1. BIOLOGICALLY MOTIVATED OSCILLATORY NETWORK MODEL FOR DYNAMICAL IMAGE SEGMENTATION
Margarita Kuzmina, Eduard Manykin
Keldysh Institute of Applied Mathematics RAS,
RRC Kurchatov Institute

2. Motivations
Synchronous cortical oscillations, experimentally discovered in the brain visual cortices of cat and monkey ( );
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). 2

3. 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

5. 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
Oscillator state is specified by a pair of real-valued variables
ODE system, governing oscillator dynamics, is written for
The oscillator is capable to demonstrate either activity state
(stable oscillations) or «silence» (quickly damping oscillations). 5

7. Dynamical connections in 3D network
The network state is defined by 3D array
of all oscillator states.
Network dynamics is governed by the system of ODE:
Here functions
define internal oscillator dynamics, and the terms
specify network
coupling. Dynamical coupling is designed in the form:
The values
defining the strength of network connections, are
constructed in the form of product of three nonlinear functions, dependent
on oscillator activities, receptive field orientations and spatial distance
between oscillator pair in the network. 7

8. Dynamical connections in 3D network
The network connectivity rule can be written as
where
and
are limit cycle radii for oscillators with indices
and
and
are RF orientations for these
oscillators,
and
are radius-vectors defining their spatial locations.
In accordance with construction of nonlinear functions
any
pair of network oscillators is proved to be connected under combination
of 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. 8

9. 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
brightness ;
network
connectivity rule includes the cofactor
, dependent
on elementary bar orientations, instead of cofactor .
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, starting
from 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.

10. Stages of 2D oscillatory network performance

11. Image versions in the process of 2D network performance

12. Texture segmentation

13. Current 2D model modifications
The following 2D model modifications have been designed and are
currently under code implementation.
A) Modifications in oscillator dynamics:
1) Replacement of former arbitrary oscillator frequency distribution
by frequences
, 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
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. 13

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

15. Conclusive remarks
The 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
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. 15