PPP (with BM3D) Image Deconvolution (ADMM Solver)

This example demonstrates the solution of an image deconvolution problem using the ADMM Plug-and-Play Priors (PPP) algorithm [50], with the BM3D [16] denoiser.

import numpy as np

from xdesign import Foam, discrete_phantom

import scico.numpy as snp
from scico import functional, linop, loss, metric, plot, random
from scico.optimize.admm import ADMM, LinearSubproblemSolver
from scico.util import device_info
plot.config_notebook_plotting()

Create a ground truth image.

np.random.seed(1234)
N = 512  # image size
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 array

Set up forward operator and test signal consisting of blurred signal with additive Gaussian noise.

n = 5  # convolution kernel size
σ = 20.0 / 255  # noise level

psf = snp.ones((n, n)) / (n * n)
A = linop.Convolve(h=psf, input_shape=x_gt.shape)

Ax = A(x_gt)  # blurred image
noise, key = random.randn(Ax.shape)
y = Ax + σ * noise

Set up ADMM solver.

f = loss.SquaredL2Loss(y=y, A=A)
C = linop.Identity(x_gt.shape)

λ = 20.0 / 255  # BM3D regularization strength
g = λ * functional.BM3D()

ρ = 1.0  # ADMM penalty parameter
maxiter = 10  # number of ADMM iterations

solver = ADMM(
    f=f,
    g_list=[g],
    C_list=[C],
    rho_list=[ρ],
    x0=A.T @ y,
    maxiter=maxiter,
    subproblem_solver=LinearSubproblemSolver(cg_kwargs={"tol": 1e-3, "maxiter": 100}),
    itstat_options={"display": True},
)

Run the solver.

print(f"Solving on {device_info()}\n")
x = solver.solve()
x = snp.clip(x, 0, 1)
hist = solver.itstat_object.history(transpose=True)

Solving on GPU (NVIDIA GeForce RTX 2080 Ti)

Iter  Time      Prml Rsdl  Dual Rsdl  CG It  CG Res
------------------------------------------------------
   0  7.79e+00  9.643e+00  1.472e+01      3  2.084e-04
   1  1.42e+01  3.772e+00  9.269e+00      3  2.291e-04
   2  2.01e+01  1.408e+00  6.593e+00      2  6.981e-04
   3  2.59e+01  1.081e+00  4.910e+00      2  4.567e-04
   4  3.18e+01  9.272e-01  3.868e+00      2  3.328e-04
   5  3.76e+01  8.377e-01  3.187e+00      2  2.492e-04
   6  4.35e+01  7.924e-01  2.704e+00      2  1.942e-04
   7  4.93e+01  6.816e-01  2.305e+00      1  9.075e-04
   8  5.50e+01  7.081e-01  2.061e+00      1  6.446e-04
   9  6.08e+01  6.930e-01  1.859e+00      1  6.303e-04

Show the recovered image.

fig, ax = plot.subplots(nrows=1, ncols=3, figsize=(15, 5))
plot.imview(x_gt, title="Ground truth", fig=fig, ax=ax[0])
nc = n // 2
yc = snp.clip(y[nc:-nc, nc:-nc], 0, 1)
plot.imview(y, title="Blurred, noisy image: %.2f (dB)" % metric.psnr(x_gt, yc), fig=fig, ax=ax[1])
plot.imview(x, title="Deconvolved image: %.2f (dB)" % metric.psnr(x_gt, x), fig=fig, ax=ax[2])
fig.show()

Plot convergence statistics.

plot.plot(
    snp.vstack((hist.Prml_Rsdl, hist.Dual_Rsdl)).T,
    ptyp="semilogy",
    title="Residuals",
    xlbl="Iteration",
    lgnd=("Primal", "Dual"),
)