Curvature Correction of the HamiltonJacobi Skeleton

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Grouping with Asymmetric Affinities A
Game-Theoretic Perspective
Andrea Torsello and Marcello Pelillo
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Grouping problem

• We want to segment edges extracted from an image
  into shape and noise classes.
• Incorporate the Gestalt principles of
  good-continuation and proximity into a pairwise
  compatibility measure, turning the problem into
  one of pairwise clustering.
• Allow non-symmetric compatibilities that emerge
  from a random-walk interpretation (Williams and
  Thornber 2000)
• We take a game-theoretic approach forcing the
  hypotheses that an edge belongs to a shape class
  to compete with one-another.

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Directed compatibility

• Compatibility between i and j is a measure of the
  likelihood that a shape-boundary passes through i
  and j.
• The boundary can be formalized as a (random) walk
  from i to j. A shape boundary must be a closed
  walk.
• If direction of movement is not taken into
  account the likelihood of going from i to j is
  the same as going from j to i.
• If we take into account the direction of
  movement, going from j to i might require a
  longer walk than going from i to j.

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The role of asymmetry

• Without representing directions, the walk process
  cannot remember the direction of travel in the
  previous time-step hence, at the next time-step,
  the walk is equally likely to continue in either
  direction.
• The walks over a symmetric compatibility matrix
  may contain cusps (i.e., reversals in direction).
• Williams and Thornber (2000) proposed to take the
  direction of motion into account yielding an
  asymmetric compatibility.

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Survival of the fittest

• We take a game-theoretic perspective to
  perceptual grouping
• The hypothesis that edge-element i belongs to a
  shape boundary is forced to compete with all
  other edges, receiving
• Support from compatible edges
• Competitive pressure from all other edges.
• The surviving elements are taken to be a part of
  a highly cohesive group.

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Game theory

• Evolutionary game theory considers the scenario
  where individuals are repeatedly drawn at random
  from a population to play a symmetric two-player
  game.
• Players are not supposed to behave rationally,
  but according to a preprogrammed behavior
  pattern, or pure strategy, and some evolutionary
  selection process operates over time on the
  distribution of behaviors.
• Le aij be the payoff that a player using strategy
  i has when encountering strategy j, is the
  average payoff of strategy i if the population
  has strategy distribution x.

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Game theory

• Assume that competitive pressure between
  strategies yields an evolution dynamics
• We say that the dynamics is payoff-monotonic if
• Intuitively, for a payoff-monotonic dynamics the
  strategies associated to higher payoffs will
  increase at higher rate.
• Associating edges to strategies and
  edge-compatibility to payoff, game dynamics model
  the competition between the hypotheses.
• We use replicator dynamics

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Discrete compatibility

• Let A be a compatibility matrix with only 0-1
  elements.
• Let AT be the adjacency matrix of graph G(V,E).
• S?V is a doubly connected clique iff
• Define xS
• Theorem 1 If S is a doubly connected clique not
  fully connected to an external vertex, xS is an
  ESS for (1)
• Theorem 2 All ESS for (1) with payoff AaI, agt½
  are characteristic vectors of doubly connected
  cliques not fully connected to an external vertex.

S
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Discrete compatibility

• What is this condition not fully connected to an
  external vertex?
• Here a,b,c is a doubly connected clique, but
  every vertex is compatible to d. For this reason
  d has a high payoffand will not become extinct.
• In this new context, a, not being linked by d,
  will be compatible to fewer edges than average
  and will eventually become extinct, leaving only
  vertices belonging to the doubly connected clique
  b,c,d.
• The asymmetry can be used to eliminate solutions
  (cliques)

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Continuous compatibility

• What happens if A has continuous entries?
• Dominant sets (Pavan and Pelillo 2003) correspond
  to ESS for (1)
• A set S?V is said to be dominant if
• ws(i)gt0 for all i in S
• ws(i)lt0 for all i not in S
• What is the role of Asymmetry?
• A strong support from a to b without a
  corresponding strong support from b to a forces a
  to extinction. I.e. The walk must be closed.

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Experimental results
Noiseadded
Original
WilliamsThornber
Replicatordynamics
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Experimental results
Noiseadded
Original
WilliamsThornber
Replicatordynamics
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Experimental results
Noiseadded
Original
WilliamsThornber
Replicatordynamics
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Experimental results
Noiseadded
Original
WilliamsThornber
Replicatordynamics
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Noise sensitivity
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Experimental results
Noiseadded
Original
Replicatordynamics
WilliamsThornber
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Conclusions

• We took a game-theoretic approach to grouping to
  deal with asymmetric compatibilities
• Edges are forced to compete and surviving
  elements are taken to be a part of a cohesive
  group
• We provide a theoretical characterization of
  stable distributions both with discrete and
  continuous compatibilities
• Experimental analysis show that the approach is
  competitive and often superior to alternative
  state-of-the-art approaches