2D tomo gaussian

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diff --git a/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Tomo2D_gaussian.py b/Wrappers/Python/demos/PDHG_examples/PDHG_TV_Tomo2D_gaussian.py
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+#========================================================================
+# 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 2D Tomography Reconstruction using PDHG algorithm:
+
+
+Problem:     min_x, x>0  \alpha * ||\nabla x||_{2,1} + \frac{1}{2}||Ax - g||^{2}
+
+             \nabla: Gradient operator 
+             
+             A: Projection Matrix
+             g: Noisy sinogram corrupted with Gaussian Noise
+             
+             \alpha: Regularization parameter
+ 
+"""
+
+from ccpi.framework import ImageData, ImageGeometry, AcquisitionGeometry, AcquisitionData
+
+import numpy as np 
+import numpy                          
+import matplotlib.pyplot as plt
+
+from ccpi.optimisation.algorithms import PDHG
+
+from ccpi.optimisation.operators import BlockOperator, Gradient
+from ccpi.optimisation.functions import ZeroFunction, L2NormSquared, \
+                      MixedL21Norm, BlockFunction
+
+from ccpi.astra.ops import AstraProjectorSimple
+
+
+
+# Create phantom for TV 2D tomography 
+N = 100
+x = np.zeros((N,N))
+x[round(N/4):round(3*N/4),round(N/4):round(3*N/4)] = 0.5
+x[round(N/8):round(7*N/8),round(3*N/8):round(5*N/8)] = 1
+
+data = ImageData(x)
+ig = ImageGeometry(voxel_num_x = N, voxel_num_y = N)
+
+detectors = N
+angles = np.linspace(0, np.pi, N)
+
+ag = AcquisitionGeometry('parallel','2D',angles, detectors)
+
+device = input('Available device: GPU==1 / CPU==0 ')
+
+if device=='1':
+    dev = 'gpu'
+else:
+    dev = 'cpu'
+    
+Aop = AstraProjectorSimple(ig, ag, dev)
+sin = Aop.direct(data)
+
+# Create noisy data. Apply Poisson noise
+n1 = np.random.normal(0, 3, size=ig.shape)
+noisy_data = AcquisitionData(n1 + sin.as_array(), ag)
+
+# Show Ground Truth and Noisy Data
+plt.figure(figsize=(10,10))
+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 = 50
+
+# Create operators
+op1 = Gradient(ig)
+op2 = Aop
+
+# 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()
+
+# Compute operator Norm
+normK = operator.norm()
+    
+# Primal & dual stepsizes
+sigma = 10
+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)
+
+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
+import astra
+import numpy
+
+try:
+    from cvxpy import *
+    cvx_not_installable = True
+except ImportError:
+    cvx_not_installable = False
+    
+if cvx_not_installable:
+    
+    ##Construct problem    
+    u = Variable(N*N)
+    
+    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), DY.matrix() * vec(u)]), 2, axis = 0))
+    
+    # create matrix representation for Astra operator
+    vol_geom = astra.create_vol_geom(N, N)
+    proj_geom = astra.create_proj_geom('parallel', 1.0, detectors, angles)
+
+    proj_id = astra.create_projector('strip', proj_geom, vol_geom)
+
+    matrix_id = astra.projector.matrix(proj_id)
+
+    ProjMat = astra.matrix.get(matrix_id)
+    
+    tmp = noisy_data.as_array().ravel()
+    
+    fidelity = 0.5 * sum_squares(ProjMat * u - tmp)
+
+    solver = MOSEK
+    obj =  Minimize( regulariser +  fidelity)
+    prob = Problem(obj)
+    result = prob.solve(verbose = True, solver = solver)    
+         
+    diff_cvx = numpy.abs( pdhg.get_output().as_array() - np.reshape(u.value, (N,N) ))
+           
+    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(np.reshape(u.value, (N, N)))
+    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), np.reshape(u.value, (N,N) )[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]))
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