Image Coding. Image Coding

Transcription

1 Course INF581 Multimedia Coding and Applications Introduction and JPEG Ifi, UiO Norsk Regnesentral Vårsemester 28 Wolfgang Leister This part of the course is held at Ifi, UiO... (Wolfgang Leister) parts of this lecture were developed by Peter Oel, and Clemens Knoerzer

2 The story so far... Data compression information theory run length encoding Huffman coding Ziv-Lempel(-Welch) algorithm Arithmetic coding JPEG Joint Photographic Expert Group Developed CCITT, ISO 1918 Lossless coding (Comp. 2:1) Coding with loss (1:1-4:1) Parameters control image quality Not limited to certain image types

3 JPEG Discrete Cosine Transformation (DCT) Huffman- or Arithmetic Coding Modes: Lossless Coding Sequential Coding Progressive Coding Hierarchical Coding Not a file format!!! JFIFF Lossless Coding Using Prediction C - 1 A A X B (Vhs. Diff.)(Vhs. Diff.)(Vhs. Diff)... Nr. Prediction 2 B 3 C 4 A+B-C 5 A+(B-C)/2 6 B+(A-C)/2 7 (A+B)/2

4 Lossy Coding Subdivision in 8x8 Blocks Transformation in frequency space Quantising Coding (Huffman or Arithmetic Coding) FDCT Quantiser Encoder Why Use a Transform? Why use frequency domain? better statistic distribution many low frequency parts few high frequent parts quantising better achievable Humans see high frequencies only for high contrast values

5 Fourier Transform Fourier transform: Inverse FT: Discrete FT: Inverse DFT: See: Why DCT? Why not Fourier Transform? 8x8 Blocks FT: ringing at block edges

6 Why DCT? Why not Fourier Transform? 8x8 Blocks FT: ringing at block edges Why DCT? Why not Fourier Transform? 8x8 Blocks FT: ringing at block edges Mirroring produces smooth function Sinus coefficients disappear

7 DCT cos, sin functions cos used only in DCT f(t) = cos( u t) sampling of function DCT - basis functions

8 DCT - example The (I)DCT Formula f x =c u Pixels pixels f x,y =c u c 1 cos g u,x Coefficients v c 1 c 2 cos g u,x cos g v,y

9 DCT 1D - 2D Inverse DCT (decoding) pixels f x,y = 1 4[ u= 7 7 v= coefficients C u C v F u,v cos 2x 1 u 16 cos 2y 1 v 16 ] with C u,c v ={ 1 2 } for u,v= 1 else

10 (Forward)DCT unsigned signed FDCT: F u,v = 1 4 C u C v [ x= 7 7 y= f x,y cos 2x 1 u 16 cos 2y 1 v 16 ] with C u,c v ={ 1 } for u,v= 2 1 else FDCT Quantiser Encoder (Forward)DCT unsigned signed FDCT Quantiser Encoder FDCT

11 An example Image example from wikipedia.org % read the image RGB = imread('phalaenopsis_(aka).jpg'); % convert pixels to the [ 1] range RGB = im2double(rgb); % convert to grayscale I = rgb2gray(rgb); % evaluate DFT (using log scale) F = log(abs(fft2(i))); % evaluate DCT (using log scale) C = log(abs(dct2(i))); % normalize C: immin = min(c(:)); immax = max(c(:)); C = (C - immin)/(immax - immin); % normalize F using the same scale as C F = (F - immin)/(immax - immin); F(find(F>immax)) = 1; F(find(F<immin)) = ; % evaluate the square of each term % (to make dark darker and bright brighter) F = F.*F; C = C.*C; % save outputs imwrite(i,'flower_original.png'); imwrite(f,'flower_fft.png'); imwrite(c,'flower_dct.png'); Quantising Q F(u,v) F (u,v) =Integer Q(u, v) FDCT Quantiser Encoder

12 Quantising FDCT Quantiser Encoder Quantiser 1 DC-Coefficient Coding 63 AC-Coefficientes DC FDCT Quantiser Encoder AC

13 DC-Coding DC -1 DC DC DC FDCT Quantiser Encoder ΔDC i =DC i -DC i-1 ΔDC i =DC i -DC i-1 DC-Coding ΔDC i =(Length, Value) Length=Huffman-Coded Value=(Sign, abs Value) FDCT Quantiser Encoder

