Download Lossless coding - Hanyang University

Computer Vision –
Compression(2)
Hanyang University
Jong-Il Park
Topics in this lecture
 Practical techniques
 Lossless coding
 Lossy coding
 Optimum quantization
 Predictive coding
 Transform coding
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Lossless coding
=Error-free compression
=information-preserving coding
 General steps
1. Devising an alternative representation of the image in
which its interpixel redundancies are reduced
2. Coding the representation to eliminate coding
redundancies
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Huffman coding
 Most popular coding (Huffman[1952])
 Two step approach
1. To create a series of source reduction by ordering the
probabilities of the symbols and combining the lowest
probability symbols into a single symbol that replaces
them in the next source reduction
2. To code each reduced source, starting with the
smallest source and working back to the original
source
 Instantaneous uniquely decodable block code
 Optimal code for a set of symbols and probabilities
subject to the constraint that the symbols be coded
one at a time.
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Eg. Huffman coding
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Arithmetic coding
 Non-block code
 One-to-one correspondence between source
symbols and code words does not exist.
 an entire sequence of source symbols is assigned
a single arithmetic code word.
 As the length of the sequence increases, the
resulting arithmetic code approaches the bound
established by the noiseless coding theorem.
 Practical limiting factors
 The addition of the end-of-message indicator
 The use of finite precision arithmetic
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Eg. Arithmetic code
0.068
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LZW coding
 Lempel-Ziv-Welch coding
 Assigning fixed-length code words to variable length
sequences of source symbols but requires no a priori
knowledge of the probability of occurrence of the
symbols to be encoded
 Generating a dictionary(=codebook) as the
encoding proceeds.
 The size of the dictionary is an important parameter.
=> trade-off
 Applied to GIF, TIFF, PDF format and many zip
algorithm
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Eg. LZW coding
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2D Run-length coding
• Relative address coding(RAC)
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Lossless predictive coding
Principle: De-correlating data by prediction
= entropy reduction
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Eg. Lossless predictive coding
Histogram
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Lossy compression
 Approaches
 Predictive coding
 Transform coding
 Vector quantization
 Etc.
 Significant data reduction compared with lossless
compression at the expense of quality degradation
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Lossy predictive coding
Prevent error
accumulation
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Delta modulation(DM)
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DPCM
(Differential pulse code modulation)
 Optimal predictor:
Try to minimize the mean-square of the prediction
error
2
ˆ 2
E{en }  E{[ f n  f n ] }
subject to the constraint that
fn  en  fˆn  en  fˆn  f n
and
m
fˆn    i f n i
i 1
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Practical prediction
 Prediction for 2D Markov source
E{ f ( x, y ) f ( x  i, y  j )}   2  vi  hj
 Reduction of accumulated transmission error
m

i 1
i
1
 Typical predictors
A) fˆ ( x, y )  0.97 f ( x, y  1)
B) fˆ ( x, y )  0.5 f ( x, y  1)  0.5 f ( x  1, y )
C ) fˆ ( x, y )  0.75 f ( x, y  1)  0.75 f ( x  1, y )  0.5 f ( x  1, y  1)
0.97 f ( x, y  1) if Δh  Δv
ˆ
D ) f ( x, y )  
 0.97 f ( x  1, y ) otherwise
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Eg. Predictor
A
B
C
D
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Optimal quantization
 Minimization of the mean-square quantization error:
E{( s  ti ) }
2
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Lloyd-Max quantizer
 Optimal quantizer in the mean-square sense
 Method


Reconstruction level: centroid
Decision level: halfway
 No explicit closed-form solutions for most pdfs
 An iterative design procedure is applied in many cases
 Optimum uniform quantizer

(uniform q.+VLC) outperforms (non-uniform q.+FLC)
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Adaptive quantization
 Different quantization for each subimage(eg.block)
 improved performance
 increased complexity
Eg. Four different quantizers: Scaled version of the same quantizer
Notice: Substantial decrease in error
BUT small improvement in compression ratio
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Eg. DPCM vs. Adaptive DPCM
DPCM
Adaptive
DPCM
Substantial decrease
in perceived error
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Transform coding
 A reversible, linear transform is used
 Goal:
 to decorrelate the pixels of each subimage, or
 to pack as much information as possible into the
smallest number of transform coefficients
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Basis images: WHT
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Basis images: DCT
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Comparison: Energy compaction
DFT
• KLT is optimal BUT it is
image dependent!
•DCT is a good
compromise!
WHT
DCT
Best
performance
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DFT vs. DCT
2n-point
periodicity
Less blocking
artifact
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Effect of subimage size
•Complexity increases
•Performance enhances
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Eg. Block size
25% reduction
Error(8x8)
Org.
4x4
2x2
8x8
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Bit allocation
 Zonal coding
 Allocation of appropriate bits for each coefficient
according to the statistics
 Rate-distortion theory
1
2
 Eg. Gaussian pdf
R  log 2
2
D
 Threshold coding
 Global threshold
 Local threshold
 Fixed (N-largest coding)
 constant rate
 Variable  variable rate. Good performance
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Zonal vs. Threshold
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Eg. Zonal vs. Threshold
Threshold
 better
zonal
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Quantization table
Z
•Different scaling for each coefficient.
•The same quantization curve for all coefficients.
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Eg. Quality control by scaling Z
34:1
67:1
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Wavelet coding
• New technique in 1990s
• Computationally efficient
• No subdivision no blocking artifact
• Good performance!
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Eg. Wavelet transform
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