Note
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Denoising#
This tutorial considers the classical problem of denoising of images affected by either random noise or salt-and-pepper noise using proximal algorithms.
The overall cost function to minimize is written in the following form:
\[\argmin_\mathbf{u} \frac{1}{2}\|\mathbf{u}-\mathbf{f}\|_2^2 + \sigma J(\mathbf{u})\]
where the L2 norm in the data term can be replaced by a L1 norm for salt-and-pepper (outlier like noise).
For both examples we investigate with different choices of regularization:
L2 on Gradient \(J(\mathbf{u}) = \|\nabla \mathbf{u}\|_2^2\)
Anisotropic TV \(J(\mathbf{u}) = \|\nabla \mathbf{u}\|_1\)
Isotropic TV \(J(\mathbf{u}) = \|\nabla \mathbf{u}\|_{2,1}\)
import numpy as np
import matplotlib.pyplot as plt
import pylops
from scipy import misc
import pyproximal
plt.close('all')
Let’s start by loading a sample image and adding some noise
/home/docs/checkouts/readthedocs.org/user_builds/pyproximal/checkouts/latest/tutorials/denoising.py:36: DeprecationWarning: scipy.misc.ascent has been deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. Dataset methods have moved into the scipy.datasets module. Use scipy.datasets.ascent instead.
img = misc.ascent()
We can now define a pylops.Gradient operator that we are going to
use for all regularizers
We then consider the first regularization (L2 norm on Gradient). We expect to get a smooth image where noise is suppressed by sharp edges in the original image are however lost.
# L2 data term
l2 = pyproximal.L2(b=noise_img.ravel())
# L2 regularization
sigma = 2.
thik = pyproximal.L2(sigma=sigma)
# Solve
tau = 1.
mu = 1. / (tau*L)
iml2 = pyproximal.optimization.primal.LinearizedADMM(l2, thik,
Gop, tau=tau,
mu=mu,
x0=np.zeros_like(img.ravel()),
niter=100)[0]
iml2 = iml2.reshape(img.shape)
Let’s try now to use TV regularization, both anisotropic and isotropic
# L2 data term
l2 = pyproximal.L2(b=noise_img.ravel())
# Anisotropic TV
sigma = .1
l1 = pyproximal.L1(sigma=sigma)
# Solve
tau = 1.
mu = tau / L
iml1 = pyproximal.optimization.primal.LinearizedADMM(l2, l1, Gop, tau=tau,
mu=mu, x0=np.zeros_like(img.ravel()),
niter=100)[0]
iml1 = iml1.reshape(img.shape)
# Isotropic TV with Proximal Gradient
sigma = .1
tv = pyproximal.TV(dims=img.shape, sigma=sigma)
# Solve
tau = 1 / L
imtv = pyproximal.optimization.primal.ProximalGradient(l2, tv, tau=tau, x0=np.zeros_like(img.ravel()),
niter=100)
imtv = imtv.reshape(img.shape)
# Isotropic TV with Primal Dual
sigma = .1
l1iso = pyproximal.L21(ndim=2, sigma=sigma)
# Solve
tau = 1 / np.sqrt(L)
mu = 1. / (tau*L)
iml12 = pyproximal.optimization.primaldual.PrimalDual(l2, l1iso, Gop,
tau=tau, mu=mu, theta=1.,
x0=np.zeros_like(img.ravel()),
niter=100)
iml12 = iml12.reshape(img.shape)
fig, axs = plt.subplots(1, 5, figsize=(14, 4))
axs[0].imshow(img, cmap='gray', vmin=0, vmax=1)
axs[0].set_title('Original')
axs[0].axis('off')
axs[0].axis('tight')
axs[1].imshow(noise_img, cmap='gray', vmin=0, vmax=1)
axs[1].set_title('Noisy')
axs[1].axis('off')
axs[1].axis('tight')
axs[2].imshow(iml1, cmap='gray', vmin=0, vmax=1)
axs[2].set_title('TVaniso')
axs[2].axis('off')
axs[2].axis('tight')
axs[3].imshow(imtv, cmap='gray', vmin=0, vmax=1)
axs[3].set_title('TViso (with ProxGrad)')
axs[3].axis('off')
axs[3].axis('tight')
axs[4].imshow(iml12, cmap='gray', vmin=0, vmax=1)
axs[4].set_title('TViso (with PD)')
axs[4].axis('off')
axs[4].axis('tight')
plt.tight_layout()
Finally we consider an example where the original image is corrupted by salt-and-pepper noise.
