FILTRATION AND RESTORATION OF SATELLITE IMAGES USING DOUBLY STOCHASTIC RANDOM FIELDS

Professor, Doctor of Engineering Konstantin Vasiliev,

PhD, Assistant ProfessorVitaliy Dementiev,and PhD Student Nikita Andriyanov

Ulyanovsk State Technical (Russia)

RELEVANCE

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PROBLEM

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SOLUTIONS

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IMAGE RESTORATION

FILTERING

???MODELLING

SINGULAR VALUE DECOMPOSITION OF MATRICES WITH GAPS

KOHONEN MAPS

GOAL AND TASKS

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.

DOUBLY STOCHASTIC MODEL

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Consider the following modification of a doubly stochastic model

where is the random field of correlation parameters by the row; is the random field of correlation parameters by the column; is the random field of independent Gaussian random values with and ; is the base random field dispersion.

ijjiijijjiijjiijij xxxx 11211211 ,,,

ij1 ij2 ij

0 mM ij

))(( 22

21

222 11 ijijxijijM 2x

IMAGES FITTING

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a) b) c)

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PARAMETERS ESTIMATION USING SLIDING WINDOW

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EXAMPLES OF IMAGE RESTORATION

Restoration of the area of the image on the border of two dissimilar surfaces

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EXAMPLES OF IMAGE RESTORATION

Restoration of the image area close to uniform

EXAMPLES OF IMAGE RESTORATION

Restoration of the image area limited by different structures

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RESTORATION DURING FILTRATION

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We carry out a similar study, when to estimate the parameters of a doubly stochastic model we use the Kalman filter. To filter flat images we will use vector (interline) nonlinear Kalman filter. To do this, combine the elements of the image into a vector line . Then the model image can be written as

TiNiii xxxx ,,, 21

iyixiixii xdiagx ,)( 1 xixixxxi r )1(1 yiyiyyyi r )1(1

xiN

xi

xi

xidiag

0......

...0

......0...0...

0...

)( 2

1

In this progressive evaluation process can be described by the known nonlinear Kalman filter equations:

эpiinpi

T

iэpipi xzVx

Pxx ˆˆˆ 1

PSEUDOGRADIENT SEARCH

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)ˆ,((ˆˆ 1111 tttttt ZJ

Pseudogradient estimation procedure will be carried out in accordance with the following general expression

where is vector of parameters to estimate; t is an iteration number; is the approximation matrix; is the pseudogradient of objective function J, that characterizes the quality of estimation; Zt is the local sample of observations using at t-th iteration.

1111111111 2khgfedcba

xzij )ˆ(min}~{

2222222222 2khgfedcba

xzij )ˆ(min}~{

Thus, we select the coefficients by minimizing each of the possible directions of joint changes

PSEUDOGRADIENT AGAINST SLIDING WINDOW

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THE RESTORATION

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CONCLUSION- we have synthesized image reconstruction algorithms based on models with a complex structure;-we have obtained the gain in comparison with the AR models (from 1.5 to 6 times depending on the image type);-we have suggested combine using of the pseudogradient search procedures and Kalman filter for image restoration;-processing of various images has been investigated. Doubly stochastic models provides gain to 5 times compared with the AR models.

THANK YOU FOR YOUR ATTENTION!

Nikita Andriyanov,Ulyanovsk State Technical University (Russia)

email: nikita-and-nov@mail.ru