fix PDHG denoising examples

6 files changed, 1136 insertions, 45 deletions

diff --git a/Wrappers/Python/demos/PDHG_TV_Denoising_Poisson.py b/Wrappers/Python/demos/PDHG_TV_Denoising_Poisson.py
index 70f6b9b..0db8c29 100644
--- a/Wrappers/Python/demos/PDHG_TV_Denoising_Poisson.py
+++ b/Wrappers/Python/demos/PDHG_TV_Denoising_Poisson.py
@@ -1,41 +1,43 @@
-# -*- coding: utf-8 -*-
-#   This work is part of the Core Imaging Library developed by
-#   Visual Analytics and Imaging System Group of the Science Technology
-#   Facilities Council, STFC
-
-#   Copyright 2018-2019 STFC,  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
-
-#   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.
+#========================================================================
+# 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:
 
-             min_{x} max_{y} < K x, y > + g(x) - f^{*}(y) 
-
-
 Problem:     min_x, x>0  \alpha * ||\nabla x||_{1} + \int x - g * log(x)
 
-             \nabla: Gradient operator              
-             g: Noisy Data with Poisson Noise
              \alpha: Regularization parameter
              
-             Method = 0:  K = [ \nabla,
-                                 Identity]
-                                                                    
-             Method = 1:  K = \nabla    
+             \nabla: Gradient operator  
+             
+             g: Noisy Data with Poisson Noise
              
              
+             Method = 0 ( PDHG - split ) :  K = [ \nabla,
+                                                 Identity]
+                          
+                                                                    
+             Method = 1 (PDHG - explicit ):  K = \nabla     
+                       
 """
 
 from ccpi.framework import ImageData, ImageGeometry
@@ -66,10 +68,25 @@ ag = ig
 n1 = random_noise(data.as_array(), mode = 'poisson', seed = 10)
 noisy_data = ImageData(n1)
 
+
+# Show Ground Truth and Noisy Data
+plt.figure(figsize=(15,15))
+plt.subplot(2,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(2,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.show()
+
+#%%
+
 # Regularisation Parameter
 alpha = 2
 
-method = '1'
+method = '0'
 
 if method == '0':
 
@@ -99,26 +116,15 @@ else:
 # Compute operator Norm
 normK = operator.norm()
 
-# Primal & dual stepsizes
+# 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
-
-def pdgap_objectives(niter, objective, solution):
-    
-
-     print( "{:04}/{:04} {:<5} {:.4f} {:<5} {:.4f} {:<5} {:.4f}".\
-                      format(niter, pdhg.max_iteration,'', \
-                             objective[0],'',\
-                             objective[1],'',\
-                             objective[2]))
 
-pdhg.run(2000, callback = pdgap_objectives)
+# Setup and Run the PDHG algorithm
+pdhg = PDHG(f=f,g=g,operator=operator, tau=tau, sigma=sigma)
+pdhg.max_iteration = 3000
+pdhg.update_objective_interval = 200
+pdhg.run(3000, verbose=False)
 