# Add salt and pepper noise
noiseperc = .1
isalt = np.random.permutation(np.arange(ny*nx))[:int(noiseperc*ny*nx)]
ipepper = np.random.permutation(np.arange(ny*nx))[:int(noiseperc*ny*nx)]
noise_img = img.copy().ravel()
noise_img[isalt] = img.max()
noise_img[ipepper] = img.min()
noise_img = noise_img.reshape(ny, nx)
Here we compare L2 and L1 norms for the data term L2 data term
l2 = pyproximal.L2(b=noise_img.ravel())
# L1 regularization (isotropic TV)
sigma = .2
l1iso = pyproximal.L21(ndim=2, sigma=sigma)
# Solve
tau = .1
mu = 1. / (tau*L)
iml12_l2 = pyproximal.optimization.primaldual.PrimalDual(l2, l1iso, Gop,
tau=tau, mu=mu, theta=1.,
x0=np.zeros_like(noise_img).ravel(),
niter=100, show=True)
iml12_l2 = iml12_l2.reshape(img.shape)
# L1 data term
l1 = pyproximal.L1(g=noise_img.ravel())
# L1 regularization (isotropic TV)
sigma = .7
l1iso = pyproximal.L21(ndim=2, sigma=sigma)
# Solve
tau = 1.
mu = 1. / (tau*L)
iml12_l1 = pyproximal.optimization.primaldual.PrimalDual(l1, l1iso, Gop,
tau=tau, mu=mu, theta=1.,
x0=np.zeros_like(noise_img).ravel(),
niter=100, show=True)
iml12_l1 = iml12_l1.reshape(img.shape)
fig, axs = plt.subplots(2, 2, figsize=(14, 14))
axs[0][0].imshow(img, cmap='gray', vmin=0, vmax=1)
axs[0][0].set_title('Original')
axs[0][0].axis('off')
axs[0][0].axis('tight')
axs[0][1].imshow(noise_img, cmap='gray', vmin=0, vmax=1)
axs[0][1].set_title('Noisy')
axs[0][1].axis('off')
axs[0][1].axis('tight')
axs[1][0].imshow(iml12_l2, cmap='gray', vmin=0, vmax=1)
axs[1][0].set_title('L2data + TViso')
axs[1][0].axis('off')
axs[1][0].axis('tight')
axs[1][1].imshow(iml12_l1, cmap='gray', vmin=0, vmax=1)
axs[1][1].set_title('L1data + TViso')
axs[1][1].axis('off')
axs[1][1].axis('tight')
plt.tight_layout()
Primal-dual: min_x f(Ax) + x^T z + g(x)
---------------------------------------------------------
Proximal operator (f): <class 'pyproximal.proximal.L2.L2'>
Proximal operator (g): <class 'pyproximal.proximal.L21.L21'>
Linear operator (A): <class 'pylops.basicoperators.gradient.Gradient'>
Additional vector (z): None
tau = 0.1 mu = 1.25
theta = 1.00 niter = 100
Itn x[0] f g z^x J = f + g + z^x
1 2.95900e-02 2.328e+04 1.474e+03 0.000e+00 2.476e+04
2 4.97650e-02 2.021e+04 1.654e+03 0.000e+00 2.186e+04
3 6.61798e-02 1.779e+04 1.606e+03 0.000e+00 1.939e+04
4 8.11699e-02 1.579e+04 1.556e+03 0.000e+00 1.735e+04
5 9.56866e-02 1.414e+04 1.539e+03 0.000e+00 1.568e+04
6 1.10187e-01 1.276e+04 1.542e+03 0.000e+00 1.430e+04
7 1.24702e-01 1.161e+04 1.556e+03 0.000e+00 1.316e+04
8 1.38884e-01 1.065e+04 1.578e+03 0.000e+00 1.222e+04
9 1.52242e-01 9.842e+03 1.603e+03 0.000e+00 1.145e+04
10 1.64381e-01 9.172e+03 1.630e+03 0.000e+00 1.080e+04
11 1.75147e-01 8.613e+03 1.656e+03 0.000e+00 1.027e+04
21 2.50045e-01 6.262e+03 1.826e+03 0.000e+00 8.088e+03
31 2.95632e-01 5.856e+03 1.892e+03 0.000e+00 7.748e+03
41 3.05298e-01 5.774e+03 1.916e+03 0.000e+00 7.690e+03
