Theory of Optimal Mass Transportation David Gu 1 1 Computer Science - PowerPoint PPT Presentation

Theory of Optimal Mass Transportation David Gu 1 1 Computer Science Department Stony Brook University, USA Center of Mathematical Sciences and Applications Harvard University Geometric Computation and Applications Trinity College, Dublin, Ireland David Gu Applications of OMT

Thanks Thanks for the invitation. David Gu Applications of OMT

Collaborators The work is collaborated with Shing-Tung Yau, Feng Luo, Jian Sun, Na Lei, Li Cui and Kehua Su etc. David Gu Applications of OMT

Motivation David Gu Applications of OMT

Mesh Parameterization David Gu Applications of OMT

Conformal Parameterization Conformal parameterization: angle-preserving Infinitesimal circles are mapped to infinitesimal circle. David Gu Applications of OMT

Area-preserving Parameterization Area-preserving parameterization Infinitesimal circles are mapped to infinitesimal ellipses, preserving the areas. David Gu Applications of OMT

Surface Parameterization Area-preserving parameterization (a) Cortical surface (b) Conformal (c) Area-preserving David Gu Applications of OMT

Surface Parameterization David Gu Applications of OMT

Surface Parameterization (a) Gargoyle model; (b) Angle-preserving; (c) Area-preserving. David Gu Applications of OMT

Volume Parameterization harmonic map volume-preserving map David Gu Applications of OMT

Volume Parameterization harmonic map volume-preserving map David Gu Applications of OMT

Registration David Gu Applications of OMT

Conformal Parameterization for Surface Matching Existing method, 3D surface matching is converted to image matching by using conformal mappings. f φ 1 φ 2 ¯ f David Gu Applications of OMT

Conformal Parameterization for Surface Matching Disadvantages: conformal parameterization may induce exponential area shrinkage, which produces numerical instability and matching mistakes. David Gu Applications of OMT

Optimal Mass Transport Map Advantage: the parameterization is area-preserving, improves the robustness. David Gu Applications of OMT

Registration based on Optimal Mass Transport Map David Gu Applications of OMT

Geometric Clustering David Gu Applications of OMT

Wasserstein Distance Given a metric surface ( S , g ) , a Riemann mapping ϕ : ( S , g ) → D 2 , the conformal factor e 2 λ gives a probability measure on the disk. The shape distance is given by the Wasserstein distance. David Gu Applications of OMT

Expression Classification David Gu Applications of OMT

Expression Classification Compute the Wasserstein distances, embed isometrically using MDS method, perform clustering. David Gu Applications of OMT

From Shape to IQ Can we tell the IQ from the shape of the cortical surface? David Gu Applications of OMT

Optimal Mass Transport Mapping David Gu Applications of OMT

Optimal Transport Problem 휙 (Ω , 휇 ) 푝 휙 ( 푝 ) ( 퐷, 휈 ) Earth movement cost. David Gu Applications of OMT

Optimal Mass Transportation Problem Setting Find the best scheme of transporting one mass distribution ( µ , U ) to another one ( ν , V ) such that the total cost is minimized, where U , V are two bounded domains in R n , such that � � U µ ( x ) dx = V ν ( y ) dy , 0 ≤ µ ∈ L 1 ( U ) and 0 ≤ ν ∈ L 1 ( V ) are density functions. 푓 푓 ( 푥 ) 푥 ( 휇, 푈 ) ( 휈, 푉 ) David Gu Applications of OMT

Optimal Mass Transportation For a transport scheme s ( a mapping from U to V ) s : x ∈ U → y ∈ V , the total cost is � U µ ( x ) c ( x , s ( x )) d x C ( s ) = where c ( x , y ) is the cost function. 푓 푓 ( 푥 ) 푥 ( 휇, 푈 ) ( 휈, 푉 ) David Gu Applications of OMT

Cost Function c ( x , y ) The cost of moving a unit mass from point x to point y . Monge ( 1781 ) : c ( x , y ) = | x − y | . This is the natural cost function. Other cost functions include | x − y | p , p � = 0 c ( x , y ) = c ( x , y ) = − log | x − y | ε + | x − y | 2 , ε > 0 � c ( x , y ) = Any function can be cost function. It can be negative. David Gu Applications of OMT

Optimal Transportation Map Problem Is there an optimal mapping T : U → V such that the total cost C is minimized, C ( T ) = inf { C ( s ) : s ∈ S } where S is the set of all measure preserving mappings, namely s : U → V satisfies � � s − 1 ( E ) µ ( x ) dx = E ν ( y ) dy , ∀ Borel set E ⊂ V David Gu Applications of OMT

Solutions Three categories: Discrete category: both ( µ , U ) and ( ν , V ) are discrete, 1 Semi-continuous category: ( µ , U ) is continuous, ( ν , V ) is 2 discrete, Continuous category: both ( µ , U ) and ( ν , V ) are 3 continuous. David Gu Applications of OMT

