def test_admm_two_prox_fn(self):
"""Test ADMM algorithm, two prox functions.
"""
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5)) * 1.
prox_fns = [px.norm1(X), px.sum_squares(X, b=B)]
sltn = admm.solve(prox_fns, [], 1.0, eps_rel=1e-5, eps_abs=1e-5)
cvx_X = cvx.Variable((10, 5))
cost = cvx.sum_squares(cvx_X - B) + cvx.norm(cvx_X, 1)
prob = cvx.Problem(cvx.Minimize(cost))
prob.solve()
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-1)
self.assertAlmostEqual(sltn, prob.value)
psi_fns, omega_fns = admm.partition(prox_fns)
sltn = admm.solve(psi_fns, omega_fns, 1.0, eps_rel=1e-5, eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value)
prox_fns = [px.norm1(X)]
quad_funcs = [px.sum_squares(X, b=B)]
sltn = admm.solve(prox_fns,
quad_funcs,
1.0,
eps_rel=1e-5,
eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value)
def test_admm_extra_param(self):
''' With parameters for px.sum_squares '''
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5)) * 1.
prox_fns = [px.norm1(X)]
quad_funcs = [px.sum_squares(X, b=B, alpha=0.1, beta=2., gamma=1, c=B)]
sltn = admm.solve(prox_fns,
quad_funcs,
1.0,
eps_rel=1e-5,
eps_abs=1e-5)
cvx_X = cvx.Variable((10, 5))
cost = 0.1 * cvx.sum_squares(2 * cvx_X - B) + cvx.sum_squares(cvx_X) + \
cvx.norm(cvx_X, 1) + cvx.trace(B.T @ cvx_X)
prob = cvx.Problem(cvx.Minimize(cost))
prob.solve()
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value, eps=1e-3)
prox_fns = [px.norm1(X)]
quad_funcs = [px.sum_squares(X - B, alpha=0.1, beta=2., gamma=1, c=B)]
quad_funcs[0] = absorb_offset(quad_funcs[0])
sltn = admm.solve(prox_fns,
quad_funcs,
1.0,
eps_rel=1e-5,
eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value, eps=1e-3)
prox_fns = [px.norm1(X)]
cvx_X = cvx.Variable((10, 5))
# With linear operators.
kernel = np.array([1, 2, 3])
x = px.Variable(3)
b = np.array([-41, 413, 2])
prox_fns = [px.nonneg(x), px.sum_squares(px.conv(kernel, x), b=b)]
sltn = admm.solve(prox_fns, [], 1.0, eps_abs=1e-5, eps_rel=1e-5)
kernel_mat = np.matrix("2 1 3; 3 2 1; 1 3 2")
cvx_X = cvx.Variable(3)
cost = cvx.norm(kernel_mat * cvx_X - b)
prob = cvx.Problem(cvx.Minimize(cost), [cvx_X >= 0])
prob.solve()
self.assertItemsAlmostEqual(x.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(np.sqrt(sltn), prob.value, eps=2e-2)
prox_fns = [px.nonneg(x)]
quad_funcs = [px.sum_squares(px.conv(kernel, x), b=b)]
sltn = admm.solve(prox_fns,
quad_funcs,
1.0,
eps_abs=1e-5,
eps_rel=1e-5)
self.assertItemsAlmostEqual(x.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(np.sqrt(sltn), prob.value, eps=2e-2)
def testTotalVariation():
im = scipy.misc.ascent()
im = im[:, :, np.newaxis].astype(np.uint8)
fn = proximal.norm1(proximal.grad(proximal.Variable(im.shape)))
thresh = 6
out = fn.prox(thresh, im)
def test_lin_admm_two_prox_fn(self):
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5))
prox_fns = [px.norm1(X), px.sum_squares(X, b=B)]
sltn = ladmm.solve(prox_fns, [],
0.1,
max_iters=500,
eps_rel=1e-5,
eps_abs=1e-5)
cvx_X = cvx.Variable((10, 5))
cost = cvx.sum_squares(cvx_X - B) + cvx.norm(cvx_X, 1)
prob = cvx.Problem(cvx.Minimize(cost))
prob.solve()
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value)
psi_fns, omega_fns = ladmm.partition(prox_fns)
sltn = ladmm.solve(psi_fns,
omega_fns,
0.1,
max_iters=500,
eps_rel=1e-5,
eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value)
