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1 / Function Image Compression FIC FIC FIC Summary As the need to use various devices increases, we need to produce the compressed image at different resolutions. The development of an image compression format that compression and resolution conversion can be performed simultaneously is desired. In this research, we suggest a lossless image compression format called Function Image Compression (FIC). Since FIC has function representation of an image data, resolution conversion can be easily performed because resampling is easy once it performs function representation. We show the results of zooming images, calculation costs for resolution conversion, and image compression ratios for the evaluation of FIC system. Key words: Lossless compression, Resolution conversion, Function representation, Shifted linear interpolation 1. PC / 1),2) Study on a Lossless Image Compression Format Suitable for Resolution Conversion by Masaki KITAGO and Ichiro HAGIWARA (Member) (Tokyo Institute of Technology Graduate School of Science and Engineering). JPEG TIFF PNG 3),4) JPEG 2

450 450[pixel] (b) JPEG (c) FIC (b) JPEG (c) FIC 2 3 FIC 4 FIC 2.

2 (a) Original image (b) JPEG + Bicubic (c) FIC 1 Fig. 1 Resolution conversion Function Image Compression FIC FIC FIC 1 (a) (b) (c) [pixel] [pixel] (b) JPEG (c) FIC (b) JPEG (c) FIC 2 3 FIC 4 FIC 2. 1),2) B-spline 5),6) 7),8) 9),10) 11),12) 13),14) 15) LZ LZW LZ LHA ZIP 16) Range Coder 17) Block-Sorting 18) Prediction by Partial Matching PPM 19) 21) JPEG-LS 22),23) 3

3 FIC Fig. 2 ThestructureofFICsystem JPEG ),25) Lossless JPEG ),25) Pixel Live 26) JPEG2000 Pixel Live 1200% 2/3 3. FIC FIC FIC 2 FIC 7),8) PPM 19) 21) 3.1 FIC 3.2 PPM 3.1 FIC f(x) T f[n] =f(nt ) f L (x) f L (x) = f[n] ϕ( x n), (1) T n Z 3 ϕ(x) 3 Fig. 3 Shifted linear interpolation { 1 x, x 1 ϕ(x) = 0, x > 1. (2) f S (x) f S (x) = ( x ) c[n] ϕ T n τ, (3) n Z τ {c[n] n Z} c[n] = τ 1 c[n 1] + f[n]. (4) 1 τ 1 τ 3 3 τ =0.4 τ ) FIC 3 (4) c[n] {d[n] n Z} d[n] = c[n]+0.5, (5) Floor Function d[n] FIC {D[n] n Z} D[n] =d[n] d[n 1]. (6) FIC τ 0.21 RGB 4

4 4 FIC Fig. 4 Distribution of FIC coefficients (Before processing) 3.2 (6) FIC D[n] PPM 19) 21) FIC D[n] Lena [pixel] FIC 4 FIC FIC D[n] d[n] Lena FIC D[n] > 255 d[n] < 0 d[n] > 255 D[n] D[n] = D[n] < 0 D[n] D[n] FIC Fig. 5 Distribution of FIC coefficients (After processing) (a) D[n] (b) 2 0 d[n] > D[n] d[n] > D[n] 2 FIC 1 2 Lena [pixel] 2 197, , FIC FIC 0, 255 0, FIC Lena [pixel] 6 N 1 entropy = p i log 2, (7) p i=1 i N p i Lena

35 1 2006 3 1 2 PPM FIC 6 Lena 256 256 Fig.

5 PPM FIC 6 Lena Fig. 6 Color distribution of Lena ( ) PPM FIC RGB FIC FIC R1, G1, B1,R2,G2,B2,... RGB 4. FIC Pentium4 3.2GHz, 2GB RAM, Windows XP 4.1 FIC 7 (a) Lena [pixel] [pixel] (b) (c) (d) (e) (f) FIC (b) (c) (d) (a) Original image (b) Nearest neighbor (c) Biline (d) Bicubic (e) Shifted linear (f) FIC 7 Fig. 7 Zooming images 6

1 [s] Table 1 CPU-time [s] Lena (256 256) Nearest Bilinear Bicubic

566 0.644 1.031 0.801 0.788 Lena (512 512) 1024 1024 0.197 0.244 0.

