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 --- ot/bregman.py | 106 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 106 insertions(+) (limited to 'ot') 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 Date: Fri, 7 Sep 2018 12:04:44 +0200 Subject: pep8 normalization --- examples/plot_convolutional_barycenter.py | 70 +++++++++++++++---------------- ot/bregman.py | 68 +++++++++++++++--------------- 2 files changed, 70 insertions(+), 68 deletions(-) (limited to 'ot') diff --git a/examples/plot_convolutional_barycenter.py b/examples/plot_convolutional_barycenter.py index d231da9..7ccdbe3 100644 --- a/examples/plot_convolutional_barycenter.py +++ b/examples/plot_convolutional_barycenter.py @@ -1,4 +1,4 @@ - + #%% # -*- coding: utf-8 -*- """ @@ -32,24 +32,24 @@ 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) +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) +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)) +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)) ############################################################################## @@ -57,36 +57,36 @@ v4=np.array((0,0,0,1)) # ---------------------------------------- # -pl.figure(figsize=(10,10)) +pl.figure(figsize=(10, 10)) pl.title('Convolutional Wasserstein Barycenters in POT') -cm='Blues' +cm = 'Blues' # regularization parameter -reg=0.004 +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) - + 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') + 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.imshow(ot.convolutional_barycenter2d(A, reg, weights), cmap=cm) pl.axis('off') -pl.show() \ No newline at end of file +pl.show() diff --git a/ot/bregman.py b/ot/bregman.py index 05f4d9d..f844f03 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -918,7 +918,8 @@ 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): + +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. @@ -979,51 +980,52 @@ def convolutional_barycenter2d(A,reg,weights=None,numItermax = 10000, stopThr=1e 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 + 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 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,:,:])) + 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)) + 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 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 + log['niter'] = cpt + log['U'] = U + return b, log else: - return b - + return b + def unmix(a, D, M, M0, h0, reg, reg0, alpha, numItermax=1000, stopThr=1e-3, verbose=False, log=False): -- cgit v1.2.3 From e8c6d2fc9c6b08bbed11628326711ab29c155bac Mon Sep 17 00:00:00 2001 From: Nicolas Courty Date: Fri, 7 Sep 2018 12:34:14 +0200 Subject: pep8 fixed (contd) --- ot/bregman.py | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) (limited to 'ot') diff --git a/ot/bregman.py b/ot/bregman.py index f844f03..5327dbc 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -920,8 +920,8 @@ def barycenter(A, M, reg, weights=None, numItermax=1000, 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. + """Compute the entropic regularized wasserstein barycenter of distributions A + where A is a collection of 2D images. The function solves the following optimization problem: @@ -966,8 +966,8 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 ---------- .. [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 + Convolutional wasserstein distances: Efficient optimal transportation on geometric domains + ACM Transactions on Graphics (TOG), 34(4), 66 """ @@ -993,7 +993,8 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 [Y, X] = np.meshgrid(t, t) xi1 = np.exp(-(X - Y)**2 / reg) - def K(x): return np.dot(np.dot(xi1, x), xi1) + def K(x): + return np.dot(np.dot(xi1, x), xi1) while (err > stopThr and cpt < numItermax): -- cgit v1.2.3 From d19295b9cb29d21e09eeb28ac4b0e61990727023 Mon Sep 17 00:00:00 2001 From: Nicolas Courty Date: Fri, 7 Sep 2018 14:41:00 +0200 Subject: stabThr and pep8 --- ot/bregman.py | 13 +++++++------ 1 file changed, 7 insertions(+), 6 deletions(-) (limited to 'ot') diff --git a/ot/bregman.py b/ot/bregman.py index 5327dbc..748ac30 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -919,7 +919,7 @@ def barycenter(A, M, reg, weights=None, numItermax=1000, return geometricBar(weights, UKv) -def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1e-9, verbose=False, log=False): +def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1e-9, stabThr=1e-30, verbose=False, log=False): """Compute the entropic regularized wasserstein barycenter of distributions A where A is a collection of 2D images. @@ -948,6 +948,8 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 Max number of iterations stopThr : float, optional Stop threshol on error (>0) + stabThr : float, optional + Stabilization threshold to avoid numerical precision issue verbose : bool, optional Print information along iterations log : bool, optional @@ -983,7 +985,6 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 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 @@ -993,7 +994,7 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 [Y, X] = np.meshgrid(t, t) xi1 = np.exp(-(X - Y)**2 / reg) - def K(x): + def K(x): return np.dot(np.dot(xi1, x), xi1) while (err > stopThr and cpt < numItermax): @@ -1003,11 +1004,11 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=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, :, :])) + KV[r, :, :] = K(A[r, :, :] / np.maximum(stabThr, K(U[r, :, :]))) + b += weights[r] * np.log(np.maximum(stabThr, U[r, :, :] * KV[r, :, :])) b = np.exp(b) for r in range(A.shape[0]): - U[r, :, :] = b / np.maximum(threshold, KV[r, :, :]) + U[r, :, :] = b / np.maximum(stabThr, KV[r, :, :]) if cpt % 10 == 1: err = np.sum(np.abs(bold - b)) -- cgit v1.2.3 From dab572396be97fcf5439e4e20f887165b1ade62c Mon Sep 17 00:00:00 2001 From: Nicolas Courty Date: Fri, 7 Sep 2018 14:49:20 +0200 Subject: whitetrail pep8 --- ot/bregman.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) (limited to 'ot') diff --git a/ot/bregman.py b/ot/bregman.py index 748ac30..35e51f8 100644 --- a/ot/bregman.py +++ b/ot/bregman.py @@ -920,8 +920,8 @@ def barycenter(A, M, reg, weights=None, numItermax=1000, def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1e-9, stabThr=1e-30, verbose=False, log=False): - """Compute the entropic regularized wasserstein barycenter of distributions A - where A is a collection of 2D images. + """Compute the entropic regularized wasserstein barycenter of distributions A + where A is a collection of 2D images. The function solves the following optimization problem: @@ -949,7 +949,7 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 stopThr : float, optional Stop threshol on error (>0) stabThr : float, optional - Stabilization threshold to avoid numerical precision issue + Stabilization threshold to avoid numerical precision issue verbose : bool, optional Print information along iterations log : bool, optional @@ -967,9 +967,9 @@ def convolutional_barycenter2d(A, reg, weights=None, numItermax=10000, stopThr=1 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 + .. [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 """ -- cgit v1.2.3