Lossless compression of greyscale images

1
Introduction

• Lossless compression of grey-scale images
• TMW achieves worlds best lossless image
  compression
• 3.85 bpp on Lenna
• Reasons for performance are unclear
• No intermediate results given, only final,
  optimal values

2
Image Compression

• Compression Modelling Coding
• Model Assign probabilities to symbols
• Coding Transmit symbols
• Code Length -log2(P)

• Pixels encoded in raster order
• Conventionally reading order
• Left-to-right, Top-to-bottom
• Previously encoded pixels known to both encoder
  and decoder

3
Prediction

• Causal neighbours can be used to predict the
  value of pixel
• Standard predictor is linear combination of
  neighbour values

4
Prediction
Sample Error Distribution

• CALIC, JPEG LS use DPCM
• Prediction error is coded instead of value
• Errors tend to follow Laplacian distribution
• Have lower entropy than raw pixel values

µ
5
TMW

• Worlds best lossless image compression program
• Blending of prediction error distributions
• Use of information from local region
• Use of large number of causal neighbours
• Optimisation of parameters with respect to
  message length
• We examine the first three points

6
Blending

• Test predictors on local window of pixels
• Calculate what the error would be for causal
  neighbours
• Spread of errors gives a measure of how effective
  the predictor has been in the region

7
Blending

• Generate Laplacian distributions
• Mean is predicted value plus mean error in window
• Variance calculated from spread of errors in
  window
• Blend probability distributions to give final
  probability distribution
• 0.7 0.3
  

8
Window Size

• Window size involves trade off
• Larger window size means using more pixels from
  current segments
• Also increases likelihood of using pixels from
  neighbouring segments

9
Window Size

• 2 windows for calculating
• Blending weights
• Mean/spread of distribution

Blending Window Size
Distribution Window Size
4.05 bpp
4.05 bpp
36 Pixels
78 Pixels
10
Locally Trained Predictor

• Blending scheme rewards predictors with low error
  norm in local window
• Find predictor to minimise this norm
• Local window contains N pixels and N
  corresponding causal neighbour vectors
• Perform Multiple Linear Regression to find
  optimal coefficients
• Can minimise e2 or e

11
Neighbourhood Size

• Noise plays large part in code length
• Consider p p N(µ, s)
• More neighbours tends to reduce impact of noise
• More neighbours means more noise terms in
  predicted value
• Noise terms have smaller coefficients, and tend
  to cancel out more

12
Future Work

• Global Optimisation
• Full investigation of benefits yet to take place
• Texture modelling
• Segmentation
• Transmit predictors and segment maps before pixel
  values
• Send the most important information first!

13
Conclusions

• Blending PDFs produces slight benefit
• Local information in prediction much more
  valuable than global information
• Global optimisation and large neighbourhood sizes
  also important to TMWs success