Image Segmentation by Complex-Valued Units Cornelius Weber and Stefan Wermter Hybrid Intelligent Systems School of Computing and Technology University

Image Segmentation by Complex-Valued UnitsImage Segmentation by Complex-Valued Units

Cornelius Weber and Stefan Wermter

Hybrid Intelligent SystemsSchool of Computing and Technology

University of Sunderland

ICANN Conference, September 2005

Project funded by the Future and Emerging Technologies arm of the IST ProgrammeProject funded by the Future and Emerging Technologies arm of the IST ProgrammeFET-Open schemeFET-Open scheme


ContentsContentsContentsContents

• Attractor Network which Converges

• Non-Convergence and Spike Synchrony

• Coupled Oscillators for Spike Phases

• Outlook







• Outlook



Attractor Network:Attractor Network:Competition via RelaxationCompetition via Relaxation

Attractor Network:Attractor Network:Competition via RelaxationCompetition via Relaxation

weight profile rate profile

rate update

ri(t+1) = f (Σij wij rj(t))



winner

Response CharacteristicsResponse CharacteristicsResponse CharacteristicsResponse Characteristics

linear sparse competitive

Weber , C. Self-Organization of Orientation Maps, Lateral Connections, and Dynamic Receptive Fields in the Primary Visual Cortex. Proc. ICANN (2001)

Weber , C. Self-Organization of Orientation Maps, Lateral Connections, and Dynamic Receptive Fields in the Primary Visual Cortex. Proc. ICANN (2001)



Learning Object RecognitionLearning Object RecognitionLearning Object RecognitionLearning Object Recognition

attractor network Active units (features) not separated

Binding- and learning problem?green

redbackground

apple

Learning objects in cluttered background is difficult

Stringer, S.M. and Rolls, E.T. Position invariant recognition in the visual system with cluttered environments. Neural Networks 13, 305-15 (2000)

Stringer, S.M. and Rolls, E.T. Position invariant recognition in the visual system with cluttered environments. Neural Networks 13, 305-15 (2000)







• Outlook



Neckar CubeNeckar CubeNeckar CubeNeckar Cube

Attractor networks that minimize an energy function do not account for bi-stability



Neuronal Spike ChaosNeuronal Spike ChaosNeuronal Spike ChaosNeuronal Spike Chaos

A wide range of spiking neuron models displays three distinct categories of behaviour:

- quiescence

- intense periodic seizure-like activity

- sustained chaos in normal operational conditionsBanerjee, A. On the Phase-Space Dynamics of Systems of Spiking Neurons. I: Model and Experiments. Neural Computation, 13(1), 161-93 (2001)

Banerjee, A. On the Phase-Space Dynamics of Systems of Spiking Neurons. I: Model and Experiments. Neural Computation, 13(1), 161-93 (2001)



Neuronal SynchronyNeuronal SynchronyNeuronal SynchronyNeuronal Synchrony

“cortical neurons often engage in oscillatory activity which is not stimulus locked but caused by internal interactions”

“activity synchronization was present in the expectation period before stimulus presentation and could not be induced de novo by the stimulus”

Singer, W. Synchronization, Bining and Expectancy. In: The Handbook of Brain Theory and Neural Networks, pp. 1136-43 (2003)

Singer, W. Synchronization, Bining and Expectancy. In: The Handbook of Brain Theory and Neural Networks, pp. 1136-43 (2003)

Cardoso de Oliviera, S., Thiele, A. and Hoffmann, K.P. Synchronization of neuronal activity during stimulus expectation in a direction discrimination task. J. Neurosci., 17, 9248-60 (1997)

Cardoso de Oliviera, S., Thiele, A. and Hoffmann, K.P. Synchronization of neuronal activity during stimulus expectation in a direction discrimination task. J. Neurosci., 17, 9248-60 (1997)



Neuronal Spike ChaosNeuronal Spike ChaosNeuronal Spike ChaosNeuronal Spike Chaos

We need a method to:

- create patterns of synchronization

- avoid long-term stabilization (bi-stability is welcome!)

van Leeuven, C., Steyvers, M. and Nooter, M. Stability and Intermittency in Large-Scale Coupled Oscillator Models for Perceptual Segmentation. J. Mathematical Psychology, 41(4), 319-44 (1997)

van Leeuven, C., Steyvers, M. and Nooter, M. Stability and Intermittency in Large-Scale Coupled Oscillator Models for Perceptual Segmentation. J. Mathematical Psychology, 41(4), 319-44 (1997)







• Outlook



Complex NumberComplex NumberComplex NumberComplex Number

φr r rate

φ phase

z = r eiφ

zi

1

= r cos φ + i r sin φ



Deterministic ChaosDeterministic ChaosDeterministic ChaosDeterministic Chaos

Logistic map:

Ф(t+1) = A Ф(t) (1- Ф(t))

Phase φ = 2π Ф



Coupling of the PhasesCoupling of the PhasesCoupling of the PhasesCoupling of the Phases

For phases: Σkj wkj rj eiφ ≡ zkwf}

coupling strength for phasescomplex number

}

“Net input” to neuron k:

For rates: Σkj wkj rj

j

weighted field



Relaxation of the PhasesRelaxation of the PhasesRelaxation of the PhasesRelaxation of the Phases

Compute “net input”: zkwf = Σkj wkj rj eiφ

Compute new phase: Фk(t+1) = A Фkwf(t) (1- Фk

wf(t))

(remember: φ = 2π Ф)

From zwf, take phase φwf

j



Relaxation of Rates and PhasesRelaxation of Rates and PhasesRelaxation of Rates and PhasesRelaxation of Rates and Phases

Phase of any neuron behaves chaoticallyCoupled neurons have similar phases



Phase Separation HistogramPhase Separation HistogramPhase Separation HistogramPhase Separation Histogram

Large phase differences at boundary of activation hill



Toward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object Recognition

attractor network



Toward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object Recognition

attractor network







• Outlook



Plans and QuestionsPlans and QuestionsPlans and QuestionsPlans and Questions

- The higher hierarchical level shall benefit!

- Should the rates depend on the phases?

→ This would influence learning!

- Learning with Phase Timing Dependent Plasticity?

Documents

Image Segmentation by Complex-Valued Units Cornelius Weber and Stefan Wermter Hybrid Intelligent Systems School of Computing and Technology University