From d99abf078537acf6cf49480b9790a9c450889031 Mon Sep 17 00:00:00 2001 From: Nicolas Courty Date: Fri, 7 Sep 2018 11:58:42 +0200 Subject: Wasserstein convolutional barycenter --- README.md | 4 +- data/duck.png | Bin 0 -> 5112 bytes data/heart.png | Bin 0 -> 5225 bytes data/redcross.png | Bin 0 -> 1683 bytes data/tooth.png | Bin 0 -> 4931 bytes examples/plot_convolutional_barycenter.py | 92 ++++++++++++++++++++++++++ ot/bregman.py | 106 ++++++++++++++++++++++++++++++ 7 files changed, 201 insertions(+), 1 deletion(-) create mode 100644 data/duck.png create mode 100644 data/heart.png create mode 100644 data/redcross.png create mode 100644 data/tooth.png create mode 100644 examples/plot_convolutional_barycenter.py diff --git a/README.md b/README.md index dded582..1105362 100644 --- a/README.md +++ b/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 \ No newline at end of file +[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. \ No newline at end of file diff --git a/data/duck.png b/data/duck.png new file mode 100644 index 0000000..9181697 Binary files /dev/null and b/data/duck.png differ diff --git a/data/heart.png b/data/heart.png new file mode 100644 index 0000000..44a6385 Binary files /dev/null and b/data/heart.png differ diff --git a/data/redcross.png b/data/redcross.png new file mode 100644 index 0000000..8d0a6fa Binary files /dev/null and b/data/redcross.png differ diff --git a/data/tooth.png b/data/tooth.png new file mode 100644 index 0000000..cd92c9d Binary files /dev/null and b/data/tooth.png differ diff --git a/examples/plot_convolutional_barycenter.py b/examples/plot_convolutional_barycenter.py new file mode 100644 index 0000000..d231da9 --- /dev/null +++ b/examples/plot_convolutional_barycenter.py @@ -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 +# +# 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() \ No newline at end of file diff --git a/ot/bregman.py b/ot/bregman.py index c755f51..05f4d9d 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -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