Colorado School of Mines
Image and Multidimensional Signal
Processing
Professor William Hoff
Dept of Electrical Engineering &Computer Science
Colorado School of Mines
Image and Multidimensional Signal Processing
http://inside.mines.edu/~whoff/
Image Compression
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Image and Multidimensional Signal Processing
2
Image Compression
• Goal: Reduce amount of data, for transmission or storage
• Try to preserve information, so that image can be
reconstructed:
– Exactly (loss-less compression)
– Approximately (lossy compression)
Compression ratio
C
b
b'
where
b = # bits uncompressed
b’ = #bits compressed
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Image and Multidimensional Signal Processing
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Redundancy
• We take advantage of redundancy in the input image:
– Coding redundancy
– Spatial and temporal redundancy
– Irrelevant information
Coding redundancy: Only a few
gray values are present; we can
represent using short code words
(with few bits)
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Spatial redundancy: All gray values
are present, but pixels along each
row are the same; if you know the
value of the first pixel you know the
whole row
Image and Multidimensional Signal Processing
Irrelevant information: Only a few
gray values are present, clustered
around value 128; could represent
using a constant 128 and would
look the same
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Coding Redundancy
• Use short code words instead of long ones
• Can use variable length codes, so that most common values have shortest
codes
• Average bit length
L 1
Lavg   l (rk ) pr (rk )
k 0
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Image and Multidimensional Signal Processing
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Measuring Image Information
• We can compute the theoretical minimum number of bits needed to code
an image
• First, consider an “event” E
– If the event is unlikely to happen, then its probability P(E) is small, and 1/P(E) is large
– When the event does occur, it is significant and provides us with a lot of information
• We define the “self information” of event E
 1 
   log 2 ( P( E ))
I ( E )  log 2 
 P( E ) 
• This is how much information is attached to E
• Example:
– If P(E) = 1, I = 0 bit
– If P(E) = ½, I = 1 bit
– If P(E) = ¼, I = 2 bits
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Measuring Image Information
• Say we have a set of random events, drawn from a possible set of values
{a1, a2, …, aJ}
• We’ll call these the “source symbols”
– They could be randomly generated from a transmitter
• The probabilities of generating these symbols are
{P(a1), P(a2), …, P(aJ)}
• The information carried by a single symbol is -log2(P(aj))
• Thus, the average information per symbol is
J
H   P(a j ) log P(a j )
j 1
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Image and Multidimensional Signal Processing
Assumes that
events are
independent
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Entropy of an image
• For images, the “events” {r0, r1, …, rL-1} are possible gray level values
• We can use the histogram to estimate the probabilities of the symbols
• The average information per pixel is (also called the entropy of the image)
L 1
H   pr (rk ) log 2 pr (rk )
k 0
This is the best you can do
(for uncorrelated values)
• Examples:
– 8 gray levels, same probability for all values: p0=p1= … = p7=1/8. H=?
– 8 gray levels, but only one has nonzero probability, say level 1: p0=p2= … = p7 = 0,
p1=1. H=?
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Image and Multidimensional Signal Processing
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Entropy - example
rk
L 1
H   pr (rk ) log 2 pr (rk )
k 0
P(ai)
87
128
186
255
0.25
0.47
0.25
0.03
log P(ai) -P log P
-2
0.5
-1.08927 0.511956
-2
0.5
-5.05889 0.151767
H (bits)
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Image and Multidimensional Signal Processing
1.663722
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Fidelity Criteria
• RMS (root mean square) error – an objective measure of error between
original and the compressed image
erms
 1

 MN
 
M 1 N 1
x 0 y 0

ˆf ( x, y)  f ( x, y) 


• Can also use SNR
M 1 N 1
SNRms 
x 0 y 0

1/ 2
2
fˆ ( x, y ) 2
M 1 N 1
 fˆ ( x, y )  f ( x, y ) 




