Entropic Regularization of Wasserstein Barycenters
Talks given at the "Optimal Transport Workshop" in Toulouse, updated for a talk at "Workshop on Computational Information Geometry for Image and Signal Processing" in Edinburg.
(X) Style image (Y) Source image after color transfer J. Rabin Wasserstein Regularization Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Sliced Wasserstein projection of X to style image color statistics Y Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Style image (Y) Sliced Wasserstein projection of X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization Colors distribution: each pixel point in R3
Source image (X) Style image (Y) Source image after color transfer J. Rabin Wasserstein Regularization Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Sliced Wasserstein projection of X to style image color statistics Y Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Style image (Y) Sliced Wasserstein projection of X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization Colors distribution: each pixel point in R3 Texture synthesis, segmentation, . . . Classiﬁcation, clustering . . . Surface processing, reﬂectance modeling . . .
p i xi xi 2 Rd Ground cost: c 2 (R +)N⇥N . p-Wasserstein transport: ci,j = || xi xj ||p q p ⇡ C(p, q) def. = ⇡ 2 (R +)N⇥N ; ⇡1 = p, ⇡T 1 = q ⌃N def. = p 2 RN + ; P i pi = 1
Point clouds: N = P, pi = qj = 1/N. W(p, q) = min 2 PermN P i ci, (i) ! Hungarian/auction algorithms, complexity O(N3). µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj ! Linear program interior points (polynomial) transportation simplex µ ⌫
Point clouds: N = P, pi = qj = 1/N. W(p, q) = min 2 PermN P i ci, (i) 1-D and convex cost: ci,j = | xi xj |p , p > 1. ! Hungarian/auction algorithms, complexity O(N3). µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj ! Linear program interior points (polynomial) transportation simplex µ ⌫ sorting the values, O(N log(N)) operations. µ ⌫
Point clouds: N = P, pi = qj = 1/N. W(p, q) = min 2 PermN P i ci, (i) 1-D and convex cost: ci,j = | xi xj |p , p > 1. ! Hungarian/auction algorithms, complexity O(N3). µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj ! Linear program interior points (polynomial) transportation simplex µ ⌫ sorting the values, O(N log(N)) operations. µ ⌫ Need for fast approximate algorithms for generic c .
W 2 (µ 2 ,µ ) W2 (µ3 ,µ ) µ2 µ? 2 argmin µ P k k W2(µk, µ) Barycenters of measures ( µk)k: P k k = 1 For µ = P i p i xi , ⌫ = P j q j yj , W2(µ, ⌫) = W(p, q) for ci,j = || xi yj ||2 W2 def. = Wasserstein distance for measures.
W 2 (µ 2 ,µ ) W2 (µ3 ,µ ) µ2 µ? 2 argmin µ P k k W2(µk, µ) Barycenters of measures ( µk)k: P k k = 1 If µ k = xk then µ? = P k kxk Generalizes Euclidean barycenter: For µ = P i p i xi , ⌫ = P j q j yj , W2(µ, ⌫) = W(p, q) for ci,j = || xi yj ||2 W2 def. = Wasserstein distance for measures.
vanish on small sets, Wasserstein Barycenters [Agueh, Carlier, 2010] µ µ1 µ3 W2 (µ1 , µ ) W 2 (µ 2 ,µ ) W2 (µ3 ,µ ) µ2 µ? 2 argmin µ P k k W2(µk, µ) Barycenters of measures ( µk)k: P k k = 1 If µ k = xk then µ? = P k kxk Generalizes Euclidean barycenter: For µ = P i p i xi , ⌫ = P j q j yj , W2(µ, ⌫) = W(p, q) for ci,j = || xi yj ||2 W2 def. = Wasserstein distance for measures.
Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing
Color Harmonization . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing erstein Barycenter Sliced Wasserstein Barycenter Experimental results Applications Conclusion olor transfer Color harmonization of several images . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; stein Barycenter Sliced Wasserstein Barycenter Experimental results Applications Conclusion lor transfer Color harmonization of several images . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; in Barycenter Sliced Wasserstein Barycenter Experimental results Applications Conclusion or transfer Color harmonization of several images . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter;
for optimal transport. Source image (X) Style image (Y) Source ima J. Rabin Wasserstein Regu Histogram features in imaging and machine learning. ! histograms are now trendy!
for optimal transport. Source image (X) Style image (Y) Source ima J. Rabin Wasserstein Regu Histogram features in imaging and machine learning. ! histograms are now trendy! Barycenters in Wasserstein space: Figure 4: Simulation results with focal random signals generated in are