894 (a) Lena (b) Peppers (c) Fruits (d) Airplane (e) Barbara (f)

8 8 Original images for the evaluation of compression ratio (e) (f)

2 FIC 1 Lena 256 256 [pixel] Lena 512 512 [pixel] 1024 1024[pixel]

6 1 [s] Table 1 CPU-time [s] Lena ( ) Nearest Bilinear Bicubic Shifted FIC Lena ( ) (a) Lena (b) Peppers (c) Fruits (d) Airplane (e) Barbara (f) Goldhill Fig. 8 8 Original images for the evaluation of compression ratio (e) (f) FIC (e) (f) FIC (d) (f) FIC FIC FIC 4.2 FIC 1 Lena [pixel] Lena [pixel] [pixel] [pixel] FIC FIC FIC FIC 4.3 FIC 8 (a) Lena (b) Peppers (c) Fruits (d) Airplane (e) Barbara (f) Goldhill (a) (d) [pixel] (e) (f) [pixel] 2 8 LZW TIFF ZIP 7

7 [%] Table 2 Lossless compression ratio [%] Lena Peppers Fruits Airplane Barbara Goldhill ( ) ( ) ( ) ( ) ( ) ( ) TIFF(LZW) TIFF(ZIP) PNG JPEG-LS JPEG FIC TIFF PNG JPEG JPEG-LS JPEG2000 FIC = 100 [%]. (8) LZW TIFF ZIP TIFF PNG Peppers Airplane Barbara Lena Fruits JPEG-LS JPEG2000 Lena Fruits Airplane Goldhill Barbara FIC Peppers 5. Function Image Compression FIC FIC FIC FIC 6. FIC FIC 3.2 D[n] d[n] D[n] 1) P. Thévenaz, T. Blu, and M. Unser: Image Interpolation and Resampling, Handbook of Medical Imaging, Processing and Analysis, I.N. Bankman, Ed., Academic Press, San Diego, CA, USA, pp (2000). 2) P. Thévenaz, T. Blu, andm. Unser: InterpolationRevisited, IEEE Transaction on Medical Imaging, Vol.19, No.7, pp Jul. (2000). 3) R.C. Gonzalez and R.E. Woods: Digital Image Pro- 8

cessing Second Edition. Prentice Hall, New Jersey, USA (2002). 4) G.K. Wallace: The JPEG Still Picture Compression Standard, Communications of the ACM, Vol.34, No.4, pp.30 44 Apr. (1991). 5) M.

Unser: Splines: A Perfect Fit for Medical Imaging, In Proceedings of the SPIE International Symposium on Medical Imaging: Image Processing (MI 02), Vol.4684, Part I, pp.