# With linear operators.
kernel = np.array([1, 2, 3])
kernel_mat = np.matrix("2 1 3; 3 2 1; 1 3 2")
x = px.Variable(3)
b = np.array([-41, 413, 2])
prox_fns = [px.nonneg(x), px.sum_squares(px.conv(kernel, x), b=b)]
sltn = ladmm.solve(prox_fns, [],
0.1,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5)
cvx_X = cvx.Variable(3)
cost = cvx.norm(kernel_mat * cvx_X - b)
prob = cvx.Problem(cvx.Minimize(cost), [cvx_X >= 0])
prob.solve()
self.assertItemsAlmostEqual(x.value, cvx_X.value, eps=2e-2)
psi_fns, omega_fns = ladmm.partition(prox_fns)
sltn = ladmm.solve(psi_fns,
omega_fns,
0.1,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5)
self.assertItemsAlmostEqual(x.value, cvx_X.value, eps=2e-2)
def testL1NormProx():
f = FlexISP()
v = np.arange(start=-12, stop=12, dtype=float)
thresh = 6
f_out = f.L1_norm_prox(v, thresh)
prox_fn = proximal.norm1(proximal.Variable(v.shape))
prox_out = prox_fn.prox(1 / thresh, v)
np.testing.assert_almost_equal(prox_out, f_out)
def test_admm_single_prox_fn(self):
"""Test ADMM algorithm, single prox function only.
"""
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5)) * 1.
prox_fns = [px.sum_squares(X, b=B)]
sltn = admm.solve(prox_fns, [], 1.0, eps_abs=1e-4, eps_rel=1e-4)
self.assertItemsAlmostEqual(X.value, B, eps=2e-2)
self.assertAlmostEqual(sltn, 0)
prox_fns = [px.norm1(X, b=B, beta=2)]
sltn = admm.solve(prox_fns, [], 1.0)
self.assertItemsAlmostEqual(X.value, B / 2., eps=2e-2)
self.assertAlmostEqual(sltn, 0)
def solver(f, kernel_img, metric, cnn_func, elemental):
"""
Solves the deblurring problem for the given input and kernel image.
:param f: Corrupted input image
:type f: np.ndarray
:param kernel_img: Blur kernel
:type kernel_img: np.ndarray
:param metric: Preinitialized metric
:type metric: proximal.utils.metrics
:param cnn_func: Preinitialized deployment CNN
:type cnn_func: function
:param elemental: General experiment configuration parameters
:type elemental: Dict
:returns: Reconstructed output image
:rtype: np.ndarray
"""
# pylint:disable=no-value-for-parameter
options = px.cg_options(tol=1e-4, num_iters=100, verbose=True)
u = px.Variable(f.shape)
alpha_sumsquare = elemental['alpha_data'] / 2.0
A_u = px.conv(kernel_img, u)
prox_fns = px.sum_squares(A_u - f, alpha=alpha_sumsquare)
if elemental['alpha_tv'] > 0.0:
prox_fns += px.norm1(elemental['alpha_tv'] * px.grad(u))
prox_fns += init_denoising_prior(u,
cnn_func,
sigma=elemental['sigma'],
sigma_scale=elemental['sigma_scale'])
prob = init_problem(prox_fns)
solve_problem(prob,
x0=f.copy(),
metric=metric,
sigma=elemental['sigma'],
lin_solver_options=options)
return np.clip(u.value, 0.0, 1.0)
def test_half_quadratic_splitting_single_prox_fn(self):
"""Test half quadratic splitting.
