MAP Problems Probabilistic Graphical Models Inference Correspondence data association Find highest scoring matching maximize ij ij X ij subject to mutual exclusion constraint ID: 424700
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Slide1
MAP
Tractable
MAP Problems
ProbabilisticGraphicalModels
InferenceSlide2
Correspondence /data associationFind highest scoring matchingmaximize ij
ij X
ijsubject to mutual exclusion constraintEasily solved using matching algorithmsMany applicationsmatching sensor readings to objectsmatching features in two related imagesmatching mentions in text to entities
X
ij
=
1 if
i
matched to j
0 otherwise
ij
= quality of “match”
between
i
and jSlide3
Snavely
, Seitz, SzeliskiSlide4
Word Alignment for Translation
WikipediaSlide5
Associative potentialsArbitrary network over binary variables using only singleton i and
supermodular pairwise potentials ij
Exact solution using algorithms for finding minimum cuts in graphs Many related variants admit efficient exact or approximate solutionsMetric MRFs
010
a
b1c
dSlide6
Example: Depth Reconstruction
running time
negative score
Scharstein & SzeliskiSlide7
Cardinality FactorsA factor over arbitrarily many binary variables X1, …,
Xk Score(
X1, …,Xk) = f(iXi)Example applications:soft parity constraintsprior on # pixels in a given categoryprior on # of instances assigned to a given cluster
A
BCD
score0000
0
0
01
0010001101000
1
0
1
0
1
1
0
0
1
1
1
1
0
0
0
1
0
0
1
1
0
1
0
1
011
11
00
110
1
11
10
111
1Slide8
Sparse Pattern FactorsA factor over variables X1,…,
Xk Score(
X1, …,Xk) specified for some small # of assignments x1,…,xk Constant for all other assignmentsExamples: give higher score to combinations that occur in real dataIn spelling, letter combinations that occur in dictionary
55 image patches that appear in natural imagesA
BC
Dscore0000
0
0
0
10010001101000
1
0
1
0
1
1
0
0
1
1
1
1
0
0
0
1
0
0
1
1
0
1
0
101
11
100
11
01
1
110
111
1Slide9
Convexity FactorsOrdered binary variables X1,…,X
k Convexity constraints
Examples: Convexity of “parts” in image segmentationContiguity of word labeling in textTemporal contiguity of subactivitiesSlide10
SummaryMany specialized models admit tractable MAP solutionMany do not have tractable algorithms for computing marginalsThese specialized models are usefulOn their own
As a component in a larger model with other types of factorsSlide11
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