Lossless Bit-plane Compression of Images with Context Tree Modeling

1. Lossless Bit-plane Compression of Images with Context Tree Modeling

2. Bit-plane coding • Bit-plane coding has been widely used in lossless compression of gray-scale images or color palette images • Progressive transmission can be used in bit-plane coding strategy

3. Bit-plane coding (cont.) Step 1: decomposition into binary layers • simple bit-plane separation (BPS) • gray code separation (GCS) • prediction error separation (PCS) • gray code prediction error separation (GCPES).

4. Bit-plane separation • simple bit-plane separation (BPS) • Not all the bit-plane is needed for encoding. This is implemented by pre-calculating the histogram of the image • For example, if we know a pixel with value x7,x6,x5,x4 in four MSBs, its possible value for this pixel is x727+x626+x525+x424+[0, 24-1]. When in image’s histogram there’s only one value in this range, encoding for this pixel for following LSBs can be omitted

5. Bit-plane coding (cont.) Step 2: Lossless compression Coding binary layers separately • JBIG • Context tree modeling Consider the correlation of binary layers • Multi-layer context tree modeling • Expectation-based bit-plane coding

6. Bit-plane coding (example)

7. Problem of bit-plane coding • Only efficient for the most significant bit-planes (MSB). • Low correlation of the pixels on the less significant bit-planes (LSB) and the bit-rate is close to 1bit/pixel.

8. Multi-layer context tree modeling • The value of previous encoding layers can be used in probability estimation, • The algorithm has higher time complexity Proposed by Kopylovetal. on IEEE Trans. on Image Processing (2005)

9. Expectation-based bit-plane coding (EBC) • Suppose simple bit-plane separation (BPS) is used. x =(x7,x6,x5,x4,x3,x2,x1,x0), xi: value at bit-plane i. When the nth bit-plane is encoded, expectation values are calculated by: • The context value of neighbor pixels is then determined by:

10. Expectation-based bit-plane coding (EBC) • Fixed template was used and the number of template pixels is decreased to 8, 7, 6 and 5 for four LSBs • Future pixel can also be used in context Proposed by Kikuchi etal. on Picture Coding Symposium (PCS’09)

11. 64 5097 66 73 Expectation-based bit-plane coding (EBC) (example) Encoding pixel fragment of original image bit-plane 4 expectation value Poor probablity estimation if value of current bit-plane is used as context • Better performance by EBC • The distribution of this context means that image is homogeneous in local region

12. Improvement of EBC • Context weighting • Probability estimation • Context tree modeling

13. Expectation-based bit-plane coding weighted (EBCW) • Two context template are used: • Context weighting is then considered in probability estimation: • α and β are the weight of two context model, updated by: μ2 =0.975 is the forgetting factor Context weighting is proposed by Xiao etal. on IEEE Trans. on Image Processing (2006)

14. Probability estimation by BACIC • A forgetting factor μ is incorporated which gives higher influence for recently encoded pixels in probability estimation. rc and sc are updated by: Δ is a bias factor (Δ = 0.006) rc(0) = 1 and sc(0)=2. • When μ=1, it is the common probability estimation method based on global statistics BACIC is proposed by Reavyetal. on IEEE Int. Conf. on Image Processing(1997)

15. Example of BACIC Sample image Context template Probability estimation Bit-rate

16. Optimize context template by context tree modeling • Calculate the order of context template A greedy context reordering process is used proposed by Martins etal.[1998] in each bit-plane • Tree pruning process Tree pruning starts from the leaves of the full grown and reordered tree, evaluating a recursively defined cost by bottom-up strategy, by Mrak etal.[2003] • For gray-scale image, a pre-calculated context tree is used, for color palette image, context tree is optimized by its only statistics

17. 6 7 5 1 2 4 7 3 2 4 8 3 4 6 1 3 5 1 5 2 Calculate the order of context template • Given a predefined search area, recursively search the context minimizing the sum of adaptive code lengths after splitting. optimized context order on bit-plane 7 optimized context order on bit-plane 3 optimized context order on bit-plane 0 predefined search area

18. Tree pruning Using a bottom-up strategy, pruning is determined by evaluating cost value J. Suppose ad-depth context treewith node s at depth d0, it has two child node sch0 and sch1: Pruning is done No Pruning where mc(d0) = log2(d - d0+ 1) is the model cost. l is the total coding cost by arithmetic coding

19. Result Gray-scale image Color palette image

20. Result (cont.) Significance of every component. Forgetting factor (BACIC), context weighting (CW) and context tree modeling (CT).

21. Conclusion • Expectation-based bit-plane coding algorithm is improved in three aspects: context weighting, probability estimation and context tree modeling • Similarperformance with JPEG-LS on gray-scale images, better performance on color palette images • Progressive transmission is also available

22. Reference • H. Kikuchi, K. Funahashi, and S. Muramatsu, "Simple bit-plane coding for lossless image compression and extended functionalities", Picture Coding Symposium (PCS’09), Chicago, USA, May 2009. • H. Xiao and C. G. Boncelet, “On the Use of Context-Weighting in Lossless Bilevel Image Compression”, IEEE Trans. ImageProcessing, 15(11), 3253 – 3260, 2006. • M. D. Reavy and C. G. Boncelet, “BACIC: a new method for lossless bi-level and grayscale image compression”, IEEE Int. Conf. on Image Processing, vol.2, 282-285, 1997. • B. Martins and S. Forchhammer, “Bi-level image compression with tree coding,” IEEE Trans. Image Process., vol. 7, no. 4, 517-528, Apr. 1998. • M. Mrak, D. Marpe and T. Wiegand, “A context modeling algorithm and its application in video compression”, Proc. of IEEE Int. Conf. on Image Processing, vol.3, 845-848, 2003. • P. Kopylov and P. Fränti, "Compression of map images by multilayer context tree modeling", IEEE Trans. on Image Processing, 14 (1), 1-11, January 2005. • A. Podlasov, P. Fränti, "Lossless image compression via bit-plane separation and multi-layer context tree modeling", Journal of Electronic Imaging, 14 (5), 043009, October-December 2006.