 3/28 [22].
3. Proposed Compression Scheme 3.1. Fuzzy Thresholding in Bandelet Domain Fuzzy set theory and Fuzzy logic [23] offer us powerful tools to represent and process knowledge represented as fuzzy if-then rules. Using fuzzy set theory, bandelet thresholding can be expressed as fuzzy bandelet thresholding. Let be the absolute value of the bandelet coefficient, , at location for the scale and direction and be the threshold value. Any one of the membership functions viz., standard-S function, sigmoid function etc. is
544
1
2
3
4
5
6
7
8
9
10
11
Figure 1. S-Membership Function (smf in matlab) with P=[1 8]
selected. We use a standard S-membership function which is shown in figure 1. The fequency histogram for the bandelet , is computed. The threshold is moved coefficients, and in each position the amount from to of fuzziness is computed. We use linear index of fuzziness [24] which is given by: (3) , is the size of the 2D image, is the where is the membership value of the histogram value and bandelet coefficient . The position where the amount of index of fuzziness is minimum is used as the threshold. Let this threshold value be denoted by . The bandelet coefficients whose absolute values are greater than will remain as such and the remaining coefficients will be set to 0. This leads to compression.
3.2. Compression Scheme Let denote a 3D volumetric image of is a 2D image n two dimensional (2D) images where is the pizel value at pixel coordinate. Let and represent a 4D image where , , represent the image acquired in the time series of discrete . An fMRI dataset consists of time intervals a time-series of volume data sets as explained above. Each point in an fMRI dataset has a unique coordinate in 4D space where are spatial coordinate and is time. The proposed compression scheme consists of three steps. In the first step the 2D image is forward bandelet transformed to obtain the bandelet coefficients. Let the coefficients of 2D discrete bandelet transform of image
2009 World Congress on Nature & Biologically Inspired Computing (NaBIC 2009)
105 Bandelet+Thresholding Bandelet+Fuzzy Thresholding 100
95
PSNR
90
85
80
75
70
65 2.5
Figure 2. Block Diagram of the Proposed Compression Scheme with Bandelet Transform and Fuzzy Thresholding
3
3.5
4 4.5 Compression Ratio
5
5.5
6
Figure 3. PSNR values for Different Compression Schemes at Different Compression Ratios for image1 75 Bandelet + Thresholding Bandelet + Fuzzy Thresholding
70
4. Experiments and Results We have used two 4D fMRI dataset of a brain. First one is of a brain tumour subject obtained from Sri Chitra Tirunal Institute of Medical Science and Technology, Thiruvanthapuram, India. The second one is obtained from the website of Statistical Parametric Mapping [25]. The second dataset was collected by Christian Buchel and is described in the paper [26]. The dimension for the first 4D fMRI image (image1) is 64*64*36 and the dimension for the second image (swaus) is 64*64*64. For our experiment we have taken 12 volumes of image1 and 2 volumes of swaus. We have used arithmetic coding as the variable length
65
PSNR
be represented as . Orthogonal bandelets uses an adaptive segmentation and a local geometric flow and is thus able to capture the anisotropic regularity of edge structures. The second step is to perform fuzzy bandelet thresholding as discussed in the previous section. The standard-S membership function is selected. The histogram of the bandelet coefficients is computed. The threshold is moved and in each position the linear index of fuzziness where the linear index of is computed. The threshold fuzziness is minimum is used as the threshold. The bandelet coefficients which are greater than T remain as such and the remaining coefficients are set as zero. Let be the fuzzy bandelet thresholded image. The third step is to perform lossless arithmetic entropy coding of the thresholded coefficients. Decompression is the inverse of the compression stage. Arithmetic decoding is done to extract the fuzzy thresholded . Next step is to perform bandelet coefficients, inverse bandelet transform to obtain the reconstructed image . The block diagram of this scheme is given in figure 2.
