Greyscale ℓ1-TV Denoising¶
This example demonstrates the use of class tvl1.TVL1Deconv
for removing salt & pepper noise from a greyscale image using Total
Variation regularization with an ℓ1 data fidelity term (ℓ1-TV
denoising). (This class is primarily intended for deconvolution
problems, but can be applied to denoising problems by choosing an
impulse filter as the blurring kernel.)
from __future__ import print_function
from builtins import input
import numpy as np
from sporco.admm import tvl1
from sporco import util
from sporco import signal
from sporco import metric
from sporco import plot
plot.config_notebook_plotting()
Load reference image.
img = util.ExampleImages().image('monarch.png', scaled=True,
idxexp=np.s_[:,160:672], gray=True)
Construct test image corrupted by 20% salt & pepper noise.
np.random.seed(12345)
imgn = signal.spnoise(img, 0.2)
Set regularization parameter and options for ℓ1-TV deconvolution solver. The regularization parameter used here has been manually selected for good performance.
lmbda = 8e-1
opt = tvl1.TVL1Deconv.Options({'Verbose': True, 'MaxMainIter': 200,
'RelStopTol': 5e-3, 'gEvalY': False,
'AutoRho': {'Enabled': True}})
Create solver object and solve, returning the the denoised image
imgr
.
b = tvl1.TVL1Deconv(np.ones((1,1)), imgn, lmbda, opt)
imgr = b.solve()
Itn Fnc DFid RegTV r s ρ
----------------------------------------------------------------
0 4.69e+04 2.97e+04 2.15e+04 3.18e-01 2.76e-01 1.70e+00
1 4.03e+04 2.95e+04 1.35e+04 2.47e-01 4.19e-01 1.70e+00
2 6.51e+04 4.39e+04 2.64e+04 2.67e-01 4.06e-01 1.31e+00
3 4.59e+04 3.08e+04 1.89e+04 2.35e-01 1.97e-01 1.06e+00
4 4.48e+04 3.02e+04 1.82e+04 2.03e-01 2.44e-01 1.06e+00
5 5.09e+04 3.55e+04 1.93e+04 1.90e-01 1.85e-01 9.67e-01
6 4.29e+04 3.12e+04 1.45e+04 1.57e-01 1.44e-01 9.67e-01
7 4.12e+04 2.97e+04 1.44e+04 1.34e-01 1.30e-01 9.67e-01
8 4.18e+04 3.08e+04 1.37e+04 1.17e-01 1.13e-01 9.67e-01
9 3.92e+04 2.97e+04 1.19e+04 9.93e-02 8.77e-02 9.67e-01
10 3.81e+04 2.88e+04 1.17e+04 8.67e-02 7.93e-02 9.67e-01
11 3.79e+04 2.93e+04 1.08e+04 7.50e-02 6.41e-02 9.67e-01
12 3.67e+04 2.86e+04 1.01e+04 6.51e-02 5.83e-02 9.67e-01
13 3.61e+04 2.83e+04 9.87e+03 5.74e-02 4.86e-02 9.67e-01
14 3.58e+04 2.84e+04 9.23e+03 5.00e-02 4.02e-02 9.67e-01
15 3.51e+04 2.81e+04 8.74e+03 4.21e-02 4.06e-02 1.08e+00
16 3.49e+04 2.79e+04 8.71e+03 3.80e-02 3.28e-02 1.08e+00
17 3.44e+04 2.78e+04 8.23e+03 3.31e-02 2.76e-02 1.08e+00
18 3.43e+04 2.78e+04 8.13e+03 2.91e-02 2.79e-02 1.18e+00
19 3.40e+04 2.76e+04 8.01e+03 2.63e-02 2.26e-02 1.18e+00
20 3.38e+04 2.76e+04 7.81e+03 2.37e-02 1.98e-02 1.18e+00
21 3.38e+04 2.76e+04 7.76e+03 2.18e-02 1.80e-02 1.18e+00
22 3.36e+04 2.75e+04 7.63e+03 1.94e-02 1.71e-02 1.30e+00
23 3.35e+04 2.75e+04 7.56e+03 1.80e-02 1.45e-02 1.30e+00
24 3.34e+04 2.74e+04 7.47e+03 1.64e-02 1.46e-02 1.45e+00
25 3.33e+04 2.74e+04 7.46e+03 1.54e-02 1.25e-02 1.45e+00
26 3.33e+04 2.73e+04 7.39e+03 1.43e-02 1.26e-02 1.61e+00
