Optimal mass transport as a distance measure between images Axel - PowerPoint PPT Presentation

Optimal mass transport as a distance measure between images Axel Ringh 1 1 Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden. 21 st of June 2018 INRIA, Sophia-Antipolis, France

Acknowledgements This is based on joint work with Johan Karlsson 1 . [1] J. Karlsson, and A. Ringh. Generalized Sinkhorn iterations for regularizing inverse problems using optimal mass transport. SIAM Journal on Imaging Sciences , 10(4), 1935-1962, 2017. I acknowledge financial support from Swedish Research Council (VR) Swedish Foundation for Strategic Research (SSF) Code: https://github.com/aringh/Generalized-Sinkhorn-and-tomography The code is based on ODL: https://github.com/odlgroup/odl 1 Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden 2 / 24

Outline Background Inverse problems Optimal mass transport Sinkhorn iterations - solving discretized optimal transport problems Sinkhorn iterations as dual coordinate ascent Inverse problems with optimal mass transport priors Example in computerized tomography 3 / 24

Background Inverse problems Consider the problem of recovering f ∈ X from data g ∈ Y , given by g = A ( f ) + ’noise’ Notation: X is called the reconstruction space. Y is called the data space. A : X → Y is the forward operator. A ∗ : Y → X denotes the adjoint operator 4 / 24

Background Inverse problems Consider the problem of recovering f ∈ X from data g ∈ Y , given by g = A ( f ) + ’noise’ Notation: X is called the reconstruction space. Y is called the data space. A : X → Y is the forward operator. A ∗ : Y → X denotes the adjoint operator Problems of interest are ill-posed inverse problems: a solution might not exist, the solution might not be unique, the solution does not depend continuously on data. Simply put: A − 1 does not exist as a continuous bijection! Comes down to: find approximate inverse A † so that ⇒ A † ( g ) ≈ f . g = A ( f ) + ’noise’ = 4 / 24

Background Variational regularization A common technique to solve ill-posed inverse problems is to use variational regularization: arg min G ( A ( f ) , g ) + λ F ( f ) f ∈ X G : Y × Y → R , data discrepancy functional. F : X → R , regularization functional. λ is the regularization parameter. Controls trade-off between data matching and regularization. Common example in imaging is total variation regularization: G ( h , g ) = � h − g � 2 2 , F ( f ) = �∇ f � 1 . If A is linear this is a convex problem! 5 / 24

Background Incorporating prior information in variational schemes How can one incorporate prior information in such a scheme? 6 / 24

Background Incorporating prior information in variational schemes How can one incorporate prior information in such a scheme? One way: consider G ( A ( f ) , g ) + λ F ( f ) + γ H (˜ arg min f , f ) f ∈ X ˜ f is prior/template H defines “closeness” to ˜ f . What is a good choice for H ? 6 / 24

Background Incorporating prior information in variational schemes How can one incorporate prior information in such a scheme? One way: consider G ( A ( f ) , g ) + λ F ( f ) + γ H (˜ arg min f , f ) f ∈ X ˜ f is prior/template H defines “closeness” to ˜ f . What is a good choice for H ? Scenarios where potentially of interest. incomplete measurements, e.g. limited angle tomography. spatiotemporal imaging: data is a time-series of data sets: { g t } T t =0 . For each set, the underlying image has undergone a deformation. each data set g t normally “contains less information”: A † ( g t ) is a poor reconstruction. Approach: solve coupled inverse problems T T � � � � arg min G ( A ( f j ) , g j ) + λ F ( f j ) + γ H ( f j − 1 , f j ) f 0 ,..., f T ∈ X j =0 j =1 6 / 24

Background Measuring distances between functions: the L p metrics Given two functions f 0 ( x ) and f 1 ( x ), what is a suitable way to measure the distance between the two? 7 / 24

Background Measuring distances between functions: the L p metrics Given two functions f 0 ( x ) and f 1 ( x ), what is a suitable way to measure the distance between the two? One suggestion: measure it pointwise, e.g., using an L p metric � 1 / p �� | f 0 ( x ) − f 1 ( x ) | p dx � f 0 − f 1 � p = . D 7 / 24

Background Measuring distances between functions: the L p metrics Given two functions f 0 ( x ) and f 1 ( x ), what is a suitable way to measure the distance between the two? One suggestion: measure it pointwise, e.g., using an L p metric � 1 / p �� | f 0 ( x ) − f 1 ( x ) | p dx � f 0 − f 1 � p = . D Draw-backs: for example unsensitive to shifts. Example: It gives the same distance from f 0 to f 1 and f 2 : � f 0 − f 1 � 1 = � f 0 − f 2 � 1 = 8. 7 / 24

Background Optimal mass transport - Monge formulation Gaspard Monge: formulated optimal mass transport 1781. Optimal transport of soil for construction of forts and roads. Gaspard Monge 8 / 24