60
55
50
45 1.2
1.4
1.6
1.8 2 Compression Ratio
2.2
2.4
2.6
Figure 4. PSNR values for Different Compression Schemes at Different Compression Ratios for swaus
encoding technique. The results of the compression of the 4D fMRI images using the proposed compression scheme are shown in table I. The table shows the Mean Squared Error (MSE) values, Peak Signal to Noise Ratio (PSNR) values and the compression ratios (CR). For comparison we have used another compression technique where normal thresholding is applied to the bandelet coefficients. The results for PSNR values versus compression ratio are plotted in figures 3 and 4 respectively for image1 and swaus. It can be seen that the proposed scheme using bandelet and fuzzy thresholding technique is better than the bandelet thresholding in terms of Peak Signal to Noise Ratio (PSNR) and compression ratio. For medical images PSNR values are very important, as the decompressed medical image must contain all the information present in the original image for effective diagnosis of the medical images. Further work is being done in this direction to improve the compression ratio without affecting the PSNR value. This can be performed by including more fuzzy linguistic variables and fuzzy rules to incorporate other information like, interscale or interband information. Incorporating these information and rules in the proposed compression scheme will definitely improve compression
2009 World Congress on Nature & Biologically Inspired Computing (NaBIC 2009)
545
Table 1. PSNR, MSE and Compression Ratio for 4D fMRI Images using Bandelet Transform a d Fuzzy Thresholding Image image1
swaus
Original Size (Bytes) 3538944
1048576
CR MSE PSNR CR MSE PSNR CR MSE PSNR CR MSE PSNR CR MSE PSNR CR MSE PSNR CR MSE PSNR CR MSE PSNR
Bandelet Transform + Thresholding 2.60 0.34 101.06 4.06 89.03 76.83 4.68 312.6492 71.38 5.33 544.95 68.97 1.39 376.53 70.57 1.51 4.82e+003 59.50 1.87 2.83e+004 51.82 2.10 5.64e+004 48.81
ratio without affecting the PSNR values.
5. Conclusion and Future Work This paper presents a compression scheme based on bandelet transform and fuzzy thresholding technique. The image is bandelet transformed to obtain bandelet coefficients and then apply fuzzy thresholding. Further arithmetic coding is used to compress the image. The above scheme has been applied to medical images. It can also be applied to other images. The proposed scheme gives better results for the medical images in terms of PSNR and compression ratio. Further work is being done to improve the compression scheme in terms of compression ratio also without affecting PSNR values by incorporating more fuzzy linguistic variables and rules with information like interscale and interband information.
Acknowledgement The first author and second author are thankful to UGC grant in terms of Minor Project and Major Project respectively. The authors are thankful to Bharathiar University for valuable support. The authors are thankful to Mr. J. Satheesh Kumar, Lecturer, Bharathiar University and Sri Chitra Tirunal Institute of Medical Science and Technology, Thiruvananthapuram, India for providing data for this research work.
546
Bandelet Transform + Fuzzy Thresholding 2.67 0.17 104.09 4.26 45.20 79.78 4.90 152.47 74.50 5.78 283.46 71.80 1.49 224.93 72.81 1.54 495.53 69.38 1.94 5.79e+003 58.70 2.38 2.06e+004 53.19
References [1] M. Zagar, M. Kovac, and I. Bcenic, “Lossless and lossy compression in 4d bio-modeling,” in Proceedings of CTA, Hammamet, Tunisia, Apr.12–14, 2007, pp. 271–276. [2] A. A. Kassim, P. Yan, W. S. Lee, and K. Sengupta, “Motion compensated lossy to lossless compression of 4d medical images using integer wavelet transform,” IEEE Transactions on Information Technology in Biomedicine, vol. 9, no. 1, pp. 132–138, Mar. 2005. [3] Y. Liu and W. Pearlman, “Four dimensional wavelet compression of 4d medical images using scalable 4d sbhp,” in Proceedings of IEEE Data Compression Conference, Mar.27– 29, 2007, pp. 233–242. [4] V. Sanchez, P. Nasiopoulos, and R. Abugharbeigh, “Lossless compression of 4d medical images using h.262/avc,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, May14–19, 2006. [5] M. Cohen, “A data compression method for image time series,” Journal of Human Brain Mapping, vol. 12, pp. 20–24, 2001. [6] H. Lalgudi, A. Bilgin, M. Marcellin, and M. Nadar, “Compression of fmri and ultrasound images using 4d spiht,” in Proceedings of IEEE International Conference on Image Processing, Sep. 11–14, 2005, pp. 746–749. [7] C. Adamson, “Lossless compression of magnetic resonance imaging data,” Master’s thesis, Monash University, Melbourne, Australia, 2002.
