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Entropy and some applications in image processing
Neucimar J. LeiteInstitute of Computing
neucimar@ic.unicamp.br
Outline
• Introduction– Intuitive understanding
• Entropy as global information• Entropy as local information
– edge detection, texture analysis
• Entropy as minimization/maximization constraints– global thresholding
– deconvolution problem
Information Entropy (Shannon´s entropy)
An information theory concept closely related to the following question:
- What is the minimum amount of data needed to represent aninformation content?
• For images (compression problems):
- How few data are sufficient to completely describe an imageswithout (much) loss of information?
Intuitive understanding:
- relates the amount of uncertainty about an event with a given probability distribution
Event: randomly draw out a ball
high uncertainty
lowuncertainty
no uncertainty
entropy = max min(uncertainty)
Example 1:
Event: a coin flipping = { heads, tails }
Probability: P(heads) = P(tails) = 1/2
bit 1 )(log )(
1log 22 EP
EP0 heads1 tails
self-information: inversely related to the probability of E
Self-information:
- Units of information used to represent an event E
Example 2:
bits 38
1log- )(log ninformatio-self 22 EP
amount of conveyed information of
event EEntropy: average information
8
1)( kballP
k kk ballPballPH
)(
1log
)(
1
8
1)( kballP
0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
8
3
8
1log
8
1
coding the balls (3 bits/ball)
8
3
8
1log
8
1
8
3
8
1log
8
1
Entropy: = 3 bits/ball
Degree of information compression:
balls 8log2
equa
l len
gth
bin
ary
code
18log
length code optimized
length code original 2 EntropyR
C
for independent data: EntropyR
H= -( 5/8 log2(5/8) + 1/8 log2(1/8) + 1/8 log2(1/8) + 1/8 log2(1/8) ) = 1.54
H = -8log21 = 0
medium uncertainty:
no uncertainty:
code0 00 1
1 01 1
H= -( 5/8 log2(5/8) + 1/8 log2(1/8) + 1/8 log2(1/8) + 1/8 log2(1/8) ) = 1.54
medium uncertainty:
code0 00 1
1 01 1
• 2 bits/ball > 1.54 bit/ball code redundancy !!!and 29.154.1
2
R
C
We need an encoding method for eliminating this code redundancy
22%
Ball Probability Reduction 1 Reduction 2
red
black
blue
green
5/8
1/8
1/8
1/8
5/8
2/8
1/8
5/8
3/8
(1)
(0)
Ball Probability Reduction 1 Reduction 2
red
black
blue
green
5/8
1/8
1/8
1/8
5/8
2/8
1/8
5/8
3/8
(1)
(0)(01)
(00)
(1)
Ball Probability Reduction 1 Reduction 2
red
black
blue
green
5/8
1/8
1/8
1/8
5/8
2/8
1/8
5/8
3/8
(1)
(0)(01)
(00)
(1)
(011)
(010)
(1)
(00)
variable length code
red
black
blue
green
ball
100
011010
1.62 8
13
8
13
8
12
8
51
R
23.162.1
2
R
Cand (18,6%)
Entropy: 4.11
512 x 512 8-bit image:
After Huffman encoding:
93.114.4
8
R
C
bits/pixel
Variable length coding does not take advantage of the highimages pixel-to-pixel correlation: a pixel can be predicted from the values of its neighbors more redundancy lower entropy (bits/pixel)
Entropy: 7.45
After Huffman encoding:
07.146.7
8
R
C
Entropy: 7.35
08.139.7
8
R
C
After Huffman encoding:
Coding the interpixel difference highlighting redundancies:
Entropy: 4.73 instead of 7.45
56.111.5
8
R
C
After Huffman encoding:
Entropy: 5.97 instead of 7.35
34.197.5
8
R
C
After Huffman encoding:
instead of 1.07
instead of 1.08
Entropy-based edge detection
• Low entropy values low frequencies uniform image regions
• High entropy values high frequencies image edges
Texture Analysis
• Similarity grouping based on brightness, colors, slopes, sizes etc
• The perceived patterns of lightness, directionality, coarseness, regularity, etc can be used to describe and segment an image
Texture description: statistical approach
• Characterizes textures as smooth, coarse, periodic, etc
- Based on the intensity histogram prob. density function
Descriptors examples:
)(1
0i
L
ii zpzm
zi = random variable denoting gray levelsp(zi) = the intensity histogram in a region
• Mean: a measure of average intensity
)()(1
0i
L
i
nin zpmz
• Other moments of different orders:
22 )( z
- e.g., standard deviation: a measure of average contrast
Entropy: a measure of randomness
1
0
)(log)(L
iii zpzpe
Texture Average Intensity Average contrast Entropy
smooth 87.5 10.8 5.3
coarse 121.2 74.2 7.8
periodic 99.6 34.0 6.5
smooth
coarse
periodic
Gray-level co-occurrence matrix: Haralick´s descriptors
• Conveys information about the positions of pixels having similar gray level values.
1 2 1 3 3 2 12 3 2 2 2 1 13 3 2 2 1 1 31 3 1 2 1 1 3
d=1
0 1 2 3 4
0 0 0 0 0 01 0 4 2 1 02 0 3 3 2 03 0 1 2 3 04 0 0 0 0 0
Md(a,b)
For the descriptor H:
large empty spaces in M little information
contentcluttered areas large information content
i j
dd jiMjiMH ],[log],[
Md = the probability that a pixel with gray level i will have a pixel with level j a distance of d pixels away in a given direction
d = 2, horizontal direction
Obviously, more complex texture analysis based on statistical descriptors should consider combination of information related toimage scale, moments, contrast, homogeneity, directionality, etc
For images with levels 0-255:
The probability that a given pixel will have value less than or equal t is:
t
iit pP
0
Now considering:
Class A: t
t
tt P
p
P
p
P
p,,, 10
Class B: tt
t
t
t
P
p
P
p
P
p
1,,
1,
125521
The optimal threshold is the value of t that maximizes
),()()( tHtHtH wb
where
t
it
oi t
ib P
p
P
pH log
t
i
ti t
iw P
p
P
pH
1log
1
255
1
Entropy as a fuzziness measure
In fuzzy set theory an element x belongs to a set S with a certain probability px defined by a membership function px(x)
Example of a membership function for a given threshold t:
t x if /|-x|1
1
t x if /||1
1
)(
1
0
C
Cxxpx
px(x) gives the degree to which x belongs to the object or background with gray-level average and , respectively.0 1
),1log()1()log()( xxxxxf pppppH
Using the Shannon´s function (for two classes):
the entropy of an entire fuzzy set of dimension MxN is
i
xf ihistipHMN
tE )())((1
)(
and for segmentation purpose, the threshold t is such that E(t) isminimum t minimizes fuzziness
The restoration problem:
Given g, h, and we can find an estimate such that
the residual
f̂ fhg ˆ
• Since there may exist many functions such that the above constraint is satisfied, we can consider the maximization entropy as an additional constraint for “optimum” restoration
f̂
original degraded restored
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