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L. Yaroslavsky. Course 0510.7211 “Digital Image Processing Applications “ Lect.5. Statistical Image and Noise Models
Statistical models of random interferences: (i) additive noise: nsr += ; (ii ) multiplicative noise nsr = ; (iii)composite noise: am nsnr += ;
(iiii) impulse noise: ;)1( enser +−= .⎩⎨⎧
=otherwise
Pprobabitycertainawithe e
,0,1
Examples: sensor’s “white” noise; narrow-band (“moire”) noise; speckle noise; quantization noise; Basic statistical characteristics of random interferences: Probability distribution/density; probability density moments: mean value, standard deviation; order statistics, autocorrelation functions/spectra. Measuring statistical characteristics of signals:
Measuring probability density by means of histogram: ( ) ( )∑−
=−=
1
0
1 N
kka am
Nmh δ .
Evaluating autocorrelation functions and spectra by FFT. Diagnostics of random interferences in images Measuring noise level in images: basic principle. Prediction method and voting method for detecting anomalies in image statistical characteristics. Measuring variance of zero-mean additive white noise in images
( ) ( )( ) =++=+= ∗nsnsnsrCF ( ) ( ) ( ) ( )nCFsCFnsnsnCFsCF +≈+++ ∗∗ Measuring intensity of “moire” noise components. Measuring variance of zero-mean additive white noise in interferograms. Measuring non-stationary and multiplicative noise: local correlation and spectral analysis Measuring probability of errors for impulse noise. Measuring the quantization noise. Generating pseudo-random signals and images. Textures: An algorithmic approach and algorithmic models. Primary random number generator. Generating uniformly distributed uncorrelated pseudo-random numbers: ( ) ( )3211 mod ccrandcrand kk += −
Point-wise nonlinearity (PWT)-model: control of the probability density: ( );ξη F= ; ( ) ( )ξη
η pddFp =
Linear filter (LF-) model. ),( hrandconv=η . Normalization effect of the distribution density. Control of the signal correlation function. An algorithm for generating correlated pseudo-random numbers with Gaussian distribution:
( ) ⎟⎠
⎞⎜⎝
⎛+= ∑−
= Nkliiw
N
N
k
imk
rekkl πξξη 2exp1 1
0.; kNk ww −=
( )∑−
=⎟⎠
⎞⎜⎝
⎛ −==
1
0
22
2exp1 N
kk
imm
iml
rem
rel N
mlkiwNN
πξ
ηηηη
LF-model and generating “natural” textures. Combined PWT-LF-and LF-PWN-models. Evolutionary models: nonlinear models with feedback. Generating “natural” textures and growth models. Eden’s cell growth model: probability of birth is proportional to the number of “alive” cells:
( )( ) ( )( ) ( )( )118 ,8/randb, −− ⊕= ttt lkSlk ξξ ; randb(x)- binary p/r numbers with probability of “ones” x
Conway’s Game of Life: ( ) ( )32 88,1
, −Σ+−Σ=+ δδtlk
tlk aa
Gray-scale model modification: ( ) ( );LOLO 21,1
, Δ+Δ=+ tlk
tlk aa Δ -“fuzzy” delta function; LO –linear operator.
