Pattern Recognition by Wave-Matching - René Doursat

Jul 2, 2006 - bottom-up self-organization of low-level features → Gestalt principles. ▫ top-down pattern ... table chair corner chair. ✓ graphs are “constellations of features”, in which. ▫ nodes carry .... b) quasi-periodic → oscillatory z = −.36.
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Coding Positional Information with Phases:

Pattern Recognition by Wave-Matching

René Doursat Brain Computation Laboratory Department of Computer Science and Engineering University of Nevada, Reno

Pattern Recognition by Wave-Matching 1. Active Perception 2. Graph Matching 3. Phase Tagging 4. Dynamic Temporal Matching 5. Lattice Wave Induction

July 2006

Doursat, R. - Pattern Recognition by Wave Matching

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Pattern Recognition by Wave-Matching 1. Active Perception ¾ ¾ ¾ ¾

Ascribing structure to data Gestalt: bottom-up self-organization Schemas: top-down guided organization Perceptual and cognitive schemas

2. Graph Matching 3. Phase Tagging 4. Dynamic Temporal Matching 5. Lattice Wave Induction July 2006

Doursat, R. - Pattern Recognition by Wave Matching

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1. Active Perception ¾ Ascribing structure to data 9 active perception means actively organizing raw sensory data 9 feature grouping & segmentation proceed on two levels ƒ bottom-up self-organization of low-level features → Gestalt principles ƒ top-down pattern recognition → pre-recorded schemas

data July 2006

shape Doursat, R. - Pattern Recognition by Wave Matching

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1. Active Perception ¾ Gestalt: bottom-up self-organization 9 low-level organizational principles group features locally 9 proximity

9 similarity

9 closure/continuity

9 symmetry 9 common fate 9 etc.

July 2006

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1. Active Perception ¾ Schemas: top-down guided organization 9 high-level stored patterns finish grouping features globally 9 local regularities or statistical properties are not enough: recognition must be guided by schemas ƒ schemas are constrained: specific assemblage of components ƒ yet also flexible: invariant by rotation, translation, scaling, distortions

July 2006

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1. Active Perception ¾ Perceptual and cognitive schemas 9 schemas (“mental representations”) are simplified but representative models of cognitive categories 9 they contain no details but have an overall resemblance with their object, mixing analogic and symbolic information 9 ex: “mental imagery”, “geons”, “cognitive linguistic icons”, etc. → compositionality: components, modules, building blocks schematic representation

photographic representation July 2006

symbolic representation

Doursat, R. - Pattern Recognition by Wave Matching

1.e4 c5 2.Nf3 e6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 d6 6.g4 Be7 7.g5 Nfd7 8.Rg1 Nc6 9.Be3 Nb6 10.Qd2 Bd7 11.00-0 Nxd4 12.Bxd4 0-0 13.f4 Rc8 14.h4 Nc4 15.Bxc4 Rxc4 16.Qd3 b5 17.f5 e5 18.Be3 Re8 19.h5 Qa5 20.g6 hxg6 21.hxg6 Bc6 22.Bh6 Bf8 23.gxf7+ Kxf7 24.Qxd6 Kg8 25.Rxg7+ 1-0 7

Pattern Recognition by Wave-Matching 1. Active Perception 2. Graph Matching ¾ Schemas as graph templates ¾ Top-down schema application as graph matching ¾ Elastic graph matching

3. Phase Tagging 4. Dynamic Temporal Matching 5. Lattice Wave Induction July 2006

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2. Graph Matching ¾ Schemas as graph templates 9 graphs provide a general relational format of representation especially appropriate for modeling schemas 9 graphs are “constellations of features”, in which ƒ nodes carry labels → symbolic information ƒ links carry geometrical relationships → analogic information ∨

∠

pixels July 2006

⎛

⎤ ⎝

wall

∧

⎡

lamp

contours Doursat, R. - Pattern Recognition by Wave Matching

table chair

chair corner

floor

symbols/objects 9

2. Graph Matching ¾ Schemas as graph templates 9 information tradeoff between labels and links 9 examples of graphs ƒ objects

ARMCHAIR

ƒ faces ƒ characters ƒ etc.