 
 plt.figure(figsize=(15,15))
diff --git a/Wrappers/Python/demos/PDHG_examples/PDHG_TGV_Denoising_SaltPepper.py b/Wrappers/Python/demos/PDHG_examples/PDHG_TGV_Denoising_SaltPepper.py
new file mode 100644
index 0000000..57f6fcd
--- /dev/null
+++ b/Wrappers/Python/demos/PDHG_examples/PDHG_TGV_Denoising_SaltPepper.py
@@ -0,0 +1,245 @@
+#========================================================================
+# 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} +
+                     \frac{1}{2} * || x - g ||_{2}^{2}
+
+             \alpha: Regularization parameter
+             \alpha: 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
+
+from skimage.util import random_noise
+
+# Create phantom for TGV SaltPepper denoising
+
+N = 100
+
+data = np.zeros((N,N))
+
+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
+data = ImageData(data/data.max())
+
+ig = ImageGeometry(voxel_num_x = N, voxel_num_y = N)
+ag = ig
+
+# Create noisy data. Add Gaussian noise
+n1 = random_noise(data.as_array(), mode = 's&p', salt_vs_pepper = 0.9, amount=0.2)
+noisy_data = ImageData(n1)
+
+# Show Ground Truth and Noisy Data
+plt.figure(figsize=(15,15))
+plt.subplot(2,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(2,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.show()
+
+# Regularisation Parameters
+alpha = 0.8
+beta = numpy.sqrt(2)* alpha
+
+method = '1'
+
+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 = L1Norm(b=noisy_data)    
+    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(L1Norm(b=noisy_data), 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 = 50
+pdhg.run(2000, verbose = False)
+
+#%%
+plt.figure(figsize=(15,15))
+plt.subplot(3,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(3,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.subplot(3,1,3)
+plt.imshow(pdhg.get_output()[0].as_array())
+plt.title('TGV Reconstruction')
+plt.colorbar()
+plt.show()
+## 
+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 = '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:    
+    
+    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_Gaussian.py b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising_Gaussian.py
new file mode 100644
index 0000000..afdb6a2
--- /dev/null
+++ b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising_Gaussian.py
@@ -0,0 +1,204 @@
+#========================================================================
+# 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} \alpha * ||\nabla x||_{2,1} + \frac{1}{2} * || x - g ||_{2}^{2}
+
+             \alpha: Regularization parameter
+             
+             \nabla: Gradient operator 
+             
+             g: Noisy Data with Gaussian 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, L2NormSquared, \
+                      MixedL21Norm, BlockFunction
+      
+# Load Data                      
+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)
+ag = ig
+                      
+# Create Noisy data. Add Gaussian noise
+np.random.seed(10)
+noisy_data = ImageData( data.as_array() + np.random.normal(0, 0.1, size=ig.shape) )
+
+# Show Ground Truth and Noisy Data
+plt.figure(figsize=(15,15))
+plt.subplot(2,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(2,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.show()
+
+# Regularisation Parameter
+alpha = 0.2
+
+method = '0'
+
+if method == '0':
+
+    # Create operators
+    op1 = Gradient(ig)
+    op2 = Identity(ig, ag)
+
+    # Create BlockOperator
+    operator = BlockOperator(op1, op2, shape=(2,1) ) 
+
+    # Create functions      
+    f1 = alpha * MixedL21Norm()
+    f2 = 0.5 * L2NormSquared(b = noisy_data)    
+    f = BlockFunction(f1, f2)  
+                                      
+    g = ZeroFunction()
+    
+else:
+    
+    # Without the "Block Framework"
+    operator =  Gradient(ig)
+    f =  alpha * MixedL21Norm()
+    g =  0.5 * L2NormSquared(b = noisy_data)
+        
+# 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 = 3000
+pdhg.update_objective_interval = 200
+pdhg.run(3000, verbose=False)
+
+# Show Results
+plt.figure(figsize=(15,15))
+plt.subplot(3,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(3,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.subplot(3,1,3)
+plt.imshow(pdhg.get_output().as_array())
+plt.title('TV Reconstruction')
+plt.colorbar()
+plt.show()
+
+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 = 0.5 * sum_squares(u - noisy_data.as_array())
+    
+    # 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 = MOSEK)
+    
+    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.plot(np.linspace(0,N,N), data.as_array()[int(N/2),:], label = 'Truth')
+    
+    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_Poisson.py b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising_Poisson.py
new file mode 100644
index 0000000..4d53635
--- /dev/null
+++ b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising_Poisson.py
@@ -0,0 +1,213 @@
+#========================================================================
+# 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} + \int x - g * log(x)
+
+             \alpha: Regularization parameter
+             
+             \nabla: Gradient operator  
+             
+             g: Noisy Data with Poisson 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, KullbackLeibler, \
+                      MixedL21Norm, BlockFunction
+
+from skimage.util import random_noise
+
+# Create phantom for TV Poisson 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)
+ag = ig
+
+# Create noisy data. Apply Poisson noise
+n1 = random_noise(data.as_array(), mode = 'poisson', seed = 10)
+noisy_data = ImageData(n1)
+
+
+# Show Ground Truth and Noisy Data
+plt.figure(figsize=(15,15))
