Denoising
In this simple notebook, we look at a denoising problem and how the linearized bregman iterations provide a simple and efficient solution.
Packages setup
# ] registry add https://github.com/slimgroup/SLIMregistryJL.git
# ] add LinearAlgebra, JOLI, TestImages, ImageViewusing SlimOptim, LinearAlgebra, JOLI, TestImages, PyPlotUse a standard reference image
n = 256
n1, n2 = 2*256, 256
img = Float32.(testimage("lena_gray_16bit.png"));imshow(img, cmap="magma", vmin=-.5, vmax=.5);
Measurment operator
A = [joRomberg(n, n; DDT=Float32, RDT=Float32);joRomberg(n, n; DDT=Float32, RDT=Float32)];Setup transform domain operator
# Sparse in wavelet domain
W = joDWT(n, n; DDT=Float32, RDT=Float32);
# Or with curvelet ifi nstalled
# W = joCurvelet2D(128, 128; DDT=Float32, RDT=Float32);Measurements
# Make noisy data
b = A*vec(img);imshow(reshape(b, n1, n2), cmap="magma", vmin=-.5, vmax=.5, aspect=.5)
PyObject <matplotlib.image.AxesImage object at 0x14ac18c70>Linearized bregman
We solve the following l1-l2 optimization problem:
$\begin{align} \minx & \ \ \lambda || W x||1 + \frac{1}{2} ||W x ||_2^2 \ & \text{s.t. } Ax = b \end{align} $
where $W$ is the sparsity promoting transform, $x$ is the unknown image and $A$ is a tall measuremment operator.
# setup bregman
opt = bregman_options(maxIter=200, verbose=2, quantile=.1)SlimOptim.BregmanParams(2, 1.0e-8, 200, false, true, 0.1)sol = bregman(A, W, zeros(Float32, n*n), b, opt);Running linearized bregman...
Progress tolerance: 1.00e-08
Maximum number of iterations: 200
Anti-chatter correction: 1
Iteration Step Length Bregman residual ||A*x - b||_2^2 λ
1 1.25000e-01 1.48864e+02 4.75737e+03 2.99459e-04
2 1.25000e-01 4.55724e+02 2.67720e+03 2.99459e-04
3 1.25000e-01 7.95676e+02 1.50593e+03 2.99459e-04
4 1.25000e-01 1.11241e+03 8.47091e+02 2.99459e-04
5 1.25000e-01 1.38470e+03 4.76491e+02 2.99459e-04
6 1.25000e-01 1.60846e+03 2.68028e+02 2.99459e-04
7 1.25000e-01 1.78728e+03 1.50766e+02 2.99459e-04
8 1.25000e-01 1.92757e+03 8.48065e+01 2.99459e-04
9 1.25000e-01 2.03627e+03 4.77039e+01 2.99459e-04
10 1.25000e-01 2.11975e+03 2.68337e+01 2.99459e-04
11 1.25000e-01 2.18632e+03 1.50940e+01 1.42984e-03
12 1.25000e-01 2.23547e+03 8.81974e+00 1.42984e-03
13 1.25000e-01 2.27269e+03 4.96397e+00 1.42984e-03
14 1.25000e-01 2.30081e+03 2.79504e+00 1.42984e-03
15 1.25000e-01 2.32200e+03 1.57486e+00 1.42984e-03
16 1.25000e-01 2.33796e+03 8.88252e-01 1.42984e-03
17 1.25000e-01 2.34997e+03 5.01863e-01 1.42984e-03
18 1.25000e-01 2.35899e+03 2.84326e-01 1.42984e-03
19 1.25000e-01 2.36577e+03 1.61829e-01 1.42984e-03
20 1.25000e-01 2.37086e+03 9.28416e-02 1.42984e-03
21 1.25000e-01 2.37761e+03 5.40014e-02 2.52146e-03
22 1.25000e-01 2.38118e+03 1.31801e-01 2.52146e-03
23 1.25000e-01 2.38383e+03 8.25535e-02 2.52146e-03
24 1.25000e-01 2.38582e+03 5.43265e-02 2.52146e-03
25 1.25000e-01 2.38732e+03 3.78512e-02 2.52146e-03
26 1.25000e-01 2.38844e+03 2.79190e-02 2.52146e-03
27 1.25000e-01 2.38929e+03 2.16570e-02 2.52146e-03
28 1.25000e-01 2.38993e+03 1.74611e-02 2.52146e-03
29 1.25000e-01 2.39042e+03 1.44713e-02 2.52146e-03
30 1.25000e-01 2.39080e+03 1.21932e-02 2.52146e-03
31 1.25000e-01 2.39378e+03 1.03789e-02 3.49909e-03
