Distributed Compression For Still Images

1
Distributed CompressionFor Still Images

• Kivanc Ozonat

2
Introduction

• Description of the Problem
• Related Concepts from Information Theory
• Application of Bit-Plane Encoding
• as a Possible Solution Strategy
• Proposed Solution Using Transform Coding
• - Basic Scheme
• - Relation to the Information Theory
  Concepts

3
Problem Description

• Given two still images, a noisy version, X, at
  the decoder and the original, Y, at the encoder,
  how to transmit Y with the best coding
  efficiency?
• No communication of X and Y at the encoder

Encoder
Decoder
Y
X
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Information Theory Background

• Slepian-Wolf Given the following scheme,

(X,Y)
(X,Y)
R1
Encode X
Encode Y
R2
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Information Theory Background

• Can transmit X and Y, if
• - R1 gt H(XY) , R2 gt H(YX), and
• - R1 R2 gt H(X,Y).

R2
H(Y)
H(YX)
R1
H(XY)
H(X)
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Information Theory Background

• Our problem is a special case of this

R2
H(Y)
H(YX)
H(YX)
R1
H(XY)
H(X)
H(X)
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General Solution Strategy

• Form cosets with 3 requirements
• - Members of the same coset should be
  maximally
• separated.
• - Members of the same coset should have the
  same (or
• very close) probabilities of occurrence.
• - Coset construction should be practically
  implementable.

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Underlying Approach

• Use the Idea of Jointly Typical Sets
• - Encode a long sequence (length n) of
  i.i.d. sources
• together, and form the typical set.
• - As n gets large, the typical set contains
  almost all of the
• probability of occurrence.
• - Further, the typical set has its members
  uniformly
• distributed.

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Underlying Approach

• The typical set contains most of the probability
  of occurrence,
• but it has only power (of 2) nH elements.

Typical Set
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Underlying Approach

• Given i.i.d. X and i.i.d. Y, can form long
  sequences to get
• the typical X and typical Y sequences.
• Then, there are power (of 2) H(X,Y) jointly
  typical sequences.

Typical Y
Typical X
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A Possible Scheme?

• Use bit-plane encoding (followed by gray coding)
  to divide the image at the encoder into
  bit-planes.
• Exploit the correlation between the adjacent
  pixels through
• the upper bit planes.
• The lower bit planes contain i.i.d. (or almost
  i.i.d) distribution of 0s and 1s.

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A Possible Scheme?

• Example A lower bit-plane, with i.i.d 0s and
  1s

0.7
0
0
0.3
0.3
1
1
0.7
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A Possible Scheme?

• Given X , the noisy version, the jointly typical
  set is
• (X,Y) such that X is as given, and Y is the
  set with
• a Hamming distance of (0.3)n from X.
• As n increases, Pr(X, Y(X-0.3n)) will approach
  1, with
• each (X,Y) pair having the same
  distribution.
• Hence, can perform an efficient coset
  construction

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A Possible Scheme?

• Problems
• The lower bit-planes, which are of interest, have
  error probabilities of close to 0.5, even with
  moderate noise
• variances.
• - Cannot compete with transform domain methods
  in terms of bit rates.

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Proposed Solution

• Using transform coding is better because
• - Energy compaction, resulting in lower bit
  rates
• for PSNRs of around 38-40 dB.
• - The addition / averaging process involved in
  transform coding reduces the effect of noise
  through the central limit theorem.
• - The coded sequences of coefficients are
  de-correlated to
• a significant extent.

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Proposed Solution

• Schematically,

Modified Huffman Coder
Y
DCT
Zonal Coder
Quantizer
Bit-Plane Encoder
Coset Constructer
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Proposed Solution

• The coefficients can be grouped in pairs with
  almost equal probabilities of occurrence.
  (because they are Laplacian)
• One member from each pair is selected and
  Huffman-coded.
• The other member of the pair is to have exactly
  the reverse
• (0s and 1s switched) code.
• The coded coefficients are placed in bit-planes.

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Proposed Solution

• 1, 2, 3, 4, 5
• 01, 10, 11, 000, 111 (w.p. 0.125, 0.125, 0.10 ,
  0.075, 0.075)
• -1, -2, -3, -4, -5
• 10, 01, 00, 111, 000 (w.p. 0.125, 0.125, 0.10 ,
  0.075, 0.075)
• Assume need to code 1,-4,-2, 2, -3,-1,5,2,3
• 100100111
• 010000100
• 010101111
• 111000101

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Proposed Solution

• Advantages
• - Equal likelihood of 0s and 1s. This makes
  the use of the
• error coding simple.
• - Essentially Huffman coding. Not a significant
  increase,
• if the upper bit-planes are run-length coded.
• - Maximal distribution of cosets possible, due
  to the uniformity and equality of probabilities.
  

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Conclusions

• Asymptotic Equipartition Property is essential in
  forming the typical sets.
• Having equal probabilities of occurrences make
  the error-coding simpler.
• Decorrelation is maintained through DCT.
• The Laplacian Distribution of the DCT
  coefficients is important in getting equally
  probable pairs.