+plt.subplot(2,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(2,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.show()
+
+#%%
+
+# Regularisation Parameter
+alpha = 2
+
+method = '0'
+
+if method == '0':
+
+    # Create operators
+    op1 = Gradient(ig)
+    op2 = Identity(ig, ag)
+
+    # Create BlockOperator
+    operator = BlockOperator(op1, op2, shape=(2,1) ) 
+
+    # Create functions
+      
+    f1 = alpha * MixedL21Norm()
+    f2 = KullbackLeibler(noisy_data)    
+    f = BlockFunction(f1, f2)  
+                                      
+    g = ZeroFunction()
+    
+else:
+    
+    # Without the "Block Framework"
+    operator = Gradient(ig)
+    f =  alpha * MixedL21Norm()
+    g =  KullbackLeibler(noisy_data)   
+        
+    
+# 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 = 3000
+pdhg.update_objective_interval = 200
+pdhg.run(3000, verbose=False)
+
+
+plt.figure(figsize=(15,15))
+plt.subplot(3,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(3,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.subplot(3,1,3)
+plt.imshow(pdhg.get_output().as_array())
+plt.title('TV Reconstruction')
+plt.colorbar()
+plt.show()
+## 
+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    
+    u1 = Variable(ig.shape)
+    q = Variable()
+    
+    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(u1), DY.matrix() * vec(u1)]), 2, axis = 0))
+    
+    fidelity = sum( u1 - multiply(noisy_data.as_array(), log(u1)) )
+    constraints = [q>= fidelity, u1>=0]
+        
+    solver = ECOS
+    obj =  Minimize( regulariser +  q)
+    prob = Problem(obj, constraints)
+    result = prob.solve(verbose = True, solver = solver)
+
+    
+    diff_cvx = numpy.abs( pdhg.get_output().as_array() - u1.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(u1.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), u1.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_SaltPepper.py b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising_SaltPepper.py
new file mode 100644
index 0000000..c5709c3
--- /dev/null
+++ b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Denoising_SaltPepper.py
@@ -0,0 +1,213 @@
+#========================================================================
+# 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
+
+from skimage.util import random_noise
+
+# 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)
+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)
+noisy_data = ImageData(n1)
+
+# Show Ground Truth and Noisy Data
+plt.figure(figsize=(15,15))
+plt.subplot(2,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(2,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.show()
+
+# Regularisation Parameter
+alpha = 2
+
+method = '0'
+
+if method == '0':
+
+    # Create operators
+    op1 = Gradient(ig)
+    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)
+        
+    
+# 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)
+
+
+plt.figure(figsize=(15,15))
+plt.subplot(3,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(3,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.subplot(3,1,3)
+plt.imshow(pdhg.get_output().as_array())
+plt.title('TV Reconstruction')
+plt.colorbar()
+plt.show()
+## 
+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
new file mode 100644
index 0000000..7b73c1a
--- /dev/null
+++ b/Wrappers/Python/demos/PDHG_examples/PDHG_Tikhonov_Denoising.py
@@ -0,0 +1,210 @@
+#========================================================================
+# 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.
+#
+#=========================================================================
+""" 
+
+Tikhonov Denoising using PDHG algorithm:
+
+
+Problem:     min_{x} \alpha * ||\nabla x||_{2}^{2} + \frac{1}{2} * || x - g ||_{2}^{2}
+
+             \alpha: Regularization parameter
+             
+             \nabla: Gradient operator 
+             
+             g: Noisy Data with Gaussian 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, L2NormSquared,  BlockFunction
+
+from skimage.util import random_noise
+
+# 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)
+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)
+noisy_data = ImageData(n1)
+
+# Show Ground Truth and Noisy Data
+plt.figure(figsize=(15,15))
+plt.subplot(2,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(2,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.show()
+
+# Regularisation Parameter
+alpha = 4
+
+method = '1'
+
+if method == '0':
+
+    # Create operators
+    op1 = Gradient(ig)
+    op2 = Identity(ig, ag)
+
+    # Create BlockOperator
+    operator = BlockOperator(op1, op2, shape=(2,1) ) 
+
+    # Create functions
+      
+    f1 = alpha * L2NormSquared()
+    f2 = 0.5 * L2NormSquared(b = noisy_data)    
+    f = BlockFunction(f1, f2)                                       
+    g = ZeroFunction()
+    
+else:
+    
+    # Without the "Block Framework"
+    operator = Gradient(ig)
+    f =  alpha * L2NormSquared()
+    g =  0.5 * L2NormSquared(b = noisy_data)
+        
+    
+# 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)
+
+
+plt.figure(figsize=(15,15))
+plt.subplot(3,1,1)
+plt.imshow(data.as_array())
+plt.title('Ground Truth')
+plt.colorbar()
+plt.subplot(3,1,2)
+plt.imshow(noisy_data.as_array())
+plt.title('Noisy Data')
+plt.colorbar()
+plt.subplot(3,1,3)
+plt.imshow(pdhg.get_output().as_array())
+plt.title('Tikhonov Reconstruction')
+plt.colorbar()
+plt.show()
+## 
+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.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_squares(norm(vstack([DX.matrix() * vec(u), DY.matrix() * vec(u)]), 2, axis = 0))
+    fidelity = 0.5 * sum_squares(u - noisy_data.as_array())    
+    
+    # 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]))
+
+
+
+
+