"""
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5))
prox_fns = [px.sum_squares(X, b=B)]
sltn = hqs.solve(prox_fns, [],
eps_rel=1e-4,
max_iters=100,
max_inner_iters=50)
self.assertItemsAlmostEqual(X.value, B, eps=2e-2)
self.assertAlmostEqual(sltn, 0)
prox_fns = [px.norm1(X, b=B, beta=2)]
sltn = hqs.solve(prox_fns, [],
eps_rel=1e-4,
max_iters=100,
max_inner_iters=50)
self.assertItemsAlmostEqual(X.value, B / 2., eps=2e-2)
self.assertAlmostEqual(sltn, 0)
def test_lin_admm_single_prox_fn(self):
"""Test linearized admm. algorithm.
"""
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5))
prox_fns = [px.sum_squares(X, b=B)]
sltn = ladmm.solve(prox_fns, [],
0.1,
max_iters=500,
eps_rel=1e-5,
eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, B, eps=2e-2)
self.assertAlmostEqual(sltn, 0)
prox_fns = [px.norm1(X, b=B, beta=2)]
sltn = ladmm.solve(prox_fns, [],
0.1,
max_iters=500,
eps_rel=1e-5,
eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, B / 2., eps=2e-2)
self.assertAlmostEqual(sltn, 0, eps=2e-2)
space = odl.uniform_discr([0, 0], [100, 100], [100, 100])
# Create ODL operator for the Laplacian
laplacian = odl.Laplacian(space)
# Create right hand side
phantom = odl.phantom.shepp_logan(space, modified=True)
phantom.show('original image')
rhs = laplacian(phantom)
rhs += odl.phantom.white_noise(space) * np.std(rhs) * 0.1
rhs.show('rhs')
# Convert laplacian to ProxImaL operator
proximal_lang_laplacian = odl.as_proximal_lang_operator(laplacian)
# Convert to array
rhs_arr = rhs.asarray()
# Set up optimization problem
x = proximal.Variable(space.shape)
funcs = [10 * proximal.sum_squares(proximal_lang_laplacian(x) - rhs_arr),
proximal.norm1(proximal.grad(x))]
# Solve the problem using ProxImaL
prob = proximal.Problem(funcs)
prob.solve(verbose=True)
# Convert back to odl and display result
result_odl = space.element(x.value)
result_odl.show('result from ProxImaL')
# Create ODL operator for the Laplacian
laplacian = odl.Laplacian(space)
# Create right hand side
phantom = odl.phantom.shepp_logan(space, modified=True)
phantom.show('original image')
rhs = laplacian(phantom)
rhs += odl.phantom.white_noise(space) * np.std(rhs) * 0.1
rhs.show('rhs')
# Convert laplacian to ProxImaL operator
proximal_lang_laplacian = odl.as_proximal_lang_operator(laplacian)
# Convert to array
rhs_arr = rhs.asarray()
# Set up optimization problem
x = proximal.Variable(space.shape)
funcs = [
10 * proximal.sum_squares(proximal_lang_laplacian(x) - rhs_arr),
proximal.norm1(proximal.grad(x))
]
# Solve the problem using ProxImaL
prob = proximal.Problem(funcs)
prob.solve(verbose=True)
# Convert back to odl and display result
result_odl = space.element(x.value)
result_odl.show('result from ProxImaL')
def solver(f, x0, metric, cnn_func, elemental):
"""
Solves the demosaicking problem for the given input.
