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)