Lossy Compression
• Based on spatial redundancy
• Measure of spatial redundancy: 2D covariance
• CovX(i,j)= σ2
e-α√(i*i+j*...
Rate-Distortion Function
• Tradeoff between bit rate (R) of compressed
image and distortion (D)
• R measured in its per en...
Sample vs. Block-based Coding
• Sample-based
• In spatial or frequency domain
• Like the JPEG-LS
• Make a predictor functi...
Which Transformation?
• Considerations
• Packing the most energy in the least number of
elements
• Minimizing the total en...
The JPEG standard
• Operating modes
• Lossless
• Sequential DCT-based
• Progressive DCT-based
• Hierarchical
JPEG: The Process
• Preprocess colors
• Divide the image into 8 pixel x 8 pixel non-
overlapping blocks – why 8?
• Transfo...
JPEG: Color Processing
• Maximum number of color components = 256
• Each sample may be 8 or 12 bits in precision
• Convers...
JPEG: Quantization Tables
• 8 X 8 quantization table for each image
component
• Q(i,j): quantization step for correspondin...
JPEG: Entropy Coding
• Baseline processing
• Total of 4 code tables allowed
• Different code tables for luminance and chro...
JPEG: Entropy Coding
• AC coefficients
• Take value in the range [-1023, 1023]
• 10 size categories
• Only non-zero coeffi...
JPEG: Progressive Coding
• Coding is performed sequentially but in multiple
scans
• The first scan should produce the full...
Subband Coding
• Pass an image through an n-band filter bank
• Possibly subsample each filtered output
• Encode each subba...
Wavelet Compression
• Special case of subband compression
• Space and frequency-limited mother function ψ(t)
• Function fa...
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Mmclass4

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MULTIMEDIA AND SYSTEM DESIGN

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Mmclass4

  1. 1. Lossy Compression • Based on spatial redundancy • Measure of spatial redundancy: 2D covariance • CovX(i,j)= σ2 e-α√(i*i+j*j) • Vertical correlation ρ1= • Horizontal correlation ρ2= • For images we assume equal correlations • Typically e-α = ρ1= ρ2= 0.95 • Measure of loss (or distortion): • MSE between encoded and decoded image E[X(i,j)X(i-1,j)] E[X2 (i,j)] E[X(i,j)X(i,j-1)] E[X2 (i,j)]
  2. 2. Rate-Distortion Function • Tradeoff between bit rate (R) of compressed image and distortion (D) • R measured in its per encoder output symbol • Compression ratio = encoder input bits/R • D normalized by the variance of the encoder input • Possible SNR definition = 10 log10 D-1 • For images that can be modeled as uncorrelated Gaussian • R(D)=0.5log2 D-1 • More realistic images • See graph • How do you make these graphs?
  3. 3. Sample vs. Block-based Coding • Sample-based • In spatial or frequency domain • Like the JPEG-LS • Make a predictor function (often weighted sum) • Compute and quantize residual • Encode • Block-based • Spatial: group pixels into blocks, compress blocks • Transform: group into blocks, transform, encode
  4. 4. Which Transformation? • Considerations • Packing the most energy in the least number of elements • Minimizing the total entropy of the sequence • Decorrelating elements in the input blocks maximally • Coding complexity • DFT, KLT, DCT, DHT? See the effects • DCT-based coding • High compaction efficiency for correlated data • Orthogonal and separable • Fast, approximate DCT algorithms available
  5. 5. The JPEG standard • Operating modes • Lossless • Sequential DCT-based • Progressive DCT-based • Hierarchical
  6. 6. JPEG: The Process • Preprocess colors • Divide the image into 8 pixel x 8 pixel non- overlapping blocks – why 8? • Transform each block into 2D DCT • Encode • Stage 1: predictive coder for DC or run-length coder for AC coefficients • Stage 2: Huffman or arithmetic coding
  7. 7. JPEG: Color Processing • Maximum number of color components = 256 • Each sample may be 8 or 12 bits in precision • Conversion of a decorrelated color space • Y-Cb-Cr, YUV, CIELAB • Subsample chrominance components • Interleave components • Maximum MCU components=4 • Maximum number of data units within MCU = 10
  8. 8. JPEG: Quantization Tables • 8 X 8 quantization table for each image component • Q(i,j): quantization step for corresponding DCT element • 1 ≤ Q(i,j) ≤ 255 • Psycho-visual experiments • Bit-rate control • Total number of block = B • yk[i,j] : (i,j) output of k-th block ∑∑= = = 8 1 8 164 1 i i ij rb target av. bit-rate bits per DCT coeff 64 , ])],[([ ]),[( 2log 2 1 ∏ += ji jiyVar jiyVar ij rb ijb jiQ 2 2046 ],[ = for 8-bit images
  9. 9. JPEG: Entropy Coding • Baseline processing • Total of 4 code tables allowed • Different code tables for luminance and chrominance • DC coefficients • DC differentials are computed and have the range [-2047, 2047] • The range is divided into 12 size categories • Category i needs i bits to represent the value • DC residuals are represented as [size, amplitude] pairs • Size is Huffman encoded
  10. 10. JPEG: Entropy Coding • AC coefficients • Take value in the range [-1023, 1023] • 10 size categories • Only non-zero coefficients need to be encoded • Processed in zig-zag order • More efficient run-length encoding of AC coefficients • Represented as (run/size, amplitude) • If run > 15, possibly several (15/0) symbols are used
  11. 11. JPEG: Progressive Coding • Coding is performed sequentially but in multiple scans • The first scan should produce the full image without all details, and details are provided in successive scans • Spectral selection • Each block divided into frequency bands • Each band transmitted during a different scan • Successive approximation • Given a frequency band, DCT coefficients are divided by 2k • MSBs are transmitted first
  12. 12. Subband Coding • Pass an image through an n-band filter bank • Possibly subsample each filtered output • Encode each subband separately • Compression may be achieved by discarding unimportant bands • Advantages • Fewer artifacts than block-coded compression • More robust under transmission errors • Selective encoding/decoding possible • More expensive
  13. 13. Wavelet Compression • Special case of subband compression • Space and frequency-limited mother function ψ(t) • Function family: ψmn(t) = a0 -m/2 ψ(a0 -m t - nb0) • If a0=2 and b0=1, the ψmn(t) function is an orthonormal basis function • f(t) = • The a0 -m term scales the signal • Scaling function φmn(t) = 2-m/2 φ(2-m t - n) ∑∑m n mnmn tc )(ψ nmn n mnmn n mm cdtf )1()1()1()1()( ++++ ∑∑ += ψϕ Unscaled high-pass filter downscaled low- pass filter

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