/ Function Image Compression FIC FIC

/ 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

(a) Original image (b) JPEG + Bicubic (c) FIC 1 Fig. 1 Resolution conversion Function Image Compression FIC FIC FIC 1 (a) (b) (c) 150 150 [pixel] 3 450 450[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

35 1 2006 2 FIC Fig. 2 ThestructureofFICsystem JPEG2000 24),25) Lossless JPEG2000 24),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 0.21 7) 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 FIC Fig. 4 Distribution of FIC coefficients (Before processing) 3.2 (6) FIC D[n] PPM 19) 21) FIC D[n] Lena 256 256 [pixel] FIC 4 FIC FIC 0 0 100 100 D[n] d[n] 0 255 Lena FIC 1. 1 2 2. D[n] > 255 d[n] < 0 d[n] > 255 D[n] 2 1 128 3. D[n] = 128 2 0 1 128 4. 2. D[n] < 0 D[n] 256 1 128 2 0 5. D[n] 1 1. 1 128 2 5 FIC Fig. 5 Distribution of FIC coefficients (After processing) (a) 2 0 2 D[n] (b) 2 0 d[n] > 255 256 D[n] 2. 1. d[n] > 255 256 D[n] 2 FIC 1 2 Lena 256 256[pixel] 2 197,376 1 197,376 2 244 1 5 FIC FIC 0, 255 0, 255 2 128 FIC 1 2 1 5 1 6 Lena 256 256 [pixel] 6 N 1 entropy = p i log 2, (7) p i=1 i N p i 1 5.64 Lena 7.30 6 4.95 5

35 1 2006 3 1 2 PPM FIC 6 Lena 256 256 Fig. 6 Color distribution of Lena (256 256) 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 32 32 [pixel] 8 256 256 [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 Shifted FIC 1024 1024 0.147 0.169 0.275 0.234 0.215 2048 2048 0.566 0.644 1.031 0.801 0.788 Lena (512 512) 1024 1024 0.197 0.244 0.375 0.322 0.256 2048 2048 0.656 0.766 1.200 1.031 0.894 (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 256 256 [pixel] Lena 512 512 [pixel] 1024 1024[pixel] 2048 2048[pixel] FIC FIC FIC FIC 4.3 FIC 8 (a) Lena (b) Peppers (c) Fruits (d) Airplane (e) Barbara (f) Goldhill (a) (d) 512 512 [pixel] (e) (f) 720 576 [pixel] 2 8 LZW TIFF ZIP 7

35 1 2006 2 [%] Table 2 Lossless compression ratio [%] Lena Peppers Fruits Airplane Barbara Goldhill (512 512) (512 512) (512 512) (512 512) (720 576) (720 576) TIFF(LZW) 84.4 46.0 87.9 18.6 47.5 69.0 TIFF(ZIP) 71.7 35.3 77.6 17.3 36.5 60.4 PNG 67.7 38.0 71.0 20.8 36.1 65.4 JPEG-LS 56.6 48.6 63.7 45.1 80.7 58.6 JPEG2000 56.6 55.2 47.9 47.3 85.3 54.1 FIC 67.4 34.6 67.8 34.6 66.7 62.0 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 0 255 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.393 420 (2000). 2) P. Thévenaz, T. Blu, andm. Unser: InterpolationRevisited, IEEE Transaction on Medical Imaging, Vol.19, No.7, pp.739 758 Jul. (2000). 3) R.C. Gonzalez and R.E. Woods: Digital Image Pro- 8

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