justin solomon *andy nguyen adrian butscher mirela ben ...texpoint fonts used in emf. read the...
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Justin Solomon* Andy Nguyen* Adrian Butscher Mirela Ben-Chen Leonidas Guibas
Stanford University
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M1 M2
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Á :M1 !M2
M1 M2
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M1 M2
Á :M1 !M2
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Global Ambiguity
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Local Ambiguity
M1 M2
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Represent maps probabilistically
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Represent maps probabilistically
M1
M2
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Likelihood of a matched pair
M1 M2
P12 2 Prob(M1 £M2)
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Likelihood of a matched pair
p1
p0
p01
p1
S1
S1
(p0; p0)
(p1; p01)
(p1; p1)
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M1
M2U
V
P12(U £V ) 2 [0;1]
Likelihood of a matched pair
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M1
M2U
Conditional probability distribution
PU2(B) 2 [0;1]
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M1
M2
V
P1V (A) 2 [0;1]
Conditional probability distribution
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M1 = U1 [ ¢ ¢ ¢ [UN1M2 = V1 [ ¢ ¢ ¢ [VN2
Voronoi cells
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Descriptor matching
Continuity
Sharpness
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Descriptor matching http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6130444
EÁ(A) =X
ij
aijÁij
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Continuity
Earth Mover’s Distance
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Sharpness
Es(A) =X
ij
a2ij = kAk2Fro
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Descriptor matching
Continuity
Sharpness
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Descriptor matching
Continuity
Sharpness
Linear program!
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Sharp transition
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)
Sharp transition
Want this point
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min
=
such that
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min
=
such that
Primal
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Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min
=
such that
EÁ(A¤)
Primal
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Emap(A)=EÁ(A)+¸Econt(A)+¯Es(A)
min
=
suchthat
EÁ(A¤)
Dual
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Dual
=
max
such that
EÁ(A¤)
¸
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Dual
=
max
such that
EÁ(A¤)
¸¸
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Dual
=
min
such that
EÁ(A¤)
¸¸
¸
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New primal
=
min
suchthat
EÁ(A¤)
¸
¸
¸
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New primal
=
min
suchthat
EÁ(A¤)
¸
¸
¸
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Continuous soft maps
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Basis for probability measures
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Basis for probability measures
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Validation improves
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Sharpness decreases
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Computing denser maps
Applications to graphics and other fields
Map collections and composition
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TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA
Questions?