:param f: Corrupted input image
:type f: np.ndarray
:param x0: Predemosaicked initialization image
:type x0: np.ndarray
:param metric: Preinitialized metric
:type metric: proximal.utils.metrics
:param cnn_func: Preinitialized deployment CNN
:type cnn_func: function
:param elemental: General experiment configuration parameters
:type elemental: Dict
:returns: Reconstructed output image
:rtype: np.ndarray
"""
# pylint:disable=no-value-for-parameter
options = px.cg_options(tol=1e-4, num_iters=100, verbose=True)
u = px.Variable(f.shape)
A = bayer_mask(f.shape)
A_u = px.mul_elemwise(A, u)
alpha_sumsquare = elemental['alpha_data'] / 2.0
data = px.sum_squares(A_u - f, alpha=alpha_sumsquare)
prox_fns = data
if elemental['alpha_tv'] > 0.0:
prox_fns += px.norm1(elemental['alpha_tv'] * px.grad(u, dims=2))
if elemental['alpha_cross'] > 0.0:
grad_u = px.grad(u, dims=2)
grad_x0 = px.grad(x0, dims=2).value
x0_stacked = np.array([x0, x0]).reshape(x0.shape + (2, ))
u_stacked = px.reshape(px.hstack([u, u]), x0.shape + (2, ))
cross_1 = px.vstack([
px.mul_elemwise(np.roll(x0_stacked, 1, 2), grad_u),
px.mul_elemwise(np.roll(x0_stacked, 2, 2), grad_u)
])
cross_2 = px.vstack([
px.mul_elemwise(np.roll(grad_x0, 1, 2), u_stacked),
px.mul_elemwise(np.roll(grad_x0, 2, 2), u_stacked)
])
prox_fns += px.norm1(0.5 * elemental['alpha_cross'] *
(cross_1 - cross_2))
prox_fns += init_denoising_prior(u,
cnn_func,
sigma=elemental['sigma'],
sigma_scale=elemental['sigma_scale'])
prob = init_problem(prox_fns)
solve_problem(prob,
x0=x0,
metric=metric,
sigma=elemental['sigma'],
lin_solver_options=options)
return np.clip(u.value, 0.0, 1.0)
def test_half_quadratic_splitting(self):
"""Test half quadratic splitting.
"""
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5))
prox_fns = [px.sum_squares(X, b=B)]
sltn = hqs.solve(prox_fns, [],
eps_rel=1e-4,
max_iters=100,
max_inner_iters=50)
self.assertItemsAlmostEqual(X.value, B, places=2)
self.assertAlmostEqual(sltn, 0)
prox_fns = [px.norm1(X, b=B, beta=2)]
sltn = hqs.solve(prox_fns, [],
eps_rel=1e-4,
max_iters=100,
max_inner_iters=50)
self.assertItemsAlmostEqual(X.value, B / 2., places=2)
self.assertAlmostEqual(sltn, 0)
prox_fns = [px.norm1(X), px.sum_squares(X, b=B)]
sltn = hqs.solve(prox_fns, [],
eps_rel=1e-7,
rho_0=1.0,
rho_scale=np.sqrt(2.0) * 2.0,
rho_max=2**16,
max_iters=20,
max_inner_iters=500)
cvx_X = cvx.Variable(10, 5)
cost = cvx.sum_squares(cvx_X - B) + cvx.norm(cvx_X, 1)
prob = cvx.Problem(cvx.Minimize(cost))
prob.solve()
self.assertItemsAlmostEqual(X.value, cvx_X.value, places=2)
self.assertAlmostEqual(sltn, prob.value, places=3)
psi_fns, omega_fns = hqs.partition(prox_fns)
sltn = hqs.solve(psi_fns,
omega_fns,
eps_rel=1e-7,
rho_0=1.0,
rho_scale=np.sqrt(2.0) * 2.0,
rho_max=2**16,
max_iters=20,
max_inner_iters=500)
self.assertItemsAlmostEqual(X.value, cvx_X.value, places=2)
self.assertAlmostEqual(sltn, prob.value, places=3)
# With linear operators.
kernel = np.array([1, 2, 3])
kernel_mat = np.matrix("2 1 3; 3 2 1; 1 3 2")
x = px.Variable(3)
b = np.array([-41, 413, 2])
prox_fns = [px.nonneg(x), px.sum_squares(px.conv(kernel, x), b=b)]
hqs.solve(prox_fns, [],
eps_rel=1e-9,
rho_0=4,
rho_scale=np.sqrt(2.0) * 1.0,
rho_max=2**16,
max_iters=30,
max_inner_iters=500)
cvx_X = cvx.Variable(3)
cost = cvx.norm(kernel_mat * cvx_X - b)
prob = cvx.Problem(cvx.Minimize(cost), [cvx_X >= 0])
prob.solve()
self.assertItemsAlmostEqual(x.value, cvx_X.value, places=0)
psi_fns, omega_fns = hqs.partition(prox_fns)
hqs.solve(psi_fns,
omega_fns,
eps_rel=1e-9,
rho_0=4,
rho_scale=np.sqrt(2.0) * 1.0,
rho_max=2**16,
max_iters=30,
max_inner_iters=500)
self.assertItemsAlmostEqual(x.value, cvx_X.value, places=0)
def test_pock_chambolle(self):
"""Test pock chambolle algorithm.
