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Justin Solomon* Andy Nguyen* Adrian Butscher Mirela Ben-Chen Leonidas Guibas

Stanford University

M1 M2

Á :M1 !M2

M1 M2

M1 M2

Á :M1 !M2

Global Ambiguity

Local Ambiguity

M1 M2

Represent maps probabilistically

Represent maps probabilistically

M1

M2

Likelihood of a matched pair

M1 M2

P12 2 Prob(M1 £M2)

Likelihood of a matched pair

p1

p0

p01

p1

S1

S1

(p0; p0)

(p1; p01)

(p1; p1)

M1

M2U

V

P12(U £V ) 2 [0;1]

Likelihood of a matched pair

M1

M2U

Conditional probability distribution

PU2(B) 2 [0;1]

M1

M2

V

P1V (A) 2 [0;1]

Conditional probability distribution

M1 = U1 [ ¢ ¢ ¢ [UN1M2 = V1 [ ¢ ¢ ¢ [VN2

Voronoi cells

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)

Descriptor matching

Continuity

Sharpness

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

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)

Continuity

Earth Mover’s Distance

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)

Sharpness

Es(A) =X

ij

a2ij = kAk2Fro

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)

Descriptor matching

Continuity

Sharpness

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)

Descriptor matching

Continuity

Sharpness

Linear program!

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)

Sharp transition

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)

Sharp transition

Want this point

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min

=

such that

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min

=

such that

Primal

Emap(A) =EÁ(A) +¸Econt(A) +¯Es(A)min

=

such that

EÁ(A¤)

Primal

Emap(A)=EÁ(A)+¸Econt(A)+¯Es(A)

min

=

suchthat

EÁ(A¤)

Dual

Dual

=

max

such that

EÁ(A¤)

¸

Dual

=

max

such that

EÁ(A¤)

¸¸

Dual

=

min

such that

EÁ(A¤)

¸¸

¸

New primal

=

min

suchthat

EÁ(A¤)

¸

¸

¸

New primal

=

min

suchthat

EÁ(A¤)

¸

¸

¸

Continuous soft maps

Basis for probability measures

Basis for probability measures

Validation improves

Sharpness decreases

Computing denser maps

Applications to graphics and other fields

Map collections and composition

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Questions?