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Wavelet Coding & JPEG 2000 INF5081 Multimedia Coding and Applications Vårsemester 2008, Ifi, UiO Wavelet Coding & JPEG 2000 Wolfgang Leister Contributions by Hans-Jakob Rivertz Svetlana Boudko Norsk Regnesentral Norwegian Computing Center JPEG revisited • JPEG ... – Uses DCT on 8x8 blocks – Visible blocks for high compression rates – Is not good for compression rate of > 40:1 – Low frequencies are not considered – provides competitive performance for the medium bit-rate range FDCT Quantiser Encoder 01011010.. Table Table Norsk Regnesentral Norwegian Computing Center Wavelet-Coding • Good quality up to 60:1 • Linear degradation for higher compression rates • No visible blocks • JPEG 2000 Standard - ISO 15444 • JPEG-2000 mostly provides improved performance at for low and high bit-rates WT Quantiser Encoder 01011010.. Norsk Regnesentral Norwegian Computing Center Wavelets - Introduction • Wavelets are used for – compression, – data modelling, – data analysis – signal processing • Wavelets are suitable for representation of digital data in many applications. Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Wavelets Model signals with: • Fourier row development (whole area): f x=a0∑ ai cos us xbi sin usx • DCT-based (8x8 blocks): f x=a0∑ ai cosus x • Wavelet-based (hierarchical): f x=a0∑ ai ui x • wavelets use piecewise constant base functions • Average of wavelet function is 0 Norsk Regnesentral Norwegian Computing Center Definition areas and side effects ω FFT t ω STFT DCT t ω WT t Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Wavelet Transform (1) a1 Δ a a a /2 2 1 2 Δ 1 ⋮ ⋮ ⋯ ∑ ai Δ N an−1 an−1an/2 an 9 1 8 7 2 6 6,2,1, 1 3 −1 [ − ] 4 5 Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Wavelet Transform (2) 9 1 8 7 2 6 6,2,1, 1 3 −1 [ − ] 4 5 Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Wavelet Transform (3) [6, 2, 1, -1] +1* +2* -1* Haar Wavelet Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! V²-Basis and Haar Wavelet 2 2 2 2 2 2 2 2 ℘x=c00xc1 1 xc22xc3 3 x 2 2 2 2 [c0 ,c1 ,c2 ,c3]=[9,7,3,5] Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Another Example Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Different Wavelets 1 -1 1 12 -2 -1 1 Haar Wavelet Daubechies Wavelet Lazy Wavelet = 9/7 Wavelet = 5/3 Wavelet Average of wavelet function is 0 Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Signal-Analysis / -Synthesis H0 s0↓ Y0 s0↑ G0 H1 s1↓ Y1 s1↑ G1 X ... ... ... ... + X’ HM-2 sM-2↓ YM-2 sM-2↑ GM-2 HM-1 sM-1↓ YM-1 sM-1↑ GM-1 1 ∑ =1 i si Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Analysis/Synthesis H0 2↓ Y0 2↑ G0 X + X’ H1 2↓ Y1 2↑ G1 Norsk Regnesentral Norwegian Computing Center One-dimensional Case ! Analysis/Synthesis H Y H X AS0 X’ L L Y1 H Y H A 0 S X’ X L L H Y H AS1 L L Y2 Norsk Regnesentral Norwegian Computing Center Two-dimensional Case ! Transformation of Images H H YHH AV SV L L YHL H H X AH SH X’ L L H H YLH AV SV L L YLL Norsk Regnesentral Norwegian Computing Center Two-dimensional Case ! Transformation of Images BLL BHL B BLH BHH Norsk Regnesentral Norwegian Computing Center Two-dimensional Case ! Construction of Haar Basis Standard construction Non-Standard construction of Haar basis of Haar basis Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center Norsk Regnesentral Norwegian Computing Center JPEG 2000 • ISO/IEC 15444 consists of 10 parts: – Part 1: Core coding system (JP2, code stream) – Part 2: Extensions (JPX, others) – Part 3: Motion JPEG 2000 – Part 4: Conformance Testing – Part 5: Reference software – Part 6: Compound image file format (JP2, JPX) – Part 7-10: JPSEC, JPIP, JP3D Norsk Regnesentral Norwegian Computing Center JPEG 2000 • Improved R-D performance at low and medium bitrates • Covers low bit-rate lossy coding up to lossless coding • Progressive Transmission Capability (resolution & SNR) • Region of interest (ROI) representation and editing • Error Resilience • Approach: Wavelet Transformation + EBCOT + Adaptive Arithmetic Coding • Embedded Block Coding with Optimized Truncation Source: Thomas Wiegand Norsk Regnesentral Norwegian Computing Center Part 1: Core Coding system • overview & 11 annexes: – Annex A - Code stream Syntax – Annex B - Data ordering – Annex C - Arithmetic entropy coding (MQ coder) – Annex D - Coefficient bit modeling – Annex E - Quantization – Annex F - Discrete wavelet transform of tile components – Annex G - DC level shifting and component transforms – Annex H - Coding of images with Regions of Interest – Annex I - JP2 file format syntax – Annex