"""
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5))
prox_fns = [px.sum_squares(X, b=B)]
sltn = pc.solve(prox_fns, [],
1.0,
1.0,
1.0,
eps_rel=1e-5,
eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, B, places=2)
self.assertAlmostEqual(sltn, 0)
prox_fns = [px.norm1(X, b=B, beta=2)]
sltn = pc.solve(prox_fns, [],
1.0,
1.0,
1.0,
eps_rel=1e-5,
eps_abs=1e-5)
self.assertItemsAlmostEqual(X.value, B / 2., places=2)
self.assertAlmostEqual(sltn, 0, places=2)
prox_fns = [px.norm1(X), px.sum_squares(X, b=B)]
sltn = pc.solve(prox_fns, [],
0.5,
1.0,
1.0,
eps_rel=1e-5,
eps_abs=1e-5)
cvx_X = cvx.Variable(10, 5)
cost = cvx.sum_squares(cvx_X - B) + cvx.norm(cvx_X, 1)
prob = cvx.Problem(cvx.Minimize(cost))
prob.solve()
self.assertItemsAlmostEqual(X.value, cvx_X.value, places=2)
self.assertAlmostEqual(sltn, prob.value)
psi_fns, omega_fns = pc.partition(prox_fns)
sltn = pc.solve(psi_fns,
omega_fns,
0.5,
1.0,
1.0,
eps_abs=1e-5,
eps_rel=1e-5)
self.assertItemsAlmostEqual(X.value, cvx_X.value, places=2)
self.assertAlmostEqual(sltn, prob.value)
# With linear operators.
kernel = np.array([1, 2, 3])
kernel_mat = np.matrix("2 1 3; 3 2 1; 1 3 2")
x = px.Variable(3)
b = np.array([-41, 413, 2])
prox_fns = [px.nonneg(x), px.sum_squares(px.conv(kernel, x), b=b)]
sltn = pc.solve(prox_fns, [],
0.1,
0.1,
1.0,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5)
cvx_X = cvx.Variable(3)
cost = cvx.norm(kernel_mat * cvx_X - b)
prob = cvx.Problem(cvx.Minimize(cost), [cvx_X >= 0])
prob.solve()
self.assertItemsAlmostEqual(x.value, cvx_X.value, places=2)
psi_fns, omega_fns = pc.partition(prox_fns)
sltn = pc.solve(psi_fns,
omega_fns,
0.1,
0.1,
1.0,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5)
self.assertItemsAlmostEqual(x.value, cvx_X.value, places=2)
def _test_pock_chambolle(self, impl):
"""Test pock chambolle algorithm.