27 3.33e+04 2.74e+04 7.38e+03 1.36e-02 1.11e-02 1.61e+00
28 3.32e+04 2.73e+04 7.34e+03 1.27e-02 1.14e-02 1.78e+00
29 3.32e+04 2.73e+04 7.34e+03 1.21e-02 9.95e-03 1.78e+00
30 3.31e+04 2.73e+04 7.31e+03 1.15e-02 1.03e-02 1.97e+00
31 3.31e+04 2.73e+04 7.31e+03 1.10e-02 9.39e-03 1.97e+00
32 3.31e+04 2.73e+04 7.30e+03 1.06e-02 9.01e-03 1.97e+00
33 3.31e+04 2.73e+04 7.30e+03 1.02e-02 8.35e-03 1.97e+00
34 3.31e+04 2.72e+04 7.29e+03 9.78e-03 8.62e-03 2.18e+00
35 3.31e+04 2.72e+04 7.29e+03 9.44e-03 8.13e-03 2.18e+00
36 3.30e+04 2.72e+04 7.28e+03 9.12e-03 7.70e-03 2.18e+00
37 3.30e+04 2.72e+04 7.28e+03 8.82e-03 7.35e-03 2.18e+00
38 3.30e+04 2.72e+04 7.28e+03 8.53e-03 7.07e-03 2.18e+00
39 3.30e+04 2.72e+04 7.28e+03 8.20e-03 7.23e-03 2.39e+00
40 3.30e+04 2.72e+04 7.28e+03 7.93e-03 6.94e-03 2.39e+00
41 3.30e+04 2.72e+04 7.28e+03 7.68e-03 6.64e-03 2.39e+00
42 3.30e+04 2.72e+04 7.28e+03 7.43e-03 6.42e-03 2.39e+00
43 3.30e+04 2.72e+04 7.27e+03 7.20e-03 6.22e-03 2.39e+00
44 3.30e+04 2.72e+04 7.27e+03 6.96e-03 5.95e-03 2.39e+00
45 3.30e+04 2.72e+04 7.27e+03 6.74e-03 5.77e-03 2.39e+00
46 3.30e+04 2.72e+04 7.27e+03 6.53e-03 5.54e-03 2.39e+00
47 3.30e+04 2.72e+04 7.27e+03 6.32e-03 5.31e-03 2.39e+00
48 3.30e+04 2.71e+04 7.27e+03 6.12e-03 5.13e-03 2.39e+00
49 3.30e+04 2.71e+04 7.27e+03 5.93e-03 4.95e-03 2.39e+00
50 3.30e+04 2.71e+04 7.26e+03 5.75e-03 4.74e-03 2.39e+00
51 3.29e+04 2.71e+04 7.26e+03 5.53e-03 4.94e-03 2.64e+00
52 3.29e+04 2.71e+04 7.26e+03 5.35e-03 4.77e-03 2.64e+00
53 3.29e+04 2.71e+04 7.26e+03 5.18e-03 4.56e-03 2.64e+00
54 3.29e+04 2.71e+04 7.26e+03 5.02e-03 4.44e-03 2.64e+00
55 3.29e+04 2.71e+04 7.26e+03 4.86e-03 4.33e-03 2.64e+00
----------------------------------------------------------------
Display solve time and denoising performance.
print("TVL1Deconv solve time: %5.2f s" % b.timer.elapsed('solve'))
print("Noisy image PSNR: %5.2f dB" % metric.psnr(img, imgn))
print("Denoised image PSNR: %5.2f dB" % metric.psnr(img, imgr))
TVL1Deconv solve time: 9.50 s
Noisy image PSNR: 11.32 dB
Denoised image PSNR: 28.65 dB
Display reference, corrupted, and denoised images.
fig = plot.figure(figsize=(20, 5))
plot.subplot(1, 3, 1)
plot.imview(img, title='Reference', fig=fig)
plot.subplot(1, 3, 2)
plot.imview(imgn, title='Corrupted', fig=fig)
plot.subplot(1, 3, 3)
plot.imview(imgr, title=r'Restored ($\ell_1$-TV)', fig=fig)
fig.show()
Get iterations statistics from solver object and plot functional value, ADMM primary and dual residuals, and automatically adjusted ADMM penalty parameter against the iteration number.
its = b.getitstat()
fig = plot.figure(figsize=(20, 5))
plot.subplot(1, 3, 1)
plot.plot(its.ObjFun, xlbl='Iterations', ylbl='Functional', fig=fig)
plot.subplot(1, 3, 2)
plot.plot(np.vstack((its.PrimalRsdl, its.DualRsdl)).T, ptyp='semilogy',
xlbl='Iterations', ylbl='Residual', lgnd=['Primal', 'Dual'],
fig=fig)
plot.subplot(1, 3, 3)
plot.plot(its.Rho, xlbl='Iterations', ylbl='Penalty Parameter', fig=fig)
fig.show()