Wasserstein convolutional barycenter

Author: ncourty

README.md

@@ -227,4 +227,6 @@ You can also post bug reports and feature requests in Github issues. Make sure t
 
 [19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A.& Blondel, M. [Large-scale Optimal Transport and Mapping Estimation](https://arxiv.org/pdf/1711.02283.pdf). International Conference on Learning Representation (2018)
 
-[20] Cuturi, M. and Doucet, A. (2014) [Fast Computation of Wasserstein Barycenters](http://proceedings.mlr.press/v32/cuturi14.html). International Conference in Machine Learning
+[20] Cuturi, M. and Doucet, A. (2014) [Fast Computation of Wasserstein Barycenters](http://proceedings.mlr.press/v32/cuturi14.html). International Conference in Machine Learning
+
+[21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A. & Guibas, L. (2015). [Convolutional wasserstein distances: Efficient optimal transportation on geometric domains](https://dl.acm.org/citation.cfm?id=2766963). ACM Transactions on Graphics (TOG), 34(4), 66.

data/duck.png

@@ -0,0 +1,92 @@
+    
+#%%
+# -*- coding: utf-8 -*-
+"""
+============================================
+Convolutional Wasserstein Barycenter example
+============================================
+
+This example is designed to illustrate how the Convolutional Wasserstein Barycenter
+function of POT works.
+"""
+
+# Author: Nicolas Courty <ncourty@irisa.fr>
+#
+# License: MIT License
+
+
+import numpy as np
+import pylab as pl
+import ot
+
+##############################################################################
+# Data preparation
+# ----------------
+#
+# The four distributions are constructed from 4 simple images
+
+
+f1 = 1 - pl.imread('../data/redcross.png')[:, :, 2]
+f2 = 1 - pl.imread('../data/duck.png')[:, :, 2]
+f3 = 1 - pl.imread('../data/heart.png')[:, :, 2]
+f4 = 1 - pl.imread('../data/tooth.png')[:, :, 2]
+
+A = []
+f1=f1/np.sum(f1)
+f2=f2/np.sum(f2)
+f3=f3/np.sum(f3)
+f4=f4/np.sum(f4)
+A.append(f1)
+A.append(f2)
+A.append(f3)
+A.append(f4)
+A=np.array(A)
+
+nb_images = 5
+
+# those are the four corners coordinates that will be interpolated by bilinear
+# interpolation
+v1=np.array((1,0,0,0))
+v2=np.array((0,1,0,0))
+v3=np.array((0,0,1,0))
+v4=np.array((0,0,0,1))
+
+
+##############################################################################
+# Barycenter computation and visualization
+# ----------------------------------------
+#
+
+pl.figure(figsize=(10,10))
+pl.title('Convolutional Wasserstein Barycenters in POT')
+cm='Blues'
+# regularization parameter
+reg=0.004
+for i in range(nb_images):
+    for j in range(nb_images):
+        pl.subplot(nb_images,nb_images,i*nb_images+j+1)
+        tx=float(i)/(nb_images-1)
+        ty=float(j)/(nb_images-1)
+        
+        # weights are constructed by bilinear interpolation
+        tmp1=(1-tx)*v1+tx*v2
+        tmp2=(1-tx)*v3+tx*v4
+        weights=(1-ty)*tmp1+ty*tmp2
+        
+        if i==0 and j==0:
+           pl.imshow(f1,cmap=cm)
+           pl.axis('off')
+        elif i==0 and j==(nb_images-1):
+           pl.imshow(f3,cmap=cm)
+           pl.axis('off')
+        elif i==(nb_images-1) and j==0:
+           pl.imshow(f2,cmap=cm)
+           pl.axis('off')
+        elif i==(nb_images-1) and j==(nb_images-1):
+           pl.imshow(f4,cmap=cm)
+           pl.axis('off')
+        else:
+            # call to barycenter computation
+            pl.imshow(ot.convolutional_barycenter2d(A,reg,weights),cmap=cm)
+            pl.axis('off')
+pl.show()

data/heart.png

@@ -918,6 +918,112 @@ def barycenter(A, M, reg, weights=None, numItermax=1000,
     else:
         return geometricBar(weights, UKv)
 