8 cessing Second Edition. Prentice Hall, New Jersey, USA (2002). 4) G.K. Wallace: The JPEG Still Picture Compression Standard, Communications of the ACM, Vol.34, No.4, pp Apr. (1991). 5) M. Unser: Splines: A Perfect Fit for Signal and Image Processing, IEEE Signal Processing Magazine, Vol.16, No.6, pp Nov. (1999). 6) M. Unser: Splines: A Perfect Fit for Medical Imaging, In Proceedings of the SPIE International Symposium on Medical Imaging: Image Processing (MI 02), Vol.4684, Part I, pp , San Diego, CA, USA Feb. (2002). 7) T. Blu, P. Thévenaz, and M. Unser: How a Simple Shift Can Significantly Improve the Performance of Linear Interpolation, In Proceedings of the 2002 IEEE International Conference on Image Processing (ICIP 02), pp.iii. 377 III. 380, Rochester, NY, USA (Sep. 2002). 8) T. Blu, and M. Unser: The Telecommunications Advancement Foundation, Vol.18, No.7, pp Jan. (2003). 9) S. Battiato, G. Gallo, and F. Stanco: A New Edge- Adaptive Zooming Algorithm for Digital Images, In Proceedings of 4th IASTED International Conference Signal Processing and Communications (SPC 2000), pp , Marbella, Spain Sep. (2000). 10) X. Li and M.T. Orchard: New Edge-Directed Interpolation, IEEE Transactions on Image Processing, Vol.10, No.10, pp Oct. (2001). 11) B.S. Morse and D. Schwartzwald: Isophote-Based Interpolation, In Proceedings of the 1998 IEEE International Conference on Image Processing (ICIP 98), Vol.3, pp , Chicago, Illinois, USA, Oct. (1998). 12) B.S. Morse and D. Schwartzwald: Image Magnification Using Level-Set Reconstruction, In Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 01), Vol.1, pp , Kauai, HI, USA Dec. (2001). 13) X. Yu, B.S. Morse, and T.W. Sederberg: Image Reconstruction Using Data-Dependent Triangulation, IEEE Computer Graphics and Applications, Vol.21, No.3, pp May/Jun. (2001). 14) D. Su and P. Willis: Image Interpolation by Pixel Level Data-Dependent Triangulation, Computer Graphics Forum, Vol.23, No.2, pp Jun. (2004). 15) J. Sun, N.N. Zheng, H. Tao, and H.Y. Shum: Image Hallucination with Primal Sketch Priors, In Proceedings of the 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 03), Vol.2, pp , Madison, WI, USA, Jun. (2003). 16) LHA ZIP (2003). 17) G.N.N. Martin: Range encoding: an algorithm for removing redundancy from a digitised message, Video & Data Recording Conference, Southampton, Jul. (1979). 18) M. Burrows and D.J. Wheeler: A Block-sorting Lossless Data Compression Algorithm, Digital SRC Research Report, No.124, Digital Systems Research Center, Palo Alto, CA, USA, May (1994). 19) J.G. Cleary and I.H. Witten: Data Compression Using Adaptive Coding and Partial String Matching, IEEE Transactions on Communications, Vol.32, No.4, pp , Apr. (1984). 20) D. Shkarin: PPM: one step to practicality, In Proceedings of the IEEE Data Compression Conference (DCC 02), pp , Snowbird, Utah, USA, Apr. (2002). 21) M. Drinic, D. Kirovski, and M. Potkonjak: PPM Model Cleaning, In Proceedings of the IEEE Data Compression Conference (DCC 03), pp , Snowbird, Utah, USA, Mar. (2003). 22) M.J. Weinberger, G. Seroussi, and G. Sapiro: LOCO-I: A Low Complexity, Context-Based, Lossless Image Compression Algorithm, In Proceedings of the IEEE Data Compression Conference (DCC 96), pp , Snowbird, Utah, USA, Mar. (1996). 23) M.J. Weinberger and G. Seroussi: The LOCO-I Lossless Image Compression Algorithm: Principles and Standardization into JPEG-LS, IEEE Transactions on Image Processing, Vol.9, pp , Aug. (2000). 24) M.W. Marcellin, M.J. Gormish, A. Bilgin, and M.P. Boliek: An Overview of JPEG-2000, Data Compression Conference 2000, pp (2000). 25) A.N. Skodras, C.A. Christopoulos, and T. Ebrahimi: JPEG2000: The Upcoming Still Image Compression Standard, In Proceedings of the 11th Portuguese Conference on Pattern Recognition, pp , Porto, Portugal, May (2000). 26) DAM VFZ. BP (2001) Universityof California, Irvine, Computer Graphics Lab Surface Reconstruction, GeometryCompression ACM

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