Institut fuer Neuroinformatik, Bochum July 2006

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2. Graph Matching ¾ Top-down schema application as graph matching 9 expectation: graphs representing the same object category are structurally similar → modeling schemas as deformable templates 9 graph templates can be directly compared by graph matching → establishment of a dynamical link mapping July 2006

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2. Graph Matching ¾ Elastic graph matching ƒ one link per node ƒ minimize distance ƒ minimize label difference

cost = d 2

→ link mapping equivalent to an elastic deformation

Bienenstock and Doursat (1994) A shape-recognition model using dynamical links. July 2006

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Pattern Recognition by Wave-Matching 1. Active Perception 2. Graph Matching 3. Phase Tagging ¾ ¾ ¾ ¾

Temporal coding of graphs Coupled oscillatory units Block synchronization: segmentation Traveling waves: positional information

4. Dynamic Temporal Matching 5. Lattice Wave Induction July 2006

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3. Phase Tagging ¾ Temporal coding of graphs 9 main idea: a schema is a graph, where a graph is a network of coupled temporal units — spiking, excitable, oscillatory, etc. ƒ nodes = timings (phases) ƒ links = timing (phase) differences

after Wang, DeLiang (http://www.cse.ohio-state.edu/~dwang/) July 2006

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3. Phase Tagging ¾ Temporal coding of graphs high activity rate rate coding

high activity rate high activity rate low activity rate low activity rate low activity rate temporal coding

¾ 1 and 2 more in sync than 1 and 3 ¾ 4, 5 and 6 correlated through delays July 2006

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3. Phase Tagging ¾ Coupled oscillatory units wEI

9 example: dual excitatoryinhibitory system 9 this system is a relaxation oscillator, i.e., exhibits discontinuous jumps

wEE

N excitatory neurons

M inhibitory neurons

wIE

9 different from sinusoidal or harmonic oscillations

after Wang, DeLiang (http://www.cse.ohio-state.edu/~dwang/) July 2006

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3. Phase Tagging ¾ Coupled oscillatory units

⇔

9 Van der Pol oscillator

limit cycle attractor after Wang, DeLiang (http://www.cse.ohio-state.edu/~dwang/) July 2006

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3. Phase Tagging ¾ Block synchronization: segmentation 9 a model of segmentation by sync: LEGION (Wang & Tierman)

July 2006

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3. Phase Tagging ¾ Traveling waves: positional information 9 instead of phase plateaus → phase gradients

ϕ

ϕ

π

π

x -π July 2006

x -π

Doursat, R. - Pattern Recognition by Wave Matching

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Pattern Recognition by Wave-Matching 1. Active Perception 2. Graph Matching 3. Phase Tagging 4. Dynamic Temporal Matching ¾ ¾ ¾ ¾

Excitable units & delayed coupling Onset of spatiotemporal patterns (STPs) Phases as coordinates 1-D and 2-D dynamic phase matching

5. Lattice Wave Induction July 2006

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4. Dynamic Temporal Matching ¾ Excitable units 9 a Bonhoeffer-van der Pol (BvP) oscillator has two main regimes: a) sparse, stochastic → excitable b) quasi-periodic → oscillatory (a)

2 1 0 −1.7

(b)

z = −.3 July 2006

z = −.36 Doursat, R. - Pattern Recognition by Wave Matching

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4. Dynamic Temporal Matching ¾ Delayed coupling 9 fully connected net of BvP units 9 i ← j coupling features: ƒ proportional to u-signal difference (only in spiking domain u < 0) ƒ positive connection weight kij ƒ nonzero transmission delay τij

9 delays verify a rule of transitivity: ƒ

9 . . .equivalent to per-node times:

kij [τij]

uj

ui

⇔ July 2006

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4. Dynamic Temporal Matching ¾ Onset of spatiotemporal patterns (STPs) 9 when coupling is turned on, units transition from regime (a) to (b) and exhibit delayed correlations ti − tj in accordance with τij

(a) → (b)

S(t)

C(t)

coupling onset → STP July 2006

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4. Dynamic Temporal Matching ¾ Phases as coordinates 9 individual spike times are taken as coordinates 9 1 STP can code a 1-D pattern → 2 STPs can code a 2-D pattern uj kij ui

