From 42b296291263c081603b828faa45da6f860f3b7b Mon Sep 17 00:00:00 2001 From: Edoardo Pasca Date: Wed, 22 May 2019 11:58:10 +0100 Subject: Added denoising demos which contain all noises and data selection --- .../demos/PDHG_examples/PDHG_TGV_Denoising.py | 291 +++++++++++++++++++++ .../demos/PDHG_examples/PDHG_TV_Denoising.py | 266 +++++++++++++++++++ .../demos/PDHG_examples/PDHG_Tikhonov_Denoising.py | 96 +++++-- 3 files changed, 628 insertions(+), 25 deletions(-) create mode 100755 Wrappers/Python/demos/PDHG_examples/PDHG_TGV_Denoising.py create mode 100755 Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising.py diff --git a/Wrappers/Python/demos/PDHG_examples/PDHG_TGV_Denoising.py b/Wrappers/Python/demos/PDHG_examples/PDHG_TGV_Denoising.py new file mode 100755 index 0000000..761c025 --- /dev/null +++ b/Wrappers/Python/demos/PDHG_examples/PDHG_TGV_Denoising.py @@ -0,0 +1,291 @@ +#======================================================================== +# Copyright 2019 Science Technology Facilities Council +# Copyright 2019 University of Manchester +# +# This work is part of the Core Imaging Library developed by Science Technology +# Facilities Council and University of Manchester +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0.txt +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. +# +#========================================================================= +""" + +Total Generalised Variation (TGV) Denoising using PDHG algorithm: + + +Problem: min_{x} \alpha * ||\nabla x - w||_{2,1} + + \beta * || E w ||_{2,1} + + fidelity + + where fidelity can be as follows depending on the noise characteristics + of the data: + * Norm2Squared \frac{1}{2} * || x - g ||_{2}^{2} + * KullbackLeibler + * L1Norm + + \alpha: Regularization parameter + \beta: Regularization parameter + + \nabla: Gradient operator + E: Symmetrized Gradient operator + + g: Noisy Data with Salt & Pepper Noise + + Method = 0 ( PDHG - split ) : K = [ \nabla, - Identity + ZeroOperator, E + Identity, ZeroOperator] + + + Method = 1 (PDHG - explicit ): K = [ \nabla, - Identity + ZeroOperator, E ] + +""" + +from ccpi.framework import ImageData, ImageGeometry + +import numpy as np +import numpy +import matplotlib.pyplot as plt + +from ccpi.optimisation.algorithms import PDHG + +from ccpi.optimisation.operators import BlockOperator, Identity, \ + Gradient, SymmetrizedGradient, ZeroOperator +from ccpi.optimisation.functions import ZeroFunction, L1Norm, \ + MixedL21Norm, BlockFunction, KullbackLeibler, L2NormSquared +import sys +if int(numpy.version.version.split('.')[1]) > 12: + from skimage.util import random_noise +else: + from demoutil import random_noise + + + +# user supplied input +if len(sys.argv) > 1: + which_noise = int(sys.argv[1]) +else: + which_noise = 0 +print ("Applying {} noise") + +if len(sys.argv) > 2: + method = sys.argv[2] +else: + method = '1' +print ("method ", method) +# Create phantom for TGV SaltPepper denoising + +N = 100 + +x1 = np.linspace(0, int(N/2), N) +x2 = np.linspace(int(N/2), 0., N) +xv, yv = np.meshgrid(x1, x2) + +xv[int(N/4):int(3*N/4)-1, int(N/4):int(3*N/4)-1] = yv[int(N/4):int(3*N/4)-1, int(N/4):int(3*N/4)-1].T + +#data = xv + +ig = ImageGeometry(voxel_num_x = N, voxel_num_y = N) +data = ig.allocate() +data.fill(xv/xv.max()) +ag = ig + +# Create noisy data. +# Apply Salt & Pepper noise +# gaussian +# poisson +noises = ['gaussian', 'poisson', 's&p'] +noise = noises[which_noise] +if noise == 's&p': + n1 = random_noise(data.as_array(), mode = noise, salt_vs_pepper = 0.9, amount=0.2, seed=10) +elif noise == 'poisson': + n1 = random_noise(data.as_array(), mode = noise, seed = 10) +elif noise == 'gaussian': + n1 = random_noise(data.as_array(), mode = noise, seed = 10) +else: + raise ValueError('Unsupported Noise ', noise) +noisy_data = ImageData(n1) + +# Show Ground Truth and Noisy Data +plt.figure(figsize=(10,5)) +plt.subplot(1,2,1) +plt.imshow(data.as_array()) +plt.title('Ground Truth') +plt.colorbar() +plt.subplot(1,2,2) +plt.imshow(noisy_data.as_array()) +plt.title('Noisy Data') +plt.colorbar() +plt.show() + +# Regularisation Parameters +if noise == 's&p': + alpha = 0.8 +elif noise == 'poisson': + alpha = .1 +elif noise == 'gaussian': + alpha = .3 + +beta = numpy.sqrt(2)* alpha + +# fidelity +if noise == 's&p': + f3 = L1Norm(b=noisy_data) +elif noise == 'poisson': + f3 = KullbackLeibler(noisy_data) +elif noise == 'gaussian': + f3 = L2NormSquared(b=noisy_data) + +if method == '0': + + # Create operators + op11 = Gradient(ig) + op12 = Identity(op11.range_geometry()) + + op22 = SymmetrizedGradient(op11.domain_geometry()) + op21 = ZeroOperator(ig, op22.range_geometry()) + + op31 = Identity(ig, ag) + op32 = ZeroOperator(op22.domain_geometry(), ag) + + operator = BlockOperator(op11, -1*op12, op21, op22, op31, op32, shape=(3,2) ) + + f1 = alpha * MixedL21Norm() + f2 = beta * MixedL21Norm() + # f3 depends on the noise characteristics + + f = BlockFunction(f1, f2, f3) + g = ZeroFunction() + +else: + + # Create operators + op11 = Gradient(ig) + op12 = Identity(op11.range_geometry()) + op22 = SymmetrizedGradient(op11.domain_geometry()) + op21 = ZeroOperator(ig, op22.range_geometry()) + + operator = BlockOperator(op11, -1*op12, op21, op22, shape=(2,2) ) + + f1 = alpha * MixedL21Norm() + f2 = beta * MixedL21Norm() + + f = BlockFunction(f1, f2) + g = BlockFunction(f3, ZeroFunction()) + +## Compute operator Norm +normK = operator.norm() +# +# Primal & dual stepsizes +sigma = 1 +tau = 1/(sigma*normK**2) + + +# Setup and run the PDHG algorithm +pdhg = PDHG(f=f,g=g,operator=operator, tau=tau, sigma=sigma) +pdhg.max_iteration = 2000 +pdhg.update_objective_interval = 200 +pdhg.run(2000, verbose = True) + +#%% +plt.figure(figsize=(20,5)) +plt.subplot(1,4,1) +plt.imshow(data.as_array()) +plt.title('Ground Truth') +plt.colorbar() +plt.subplot(1,4,2) +plt.imshow(noisy_data.as_array()) +plt.title('Noisy Data') +plt.colorbar() +plt.subplot(1,4,3) +plt.imshow(pdhg.get_output()[0].as_array()) +plt.title('TGV Reconstruction') +plt.colorbar() +plt.subplot(1,4,4) +plt.plot(np.linspace(0,N,N), data.as_array()[int(N/2),:], label = 'GTruth') +plt.plot(np.linspace(0,N,N), pdhg.get_output()[0].as_array()[int(N/2),:], label = 'TGV reconstruction') +plt.legend() +plt.title('Middle Line Profiles') +plt.show() + + +#%% Check with CVX solution + +from ccpi.optimisation.operators import SparseFiniteDiff + +try: + from cvxpy import * + cvx_not_installable = True +except ImportError: + cvx_not_installable = False + +if cvx_not_installable: + + u = Variable(ig.shape) + w1 = Variable((N, N)) + w2 = Variable((N, N)) + + # create TGV regulariser + DY = SparseFiniteDiff(ig, direction=0, bnd_cond='Neumann') + DX = SparseFiniteDiff(ig, direction=1, bnd_cond='Neumann') + + regulariser = alpha * sum(norm(vstack([DX.matrix() * vec(u) - vec(w1), \ + DY.matrix() * vec(u) - vec(w2)]), 2, axis = 0)) + \ + beta * sum(norm(vstack([ DX.matrix().transpose() * vec(w1), DY.matrix().transpose() * vec(w2), \ + 0.5 * ( DX.matrix().transpose() * vec(w2) + DY.matrix().transpose() * vec(w1) ), \ + 0.5 * ( DX.matrix().transpose() * vec(w2) + DY.matrix().transpose() * vec(w1) ) ]), 2, axis = 0 ) ) + + constraints = [] + fidelity = pnorm(u - noisy_data.as_array(),1) + solver = MOSEK + + # choose solver + if 'MOSEK' in installed_solvers(): + solver = MOSEK + else: + solver = SCS + + obj = Minimize( regulariser + fidelity) + prob = Problem(obj) + result = prob.solve(verbose = True, solver = solver) + + diff_cvx = numpy.abs( pdhg.get_output()[0].as_array() - u.value ) + + plt.figure(figsize=(15,15)) + plt.subplot(3,1,1) + plt.imshow(pdhg.get_output()[0].as_array()) + plt.title('PDHG solution') + plt.colorbar() + plt.subplot(3,1,2) + plt.imshow(u.value) + plt.title('CVX solution') + plt.colorbar() + plt.subplot(3,1,3) + plt.imshow(diff_cvx) + plt.title('Difference') + plt.colorbar() + plt.show() + + plt.plot(np.linspace(0,N,N), pdhg.get_output()[0].as_array()[int(N/2),:], label = 'PDHG') + plt.plot(np.linspace(0,N,N), u.value[int(N/2),:], label = 'CVX') + plt.legend() + plt.title('Middle Line Profiles') + plt.show() + + print('Primal Objective (CVX) {} '.format(obj.value)) + print('Primal Objective (PDHG) {} '.format(pdhg.objective[-1][0])) + + + + + diff --git a/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising.py b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising.py new file mode 100755 index 0000000..0f1effa --- /dev/null +++ b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising.py @@ -0,0 +1,266 @@ +#======================================================================== +# Copyright 2019 Science Technology Facilities Council +# Copyright 2019 University of Manchester +# +# This work is part of the Core Imaging Library developed by Science Technology +# Facilities Council and University of Manchester +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0.txt +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. +# +#========================================================================= + +""" + +Total Variation Denoising using PDHG algorithm: + + +Problem: min_x, x>0 \alpha * ||\nabla x||_{2,1} + ||x-g||_{1} + + \alpha: Regularization parameter + + \nabla: Gradient operator + + g: Noisy Data with Salt & Pepper Noise + + + Method = 0 ( PDHG - split ) : K = [ \nabla, + Identity] + + + Method = 1 (PDHG - explicit ): K = \nabla + + +""" + +from ccpi.framework import ImageData, ImageGeometry + +import numpy as np +import numpy +import matplotlib.pyplot as plt + +from ccpi.optimisation.algorithms import PDHG + +from ccpi.optimisation.operators import BlockOperator, Identity, Gradient +from ccpi.optimisation.functions import ZeroFunction, L1Norm, \ + MixedL21Norm, BlockFunction, L2NormSquared,\ + KullbackLeibler +from ccpi.framework import TestData +import os, sys +if int(numpy.version.version.split('.')