Problems for self-testing 1. Describe and explain mathematical models of random interferences and their main characteristics. 2. Describe and substantiate algorithms for measuring parameters of additive noise in images 3. What are practical methods for diagnostics of impulse noise, of quantization noise 4. Describe and illustrate the algorithmic approach to synthesis and analysis of texture images 5. Describe and justify the algorithm for generating correlated p/r numbers with normal distribution
Image random distortions: practical examples
Examples of images returned from space ships Mars-4, 1973, USSR
Models of random interferences in imaging systems
Noise free image
Additive noise, stdev=20/256
Impulse noise, Pe=0.06, stdev=20/256
“Moire” noise, stdev=20/256
Quantization noise, Q=4, stdev=21/256
“Speckle” noise (local mean value and standard
deviation of noise are equal to signal values)
Image noise models and noise diagnostics Diagnostics of additive Gaussian noise
Noise-less image
Noisy image: additive Gaussian noise,
Stdev=20
50 100 150 200 25020
40
60
80
100
120
140
160
1-D spectra of initial (red) and noisy (blue)
1-D row-wise spectra of noise-free (red) and noisy (blue) images
-1 5 -1 0 -5 0 5 1 0 1 50 .4
0 .5
0 .6
0 .7
0 .8
0 .9
11 -D c o r re la t io n fu n c t io n o f th e in i t ia l im a g e
-15 -10 -5 0 5 10 150 .4
0 .5
0 .6
0 .7
0 .8
0 .9
11 -D c o rre la tio n func tion o f the nois y im age
1-D row-wise correlation function of noise-less (left) and noisy (right) images
Diagnostics of narrow-band noise
Image corrupted by periodical (moiré) noise
50 100 150 200 250
1.6
1.8
2
2.2
Av. power spectrum along rows
50 100 150 200 2500
100
200
300
400Noise spectrum
Moire noise diagnostics: Average row wise image spectrum (top) and dedected noise
components (bottom)
Noisy interferogram
Spectrum of noisy interferogram
Spectrum of noisy interferogram
Noise diagnostics in interferograms
Diagnostics of impulse noise
Image corrupted by impulse noise, Perr=0.3
Histograms of 2-d prediction error for noise-less (blue) and noisy (green) images
0 50 100 150 200 250 0
0.05
0.1
0.15
0.2
0.25
Generating pseudo-random numbers
Schematic diagram of a pseudo-random number generator.
Problem: difficult to secure statistical independence of pseudo-random numbers
C3
Point-wise nonlinearity
output=(input)mod 3C
output
input
1C 2C
One sample delay unit
Initial number (seed)
3C
“Starry night-1”
Illustration of statistical imperfectness of Matlab pseudo-random number generator: Mean value and standard deviation from uniform hisorgram as a function of the amount of numbers (left) and 2-D distribution histogram of two adjacent numbers in a sequence of numbers (right image). One can clearly see diagonal in the 2-D histogram and deviation from 1/N standard deviation curve that evidence that pseudo-random numbers are not perfectly statistically independent
Generating texture images: an algorithmic approach
Point-wise nonlinearity (PWN)-model
Applications: generating pseudo-random numbers with a given distribution Pseudo-random numbers { }ξ with known distribution density ( )ξp can be converted into numbers { }η with distribution density ( )ηp by nonlinear transformation ( )ξη F= that is defined by differential equation:
( )( )ηξ
ξ pp
ddF
=
An example: Generating “Pepper&salt” noise
Primary 2-D pseudo-random
number generatorTransformation
system
Texture image
Point-wise nonlinearity
Texture image Primary 2-D pseudo-random
number generator
Linear filter (LF)-model
Applications: - Generating correlated and uncorrelated high quality pseudo-random numbers with Gaussian distribution
Flow-cart of the algorithm for generating corelated and uncorrelated pseudo-random numbers with Gaussian distribution density
Distribution function of pseudo-random Gaussian numbers generated by LF-model (FFT method, 128x128 array)
Linear filter Texture image Primary 2-D
pseudo-random number generator
FFT method features: • Output distribution tends to Gaussian with the increase of N however input distribution is • Obtaining necessary correlation function in one step • Low computational complexity (O(logN) per number) •Efficient use of initial pseudo-random numbers (one output number per one input number) •Input correlations do not propagate to output
Generating pseudo-random numbers with
arbitrary distribution
Point wise multiplication by weight coefficients defined by
the required correlation function
FFT
Texture images generated by the Linear filter (LF)-model
Examples of texture images generated by LF-models and the corresponding filter frequency responses
Filter frequency response Generated texture
Generated texture (fractal)
Clouds in the night
LF-model: natural texture images from Brodatz’s album (left column) and their synthetic copies (right column)
PWN-LF-model
b) PWN-LF-model of texture images (a) and examples of generated textures (b)
Linear filter
Texture image Threshold type
point-wise nonlinearity
Primary 2-D pseudo-random
number generator output
input
LF-PNW-model
Examples of texture images generated by the LF-PWN-model.
