fast, approximately optimal solutions for single and dynamic markov random fields (mrfs)
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CVPR, June 2007
Fast, Approximately Optimal Solutions
for Single and Dynamic Markov Random Fields (MRFs)
Nikos Komodakis (Ecole Centrale Paris)Georgios Tziritas (University of Crete)Nikos Paragios (Ecole Centrale Paris)
The MRF optimization problem vertices G = set of objects
edges E = object relationships
set L = discrete set of labels
Vp(xp) = cost of assigning label xp to vertex p(also called single node potential)
Vpq(xp,xq) = cost of assigning labels (xp,xq) to neighboring vertices (p,q) (also called pairwise potential)
Find labels that minimize the MRF energy (i.e.,
the sum of all potentials):
MRF optimization in vision MRFs ubiquitous in vision and beyond
Have been used in a wide range of problems:segmentation stereo matchingoptical flow image restorationimage completion object detection & localization...
MRF optimization is thus a task of fundamental importanceMRF optimization is thus a task of fundamental importance
Yet, highly non-trivial, since almost all interesting MRFs are actually NP-hard to optimize
Many proposed algorithms (e.g., [Boykov,Veksler,Zabih], [Kolmogorov], [Kohli,Torr], [Wainwright]…)
MRF hardness
MRF pairwise potential
MRF hardness
linear
exact global optimum
arbitrary
local optimum
metric
global optimum approximation
Move left in the horizontal axis,
But we want to be able to do that efficiently, i.e. fast
and remain low in the vertical axis (i.e., still be able to provide approximately optimal solutions)
Our contributions to MRF optimization
Can handle a very wide class of MRFs
General framework for optimizing MRFs based on dualitytheory of Linear Programming (the Primal-Dual schema)
Can guarantee approximately optimal solutions(worst-case theoretical guarantees)
Can provide tight certificates of optimality per-instance(per-instance guarantees)
Fast-PD
Provides significant speed-up for static MRFs
Provides significant speed-up for dynamic MRFs
Presentation outline
The primal-dual schema
Applying the schema to MRF optimization
Algorithmic properties Worst-case optimality guarantees Per-instance optimality guarantees Computational efficiency for static MRFs Computational efficiency for dynamic MRFs
The primal-dual schema Highly successful technique for exact algorithms.
Yielded exact algorithms for cornerstone combinatorial problems:
matching network flow minimum spanning tree minimum branching
shortest path ...
Soon realized that it’s also an extremely powerful tool for deriving approximation algorithms:
set cover steiner treesteiner network feedback vertex setscheduling ...
The primal-dual schema Say we seek an optimal solution x* to the following
integer program (this is our primal problem):
(NP-hard problem)
To find an approximate solution, we first relax the integrality constraints to get a primal & a dual linear program:
primal LP: dual LP:
The primal-dual schema Goal: find integral-primal solution x, feasible dual solution y such that their primal-
dual costs are “close enough”, e.g.,
Tb y Tc x
primal cost of solution x
primal cost of solution x
dual cost of solution y
dual cost of solution y
*Tc x
cost of optimal integral solution x*
cost of optimal integral solution x*
*fT
T
c x
b y
**f
T
T
c x
c x
Then x is an f*-approximation to optimal solution x*
Then x is an f*-approximation to optimal solution x*
The primal-dual schema
1Tb y 1Tc x
sequence of dual costssequence of dual costs sequence of primal costssequence of primal costs
2Tb y … kTb y*Tc x unknown optimumunknown optimum
2Tc x…kTc x
k*
kf
T
T
c x
b y
The primal-dual schema works iteratively
Global effects, through local improvements!
Instead of working directly with costs (usually not easy), use RELAXED complementary slackness conditions (easier)
Different relaxations of complementary slackness Different approximation algorithms!!!
