Transcript Lossy Compression - San Diego Supercomputer Center

Lossy Compression
• Based on spatial redundancy
• Measure of spatial redundancy: 2D covariance
• CovX(i,j)= 2e-a(i*i+j*j)
E[X(i,j)X(i-1,j)]
• Vertical correlation r1=
E[X2(i,j)]
E[X(i,j)X(i,j-1)]
• Horizontal correlation r2=
E[X2(i,j)]
• For images we assume equal correlations
• Typically e-a = r1= r2= 0.95
• Measure of loss (or distortion):
• MSE between encoded and decoded image
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?
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
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
The JPEG standard
• Operating modes
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Lossless
Sequential DCT-based
Progressive DCT-based
Hierarchical
JPEG: The Process
• Preprocess colors
• Divide the image into 8 pixel x 8 pixel nonoverlapping 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
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
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
bits per DCT coeff
target av. bit-rate
• Total number of block = B
• yk[i,j] : (i,j) output of k-th block
for 8-bit images
JPEG: Entropy Coding
• Baseline processing
• Total of 4 code tables allowed
• Different code tables for luminance and chrominance
• DC coefficients
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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
JPEG: Entropy Coding
• AC coefficients
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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
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
Subband Coding
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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
Wavelet Compression
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Special case of subband compression
Space and frequency-limited mother function y(t)
Function family: ymn(t) = a0-m/2 y(a0-mt - nb0)
If a0=2 and b0=1, the ymn(t) function is an orthonormal basis
function
• f(t) =
• The a0-m term scales the signal
• Scaling function fmn(t) = 2-m/2 f(2-mt - n)
downscaled low-pass
filter
Unscaled high-pass
filter