TV-Regularized Sparse-View CT Reconstruction (ASTRA Projector)#
This example demonstrates solution of a sparse-view CT reconstruction problem with isotropic total variation (TV) regularization
\[\mathrm{argmin}_{\mathbf{x}} \; (1/2) \| \mathbf{y} - A \mathbf{x}
\|_2^2 + \lambda \| C \mathbf{x} \|_{2,1} \;,\]
where \(A\) is the X-ray transform (the CT forward projection operator), \(\mathbf{y}\) is the sinogram, \(C\) is a 2D finite difference operator, and \(\mathbf{x}\) is the desired image. This example uses the CT projector provided by the astra package, while the companion example script uses the projector integrated into scico.
[1]:
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable
from xdesign import Foam, discrete_phantom
import scico.numpy as snp
from scico import functional, linop, loss, metric, plot
from scico.linop.xray.astra import XRayTransform
from scico.optimize.admm import ADMM, LinearSubproblemSolver
from scico.util import device_info
plot.config_notebook_plotting()
Create a ground truth image.
[2]:
N = 512 # phantom size
np.random.seed(1234)
x_gt = discrete_phantom(Foam(size_range=[0.075, 0.0025], gap=1e-3, porosity=1), size=N)
x_gt = snp.array(x_gt) # convert to jax type
Configure CT projection operator and generate synthetic measurements.
[3]:
n_projection = 45 # number of projections
angles = np.linspace(0, np.pi, n_projection) # evenly spaced projection angles
A = XRayTransform(x_gt.shape, 1, N, angles) # CT projection operator
y = A @ x_gt # sinogram
Set up ADMM solver object.
[4]:
λ = 2e0 # ℓ1 norm regularization parameter
ρ = 5e0 # ADMM penalty parameter
maxiter = 25 # number of ADMM iterations
cg_tol = 1e-4 # CG relative tolerance
cg_maxiter = 25 # maximum CG iterations per ADMM iteration
# The append=0 option makes the results of horizontal and vertical
# finite differences the same shape, which is required for the L21Norm,
# which is used so that g(Cx) corresponds to isotropic TV.
C = linop.FiniteDifference(input_shape=x_gt.shape, append=0)
g = λ * functional.L21Norm()
f = loss.SquaredL2Loss(y=y, A=A)
x0 = snp.clip(A.fbp(y), 0, 1.0)
solver = ADMM(
f=f,
g_list=[g],
C_list=[C],
rho_list=[ρ],
x0=x0,
maxiter=maxiter,
subproblem_solver=LinearSubproblemSolver(cg_kwargs={"tol": cg_tol, "maxiter": cg_maxiter}),
itstat_options={"display": True, "period": 5},
)
Run the solver.
[5]:
print(f"Solving on {device_info()}\n")
solver.solve()
hist = solver.itstat_object.history(transpose=True)
x_reconstruction = snp.clip(solver.x, 0, 1.0)
Solving on GPU (NVIDIA GeForce RTX 2080 Ti)
Iter Time Objective Prml Rsdl Dual Rsdl CG It CG Res
-----------------------------------------------------------------
0 2.91e+00 8.859e+03 1.112e+02 3.057e+02 25 3.926e-04
5 4.99e+00 2.254e+04 2.335e+01 1.595e+01 25 1.098e-04
10 5.98e+00 2.679e+04 1.315e+01 3.201e+00 0 9.710e-05
15 9.88e+00 2.714e+04 9.564e+00 6.698e+00 17 7.857e-05
20 1.05e+01 2.755e+04 5.713e+00 9.953e-01 0 9.503e-05
24 1.09e+01 2.780e+04 4.691e+00 8.335e-01 2 9.176e-05
Show the recovered image.
[6]:
fig, ax = plot.subplots(nrows=1, ncols=3, figsize=(15, 5))
plot.imview(x_gt, title="Ground truth", cbar=None, fig=fig, ax=ax[0])
plot.imview(
x0,
title="FBP Reconstruction: \nSNR: %.2f (dB), MAE: %.3f"
% (metric.snr(x_gt, x0), metric.mae(x_gt, x0)),
cbar=None,
fig=fig,
ax=ax[1],
)
plot.imview(
x_reconstruction,
title="TV Reconstruction\nSNR: %.2f (dB), MAE: %.3f"
% (metric.snr(x_gt, x_reconstruction), metric.mae(x_gt, x_reconstruction)),
fig=fig,
ax=ax[2],
)
divider = make_axes_locatable(ax[2])
cax = divider.append_axes("right", size="5%", pad=0.2)
fig.colorbar(ax[2].get_images()[0], cax=cax, label="arbitrary units")
fig.show()
Plot convergence statistics.
[7]:
fig, ax = plot.subplots(nrows=1, ncols=2, figsize=(12, 5))
plot.plot(
hist.Objective,
title="Objective function",
xlbl="Iteration",
ylbl="Functional value",
fig=fig,
ax=ax[0],
)
plot.plot(
snp.vstack((hist.Prml_Rsdl, hist.Dual_Rsdl)).T,
ptyp="semilogy",
title="Residuals",
xlbl="Iteration",
lgnd=("Primal", "Dual"),
fig=fig,
ax=ax[1],
)
fig.show()