[1]) > 12: + from skimage.util import random_noise +else: + from demoutil import random_noise + +# user supplied input +if len(sys.argv) > 1: + which_noise = int(sys.argv[1]) +else: + which_noise = 0 +print ("Applying {} noise") + +if len(sys.argv) > 2: + method = sys.argv[2] +else: + method = '0' +print ("method ", method) +# Create phantom for TV Salt & Pepper denoising +N = 100 + +loader = TestData(data_dir=os.path.join(sys.prefix, 'share','ccpi')) +data = loader.load(TestData.SIMPLE_PHANTOM_2D, size=(N,N)) +data = loader.load(TestData.PEPPERS, size=(N,N)) +ig = data.geometry +ag = ig + +# Create noisy data. +# Apply Salt & Pepper noise +# gaussian +# poisson +noises = ['gaussian', 'poisson', 's&p'] +noise = noises[which_noise] +if noise == 's&p': + n1 = random_noise(data.as_array(), mode = noise, salt_vs_pepper = 0.9, amount=0.2) +elif noise == 'poisson': + n1 = random_noise(data.as_array(), mode = noise, seed = 10) +elif noise == 'gaussian': + n1 = random_noise(data.as_array(), mode = noise, seed = 10) +else: + raise ValueError('Unsupported Noise ', noise) +noisy_data = ig.allocate() +noisy_data.fill(n1) +#noisy_data = ImageData(n1) + +# Show Ground Truth and Noisy Data +plt.figure(figsize=(10,5)) +plt.subplot(1,2,1) +plt.imshow(data.as_array()) +plt.title('Ground Truth') +plt.colorbar() +plt.subplot(1,2,2) +plt.imshow(noisy_data.as_array()) +plt.title('Noisy Data') +plt.colorbar() +plt.show() + +# Regularisation Parameter +alpha = .2 + +# fidelity +if noise == 's&p': + f2 = L1Norm(b=noisy_data) +elif noise == 'poisson': + f2 = KullbackLeibler(noisy_data) +elif noise == 'gaussian': + f2 = L2NormSquared(b=noisy_data) + +if method == '0': + + # Create operators + op1 = Gradient(ig, correlation=Gradient.CORRELATION_SPACECHANNEL) + op2 = Identity(ig, ag) + + # Create BlockOperator + operator = BlockOperator(op1, op2, shape=(2,1) ) + + # Create functions + f1 = alpha * MixedL21Norm() + #f2 = L1Norm(b = noisy_data) + f = BlockFunction(f1, f2) + + g = ZeroFunction() + +else: + + # Without the "Block Framework" + operator = Gradient(ig) + f = alpha * MixedL21Norm() + #g = L1Norm(b = noisy_data) + g = f2 + + +# Compute operator Norm +normK = operator.norm() + +# Primal & dual stepsizes +sigma = 1 +tau = 1/(sigma*normK**2) +opt = {'niter':2000, 'memopt': True} + +# Setup and run the PDHG algorithm +pdhg = PDHG(f=f,g=g,operator=operator, tau=tau, sigma=sigma, memopt=True) +pdhg.max_iteration = 2000 +pdhg.update_objective_interval = 50 +pdhg.run(2000) + +if data.geometry.channels > 1: + plt.figure(figsize=(20,15)) + for row in range(data.geometry.channels): + + plt.subplot(3,4,1+row*4) + plt.imshow(data.subset(channel=row).as_array()) + plt.title('Ground Truth') + plt.colorbar() + plt.subplot(3,4,2+row*4) + plt.imshow(noisy_data.subset(channel=row).as_array()) + plt.title('Noisy Data') + plt.colorbar() + plt.subplot(3,4,3+row*4) + plt.imshow(pdhg.get_output().subset(channel=row).as_array()) + plt.title('TV Reconstruction') + plt.colorbar() + plt.subplot(3,4,4+row*4) + plt.plot(np.linspace(0,N,N), data.subset(channel=row).as_array()[int(N/2),:], label = 'GTruth') + plt.plot(np.linspace(0,N,N), pdhg.get_output().subset(channel=row).as_array()[int(N/2),:], label = 'TV reconstruction') + plt.legend() + plt.title('Middle Line Profiles') + plt.show() +else: + plt.figure(figsize=(20,5)) + plt.subplot(1,4,1) + plt.imshow(data.subset(channel=0).as_array()) + plt.title('Ground Truth') + plt.colorbar() + plt.subplot(1,4,2) + plt.imshow(noisy_data.subset(channel=0).as_array()) + plt.title('Noisy Data') + plt.colorbar() + plt.subplot(1,4,3) + plt.imshow(pdhg.get_output().subset(channel=0).as_array()) + plt.title('TV