Background Optimal mass transport - Monge formulation Gaspard Monge: formulated optimal mass transport 1781. Optimal transport of soil for construction of forts and roads. Gaspard Monge 8 / 24

Background Optimal mass transport - Monge formulation Gaspard Monge: formulated optimal mass transport 1781. Optimal transport of soil for construction of forts and roads. Gaspard Monge 8 / 24

Background Optimal mass transport - Monge formulation Gaspard Monge: formulated optimal mass transport 1781. Optimal transport of soil for construction of forts and roads. Gaspard Monge Let c ( x 0 , x 1 ) : X × X → R + describes the cost for transporting a unit mass from location x 0 to x 1 . 3 3 2.5 2.5 Given two functions f 0 , f 1 : X → R + , find the function φ : X → X 2 2 minimizing the transport cost 1.5 1.5 � 1 1 c ( x , φ ( x )) f 0 ( x ) dx 0.5 0.5 X 0 0 0 5 10 0 0.5 1 1.5 2 2.5 3 10 8 6 4 2 0 0 0.5 1 1.5 2 2.5 3 8 / 24

Background Optimal mass transport - Monge formulation Gaspard Monge: formulated optimal mass transport 1781. Optimal transport of soil for construction of forts and roads. Gaspard Monge Let c ( x 0 , x 1 ) : X × X → R + describes the cost for transporting a unit mass from location x 0 to x 1 . 3 3 2.5 2.5 Given two functions f 0 , f 1 : X → R + , find the function φ : X → X 2 2 minimizing the transport cost 1.5 1.5 � 1 1 c ( x , φ ( x )) f 0 ( x ) dx 0.5 0.5 X 0 0 0 5 10 0 0.5 1 1.5 2 2.5 3 where φ is mass preserving map from f 0 to f 1 : 10 8 6 � � 4 f 1 ( x ) dx = f 0 ( x ) dx for all A ⊂ X . 2 0 0 0.5 1 1.5 2 2.5 3 φ ( x ) ∈ A x ∈ A 8 / 24

Background Optimal mass transport - Monge formulation Gaspard Monge: formulated optimal mass transport 1781. Optimal transport of soil for construction of forts and roads. Gaspard Monge Let c ( x 0 , x 1 ) : X × X → R + describes the cost for transporting a unit mass from location x 0 to x 1 . 3 3 2.5 2.5 Given two functions f 0 , f 1 : X → R + , find the function φ : X → X 2 2 minimizing the transport cost Nonconvex problem! 1.5 1.5 � 1 1 c ( x , φ ( x )) f 0 ( x ) dx 0.5 0.5 X 0 0 0 5 10 0 0.5 1 1.5 2 2.5 3 where φ is mass preserving map from f 0 to f 1 : 10 8 6 � � 4 f 1 ( x ) dx = f 0 ( x ) dx for all A ⊂ X . 2 0 0 0.5 1 1.5 2 2.5 3 φ ( x ) ∈ A x ∈ A 8 / 24

Background Optimal mass transport - Kantorovich formulation Leonid Kantorovich: convex formulation and duality theory 1942. Nobel Memorial Prize 1975 in Economics for “contributions to the theory of optimum allocation of resources.” Leonid Kantorovich 9 / 24

Background Optimal mass transport - Kantorovich formulation Leonid Kantorovich: convex formulation and duality theory 1942. Nobel Memorial Prize 1975 in Economics for “contributions to the theory of optimum allocation of resources.” Again, let c ( x 0 , x 1 ) denote the cost of transporting a unit mass from Leonid Kantorovich the point x 0 to the point x 1 . 3 3 Given two functions f 0 , f 1 : X → R + , find a transport plan M : 2.5 2.5 X × X → R + , where M ( x 0 , x 1 ) is the amount of mass moved between 2 2 x 0 to x 1 . 1.5 1.5 1 1 0.5 0.5 0 0 0 5 10 0 0.5 1 1.5 2 2.5 3 10 8 6 4 2 0 0 0.5 1 1.5 2 2.5 3 9 / 24

Background Optimal mass transport - Kantorovich formulation Leonid Kantorovich: convex formulation and duality theory 1942. Nobel Memorial Prize 1975 in Economics for “contributions to the theory of optimum allocation of resources.” Again, let c ( x 0 , x 1 ) denote the cost of transporting a unit mass from Leonid Kantorovich the point x 0 to the point x 1 . Transportation plan (M) 3 Given two functions f 0 , f 1 : X → R + , find a transport plan M : 2.5 X × X → R + , where M ( x 0 , x 1 ) is the amount of mass moved between 2 x 0 to x 1 . 1.5 1 0.5 0 0 5 10 10 8 6 4 2 0 0 0.5 1 1.5 2 2.5 3 9 / 24