(only one label assigned per vertex)
enforce consistency between variables xp,a, xq,b and variable xpq,ab
The primal-dual schema for MRFs
Binaryvariables
xp,a=1 label a is assigned to node p
xpq,ab=1 labels a, b are assigned to nodes p, q
xp,a=1 label a is assigned to node p
xpq,ab=1 labels a, b are assigned to nodes p, q
The primal-dual schema for MRFs During the PD schema for MRFs, it turns out that:
each update of primal and dual
variables
each update of primal and dual
variables
solving max-flow in appropriately
constructed graph
solving max-flow in appropriately
constructed graph
Max-flow graph defined from current primal-dual pair (xk,yk) (xk,yk) defines connectivity of max-flow graph (xk,yk) defines capacities of max-flow graph
Max-flow graph is thus continuously updated
Resulting flows tell us how to update both: the dual variables, as well as the primal variables
for each iteration of primal-dual schema
The primal-dual schema for MRFs Very general framework. Different PD-algorithms by RELAXING complementary slackness conditions differently.
Theorem: All derived PD-algorithms shown to satisfy certain relaxed complementary slackness conditions
Worst-case optimality properties are thus guaranteed
E.g., simply by using a particular relaxation of complementary slackness conditions (and assuming Vpq(·,·) is a metric) THEN resulting algorithm shown equivalent to a-expansion!
PD-algorithms for non-metric potentials Vpq(·,·) as well
Per-instance optimality guarantees Primal-dual algorithms can always tell you (for free)
how well they performed for a particular instance
2Tb y *Tc x
unknown optimumunknown optimum
2Tc x1Tb y 1Tc x… kTb y …kTc x
T
T
c x
b y
2
2 2r
per-instance approx. factorper-instance approx. factor
per-instance lower bound (per-instance certificate)
per-instance lower bound (per-instance certificate)
Computational efficiency (static MRFs) MRF algorithm only in the primal domain (e.g., a-expansion)
primalk primalk-
1
primal1…
primal costs
dual1
fixed dual costgapk
STILL BIG Many augmenting paths per max-flowMany augmenting paths per max-flow
Theorem: primal-dual gap = upper-bound on #augmenting paths(i.e., primal-dual gap indicative of time per max-flow)
Theorem: primal-dual gap = upper-bound on #augmenting paths(i.e., primal-dual gap indicative of time per max-flow)
dualkdual1 dualk-1
…
dual costs gap
kprimalk primalk-
1
primal1…
primal costs
SMALL Few augmenting paths per max-flowFew augmenting paths per max-flow
MRF algorithm in the primal-dual domain (Fast-PD)
Computational efficiency (static MRFs)
dramatic decreasedramatic decrease
always very highalways very high
Incremental construction of max-flow graphs(recall that max-flow graph changes per iteration)
This is possible only because we keep both primal and dual informationThis is possible only because we keep both primal and dual information
Our framework provides a principled way of doing this incremental graph construction for general MRFs
noisy image
denoised image
Computational efficiency (static MRFs)penguin Tsukub
aSRI-tree
almost constantalmost constant
dramatic decreasedramatic decrease
Computational efficiency (dynamic MRFs) Fast-PD can speed up dynamic MRFs [Kohli,Torr] as well
(demonstrates the power and generality of our framework)
gap
primalxdualy
SMALL
primalx
gap
dualy
SMALLfew path augmentationsfew path augmentations
primalxSMALL
gap
dual1fixed dual cost
primalx
gap LARGEmany path augmentationsmany path augmentations
Our framework provides principled (and simple) way to update dual variables when switching between different MRFs
Fast-PD algorithm
primal-basedalgorithm
Computational efficiency (dynamic MRFs) Essentially, Fast-PD works along 2 different “axes” reduces augmentations
across different iterations of the same MRF
reduces augmentations across different MRFs
Handles general (multi-label) dynamic MRFs
Time per frame for SRI-tree stereo sequence
primal-dual framework
primal-dual framework
Handles wide class of MRFsHandles wide class of MRFs
Approximatelyoptimal solutions
Approximatelyoptimal solutions
Theoretical guarantees AND tight certificates
per instance
Theoretical guarantees AND tight certificates
per instance
Significant speed-up
for static MRFs
Significant speed-up
for static MRFs
Significant speed-up
for dynamic MRFs
Significant speed-up
for dynamic MRFs
- New theorems- New insights into
existing techniques- New view on MRFs
- New theorems- New insights into
existing techniques- New view on MRFs
Take-home message:try to take advantage of duality, whenever you can
Take-home message:try to take advantage of duality, whenever you can
Thankyou!
Thankyou!
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