July 2006

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4. Dynamic Temporal Matching ¾ Phases as coordinates 9 the simultaneous onset of a pair of STPs is graphically equivalent to the unfolding of a 2-D constellation of dots

9 note: the two STPs can be on two different networks or they can alternate on the same network (see waves on a lattice in 5.) uj kij ui

July 2006

u'j kij u'i

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4. Dynamic Temporal Matching ¾ 1-D dynamic phase matching 9 graph matching implemented as dynamical link matching between two pairs of STPs

+ Wi Wi = ∑ wii' (ui' − ui)

STP 1y

graph 2 nodes i

STP 1x July 2006

graph 2

graph 1 nodes i'

link matrix

STP 2y

graph 1

wii'

Doursat, R. - Pattern Recognition by Wave Matching

STP 2x 26

4. Dynamic Temporal Matching ¾ 1-D dynamic phase matching 9 additional coupling term: 9 where wii' varies according to 1. Hebbian-type synaptic plasticity based on temporal correlations with and 2. competition: renormalize efferent links

wii' → wii' / ∑j wji' 3. label-matching constraint STP 1x July 2006

STP 2x Doursat, R. - Pattern Recognition by Wave Matching

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4. Dynamic Temporal Matching ¾ 1-D dynamic phase matching 9 labels and positions not constraining enough in 1-D: several possible partial matches (local minima) 1→2 dynamical mapping

July 2006

1→2 correlations

Doursat, R. - Pattern Recognition by Wave Matching

1→2 weights

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4. Dynamic Temporal Matching ¾ 2-D dynamic phase matching 9 Hebbian rule in 2-D:

July 2006

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4. Dynamic Temporal Matching ¾ 2-D dynamic phase matching 9 labels and positions more constraining in 2-D: less ambiguities 9 however, to definitely find the best match (global minimum), we regularly drop and raise coupling strength within graph 2 layer ƒ if match is weak, this will perturb STP 2 and undo matching links ƒ if match is strong, this will not perturb STP 2 because it will be sustained by matching links → resonance between links and STPs

S(t)

S(t)

C(t)

C(t)

weak (mis)match → undone by uncoupling

strong match → resistant to uncoupling

July 2006

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Pattern Recognition by Wave-Matching 1. Active Perception 2. Graph Matching 3. Phase Tagging 4. Dynamic Temporal Matching 5. Lattice Wave Induction ¾ Traveling waves ¾ Wave induction ¾ Dynamic wave mapping ¾ Phase matching / elastic matching equivalence July 2006

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5. Lattice Wave Induction ¾ Traveling waves 9 a constellation of dots can be a subset of a larger medium, analogous to a group of buoys on the water surface → STP as a subset of a traveling wave on a lattice 9 instead of fully connected, transitive delays: locally connected, uniform delays

July 2006

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5. Lattice Wave Induction ¾ Wave induction 9 through mapping links, a traveling wave on one layer can induce a wave on the other layer; all directions are possible

July 2006

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5. Lattice Wave Induction ¾ Dynamic wave mapping 9 lattice wave induction is graphically equivalent to the unfolding of a 2-D mesh → elastic matching fashion 9 each graphical point corresponds to a neighborhood average of destination phases, compounded by efferent mapping weights

Schwarz, Andreas (1995), Technical Report (supervised by R. Doursat & L. Wiskott), Institut fuer Neuroinformatik, Bochum July 2006

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5. Lattice Wave Induction ¾ Dynamic wave mapping 9 ex: application to face recognition 9 labels are Gabor-filter “jets”

Schwarz, Andreas (1995), Technical Report (supervised by R. Doursat & L. Wiskott), Institut fuer Neuroinformatik, Bochum July 2006

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5. Lattice Wave Induction ¾ Phase matching / elastic matching equivalence 9 similarity between Kuramoto’s phase equation and elastic matching ƒ Kuramoto: phases attract each other, trying to minimize discrepancy with given delay (generally through sine function)

ƒ elastic matching: link destinations attract each other, trying to minimize discrepancy with given rigid length

⇔ July 2006

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Pattern Recognition by Wave-Matching 1. Active Perception 2. Graph Matching 3. Phase Tagging 4. Dynamic Temporal Matching 5. Lattice Wave Induction

July 2006

Doursat, R. - Pattern Recognition by Wave Matching

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