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ECE 6258 Russell M. Mersereau
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ECE6258 Lecture 18^ Coding Binary ImagesQuantization
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ECE 6258 Russell M. Mersereau
Announcements ^ Problem Set #4 is posted on the web site. ^ Due: Friday, October 10, 2003
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Coding Binary Images ^ Binary images, such as printed documents contain,on average many fewer than 1 bit/sample. ^ They are usually coded losslessly (within samplinglimits). ^ Markov models can be successfully applied. ^ Examples,^ ^ Group 3 and Group 4 fax standards^ ^ JBIG (Joint Binary Images Group) standard (ITU).
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ECE 6258 Russell M. Mersereau
Run-length coding ^ For sources that emit “runs” of identical symbols(e.g. line art) ^ Replace a sequence {
x^ } by a shorter sequence of n symbol pairs {
a^ , r } such that kk ^ Entropy coding of new symbol pairs {
a ,^ r } kk
^ For binary images,
a^ , can usually be omitted. k
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^ Markov model [Capon, 1959] ^ State probabilitiesPr{ k^ successive white pixels} = (1-
k -1^ p )p^ forWBWB^ k =1,2,3,…
Pr{ k^ successive black pixels} = (1-
k -1^ p )p^ forBW BW^ k =1,2,3,…
Statistical model for binary images of line art
B p WB W^ p BW
1- p BW
1- p WB
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ECE 6258 Russell M. Mersereau Measured parameters for Capon model
0.9350.0650.0240.3470.215 bpp 0.8870.1130.0270.2140.241 bpp Pr{ W } Pr{ B }^ p WB^ p BW^ H (SS)
Printed Text Weather Map Document
[Kunt, 1974]
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Facsimile compression standards ^ Standards by the ITU-T^ ^ T4 (Group 3)^ ^ Used by all fax machines over PSTN^ ^ 1-D modified Huffman code (MH) or 2-D MMR code^ ^ T6 (Group 4)^ ^ Fax over digital networks^ ^ Always uses 2-D MMR ^ Format^ ^ Horizontal resolution: 1728 pixels/line^ ^ Vertical resolution: 3.85 lines/mm (standard)
7.70 lines/mm (fine)
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ECE 6258 Russell M. Mersereau Group 3 fax: modified Huffman code ^ Lengths of white pixels and black pixels encoded within ascan line ^ Each run represented aswhite runs:
r =64 x^ r w^ w/make-up
+^ r similar for black w/term^ ^ Two separate Huffman code tables for white and black runsbased on the statistics of 8 representative documents. ^ Shortest code words (2 bits) for black runs of length 2 and 3 ^ Shortest code words (4 bits) for white runs of lengths 2…7. ^ Special EOL codework for each line, 6x EOL as end of page.
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Quantization ^ The output of the quantizer is a sequence ofsymbols that is fed to an entropy encoder. ^ If the pdf (probability density function) of
x [ n ]
is known, the probabilities of the quantizedvalues can be calculated.
Quantizer
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ECE 6258 Russell M. Mersereau Quantization (cont’d) ^ If
t^ t^ i^ i+
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Common pdf models for Images ^ Gaussian ^ Laplacian ^ Generalized Gaussian
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ECE 6258 Russell M. Mersereau Quantization Intervals
^ Interval boundaries-∞,^ x ,^ x^1
,^ x ,^ x ,^ ∞ 234 ^ Representationvalues^ y ,^ y ,^ y^123
,^ y ,^ y 4 5 Q [ r ]^ r xx^1 2^ x^3 y^5 y^4 x^4^ y^2^ y^1
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Formalizing ^ An^ N -point scalar quantizer
Q^ is a mapping Q :^ R Æ C , where
R^ is the real line and C ={ y , y ,…,^12
y^ }^ ⊂^ R^ is the output set or N codebook
with size^
N.
^ Associated with every
N -point quantizer is a partition of the real line
R^ into^ N^ cells or atoms. The
th^ i cell is given by R^ = { x^ ∈ R : i^
Q ( x )= y^ } = i
-1 Q ( y^ ) i
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ECE 6258 Russell M. Mersereau Quantization Error ^ There are a number of measures of thedistortion introduced by a quantizer.^ ^ d ( x , y^ ) = | i
(^2) x - y | (^) i (squared error) ^ d ( x , y^ ) = | i x - y^ |^ i (absolute error) ^ d ( x , y^ ) = | i
m^ x - y | i th^ ( m power) ^ Mean-Squared Error
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Quantizer Design ^ Design Problem #1Given^ p^ X
( x ),^ N , and distortion criterion, find {
y^ }, { x^ } to ii
minimize^ D. Design Problem #2Assume knowledge of interval can be encoded using
l^ bits. i^
Subject to^ R ≤
R *, find { l^ }, { xi
}, { y },^ N^ that minimize ii
D.
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ECE 6258 Russell M. Mersereau Uniform Quantizers ^ Regular quantizers ^ Partitions of same size ^ Reconstruction levels are midpoints ofintervals. ^ Implemented by most A/D converters ^ Uniform quantizers minimize the maximumerror
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ECE 6258 Russell M. Mersereau The Lloyd algorithm 1.^ Begin with an initial codebook,
C. Set^ m^1
2.^ Given codebook
C^ , perform a Lloyd m iteration to generate an improved codebook C^. m +13. Compute the average distortion for
C^. If it m +
has changed by a small enough amountsince the last iteration, stop. Else
m +1Æ m
and go to Step 2.