2009 World Congress on Nature & Biologically Inspired Computing (NaBIC 2009)
[8] W. Chen and W. Pratt, “Scene adaptive coder,” IEEE Transactions on Communications, vol. 32, pp. 225–232, 1984. [9] S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, 1998.
[24] A. Kaufman, Introduction to Theory of Fuzzy Subsets Fundamental Theoretical Elements. New York: Academic Press, 1975.
SanDiego: [25] The spm website. [Online]. http://www.fil.ion.ucl.ac.uk/spm/data
[10] Y. G. Wu, “Medical image compression by sampling dct coefficients,” IEEE Transactions on Information Technology in Biomedicine, vol. 6, no. 1, pp. 86–94, 2002. [11] Y. Y. Chen, “Medical image compression using dct based subband decomposition and modified spiht data organization,” International Journal of Medical Informatics, vol. 76, no. 10, pp. 717–725, 2007.
Available:
[26] C. Buchel and K. Friston, “Modulation of connectivity in visual pathways by attention: Cortical interactions evaluated with structural equation modelling and fmri,” Cerebral Cortex, vol. 7, no. 8, pp. 768–778, 1977.
[12] D. Taubman and M. Marcellin, JPEG2000: Image Compression Fundamentals: Standards and Practices. Kluwer Academic Publishers, 2002. [13] A. Munteanu, J. Cornelis, and P. Cristea, “Wavelet based lossless compression of coronary angiographic images,” IEEE Transactions on Medical Imaging, vol. 18, pp. 272–281, 1999. [14] L. Zeng, C. Jansen, M. Umer, and P. Hunzeiken, “Extension of wavelet compression algorithm to 3d and 4d image data: Exploitation of data coherence in higher dimension allows very high compression ratios,” in Proceedings of SPIE, vol. 4478, no. 55, 2001. [15] D. Huffman, “A method for the construction of minimum redundancy codes,” in Proceedings of IRE, 1951, pp. 1098– 1101. [16] N. Abramson, Information Theory and Coding. McGraw-Hill, 1963.
New York:
[17] M. Do and M. Vetterli, “The contourlet transform: An efficient directional multiresolution image representation,” IEEE Transactions on Image Processing, vol. 14, no. 12, pp. 2091– 2106, 2005. [18] G. Peyre and S. Mallat, “Surface compression with geometric bandelets,” ACM Transactions on Graphics, vol. 24, no. 3, pp. 601–608, 2005. [19] ——, “Discrete bandelets with geometric orthogonal filters,” in Proceedings of IEEE International Conference on Image Processing, Sep.11–14, 2005, pp. 165–168. [20] ——, “A review of bandelet methods for geometrical image representation,” Numerical Algorithms, vol. 44, no. 3, pp. 205–234, 2007. [21] ——, “Orthogonal bandelet bases for geometric images approximation,” Communications on Pure and Applied Mathematics, vol. 61, no. 9, pp. 1173–1212, 2008. [22] E. L. Pennec and S. Mallat, “Sparse geometrical image approximation with bandelets,” IEEE Transactions on Image Processing, vol. 14, no. 4, pp. 423–438, 2004. [23] R. Yager and L. Zadeh, An Introduction to Fuzzy Logic Applications in Intelligent Systems. Kluwer Academic Publishers, 1991.
2009 World Congress on Nature & Biologically Inspired Computing (NaBIC 2009)
547