Reconstruction') + plt.colorbar() + plt.subplot(1,4,4) + plt.plot(np.linspace(0,N,N), data.as_array()[int(N/2),:], label = 'GTruth') + plt.plot(np.linspace(0,N,N), pdhg.get_output().as_array()[int(N/2),:], label = 'TV reconstruction') + plt.legend() + plt.title('Middle Line Profiles') + plt.show() + + +##%% Check with CVX solution + +from ccpi.optimisation.operators import SparseFiniteDiff + +try: + from cvxpy import * + cvx_not_installable = True +except ImportError: + cvx_not_installable = False + + +if cvx_not_installable: + + ##Construct problem + u = Variable(ig.shape) + + DY = SparseFiniteDiff(ig, direction=0, bnd_cond='Neumann') + DX = SparseFiniteDiff(ig, direction=1, bnd_cond='Neumann') + + # Define Total Variation as a regulariser + regulariser = alpha * sum(norm(vstack([DX.matrix() * vec(u), DY.matrix() * vec(u)]), 2, axis = 0)) + fidelity = pnorm( u - noisy_data.as_array(),1) + + # choose solver + if 'MOSEK' in installed_solvers(): + solver = MOSEK + else: + solver = SCS + + obj = Minimize( regulariser + fidelity) + prob = Problem(obj) + result = prob.solve(verbose = True, solver = solver) + + diff_cvx = numpy.abs( pdhg.get_output().as_array() - u.value ) + + plt.figure(figsize=(15,15)) + plt.subplot(3,1,1) + plt.imshow(pdhg.get_output().as_array()) + plt.title('PDHG solution') + plt.colorbar() + plt.subplot(3,1,2) + plt.imshow(u.value) + plt.title('CVX solution') + plt.colorbar() + plt.subplot(3,1,3) + plt.imshow(diff_cvx) + plt.title('Difference') + plt.colorbar() + plt.show() + + plt.plot(np.linspace(0,N,N), pdhg.get_output().as_array()[int(N/2),:], label = 'PDHG') + plt.plot(np.linspace(0,N,N), u.value[int(N/2),:], label = 'CVX') + plt.legend() + plt.title('Middle Line Profiles') + plt.show() + + print('Primal Objective (CVX) {} '.format(obj.value)) + print('Primal Objective (PDHG) {} '.format(pdhg.objective[-1][0])) diff --git a/Wrappers/Python/demos/PDHG_examples/PDHG_Tikhonov_Denoising.py b/Wrappers/Python/demos/PDHG_examples/PDHG_Tikhonov_Denoising.py index 7b73c1a..f00f1cc 100644 --- a/Wrappers/Python/demos/PDHG_examples/PDHG_Tikhonov_Denoising.py +++ b/Wrappers/Python/demos/PDHG_examples/PDHG_Tikhonov_Denoising.py @@ -39,7 +39,7 @@ Problem: min_{x} \alpha * ||\nabla x||_{2}^{2} + \frac{1}{2} * || x - g ||_{ """ -from ccpi.framework import ImageData, ImageGeometry +from ccpi.framework import ImageData, ImageGeometry, TestData import numpy as np import numpy @@ -48,40 +48,83 @@ import matplotlib.pyplot as plt from ccpi.optimisation.algorithms import PDHG from ccpi.optimisation.operators import BlockOperator, Identity, Gradient -from ccpi.optimisation.functions import ZeroFunction, L2NormSquared, BlockFunction +from ccpi.optimisation.functions import ZeroFunction, L2NormSquared,\ + BlockFunction, KullbackLeibler, L1Norm -from skimage.util import random_noise +import sys, os +if int(numpy.version.version.split('.')