BIOLOGICALLY MOTIVATED OSCILLATORY NETWORK MODEL FOR DYNAMICAL IMAGE SEGMENTATION

Margarita Kuzmina, Eduard Manykin

Keldysh Institute of Applied Mathematics RAS,

RRC Kurchatov Institute

Motivations

• Synchronous cortical oscillations, experimentally discovered in the brain visual cortices of cat and monkey (1988-1989);

• Evidence on exploitation of synchronization and resonance in functioning of brain structures, different of the visual cortex (olfactory bulb and cortex, hippocampus, thalamo-cortical system, spinal cord, neocortex).

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3D oscillatory network (model of VC)

• neural oscillator is network processing unit

• network architecture imitates columnar structure of VC

• network performance consists in synchronization of network assemblies (clusters) of dynamically coupled oscillators; it imitates self-organized collective behavior of orientation-selective (simple) cells of VC in preattentive stage of image processing

• 3D network is a tunable network. The whole set of network parameters consists of 3D array of receptive field orientations (internal network parameters) and 2D array of image characteristics - pixel brightness values and elementary bar orientations. The parameters provide tuning of both internal dynamics of network oscillators and self-organized dynamical network coupling. 3

][ kjmn

)],[( jmjm sI

Single network oscillator

• The oscillator model is based on biologically motivated model of neural oscillator, formed by a pair of interconnected cortical neurons, that was designed by Z.Li in 1998.

• It is a relaxational, or limit cycle oscillator with dynamics, parametrically dependent on

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),,( nsI

Oscillator state is specified by a pair of real-valued variables ODE system, governing oscillator dynamics, is written for

,, 21 uu

21 iuuu

);,())(||(/ 220 nsIgcucuidtdu

The oscillator is capable to demonstrate either activity state

(stable oscillations) or «silence» (quickly damping oscillations).

Single network oscillator

Dynamical connections in 3D network The network state is defined by 3D array ][ k

jmu of all oscillator states. Network dynamics is governed by the system of ODE:

;),(/ kjm

kjm

kjm

kjm Sufdtdu

Here functions ,))(||(),( 22

0 cucuiuf ),;,( kjmjmjm

kjm nsIg

,1 Mj ,1 Nm .1 Kk

define internal oscillator dynamics, and the terms kjmS specify network

coupling. Dynamical coupling is designed in the form:

).(,,

kjm

kmj

kmj

kkmmjj

kjm uuWS

The values ,kkmmjjW

defining the strength of network connections, are

on oscillator activities, receptive field orientations and spatial distance constructed in the form of product of three nonlinear functions, dependent

between oscillator pair in the network.

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Dynamical connections in 3D network

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The network connectivity rule can be written as

|),(|),(),( rrDnnQPW kkmmjj

kkmmjj

kkmmjj

kkmmjj

where and are limit cycle radii for oscillators with indices

oscillators, ),,( kmj and ),,,( kmj n and n are RF orientations for these

r and r are radius-vectors defining their spatial locations. In accordance with construction of nonlinear functions ,P ,Q D anypair of network oscillators is proved to be connected under combinationof the conditions:

a) both oscillators are active; b) they possess close receptive field orientations;c) they are separated by a distance not exceeding the prescribed radius

of spatial interaction.

Otherwise dynamical connection is absent.

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2D reduced network model 2D network is a limit version of 3D oscillatory network.

The network oscillators are located in 2D spatial lattice being in one-to-one correspondence with image pixel array.

Single oscillator dynamics is tunable only by pixel

network connectivity rule includes the cofactor ),( ssQ , dependent.

2D network provides both brightness image segmentation and solving of some texture segmentation tasks, including contour integration.

In problems of brightness image segmentation the network performance is improved via simple method of network coupling adjustment, providing synchronization control. Synchronized clusters arise successfully, startingfrom the one, corresponding to the brightest image fragment. At final stage of performance the oscillatory network is decomposed into a set of internally synchronized, but mutually desynchronized clusters, corresponding to all image fragments.

brightness I

on elementary bar orientations, instead of cofactor

;

).,( nnQ

Stages of 2D oscillatory network performance

Image versions in the process of 2D network performance

Texture segmentation

Current 2D model modifications The following 2D model modifications have been designed and arecurrently under code implementation.

A) Modifications in oscillator dynamics:1) Replacement of former arbitrary oscillator frequency distribution by frequences )( jkjkjk I , dependent on pixel brightness;2) Design of new version of oscillator dynamics, providing arbitrary monotonic dependence of oscillator activity (limit cycle size ) on pixel brightness

.I

B) Modification of network connectivity rule:

1) Design of modified cofactor , providing higher segmentation accuracy;2) Replacement of former network coupling adjustment by more efficient discrete-time process of successive image fragment selection.

),( P

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Image versions in the case of oscillator frequences dependent on pixel brightness

Conclusive remarksThe following advantages of the dynamical image segmentatiom method can be marked:

parallel and «automatic» performance (similar to that inherent in VC )convenient successive image fragment selection

informative and flexibly controllable visualization of image decomposition into the set of fragments

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The following directions of further model extension look like possible: a) extension to real-time segmentation of moving images; b) development of active vision approaches.