Texture images generated by models with multiple branches
Texture image
Point-wise nonlinearity
Linear filterPrimary 2-D
pseudo-random number generator
Composite and spatially inhomogeneous textures generated by models with multiple branches
“Control” field (left) and spatially inhomogeneous texture (fur_txtr.m)
Another example of a spatially
inhomogeneous texture
“One dollar” on textile texture generated
by a multiple branch model
New Year 2000 texture
Further examples of synthetic textures
Wood
Clouds
Bricks
Mountains
Evolutionary models
a)
Images generated by the model
Initial distribution
Iterated spatially homogeneous pattern
Iterated spatially in homogeneous pattern and its edges
Natural crystall pattern: alumngranul
Primary 2-D pseudo-random number
generator
Rank filter: Texture image
( ) ( )( )lkinputMODlkoutput S ,, =
One-frame delay unit
Growth models Eden’s model-1
An example of patterns generated by the Eden’s model 1 (color reflects “age” of different
parts of the formation as it is indicated by colorbar )
11 1 1 1 0 1 1 1 1
Linear filter with
3x3 impulse response
pseudo-randomnumber
generator
Point-wise nonlinearity
1
P
randb(P)
Seed
One frame delay unit
Output
P
81
Eden’s model-2
An example of “dendrite”-patterns generated by the Eden’s model 2 ((color reflects “age” of different parts of the formation as it is indicated by colorbar ))
Seed
1 1 11 0 11 1 1
Linear filter with 3x3 impulse response
Linear filter with uniform
impulse response in the window
that exceeds the allowed size of the formation
P-W nonlinearity yx −
x
y
randb(P) P
One frame delay unit
Output
81
Conway’s game of Life model
a) b)
c) d) Evolution of pattern a) in the Game of Life model: a) – initial pattern, b) – d) – patterns on 75, 76, and 77-th iterations, correspondingly. Note “gliders” outlined by black boxes
1
1 2 3
Point wise nonlinearity
unit
randb(P)
1 1 11 0 11 1 1
Linear filter with 3x3 impulse response
1 1 11 0 11 1 1
Linear filter with 3x3 impulse
1
1 2 3
Point wise nonlinearity
unit
× +
One-frame delay unit
P
Evolution (downward in vertical direction) of a one-dimensional (in horizontal direction) modification of the Game of Life. (Initial rate of “alive” points in the first row is 0.3.)
“Oliva porphiria” see shell
Another example of a see shall
\
a)
b)
c)
d)
An example of the modified Conway’s model evolution with 25.0=dP and 1=bP : a-initial binary patterns; b), c), d) - evolution results and natural textures
Lifebin1:Initial "soup"; Plive=0.03 Image after 50th iteration
Image after 75th iteration Image after 200th iteration
Natural “labyrinth” and “zebra skin” patterns
Magnetic domain pattern
(adopted from: http://www.phys.uni.lodz.pl/kfcs/ Mat_Lab/exdsws.htm)
Fingerprint
Zebra skin
Zebra skin patterned mollusc
Examples of the evolutionary behavior of the modified Conway’s model of Eq. 7. From left to right: stable “star constellations” patterns, “clouds”, and labyrinth-alike pattern Cell value levels in the images are varying here from 0 to 255 and are coded in color as it is represented by the color bar
Natural “oncocytic papillary pattern “ (adopted from http://www.ucalgary.ca/UofC/faculties/ medicine/PATH/Banff_Path_Course/ImagesDocuments/Thurs%200715%20Rosai%20A%20Class%20Scheme.pdf)
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