Fast Primal-Dual Strategies for MRF Optimization
(Fast PD)
Robot Perception Lab
Taha Hamedani
Aug 2014
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Overview
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A new efficient MRF optimization algorithm
generalizes α-expansion
at least 3-9 times faster than α-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, θ ={ {θp}p∈V, {θpq}pq∈E} represents
a vector of MRF-parameters that consists of all unary θp =
{θp(·)} and pairwise θpq = {θpq(·, ·)}
• x ={{xp}p∈V, {xpq}pq∈E} denotes a vector of binary MRFvariables 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 MRFparameters θT.
• as well as a vector of MRF-variables xT
(these have similar form to vectors θ 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= x|T
• x|T represents the sub vector of x
containing MRF-variables only for nodes
and edges of tree T
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• all the vectors {θT} 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= x|T , 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 :
• λT 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 & learning in computer vision & image
understanding: A survey, Computer Vision and Image Understanding,
Volume 117, Issue 11, November 2013, Pages 1610-1627, ISSN 1077-3142.
Thank You
?
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