14 DC-Coding Length Value Examples: -1, 1-3,-2, 2, , , , , , , , , FDCT Quantiser Encoder , : (2) : (1).1. : ().. 5 : (3) : (7) DC-Coding Length Code Examples: -3 : (2).1.1 : : (1).1. : 1.1. : ().. :.. 5 : (3)..1 : : (7)..11: FDCT Quantiser Encoder

15 Zig-Zag Serialising Zero-Run Length Huffman-Coding AC-Coding FDCT Quantiser Encoder Zig-Zag Serialising DC FDCT Quantiser Encoder

16 AC-Coding,-2,-1,-1,-1,,,-1,,,,,... (1x),-2,(x),-1,(x),-1,(x),-1, (2x),-1,<EOB> (1),-2,(),-1,(),-1,(),-1,(2),-1,<EOB> FDCT Quantiser Encoder AC-Coding,-2,-1,-1,-1,,,-1,,,,,... (1x),-2,(x),-1,(x),-1,(x),-1, (2x),-1,<EOB> (1),-2,(),-1,(),-1,(),-1,(2),-1,<EOB> (1),((2).1.),(),((1).1),(),((1).1),(), ((1).1),(2),((1).1),<EOB> FDCT Quantiser Encoder

17 AC-Coding,-2,-1,-1,-1,,,-1,,,,,... (1x),-2,(x),-1,(x),-1,(x),-1, (2x),-1,<EOB> (1),-2,(),-1,(),-1,(),-1,(2),-1,<EOB> (1),((2).1.),(),((1).1),(),((1).1),(), ((1).1),(2),((1).1),<EOB> (1/2).1.,(/1).1,(/1).1,(/1).1,(2/1).1,<EOB> FDCT Quantiser Encoder AC-Coding <EOB> 11 1/1 11 /1 1/ / /3 1 2/1 111 / / ,-2,-1,-1,-1,,,-1,,,,,... (1x),-2,(x),-1,(x),-1,(x),-1, / (2x),-1,<EOB> / /1... (1),-2,(),-1,(),-1,(),-1,(2),-1,<EOB> / (1),((2).1.),(),((1).1),(),((1).1),(),... 15/1... ((1).1),(2),((1).1),<EOB> (1/2).1.,(/1).1,(/1).1,(/1).1,(2/1).1,<EOB> AC-Coefficients 26 Bit /1... <ZR16>

18 DeCoding Decompression Rescale by DeQuantising Inverse DCT Decoder DeQuantiser IDCT Compression Results n:1 Quality 3-2 usable - good 2-1 good - very good 1-5 excellent 5-4 not distinguishable from original

19 Progressive Mode Transfer coefficients partially in several runs. Two Possibilities: Spectral Transfer Approximated Transfer DCT Koeff. Blocks Bits. Progressive Mode Spectral Transfer DC AC

20 DCT Koeff. Blocks Bits. Progressive Mode AC DC Approximated Transfer Hierarchical Mode Code image with low resolution first Code higher resolution as difference to previous lower resolution Image is represented in several resolutions Unnecessry data are not transferred

21 Images with several channels JPEG uses several (Colour)-Channels (e.g., YC b C r ) Channels can have different resolution Resolution factor as integer number JPEG method uses each channel separately! JFIFF JPEG defines algorithm only. JPEG is not a file format JPEG is colour-blind Parameters and tables are pre-defined Based on JPEG mode of TIFF 6. Consists of segments which are defined by markers (like TIFF)

22 Is JPEG good enough? Visible blocks for high compression rates For compression rates > 4:1 JPEG does not work well. low frequencies are not taken into account. Wavelet Coding Good quality up to 6:1 Linear degradation for higher compression rates No visible blocks JPEG 2 Standard Fractal Coding Comparison

23 JPEG: Literature Pennebaker,Mitchell: JPEG, Still Image Data Compression Standard, Van Nostrand Reinhold (1993) Wavelet-Coding: Daubechies: Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics The End of this Lecture