51 3.10402e-01 5.754e+03 1.925e+03 0.000e+00 7.678e+03
61 3.12550e-01 5.748e+03 1.927e+03 0.000e+00 7.675e+03
71 3.12918e-01 5.745e+03 1.928e+03 0.000e+00 7.673e+03
81 3.13311e-01 5.744e+03 1.928e+03 0.000e+00 7.672e+03
91 3.13517e-01 5.744e+03 1.928e+03 0.000e+00 7.672e+03
92 3.13527e-01 5.744e+03 1.928e+03 0.000e+00 7.672e+03
93 3.13536e-01 5.744e+03 1.928e+03 0.000e+00 7.672e+03
94 3.13544e-01 5.744e+03 1.928e+03 0.000e+00 7.672e+03
95 3.13551e-01 5.744e+03 1.928e+03 0.000e+00 7.672e+03
96 3.13557e-01 5.744e+03 1.928e+03 0.000e+00 7.672e+03
97 3.13562e-01 5.744e+03 1.927e+03 0.000e+00 7.672e+03
98 3.13566e-01 5.744e+03 1.927e+03 0.000e+00 7.671e+03
99 3.13570e-01 5.744e+03 1.927e+03 0.000e+00 7.671e+03
100 3.13573e-01 5.744e+03 1.927e+03 0.000e+00 7.671e+03
Total time (s) = 2.73
---------------------------------------------------------
Primal-dual: min_x f(Ax) + x^T z + g(x)
---------------------------------------------------------
Proximal operator (f): <class 'pyproximal.proximal.L1.L1'>
Proximal operator (g): <class 'pyproximal.proximal.L21.L21'>
Linear operator (A): <class 'pylops.basicoperators.gradient.Gradient'>
Additional vector (z): None
tau = 1.0 mu = 0.125
theta = 1.00 niter = 100
Itn x[0] f g z^x J = f + g + z^x
1 3.25490e-01 9.646e+04 5.675e+04 0.000e+00 1.532e+05
2 3.25490e-01 9.646e+04 5.675e+04 0.000e+00 1.532e+05
3 3.25490e-01 9.461e+04 5.122e+04 0.000e+00 1.458e+05
4 3.25490e-01 9.028e+04 3.647e+04 0.000e+00 1.267e+05
5 3.25490e-01 8.691e+04 2.683e+04 0.000e+00 1.137e+05
6 3.25490e-01 8.564e+04 2.375e+04 0.000e+00 1.094e+05
7 3.25490e-01 8.555e+04 2.110e+04 0.000e+00 1.067e+05
8 3.25490e-01 8.663e+04 1.727e+04 0.000e+00 1.039e+05
9 3.25490e-01 8.783e+04 1.470e+04 0.000e+00 1.025e+05
10 3.25490e-01 8.878e+04 1.324e+04 0.000e+00 1.020e+05
11 3.25490e-01 8.933e+04 1.210e+04 0.000e+00 1.014e+05
21 3.25490e-01 8.939e+04 7.485e+03 0.000e+00 9.687e+04
31 3.25490e-01 8.950e+04 6.782e+03 0.000e+00 9.628e+04
41 3.25490e-01 8.956e+04 6.542e+03 0.000e+00 9.610e+04
51 3.25490e-01 8.959e+04 6.410e+03 0.000e+00 9.600e+04
61 3.25490e-01 8.961e+04 6.319e+03 0.000e+00 9.593e+04
71 3.25490e-01 8.962e+04 6.244e+03 0.000e+00 9.587e+04
81 3.25490e-01 8.963e+04 6.190e+03 0.000e+00 9.582e+04
91 3.25490e-01 8.963e+04 6.147e+03 0.000e+00 9.578e+04
92 3.25490e-01 8.963e+04 6.143e+03 0.000e+00 9.577e+04
93 3.25490e-01 8.963e+04 6.139e+03 0.000e+00 9.577e+04
94 3.25490e-01 8.963e+04 6.134e+03 0.000e+00 9.576e+04
95 3.25490e-01 8.963e+04 6.130e+03 0.000e+00 9.576e+04
96 3.25490e-01 8.963e+04 6.126e+03 0.000e+00 9.576e+04
97 3.25490e-01 8.963e+04 6.122e+03 0.000e+00 9.575e+04
98 3.25490e-01 8.963e+04 6.117e+03 0.000e+00 9.575e+04
99 3.25490e-01 8.963e+04 6.113e+03 0.000e+00 9.574e+04
100 3.25490e-01 8.963e+04 6.110e+03 0.000e+00 9.574e+04
Total time (s) = 2.16
---------------------------------------------------------
Total running time of the script: (0 minutes 55.593 seconds)