This document discusses different techniques for lossy image compression including JPEG. It describes lossy compression being based on spatial redundancy and using a covariance measure. It then discusses rate-distortion functions and how they relate compression ratio and distortion. It provides details on sample-based and block-based coding approaches as well as considerations for different transformations like DCT. The document outlines the JPEG encoding process including color processing, quantization tables, and entropy coding methods. It also discusses progressive coding and subband coding techniques.

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. 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. 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. 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. The JPEG standard
• Operating modes
• Lossless
• Sequential DCT-based
• Progressive DCT-based
• Hierarchical

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. 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. 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. 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. 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. 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. 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. 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