Kantorovich’s Approach Both ( µ , U ) and ( ν , V ) are discrete. µ and ν are Dirac measures. ( µ , U ) is represented as { ( µ 1 , p 1 ) , ( µ 2 , p 2 ) , ··· , ( µ m , p m ) } , ( ν , V ) is { ( ν 1 , q 1 ) , ( ν 2 , q 2 ) , ··· , ( ν n , q n ) } . A transportation plan f : { p i } → { q j } , f = { f ij } , f ij means how much mass is moved from ( µ i , p i ) to ( ν j , q j ) , i ≤ m , j ≤ n . The optimal mass transportation plan is: f ij c ( p i , q j ) min f with constraints: n m ∑ ∑ f ij = µ i , f ij = ν j . j = 1 i = 1 Optimizing a linear energy on a convex set, solvable by linear programming method. David Gu Applications of OMT

Kantorovich’s Approach Kantorovich won Nobel’s prize in economics. ( 휇 1 , 푝 1 ) 푓 푖푗 f ∑ f ij c ( p i , p j ) , min ( 휈 1 , 푞 1 ) ( 휇 2 , 푝 2 ) ij such that ( 휈 2 , 푞 2 ) ∑ f ij = µ i , ∑ f ij = ν j . j i mn unknowns in total. The ( 휈 푛 , 푞 푛 ) complexity is quite high. ( 휇 푚 , 푝 푚 ) David Gu Applications of OMT

Brenier’s Approach Theorem (Brenier) If µ , ν > 0 and U is convex, and the cost function is quadratic distance, c ( x , y ) = | x − y | 2 then there exists a convex function f : U → R unique upto a constant, such that the unique optimal transportation map is given by the gradient map T : x → ∇ f ( x ) . David Gu Applications of OMT

Brenier’s Approach Continuous Category: In smooth case, the Brenier potential f : U → R statisfies the Monge-Ampere equation � ∂ 2 f µ ( x ) � det = ν ( ∇ f ( x )) , ∂ x i ∂ x j and ∇ f : U → V minimizes the quadratic cost � U | x − ∇ f ( x ) | 2 d x . min f David Gu Applications of OMT

Semi-Continuous Category Discrete Optimal Mass Transportation Problem T ( p i , A i ) W i Ω Given a compact convex domain U in R n and p 1 , ··· , p k in R n and A 1 , ··· , A k > 0, find a transport map T : U → { p 1 , ··· , p k } with vol ( T − 1 ( p i )) = A i , so that T minimizes the transport cost � U | x − T ( x ) | 2 d x . David Gu Applications of OMT

Power Diagram vs Optimal Transport Map Theorem (Aurenhammer-Hoffmann-Aronov 1998) Given a compact convex domain U in R n and p 1 , ··· , p k in R n and A 1 , ··· , A k > 0 , ∑ i A i = vol ( U ) , there exists a unique power diagram k � U = W i , i = 1 vol ( W i ) = A i , the map T : W i �→ p i minimizes the transport cost � U | x − T ( x ) | 2 d x . David Gu Applications of OMT

Power Diagram vs Optimal Transport Map 푢 ∗ 푢 휋 ∗ 휋 푖 푖 푝푟표푗 ∗ 푝푟표푗 ∇ 푢 푝 푖 푊 푖 Ω , 풯 Ω ∗ , 풯 ∗ David Gu Applications of OMT

Power Diagram vs Optimal Transport Map 1 F. Aurenhammer, F. Hoffmann and B. Aronov, Minkowski-type theorems and least squares partitioning, in Symposium on Computational Geometry, 1992, pp. 350-357. 2 F. Aurenhammer, F. Hoffmann and B. Aronov, Minkowski-type Theorems and Least-Squares Clustering, vol 20, 61-76, Algorithmica, 1998. 3 Bruno L´ evy, “A numerical algorithm for L 2 semi-discrete optimal transport in 3D”, arXiv:1409.1279, Year 2014. 4 X. Gu, F. Luo, J. Sun and S.-T. Yau, “Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations”, arXiv:1302.5472, Year 2013. David Gu Applications of OMT

Power Diagram vs Optimal Transport Map In Aurenhammer et al.’s and Levy’s works, the main theorems are: Theorem Given a set of points P and a set of weights W = ( w i ) , the assignment T P , W defined by the power diagram is an optimal transport map. Theorem Given a measure µ with density, a set of points ( p i ) and prescribed mass ν i such that ∑ ν i = µ (Ω) , there exists a weights vector W such that µ ( Pow W ( p i )) = ν i . David Gu Applications of OMT

Power Diagram vs Optimal Transport Map In Levy’s proof, the following energy is examined: Let T : Ω → P be an arbitrary assignment, � Ω � x − T ( x ) � 2 − w T ( x ) d µ , f T ( W ) := using envelope theorem, ∂ f T W ( W ) = − µ ( Pow W ( p i )) , ∂ w i the convex energy is defined as g ( W ) = f T W ( W )+ ∑ ν i w i , i the gradient is ∂ g ( W ) = − µ ( Pow W ( p i ))+ ν i , ∂ w i David Gu Applications of OMT