[1]) > 12: + from skimage.util import random_noise +else: + from demoutil import random_noise + + +# user supplied input +if len(sys.argv) > 1: + which_noise = int(sys.argv[1]) +else: + which_noise = 0 +print ("Applying {} noise") + +if len(sys.argv) > 2: + method = sys.argv[2] +else: + method = '0' +print ("method ", method) # Create phantom for TV Salt & Pepper denoising N = 100 -data = np.zeros((N,N)) -data[round(N/4):round(3*N/4),round(N/4):round(3*N/4)] = 0.5 -data[round(N/8):round(7*N/8),round(3*N/8):round(5*N/8)] = 1 -data = ImageData(data) -ig = ImageGeometry(voxel_num_x = N, voxel_num_y = N) +loader = TestData(data_dir=os.path.join(sys.prefix, 'share','ccpi')) +data = loader.load(TestData.SIMPLE_PHANTOM_2D, size=(N,N)) +ig = data.geometry ag = ig # Create noisy data. Apply Salt & Pepper noise -n1 = random_noise(data.as_array(), mode = 's&p', salt_vs_pepper = 0.9, amount=0.2) +# Create noisy data. +# Apply Salt & Pepper noise +# gaussian +# poisson +noises = ['gaussian', 'poisson', 's&p'] +noise = noises[which_noise] +if noise == 's&p': + n1 = random_noise(data.as_array(), mode = noise, salt_vs_pepper = 0.9, amount=0.2) +elif noise == 'poisson': + n1 = random_noise(data.as_array(), mode = noise, seed = 10) +elif noise == 'gaussian': + n1 = random_noise(data.as_array(), mode = noise, seed = 10) +else: + raise ValueError('Unsupported Noise ', noise) noisy_data = ImageData(n1) +# fidelity +if noise == 's&p': + f2 = L1Norm(b=noisy_data) +elif noise == 'poisson': + f2 = KullbackLeibler(noisy_data) +elif noise == 'gaussian': + f2 = 0.5 * L2NormSquared(b=noisy_data) + # Show Ground Truth and Noisy Data -plt.figure(figsize=(15,15)) -plt.subplot(2,1,1) +plt.figure(figsize=(10,5)) +plt.subplot(1,2,1) plt.imshow(data.as_array()) plt.title('Ground Truth') plt.colorbar() -plt.subplot(2,1,2) +plt.subplot(1,2,2) plt.imshow(noisy_data.as_array()) plt.title('Noisy Data') plt.colorbar() plt.show() -# Regularisation Parameter -alpha = 4 -method = '1' +# Regularisation Parameter +# no edge preservation alpha is big +if noise == 's&p': + alpha = 8. +elif noise == 'poisson': + alpha = 8. +elif noise == 'gaussian': + alpha = 8. if method == '0': @@ -95,7 +138,8 @@ if method == '0': # Create functions f1 = alpha * L2NormSquared() - f2 = 0.5 * L2NormSquared(b = noisy_data) + # f2 must change according to noise + #f2 = 0.5 * L2NormSquared(b = noisy_data) f = BlockFunction(f1, f2) g = ZeroFunction() @@ -104,7 +148,9 @@ else: # Without the "Block Framework" operator = Gradient(ig) f = alpha * L2NormSquared() - g = 0.5 * L2NormSquared(b = noisy_data) + # g must change according to noise + #g = 0.5 * L2NormSquared(b = noisy_data) + g = f2 # Compute operator Norm @@ -119,31 +165,31 @@ opt = {'niter':2000, 'memopt': True} pdhg = PDHG(f=f,g=g,operator=operator, tau=tau, sigma=sigma, memopt=True) pdhg.max_iteration = 2000 pdhg.update_objective_interval = 50 -pdhg.run(2000) +pdhg.run(1500) -plt.figure(figsize=(15,15)) -plt.subplot(3,1,1) +plt.figure(figsize=(20,5)) +plt.subplot(1,4,1) plt.imshow(data.as_array()) plt.title('Ground Truth') plt.colorbar() -plt.subplot(3,1,2) +plt.subplot(1,4,2) plt.imshow(noisy_data.as_array()) plt.title('Noisy Data') plt.colorbar() -plt.subplot(3,1,3) +plt.subplot(1,4,3) plt.imshow(pdhg.get_output().as_array()) plt.title('Tikhonov Reconstruction') plt.colorbar() -plt.show() -## +plt.subplot(1,4,4) plt.plot(np.linspace(0,N,N), data.as_array()[int(N/2),:], label = 'GTruth') -plt.plot(np.linspace(0,N,N), pdhg.get_output().as_array()[int(N/2),:], label = 'Tikhonov reconstruction') +plt.plot(np.linspace(0,N,N), pdhg.get_output().as_array()[int(N/2),:], label = 'TV reconstruction') plt.legend() plt.title('Middle Line Profiles') plt.show() + ##%% Check with CVX solution from ccpi.optimisation.operators import SparseFiniteDiff -- cgit v1.2.3