image segmentation by complex-valued units
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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
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
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
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
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
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
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))
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
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
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)
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
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
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)
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
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
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
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
Neckar CubeNeckar CubeNeckar CubeNeckar Cube
Attractor networks that minimize an energy function do not account for bi-stability
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
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
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)
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
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
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)
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
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
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)
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
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
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
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
Complex NumberComplex NumberComplex NumberComplex Number
φr r rate
φ phase
z = r eiφ
zi
1
= r cos φ + i r sin φ
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
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
Deterministic ChaosDeterministic ChaosDeterministic ChaosDeterministic Chaos
Logistic map:
Ф(t+1) = A Ф(t) (1- Ф(t))
Phase φ = 2π Ф
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
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
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
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
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
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
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
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
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
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
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
Phase Separation HistogramPhase Separation HistogramPhase Separation HistogramPhase Separation Histogram
Large phase differences at boundary of activation hill
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
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
Toward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object Recognition
attractor network
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
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
Toward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object RecognitionToward Learning Object Recognition
attractor network
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
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
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
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
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?
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