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EEM 463 Introduction to Image Processing
Week 10: Wavelets and Multiresolution Precessing
Fall 2013
Instructor: Hatice Çınar Akakın, Ph.D.
Anadolu University
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Motivation for Multiresolution Processing
• If the objects are small in size or low in contrats: examine them in high resolutions
• If they are large in size or high in contrast: a coarse view is required
• If they are present simultaneously: study at several resolutions• Analyze large structures or overall image context at lower-resolution levels
• Analyze individual object characteristics at higher-resolution levels
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Image Pyramids
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Represents imagesat more than one
resolution
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Lower resolution is suitable for the analysisof large structures andapverall image context
High resolution is appropriate for the
analysis of individualobject characteristics!
Subband Coding• An image is decomposed into a set of bandlimited components called
subbands.
• Each subband is generated by bandpass filtering the input image
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The Haar Transform
• H2 =1
2
1 11 − 1
• T = HFHT
Where F is an NxN image
matrix
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Haar basisfunctions
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EEM 463 Introduction to Image Processing
Week 10: Image Compression
Fall 2013
Instructor: Hatice Çınar Akakın, Ph.D.
Anadolu University
24.12.2013
Goal of Image Compression
• Everyday an enormous amount of information is stored, processedand transmitted
• Goal : Reducing the amount of data required to represent a digitalimage while keeping information as much as possible• Data storage• Data transmission
• It plays an important role in video conferencing, remote sensing, medical imaging…
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Fundamentals
• Data and information are not synonymous terms!
• Data is the means by which information is conveyed
Coding Redundancy: Most 2-D intensity arrays contain more bits than are needed to represent the intensities
Spatial and temporal redundancy: Pixels of most 2-D intensity arrays are correlated spatially and video sequences are temporally correlated
Irrelevant information: Most 2-D intensity arrays contain information that is ignored by the human visual system
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• Where b an b’ denote the number of bits in two representations of the same information
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CR
11
Relative Data Redundancy
Compression Ratio
'b
bC
Coding Redundancy
84.42
1.81
1 1/ 4.42 0.774
C
R
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n
n
n
rhrp kkk
)()(
1
0
The average number of bits required to represent each pixel is
( ) ( ) 0.25(2) 0.47(1) 0.24(3) 0.03(3) 1.81L
avg k r k
k
L l r p r bits
84.42
1.81
1 1/ 4.42 0.774
C
R
Spatial and Temporal Redundancy
• All 256 intensities are equally probable (Uniform histogram)
• The pixels of each line are independent of another in the verticaldirection
• Since pixels along each line are identical, they are maximallycorrelated in the horizontal direction
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Irrelevant Information
• 256x256x8/8 = 65536:1
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Afterhistogram
equalization
• Is there a minimum amount of data that is sufficient to describe an imagewithout losing information?
• Answer: Information theory
A random event E with probability P(E) is said to contain
units of information! (Note: I(E)=0 when P(E)=1 )
(P(E)=1/2 ⇒I(E)=1 bit )
Measuring Image Information
A random event E with probability P(E) is said to contain
1 ( ) log -log ( )
( )
units of information.
I E P EP E
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• Possible events (source symbols) {a1, a2, …, aj}
• Associated probabilities {P(a1), . . . , P(aj) }
• The average information per source output
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j
jj aPaPH ))(log()(
H is the uncertainty or entropy of the source or the average amount of information (in m-ary units) obtained by observing a single source output
• If an image is considered to be the output pf an imaginary zero-memory intensity source, the intensity source’s entropy becomes
• is the normalized histogram
• For the Fig. 8.1(a),
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1
0
2 )(log)(~ L
k
krkr rprpH
)( kr rp
It is not possible to code theintensity values of the imaginary
source with fewer thanbits/pixel
H~
2 2 2 2
For the fig.8.1(a),
[0.25log 0.25 0.47 log 0.47 0.25log 0.25 0.03log 0.03]
1.6614 bits/pixel
H
H~
Fidelity Criteria
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1/221 1
0 0
Let ( , ) be an input image and ( , ) be an approximation
of ( , ). The images are of size .
The - - is
1 ( , ) ( , )
M N
rms
x y
f x y f x y
f x y M N
root mean square error
e f x y f x yMN
ms
21 1
0 0
ms 21 1
0 0
The - - - of the output image,
denoted SNR
( , )
SNR
( , ) ( , )
M N
x y
M N
x y
mean square signal to noise ratio
f x y
f x y f x y
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• The mean-square-signal to noise ratio of the output image is:
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RMSE = 5.17 RMSE = 15.67 RMSE = 14.17
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Some Basic Compression Methods• Huffman Coding : one of the most popular algorithms
• The resulting code is optimal
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1st step: source reduction
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The average length of this code is
0.4*1 0.3*2 0.1*3 0.1*4 0.06*5 0.04*5
= 2.2 bits/pixel
avgL
Arithmetic Coding• Unlike the variable
length codes, it generates nonblockcodes• There is no one to
one correspondencebetween sourcesymbols and codewords!
• Entire seuence of symbols is assigned a single arithmetic codeWord.
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How to code a1a2a3a3a4 ?
LZW Coding (formulated in 1984)
• LZW (Lempel-Ziv-Welch) coding, assigns fixed-length code words to variable length sequences of source symbols, but requires no a priori knowledge of the probability of the source symbols.
• LZW is used in:
• Tagged Image file format (TIFF)
• Graphic interchange format (GIF)
• Portable document format (PDF)
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• Example
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39 39 126 126
39 39 126 126
39 39 126 126
39 39 126 126
Run-Length Coding (RLE)
• Encodes repeating string of symbols (i.e., runs) using a few bytes: (symbol, count)
1 1 1 1 1 1 0 0 0 0 1 1 (1,6) (0, 4) (1,2)
a a a a b b b b c c c (a,4) (b, 4) (c, 3)
Can compress any type of data but cannot achieve high compression ratios compared to other compression methods.
Run-length encoding is supported by most bitmap file formats such as TIFF, BMP and PCX
Read the rest of Chapter 8 for details!
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Transform Coding
• Energy packing• 2D transforms pack most of the energy
into small number of coefficients located
at the upper left corner of the 2D array
2D Transforms
EnergyPacking
• Consider an image f(x,y) of size N x N
Forward transform
g(x,y,u,v) is the forward transformation kernel or basis functions
2D Transforms
.1,...,2,1,0,
),,,(),(),(1
0
1
0
Nvu
vuyxgyxfvuTN
x
N
y
2D Transforms
• Inverse transform
h(x,y,u,v) is the inverse transformation kernel or basis functions
.1,...,2,1,0,
),,,(),(),(1
0
1
0
Nyx
vuyxhvuTyxfN
u
N
v
Discrete Cosine Transform
• One of the most frequently used transformations for image compression is the DCT.
N
Nu
N
vy
N
uxvu
vuyxhvuyxg
2
1
)(
2
)12(cos
2
)12(cos)()(
),,,(),,,(
for u=0
for u=1, 2, …, N-1
Discrete Cosine Transform
Software Research
2D Transforms
Effect of Window Size
