Image Compression

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Image Compression
• Lecture 15
• Transform Coding KLT, DCT
• Alexander Kolesnikov

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Transformation

• The main goal of transformation is decorrelation
  of data.

C
X
Q(Y)
Y
Transformation
Quantization
Encoding
Model
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Correlation matrix

• Correlation matrix Rx

i
j
where E? is mathematical expectance (average
value)
is mean value (average) of the signal x
is variance of the input signal x
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Karhunen-Loeve Transform (KLT)

• Find eigenvectors and eigenvalues ?j for
  
• correlation matrix R

• Construct orthonormal basis with transform
  matrix T

• Transform x into new basis

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Independence of KLT coefficients

• Correlation matrix for y

KLT coefficients y are decorrelated!
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KLT Example for N2
2. Find eigenvectors
1. Find eigenvalues
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KLT Example for N2
3. Normalization
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KLT Example for N2
4. Check orthogonality
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KLT Example for N2
5. Basis
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KLT

• KLT is rotation is signal space
• Other names of KLT
• Hotelling Transform
• Principal Component Transform

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KLT is optimal, but...

• Decorrelation of data ? redundancy reduction
• Energy compactness
• but
• KLT is signal dependent, complexity of basis
  calculation
• O(N4)
• The basis has to be sent to decoder
  

Lets introduce a model!
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1st order Markov process

• Image as 1st-order Markov process
• xn?xn-1 ?n, where ?n is (white) noise
• Correlation matrix Rx

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Correlation function
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KLT Example N2
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KLT Example N8
T,Leig(Rx) for k1N subplot(2,4,k)
stem(T(,1N-k)) axis(0 N1 -0.8 0.8
) end
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KLT? DFT ? DCT

• Discrete Fourier Transform (DFT) is asimptotics
  of KLT
• for the model. In other words, the KLT is
  spectral
• decomposition of image data!
• KLT ?DFT
• Why Discrete Cosine Transform (DCT) is better
  than DFT?
• DFT?DCT

• What about Discrete Sine Transform (DST)?

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KLT DCT N8, k0..3
KLT
DCT
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KLT DCT N8, k4..7
KLT
DCT
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1-D DCT N8
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1-D DCT

• Complexity O(N2)?
• Fast algorithm O(N log2N)

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Inverse 1-D DCT
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2-D DCT

• The 2-D DCT is performed as two sequential 1-D
  DCTs
• Complexity of DCT is O(N logN) instead of O(N4)

Image
x 1-D DCT
y 1-D DCT
2-D DCT
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2-D DCT N4
2-D DCT basis functions
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Zig-zag DCT coefficients ordering
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Where is compression?

• DCT is reversible transformation
• YTX ? XT-1Y
• Where is compression?
• Data decorrelation and energy compactness
• ? Quantization (lossy operation)
• ? Statistical encoding

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1st DCT coefficient distribution
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2nd DCT coefficient distribution
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3rd DCT coefficient distribution
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Example