1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
|
from ccpi.framework import ImageData, ImageGeometry, AcquisitionGeometry, DataContainer
from ccpi.optimisation.algs import FISTA, FBPD, CGLS
from ccpi.optimisation.funcs import Norm2sq, ZeroFun, Norm1, TV2D, Identity
from ccpi.optimisation.ops import LinearOperatorMatrix
from ccpi.plugins.regularisers import _ROF_TV_, _FGP_TV_, _SB_TV_
import numpy as np
import matplotlib.pyplot as plt
#%%
# Requires CVXPY, see http://www.cvxpy.org/
# CVXPY can be installed in anaconda using
# conda install -c cvxgrp cvxpy libgcc
# Whether to use or omit CVXPY
use_cvxpy = True
if use_cvxpy:
from cvxpy import *
#%%
# Now try 1-norm and TV denoising with FBPD, first 1-norm.
# Set up phantom size NxN by creating ImageGeometry, initialising the
# ImageData object with this geometry and empty array and finally put some
# data into its array, and display as image.
N = 64
ig = ImageGeometry(voxel_num_x=N,voxel_num_y=N)
Phantom = ImageData(geometry=ig)
x = Phantom.as_array()
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
plt.imshow(x)
plt.title('Phantom image')
plt.show()
# Identity operator for denoising
I = Identity()
# Data and add noise
y = I.direct(Phantom)
np.random.seed(0)
y.array = y.array + 0.1*np.random.randn(N, N)
plt.imshow(y.array)
plt.title('Noisy image')
plt.show()
#%% TV parameter
lam_tv = 1.0
#%% Do CVX as high quality ground truth
if use_cvxpy:
# Compare to CVXPY
# Construct the problem.
xtv_denoise = Variable(N,N)
objectivetv_denoise = Minimize(0.5*sum_squares(xtv_denoise - y.array) + lam_tv*tv(xtv_denoise) )
probtv_denoise = Problem(objectivetv_denoise)
# The optimal objective is returned by prob.solve().
resulttv_denoise = probtv_denoise.solve(verbose=False,solver=SCS,eps=1e-12)
# The optimal solution for x is stored in x.value and optimal objective value
# is in result as well as in objective.value
print("CVXPY least squares plus TV solution and objective value:")
# print(xtv_denoise.value)
# print(objectivetv_denoise.value)
plt.figure()
plt.imshow(xtv_denoise.value)
plt.title('CVX TV with objective equal to {:.2f}'.format(objectivetv_denoise.value))
plt.show()
print(objectivetv_denoise.value)
#%% THen FBPD
# Data fidelity term
f_denoise = Norm2sq(I,y,c=0.5)
# Initial guess
x_init_denoise = ImageData(np.zeros((N,N)))
gtv = TV2D(lam_tv)
gtv(gtv.op.direct(x_init_denoise))
opt_tv = {'tol': 1e-4, 'iter': 10000}
x_fbpdtv_denoise, itfbpdtv_denoise, timingfbpdtv_denoise, criterfbpdtv_denoise = FBPD(x_init_denoise, None, f_denoise, gtv,opt=opt_tv)
print("CVXPY least squares plus TV solution and objective value:")
plt.figure()
plt.imshow(x_fbpdtv_denoise.as_array())
plt.title('FBPD TV with objective equal to {:.2f}'.format(criterfbpdtv_denoise[-1]))
plt.show()
print(criterfbpdtv_denoise[-1])
#%%
plt.loglog([0,opt_tv['iter']], [objectivetv_denoise.value,objectivetv_denoise.value], label='CVX TV')
plt.loglog(criterfbpdtv_denoise, label='FBPD TV')
plt.show()
#%% FISTA with ROF-TV regularisation
g_rof = _ROF_TV_(lambdaReg = lam_tv,iterationsTV=5000,tolerance=0,time_marchstep=0.0009,device='cpu')
xtv_rof = g_rof.prox(y,1.0)
print("CCPi-RGL TV ROF:")
plt.figure()
plt.imshow(xtv_rof.as_array())
valObjRof = g_rof(xtv_rof)
plt.title('ROF TV prox with objective equal to {:.2f}'.format(valObjRof[0]))
plt.show()
print(valObjRof[0])
#%% FISTA with FGP-TV regularisation
g_fgp = _FGP_TV_(lambdaReg = lam_tv,iterationsTV=5000,tolerance=0,methodTV=0,nonnegativity=0,printing=0,device='cpu')
xtv_fgp = g_fgp.prox(y,1.0)
print("CCPi-RGL TV FGP:")
plt.figure()
plt.imshow(xtv_fgp.as_array())
valObjFGP = g_fgp(xtv_fgp)
plt.title('FGP TV prox with objective equal to {:.2f}'.format(valObjFGP[0]))
plt.show()
print(valObjFGP[0])
#%% Split-Bregman-TV regularisation
g_sb = _SB_TV_(lambdaReg = lam_tv,iterationsTV=150,tolerance=0,methodTV=0,printing=0,device='cpu')
xtv_sb = g_sb.prox(y,1.0)
print("CCPi-RGL TV SB:")
plt.figure()
plt.imshow(xtv_sb.as_array())
valObjSB = g_sb(xtv_sb)
plt.title('SB TV prox with objective equal to {:.2f}'.format(valObjSB[0]))
plt.show()
print(valObjSB[0])
#%%
# Compare all reconstruction
clims = (-0.2,1.2)
dlims = (-0.2,0.2)
cols = 3
rows = 2
current = 1
fig = plt.figure()
a=fig.add_subplot(rows,cols,current)
a.set_title('FBPD')
imgplot = plt.imshow(x_fbpdtv_denoise.as_array(),vmin=clims[0],vmax=clims[1])
plt.axis('off')
current = current + 1
a=fig.add_subplot(rows,cols,current)
a.set_title('ROF')
imgplot = plt.imshow(xtv_rof.as_array(),vmin=clims[0],vmax=clims[1])
plt.axis('off')
current = current + 1
a=fig.add_subplot(rows,cols,current)
a.set_title('FGP')
imgplot = plt.imshow(xtv_fgp.as_array(),vmin=clims[0],vmax=clims[1])
plt.axis('off')
current = current + 1
a=fig.add_subplot(rows,cols,current)
a.set_title('FBPD - CVX')
imgplot = plt.imshow(x_fbpdtv_denoise.as_array()-xtv_denoise.value,vmin=dlims[0],vmax=dlims[1])
plt.axis('off')
current = current + 1
a=fig.add_subplot(rows,cols,current)
a.set_title('ROF - TV')
imgplot = plt.imshow(xtv_rof.as_array()-xtv_denoise.value,vmin=dlims[0],vmax=dlims[1])
plt.axis('off')
current = current + 1
a=fig.add_subplot(rows,cols,current)
a.set_title('FGP - TV')
imgplot = plt.imshow(xtv_fgp.as_array()-xtv_denoise.value,vmin=dlims[0],vmax=dlims[1])
plt.axis('off')
|
