Fast%20%20Primal-Dual%20%20Strategies%20%20for%20%20MRF%20%20Optimization

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Fast Primal-Dual Strategies for MRF
Optimization (Fast PD)
Robot Perception Lab  Taha Hamedani    
Aug 2014
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Overview

• A new efficient MRF optimization algorithm
• generalizes a-expansion
• at least 3-9 times faster than a-expansion
• used for boosting the performance of dynamic
  MRFs, i.e. MRFs varying over time
• guarantee an almost optimal solution for a much
  wider class of NP-hard MRF problems

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Energy Function

• weighted graph G (with nodes V , edges E and
  weights wpq), one seeks to assign a label xp
  (from a discrete set of labels L) to each p ? V ,
  so that the following cost is minimized
• p(), d(, ) determine the singleton and
  pairwise MRF potential functions

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Primal-dual MRF optimization algorithms

• Theorem 1 (Primal-Dual schema). Keep generating
  pairs of integral-primal, dual solutions (xk,
  yk), until the elements of the last pair, (say x,
  y), are both feasible and have costs that are
  close enough, e.g. their ratio is f app
• Then x is guaranteed to be an fapp-approximate
  solution to the optimal integral solution x,
  i.e. cTx fapp cTx.

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The primal dual schema
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Fast primal-dual MRF optimization

• In the above formulation, ? ?pp?V,
  ?pqpq?E represents a vector of MRF-parameters
  that consists of all unary ?p ?p() and
  pairwise ?pq ?pq(, )
• x xpp?V, xpqpq?E denotes a vector of
  binary MRF-variables consisting of all unary
  subvectors xp xp() and pairwise subvectors
  xpq xpq(, ). (0,1 variables)
• i.e, they satisfy xp(l) 1 ? label l is assigned
  to p,
• while xpq(l, l') 1 ? labels l, l' are assigned
  to p, q

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MRF constraints

• (first constraint) simply express the fact that a
  unique label must be assigned to each node p
• (second constraint) since they ensure that if
  xp(l) xq(l') 1, then xpq(l, l') 1 as well
• (marginal polytope)

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• local marginal polytope
• connected with the linear programming (LP)
  relaxation, which is formed by replacing the
  integer constraints xp(), xpq(, ) ? 0, 1
  with the relaxed constraints
• xp(), xpq(, ) 0

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• The original (possibly difficult) optimization
  problem decomposes into easier sub problems
  (called the slaves) that are coordinated by a
  master problem via message exchanging

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• decompose the original MRF optimization problem,
  which is NP-hard (since it is defined on a
  general graph G )
• decompose into a set of easier MRF sub problems,
  each one defined on a tree T ? G .
• Needed to transform our problem into a more
  appropriate form by introducing a set of
  auxiliary variables.
• let T (G ) be a set of sub trees of graph G
  (cover at least one node and edge of graph G)

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• For each tree T ? T (G ) we will then imagine
  that there is a smaller MRF defined just on the
  nodes and edges of tree T
• We will associate to it a vector of
  MRF-parameters ?T.
• as well as a vector of MRF-variables xT (these
  have similar form to vectors ? and x of the
  original MRF, except that they are smaller in
  size) (Decomposition)

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Redundancy

• MRF-variables contained in vector xT will be
  redundant
• initially assume that they are all equal to the
  corresponding MRF-variables in vector x, i.e it
  will hold xT xT
• xT represents the sub vector of x containing
  MRF-variables only for nodes and edges of tree T

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• all the vectors ?T will be defined so that they
  satisfy the following conditions
• Here, T (p) and T (pq) denote the set of all
  trees of T (G ) that contain node p and edge pq
  respectively.

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Energy Decomposition

• The first constraints can reduced by
• MRF problem can decomposed as

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• It is clear that without constraints xT xT ,
  this problem would decouple into a series of
  smaller MRF problems (one per tree T )
• Lagrangian dual form
• Eliminate vector x by minimizing over it

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• The resulting lagrangian dual form is simplified
  as
• Dual from by maximizing over feasible set
• Master
• Slave

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• According to Lemma 1
• ?T must first be updated as
• Sub gradient of gt is equal to optimal solution
  of slave problem

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Fast PD procedure
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References

• 1 Komodakis, N. Paragios, N. Tziritas, G.,
  "MRF Energy Minimization and Beyond via Dual
  Decomposition," Pattern Analysis and Machine
  Intelligence, IEEE Transactions on , vol.33,
  no.3, pp.531,552, March 2011.
• 2 Chaohui Wang, Nikos Komodakis, Nikos
  Paragios, Markov Random Field modeling, inference
  amp learning in computer vision amp image
  understanding A survey, Computer Vision and
  Image Understanding, Volume 117, Issue 11,
  November 2013, Pages 1610-1627, ISSN 1077-3142.

Thank You ?