x 0 y 0
2
• And subjective measures
– E.g., rate quality on a scale of 1 to 10
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Image Compression & Decompression Model
• Mapper: Transforms data to a form that can be more easily compressed
(eg, Fourier or wavelet transform)
• Quantizer: Reduces amount of data (eg., throws away smallest transform
coefficients)
• Symbol coder: Codes the resulting data using the shortest code words
(eg., variable length coding)
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Image and Multidimensional Signal Processing
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Loss-less Compression Methods
• “Loss-less”: the compressed image can be reconstructed
exactly
• We’ll look at these methods:
–
–
–
–
Huffman coding
Arithmetic coding
LZW coding
Run-length coding
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Huffman Coding
• Takes advantage of coding redundancy
• Generates a variable length code, as close as possible to the theoretical
minimum length
• Doesn’t take advantage of inter-pixel redundancy
• Widely used as a component (the symbol coder) in many compression
methods
Algorithm:
(1) Find the gray level probabilities
(2) Order the probabilities, from smallest to largest
(3) Combine the smallest two by addition
(4) Repeat steps 2-3 until only two probabilities are left
(5) By working backward along the tree, generate code by alternating
assignment of 0 and 1
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Example
Start with gray level
probabilities
Symbol
Probability
a1
0.1
a2
0.4
a3
0.06
a4
0.1
a5
0.04
a6
0.3
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Sort the probabilities, from
largest to smallest
Image and Multidimensional Signal Processing
Combine the smallest
two by addition
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Example
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Example
Work backward along the tree, generate code by alternating assignment of 0 and 1
Final result:
Sym
Code
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Example
• Example – generate code sequence
for symbols
a2 a1 a3 a1
• Example – decode the sequence
010100111100
• Average length of code?
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Example
• Compare to theoretical minimum
J
H (z )    P( a j ) log2 P( a j )
j 1
P(ai)
a2
a6
a1
a4
a3
a5
log P(ai)
-P log P
0.4
0.3
0.1
0.1
0.06
0.04
H
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Example
• Compare to theoretical minimum
J
H (z )    P( a j ) log2 P( a j )
j 1
P(ai)
a2
a6
a1
a4
a3
a5
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0.4
0.3
0.1
0.1
0.06
0.04
log P(ai)
-1.32193
-1.73697
-3.32193
-3.32193
-4.05889
-4.64386
-P log P
0.528771
0.52109
0.332193
0.332193
0.243534
0.185754
H
2.143534
Image and Multidimensional Signal Processing
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Example
• Find Huffman code for
Gray Level
Probability
00
0.2
01
0.3
10
0.1
11
0.4
Algorithm:
(1) Find the gray level probabilities
(2) Sort the probabilities
(3) Combine the smallest two by addition
(4) Repeat steps 2-3 until only two are left
(5) Work backward, generate code
Lavg = ?
H=?
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Example
• Find Huffman code for
Algorithm:
(1) Find the gray level probabilities
(2) Sort the probabilities
(3) Combine the smallest two by addition
(4) Repeat steps 2-3 until only two are left
(5) Work backward, generate code
Gray Level
Probability
00
0.2
010
0.4 1
0.4 1
0.6 0
01
0.3
00
0.3 00
0.3 00
0.4 1
10
0.1
011
0.2
010
11
0.4
1
0.1
011
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0.3 01
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Arithmetic Coding
• A sequence of values is assigned a single arithmetic code
word
• The code word is a fractional number between 0 and 1 (e.g.,
0.4731)
• Each symbol is assigned an interval based on its probability of
occurrence
• Code words are fixed length
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Example
Message:
a1a2a3a3a4
With each new
interval, you find
the subintervals
by multiplying the
total interval by
the probability of
each symbol, and
then add it to the
lower bound of
the interval to get
the upper bound.
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Image and Multidimensional Signal Processing
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Example
Message:
a1a2a3a3a4
(0.08-0.04)(0.8)+0.04
0.8
0.072
(0.072-0.056)(0.8)+0.056
0.0688
Can
use
0.068
0.2(0.4)+0
0.4
0.08
0.2
0.04
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Example
• Final code word: 0.068
• Three decimal digits for five symbols, or 3/5 = 0.6 digits per symbol
– Equivalently, 1.99 bits
– 100.6 = 2x -> x = 1.99 bits
• Theoretical minimum: 1.92 bits or 0.58 digits
a1
a2
a3
a4
P(ai) log P(ai)
0.2 -2.3219
0.2 -2.3219
0.4 -1.3219
0.2 -2.3219
H
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-P log P
0.46439
0.46439
0.52877
0.46439
1.92193
Image and Multidimensional Signal Processing
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Example
•
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Image and Multidimensional Signal Processing
The sequence “a1 a1
a1” can be encoded
as what number?
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LZW Coding
• Stands for Lempel-Ziv-Welch
• Works by coding short strings of data
• Used in GIF, TIFF, and PDF file formats
• Creates a “dictionary” of code words
– For an 8-bit image, the first 256 words are assigned to the gray values
0,1,2, … , 255
– As sequences are discovered, new code words (i.e., 256 through 511)
are assigned to represent them
– Eg: The sequence 126-126 may be assigned to code word 256
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Example
• Original image
39
39
126
126
39
39
126
126
39
39
126
126
39
39
126
126
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Initial Dictionary
Dictionary Index
Entry
0
0
1
1
:
:
255
255
256
?
:
:
511
?
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Final Dictionary
• Original image
39
39
126
126
39
39
126
126
39
39
126
126
39
39
126
126
• Coded sequence
39
39
256
126
258
260
259
257
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126
126
Dictionary Index
Entry
0
0
1
1
:
:
255
255
256
39-39
257
39-126
258
126-126
259
126-39
260
39-39-126
261
126-126-39
262
39-39-126-126
263
126-39-39
264
39-126-126
Image and Multidimensional Signal Processing
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Run Length Coding
• Inter-pixel redundancy
• Usually used for binary images
• Output the number of consecutive 0’s along a row, then the # of 1’s, etc
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
1
1
0
0
0
0
0
0
1
1
1
1
1
0
0
0
1
1
1
0
0
1
0
0
0
1
0
0
1
1
0
1
1
1
1
0
1
0
0
0
0
0
0
0
0
0
0
8
044
125
:
Output code
• Best case for compression: a row is all zeros (or all ones)
• Worst case for compression?
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RLC Applied to Non-binary Images
• For an m-bit image, the gray values are
am-12m-1 + am-22m-2 + … + a121 + a020
• Apply RLC separately to each bit plane
• This can be a problem in areas where values fluctuate about
certain transition points
– Example: 12710 = 011111112
– But 12810 = 100000002
– So each bit keeps flipping from 0 to 1
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Example
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Better: Use Gray Code
• First code the image using the “Gray Code” rather
than the normal binary code
• Gray code:
– Each adjacent code word differs only by one bit
– So if a pixel differs from its neighbor by one gray level, only
one bit in the gray code is different
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Gray Codes
2 bit
binary
2 bit
Gray
0
00
00
1
01
01
10
11
11
10
2
3
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3 bit
Gray
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