Tutorial on Optimal Mass Transprot for Computer Vision

Transcription

1 Tutorial on Optimal Mass Transprot for Computer Vision David Gu State University of New York at Stony Brook Graduate Summer School: Computer Vision, IPAM, UCLA July 31, 2013

2 Joint work with Shing-Tung Yau, Feng Luo and Jian Sun

3 Optimal Mass Transportation Problem

4 Motivation Tannenbaum: Medical image registration

5 Motivation: Surface registration

6 Optimal Mass Transportation Problem

7 Optimal Mass Transportation Problem

8 Cost Functions

9 Optimal Mass Transportation Problem

10 Applications

11 Duality and potential functions

12 Duality and potential functions

13 Brenier s Approach

14 Brenier s Approach

15 Brenier s Approach

16 Minkowski problem and several related problems Eg. A convex polygon P in R 2 is determined by its edge lengths A i and unit normal vectors n i.

17 THM (Minkoswki) P exists and is unique up to translations. Minkowski s proof is variational and constructs P.

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19 Q1. What is Minkowski problem for non-compact polyhedra? P.S. Alexandrov: Polyhedron P Discrete optimal transport Pogorelov: Their results: MP solvable for bound faces with unbounded faces fixed. Discrete Monge-Ampere equation

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21 PL convex function W i

22 Alexandrov s proof is not variational and is topological. On page 321 of his book Convex polyhedra, he asked if there exists a variational proof of his thm. He said such a proof is of prime importance by itself.

23 Pogorelov theorem max{x.v j +g j } unbounded faces max{x.p i +h i }, bounded faces

24 Our main result: there exist variational proofs of Alexandrov s and Pogorelov s theorems. Basically the same as Minkowski s original proof. We are motivated by computational problems from computer graphics, discrete optimal transportation and discrete Monge-Ampere equation.

25 Voronoi decomposition and power diagrams Given p 1,, p k in R N, the Voronoi cell V i at p i is: V i ={x x-p i 2 x-p j 2, all j} A generalization: power diagram, given p 1,, p k in R N and weights a 1,,a k in R, the power diagram at p i is W i ={x x-p i 2 +a i x-p j 2 +a j, all j}

26 PL convex function f(x)=max{x. p i +h i } and power diagram x. p i +h i x.p j +h j is the same as x.x -2x.p i +p i.p i -2h i -p i.p i x.x -2x.p j +p j.p j - 2h j -p j.p j, i.e., x i - p i 2-2h i -p i.p i x-p j 2-2h j -p j.p j for all j

27 Proof. Take x in X, say x in X j and also in W i. Then LHS= b j (x) RHS b i (x) b j (x) =LHS.

28 Discrete optimal transport problem (Monge) Given a compact convex domain X In R N and p 1,, p k in R N and A 1,, A k >0, find a transport map T: X {p 1,, p k } with vol(t -1 (p i ))=A i so that T minimizes the cost X x T(x) 2 dx. (Y. Brenier)

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30 Theorem(Aurenhammer- Hoffmann- Aronov, (1998))

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32 Recall

33 Mikowski s proof of his thm Given h=(h 1,, h k ), h i >0, define cpt convex polytope P(h)={x x. n i h i, all i}. Let Vol: R + k R be vol(h)=vol(p(h)). F i The solution h (up to scaling) to MP is the critical point of Vol on { h h i 0, h i A i =1}, using Lagrangian multiplier. Uniqueness part is proved using Brunn-Minkowski inequality which implies (Vol(h)) 1/N is concave in h. So far, this is the ONLY proof of uniqueness.

34 Our Proof. For h =(h 1,, h k ) in R k, define f as above and let W i (h)={ x x. p i +h i x.p j + h j, all j} and w i (h)=vol(w i (h)).

35 This shows the uniqueness part of Alexandrov s thm.

36 We show that the concave function G(h) = F(h) - h i A i has a minimum point in H 0. The min point h is the solution to Alexandrov s them. Exactly the same proof works for Pogorelov s thm. Alexandrov thm corresponds to s(x)=1. Y. Brenier proved a more general form.

37 Discrete Monge-Ampere Eq (DMAE) This is related to Monge s optimal transport problem:

38 Q2: Given A, g how to compute f?

39 Indeed, w(y)=sup{x.y-f(x) x} is the Fenchel-Legendre dual of the solution to Pogorelov s thm: f(x)=max{max{x.p i +h i }, max{x.v j +g j }}. Our result shows that w can be constructed by a finite dim variational principle since dual of PK convex function is computable using linear programming.

40 Algorithm Convex Hull Delaunay Triangulation Vornoi diagram Power Diagram upper envelope Optimal Transportation Map

41 Computational Algorithm

42 Computational Algorithm

43 Computational Algorithm

44 Computational Algorithm

45 Examples

46 Examples

47 Examples

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56 Thank you.