J - Examples and guidelines – Annex K - Bibliography Norsk Regnesentral Norwegian Computing Center JPEG 2000 Part 1 Norsk Regnesentral Norwegian Computing Center Encoding in JPEG 2000 • DC level component transform (G) • image decomposition (tiles, sub-bands) (B) • DFWT (F.3) • Quantization (E) • Coefficient bit modeling (D) • Arithmetic coding (C.1, C.2) (MQ coder) • Code stream Syntax (A) • Encoding to JP2 file format (I) Norsk Regnesentral Norwegian Computing Center Image Decomposition • components • tiles • sub-bands • precincts • code blocks • layer • packet Norsk Regnesentral Norwegian Computing Center DC level component transform • components • unsigned → signed values • RCT: reversible component • tiles transform (used with 5-3 • sub-bands reversible WT) • precincts • ICT: irreversible component transform (used with 9-7 • code blocks irreversible WT) • layer • When first components are RGB: ICT is approximation to • packet YCbCr transform. Norsk Regnesentral Norwegian Computing Center Image Decomposition • components Component • tiles • Two-dimensional array of samples (1D) • sub-bands • Color images consists • precincts of several • code blocks components • layer • packet Norsk Regnesentral Norwegian Computing Center Reference Grid • components • Component has • tiles reference grid • (0,0) upper left • sub-bands • precincts • code blocks • layer • packet Norsk Regnesentral Norwegian Computing Center Reference Grid • components • Rectangular array of • tiles tiles • Row by row • sub-bands • precincts • code blocks • layer • packet Norsk Regnesentral Norwegian Computing Center Sub-Bands • components • tiles • sub-bands • precincts • code blocks • layer • packet Norsk Regnesentral Norwegian Computing Center Precinct Partition • components • Tiles are divided into • tiles precincts of size PP PP • sub-bands 2 x ,2 y • precincts • code blocks • layer • packet Norsk Regnesentral Norwegian Computing Center • Each layer divided into code blocks Code Blocks • Code blocks equal size for entire tile • components • Coefficient bit modeling • tiles • sub-bands • precincts • code blocks • layer • packet Norsk Regnesentral Norwegian Computing Center Layers • components • One or more layers in • tiles code stream • Each layer consists of • sub-bands consecutive bit-plane • precincts passes from each code • code blocks block in the tile • layer • Each layer successively • packet and monotonically improves image quality Norsk Regnesentral Norwegian Computing Center Packet • components • 8 bit packet boundaries • tiles • Contributions in raster order • For resolution r=0: • sub-bands – LL band only • precincts • For resolution r>0: • code blocks – HL, LH, HH bands only • layer • Progression order • packet Norsk Regnesentral Norwegian Computing Center FDWT Fast Discrete Wavelet Transform Norsk Regnesentral Norwegian Computing Center Wavelet transforms in JPEG2000 5/3 wavelet transform 9/7 wavelet transform Norsk Regnesentral Norwegian Computing Center Extension of signal Norsk Regnesentral Norwegian Computing Center Extension for Haar Wavelet • Extend with mirrored pixel 17 21 9 5 7 5 • Perform wavelet transform • The 1 is redundant 19 7 6 -2 2 1 Reconstruction • Perform IWT 17 21 9 5 7 5 • The 1 is reconstructed from 6 and 5 Norsk Regnesentral Norwegian Computing Center Decomposition into sub-bands Norsk Regnesentral Norwegian Computing Center IDWT Inverse Discrete Wavelet Transform Norsk Regnesentral Norwegian Computing Center Quantization • Divide by factor and use floor-function • Irreversible case: Scalar Quantization with dead zone • separate approach for every sub-band possible Norsk Regnesentral Norwegian Computing Center Arithmetic entropy coding • Compatible with ISO/IEC 14492 • MQ-Coder Norsk Regnesentral Norwegian Computing Center Codestream syntax • Markers: 0xFF + Opcode (0x01-0xFE) • Marker segments: marker + parameter • Marker segments include a length-field • Headers: Collections of markers and marker segments • Codestream: SOC – SIZ - ... - EOC • Part: SOT - ... - SOD ... bitstream Norsk Regnesentral Norwegian Computing Center Construction of codestream Norsk Regnesentral Norwegian Computing Center Coding into JP2 File Format • Sequence of boxes • Simple boxes (containing payload) • Superboxes (containing other boxes) • optional/mandatory boxes Norsk Regnesentral Norwegian Computing Center JPEG 2000 Process Source: Thomas Wiegand Norsk Regnesentral Norwegian Computing Center JPEG vs JPEG2000 (Original) Norsk Regnesentral Norwegian Computing Center
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