def test_CGLS(self):
print ("Test CGLS")
ig = ImageGeometry(124,153,154)
x_init = ImageData(geometry=ig)
b = x_init.copy()
# fill with random numbers
b.fill(numpy.random.random(x_init.shape))
identity = TomoIdentity(geometry=ig)
alg = CGLS(x_init=x_init, operator=identity, data=b)
alg.max_iteration = 1
alg.run(20, verbose=True)
self.assertNumpyArrayAlmostEqual(alg.x.as_array(), b.as_array())
def test_GradientDescent(self):
print ("Test GradientDescent")
ig = ImageGeometry(12,13,14)
x_init = ImageData(geometry=ig)
b = x_init.copy()
# fill with random numbers
b.fill(numpy.random.random(x_init.shape))
identity = TomoIdentity(geometry=ig)
norm2sq = Norm2sq(identity, b)
alg = GradientDescent(x_init=x_init,
objective_function=norm2sq,
rate=0.3)
alg.max_iteration = 20
alg.run(20, verbose=True)
self.assertNumpyArrayAlmostEqual(alg.x.as_array(), b.as_array())
def test_FISTA(self):
print ("Test FISTA")
ig = ImageGeometry(127,139,149)
x_init = ImageData(geometry=ig)
b = x_init.copy()
# fill with random numbers
b.fill(numpy.random.random(x_init.shape))
x_init = ImageData(geometry=ig)
x_init.fill(numpy.random.random(x_init.shape))
identity = TomoIdentity(geometry=ig)
norm2sq = Norm2sq(identity, b)
opt = {'tol': 1e-4, 'memopt':False}
alg = FISTA(x_init=x_init, f=norm2sq, g=None, opt=opt)
alg.max_iteration = 2
alg.run(20, verbose=True)
self.assertNumpyArrayAlmostEqual(alg.x.as_array(), b.as_array())
alg.run(20, verbose=True)
self.assertNumpyArrayAlmostEqual(alg.x.as_array(), b.as_array())
# 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 = TomoIdentity(ig)
# Data and add noise
y = I.direct(Phantom)
y.array = y.array + 0.1*np.random.randn(N, N)
plt.imshow(y.array)
plt.title('Noisy image')
plt.show()
###################
# Data fidelity term
f_denoise = Norm2sq(I,y,c=0.5,memopt=True)
# 1-norm regulariser
def test_FISTA_denoise_cvx(self):
if not cvx_not_installable:
opt = {'memopt': True}
N = 64
ig = ImageGeometry(voxel_num_x=N, voxel_num_y=N)
Phantom = ImageData(geometry=ig)
x = Phantom.as_array()
x[int(round(N / 4)):int(round(3 * N / 4)),
int(round(N / 4)):int(round(3 * N / 4))] = 0.5
x[int(round(N / 8)):int(round(7 * N / 8)),
int(round(3 * N / 8)):int(round(5 * N / 8))] = 1
# Identity operator for denoising
I = TomoIdentity(ig)
# Data and add noise
y = I.direct(Phantom)
y.array = y.array + 0.1 * np.random.randn(N, N)
# Data fidelity term
f_denoise = Norm2sq(I, y, c=0.5, memopt=True)
# 1-norm regulariser
lam1_denoise = 1.0
g1_denoise = Norm1(lam1_denoise)
# Initial guess
x_init_denoise = ImageData(np.zeros((N, N)))
# Combine with least squares and solve using generic FISTA implementation
x_fista1_denoise, it1_denoise, timing1_denoise, \
criter1_denoise = \
FISTA(x_init_denoise, f_denoise, g1_denoise, opt=opt)
print(x_fista1_denoise)
print(criter1_denoise[-1])
# Now denoise LS + 1-norm with FBPD
x_fbpd1_denoise, itfbpd1_denoise, timingfbpd1_denoise,\
criterfbpd1_denoise = \
FBPD(x_init_denoise, I, None, f_denoise, g1_denoise)
print(x_fbpd1_denoise)
print(criterfbpd1_denoise[-1])
# Compare to CVXPY
# Construct the problem.
x1_denoise = Variable(N**2, 1)
objective1_denoise = Minimize(
0.5 * sum_squares(x1_denoise - y.array.flatten()) +
lam1_denoise * norm(x1_denoise, 1))
prob1_denoise = Problem(objective1_denoise)
# The optimal objective is returned by prob.solve().
result1_denoise = prob1_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 1-norm solution and objective value:"
)
print(x1_denoise.value)
print(objective1_denoise.value)
self.assertNumpyArrayAlmostEqual(x_fista1_denoise.array.flatten(),
x1_denoise.value, 5)
self.assertNumpyArrayAlmostEqual(x_fbpd1_denoise.array.flatten(),
x1_denoise.value, 5)
x1_cvx = x1_denoise.value
x1_cvx.shape = (N, N)
# Now TV with FBPD
lam_tv = 0.1
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, gtv.op, None, f_denoise, gtv, opt=opt_tv)
print(x_fbpdtv_denoise)
print(criterfbpdtv_denoise[-1])
# 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 1-norm solution and objective value:"
)
print(xtv_denoise.value)
print(objectivetv_denoise.value)
self.assertNumpyArrayAlmostEqual(x_fbpdtv_denoise.as_array(),
xtv_denoise.value, 1)
else:
self.assertTrue(cvx_not_installable)