+def convolutional_barycenter2d(A,reg,weights=None,numItermax = 10000, stopThr=1e-9, verbose=False, log=False):
+    """Compute the entropic regularized wasserstein barycenter of distributions A
+       where A is a collection of 2D images. 
+
+     The function solves the following optimization problem:
+
+    .. math::
+       \mathbf{a} = arg\min_\mathbf{a} \sum_i W_{reg}(\mathbf{a},\mathbf{a}_i)
+
+    where :
+
+    - :math:`W_{reg}(\cdot,\cdot)` is the entropic regularized Wasserstein distance (see ot.bregman.sinkhorn)
+    - :math:`\mathbf{a}_i` are training distributions (2D images) in the mast two dimensions of matrix :math:`\mathbf{A}`
+    - reg is the regularization strength scalar value
+
+    The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [21]_
+
+    Parameters
+    ----------
+    A : np.ndarray (n,w,h)
+        n distributions (2D images) of size w x h
+    reg : float
+        Regularization term >0
+    weights : np.ndarray (n,)
+        Weights of each image on the simplex (barycentric coodinates)
+    numItermax : int, optional
+        Max number of iterations
+    stopThr : float, optional
+        Stop threshol on error (>0)
+    verbose : bool, optional
+        Print information along iterations
+    log : bool, optional
+        record log if True
+
+
+    Returns
+    -------
+    a : (w,h) ndarray
+        2D Wasserstein barycenter
+    log : dict
+        log dictionary return only if log==True in parameters
+
+
+    References
+    ----------
+
+    .. [21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A. & Guibas, L. (2015). 
+        Convolutional wasserstein distances: Efficient optimal transportation on geometric domains
+        ACM Transactions on Graphics (TOG), 34(4), 66 
+
+
+    """
+
+    if weights is None:
+        weights = np.ones(A.shape[0]) / A.shape[0]
+    else:
+        assert(len(weights) == A.shape[0])
+
+    if log:
+        log = {'err': []}
+
+    b=np.zeros_like(A[0,:,:])
+    U=np.ones_like(A)
+    KV=np.ones_like(A)
+    threshold = 1e-30 # in order to avoids numerical precision issues
+
+    cpt = 0
+    err=1
+    
+    # build the convolution operator 
+    t = np.linspace(0,1,A.shape[1])
+    [Y,X] = np.meshgrid(t,t)
+    xi1 = np.exp(-(X-Y)**2/reg)
+    K = lambda x: np.dot(np.dot(xi1,x),xi1)
+
+    while (err>stopThr and cpt<numItermax):
+        
+        bold=b
+        cpt = cpt +1
+        
+        b=np.zeros_like(A[0,:,:])
+        for r in range(A.shape[0]):
+            KV[r,:,:]=K(A[r,:,:]/np.maximum(threshold,K(U[r,:,:])))
+            b += weights[r] * np.log(np.maximum(threshold, U[r,:,:]*KV[r,:,:]))
+        b = np.exp(b)
+        for r in range(A.shape[0]):
+            U[r,:,:]=b/np.maximum(threshold,KV[r,:,:])
+    
+        if cpt%10==1:
+            err=np.sum(np.abs(bold-b))
+            # log and verbose print
+            if log:
+                log['err'].append(err)
+
+            if verbose:
+                if cpt%200 ==0:
+                    print('{:5s}|{:12s}'.format('It.','Err')+'\n'+'-'*19)
+                print('{:5d}|{:8e}|'.format(cpt,err))
+
+    if log:
+        log['niter']=cpt
+        log['U']=U
+        return b,log
+    else:
+        return b   
+    
 
 def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000,
           stopThr=1e-3, verbose=False, log=False):