"""
#print()
#print("----------------------",impl,"-------------------------")
if impl == 'pycuda':
kw = {'adapter': PyCudaAdapter()}
else:
kw = {}
X = px.Variable((10, 5))
B = np.reshape(np.arange(50), (10, 5))
prox_fns = [px.sum_squares(X, b=B)]
sltn = pc.solve(prox_fns, [],
1.0,
1.0,
1.0,
eps_rel=1e-5,
eps_abs=1e-5,
**kw)
self.assertItemsAlmostEqual(X.value, B, eps=2e-2)
self.assertAlmostEqual(sltn, 0)
prox_fns = [px.norm1(X, b=B, beta=2)]
sltn = pc.solve(prox_fns, [],
1.0,
1.0,
1.0,
eps_rel=1e-5,
eps_abs=1e-5,
**kw)
self.assertItemsAlmostEqual(X.value, B / 2., eps=2e-2)
self.assertAlmostEqual(sltn, 0, eps=2e-2)
prox_fns = [px.norm1(X), px.sum_squares(X, b=B)]
#print("----------------------------------------------------")
#print("----------------------------------------------------")
sltn = pc.solve(prox_fns, [],
0.5,
1.0,
1.0,
eps_rel=1e-5,
eps_abs=1e-5,
conv_check=1,
**kw)
cvx_X = cvx.Variable((10, 5))
cost = cvx.sum_squares(cvx_X - B) + cvx.norm(cvx_X, 1)
prob = cvx.Problem(cvx.Minimize(cost))
prob.solve()
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value)
psi_fns, omega_fns = pc.partition(prox_fns)
sltn = pc.solve(psi_fns,
omega_fns,
0.5,
1.0,
1.0,
eps_abs=1e-5,
eps_rel=1e-5,
**kw)
self.assertItemsAlmostEqual(X.value, cvx_X.value, eps=2e-2)
self.assertAlmostEqual(sltn, prob.value)
# With linear operators.
kernel = np.array([1, 2, 3])
kernel_mat = np.matrix("2 1 3; 3 2 1; 1 3 2")
x = px.Variable(3)
b = np.array([-41, 413, 2])
prox_fns = [px.nonneg(x), px.sum_squares(px.conv(kernel, x), b=b)]
sltn = pc.solve(prox_fns, [],
0.1,
0.1,
1.0,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5,
**kw)
cvx_X = cvx.Variable(3)
cost = cvx.norm(kernel_mat * cvx_X - b)
prob = cvx.Problem(cvx.Minimize(cost), [cvx_X >= 0])
prob.solve()
self.assertItemsAlmostEqual(x.value, cvx_X.value, eps=2e-2)
psi_fns, omega_fns = pc.partition(prox_fns)
sltn = pc.solve(psi_fns,
omega_fns,
0.1,
0.1,
1.0,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5,
**kw)
self.assertItemsAlmostEqual(x.value, cvx_X.value, eps=2e-2)
# TODO
# Multiple variables.
x = px.Variable(1)
y = px.Variable(1)
prox_fns = [
px.nonneg(x),
px.sum_squares(vstack([x, y]), b=np.arange(2))
]
sltn = pc.solve(prox_fns, [prox_fns[-1]],
0.1,
0.1,
1.0,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5,
try_diagonalize=False)
self.assertItemsAlmostEqual(x.value, [0])
self.assertItemsAlmostEqual(y.value, [1])
sltn = pc.solve(prox_fns, [prox_fns[-1]],
0.1,
0.1,
1.0,
max_iters=3000,
eps_abs=1e-5,
eps_rel=1e-5,
try_diagonalize=True)
self.assertItemsAlmostEqual(x.value, [0])
self.assertItemsAlmostEqual(y.value, [1])
# Convert ray transform to proximal language operator
proximal_lang_ray_trafo = odl.as_proximal_lang_operator(ray_trafo)
# Create sinogram of forward projected phantom with noise
phantom = odl.phantom.shepp_logan(reco_space, modified=True)
phantom.show('phantom')
data = ray_trafo(phantom)
data += odl.phantom.white_noise(ray_trafo.range) * np.mean(data) * 0.1
data.show('noisy data')
# Convert to array for ProxImaL
rhs_arr = data.asarray()
# Set up optimization problem
# Note that proximal is not aware of the underlying space and only works with
# matrices. Hence the norm in proximal does not match the norm in the ODL space
# exactly.
x = proximal.Variable(reco_space.shape)
funcs = [proximal.sum_squares(proximal_lang_ray_trafo(x) - rhs_arr),
0.2 * proximal.norm1(proximal.grad(x)),
proximal.nonneg(x),
proximal.nonneg(1 - x)]
# Solve the problem using ProxImaL
prob = proximal.Problem(funcs)
prob.solve(verbose=True)
# Convert back to odl and display result
result_odl = reco_space.element(x.value)
result_odl.show('ProxImaL result')