32 1.25000e-01 2.39453e+03 9.09539e-02 3.49909e-03
33 1.25000e-01 2.39511e+03 6.70223e-02 3.49909e-03
34 1.25000e-01 2.39556e+03 5.15073e-02 3.49909e-03
35 1.25000e-01 2.39591e+03 4.11505e-02 3.49909e-03
36 1.25000e-01 2.39618e+03 3.39748e-02 3.49909e-03
37 1.25000e-01 2.39639e+03 2.88185e-02 3.49909e-03
38 1.25000e-01 2.39656e+03 2.49902e-02 3.49909e-03
39 1.25000e-01 2.39669e+03 2.20824e-02 3.49909e-03
40 1.25000e-01 2.39679e+03 1.98347e-02 3.49909e-03
41 1.25000e-01 2.39923e+03 1.80850e-02 4.33748e-03
42 1.25000e-01 2.39967e+03 9.52648e-02 4.33748e-03
43 1.25000e-01 2.40002e+03 7.99760e-02 4.33748e-03
44 1.25000e-01 2.40031e+03 6.95839e-02 4.33748e-03
45 1.25000e-01 2.40054e+03 6.21067e-02 4.33748e-03
46 1.25000e-01 2.40073e+03 5.63292e-02 4.33748e-03
47 1.25000e-01 2.40090e+03 5.15754e-02 4.33748e-03
48 1.25000e-01 2.40105e+03 4.74253e-02 4.33748e-03
49 1.25000e-01 2.40118e+03 4.36792e-02 4.33748e-03
50 1.25000e-01 2.40129e+03 4.01961e-02 4.33748e-03
51 1.25000e-01 2.40320e+03 3.69216e-02 4.96721e-03
52 1.25000e-01 2.40359e+03 9.53462e-02 4.96721e-03
53 1.25000e-01 2.40391e+03 8.20759e-02 4.96721e-03
54 1.25000e-01 2.40419e+03 7.17705e-02 4.96721e-03
55 1.25000e-01 2.40442e+03 6.33917e-02 4.96721e-03
56 1.25000e-01 2.40463e+03 5.63082e-02 4.96721e-03
57 1.25000e-01 2.40481e+03 5.01847e-02 4.96721e-03
58 1.25000e-01 2.40497e+03 4.47861e-02 4.96721e-03
59 1.25000e-01 2.40511e+03 3.99913e-02 4.96721e-03
60 1.25000e-01 2.40525e+03 3.57015e-02 4.96721e-03
61 1.25000e-01 2.40736e+03 3.18678e-02 5.65533e-03
62 1.25000e-01 2.40780e+03 9.48295e-02 5.65533e-03
63 1.25000e-01 2.40817e+03 7.93026e-02 5.65533e-03
64 1.25000e-01 2.40849e+03 6.75133e-02 5.65533e-03
65 1.25000e-01 2.40875e+03 5.82329e-02 5.65533e-03
66 1.25000e-01 2.40898e+03 5.06981e-02 5.65533e-03
67 1.25000e-01 2.40918e+03 4.44525e-02 5.65533e-03
68 1.25000e-01 2.40936e+03 3.91909e-02 5.65533e-03
69 1.25000e-01 2.40951e+03 3.47124e-02 5.65533e-03
70 1.25000e-01 2.40965e+03 3.08569e-02 5.65533e-03
71 1.25000e-01 2.41192e+03 2.75181e-02 6.38341e-03
72 1.25000e-01 2.41240e+03 9.30221e-02 6.38341e-03
73 1.25000e-01 2.41281e+03 7.65585e-02 6.38341e-03
74 1.25000e-01 2.41314e+03 6.43722e-02 6.38341e-03
75 1.25000e-01 2.41343e+03 5.50533e-02 6.38341e-03
76 1.25000e-01 2.41367e+03 4.77050e-02 6.38341e-03
77 1.25000e-01 2.41388e+03 4.17815e-02 6.38341e-03
78 1.25000e-01 2.41406e+03 3.69022e-02 6.38341e-03
79 1.25000e-01 2.41422e+03 3.28233e-02 6.38341e-03
80 1.25000e-01 2.41436e+03 2.93602e-02 6.38341e-03
81 1.25000e-01 2.41674e+03 2.63915e-02 7.13889e-03
82 1.25000e-01 2.41725e+03 9.33843e-02 7.13889e-03
83 1.25000e-01 2.41768e+03 7.64113e-02 7.13889e-03
84 1.25000e-01 2.41803e+03 6.40490e-02 7.13889e-03
85 1.25000e-01 2.41833e+03 5.47431e-02 7.13889e-03
86 1.25000e-01 2.41858e+03 4.75155e-02 7.13889e-03
87 1.25000e-01 2.41880e+03 4.17613e-02 7.13889e-03
88 1.25000e-01 2.41898e+03 3.70686e-02 7.13889e-03
89 1.25000e-01 2.41914e+03 3.31771e-02 7.13889e-03
90 1.25000e-01 2.41928e+03 2.98934e-02 7.13889e-03
91 1.25000e-01 2.42174e+03 2.70862e-02 7.91021e-03
92 1.25000e-01 2.42228e+03 9.49944e-02 7.91021e-03
93 1.25000e-01 2.42273e+03 7.76408e-02 7.91021e-03
94 1.25000e-01 2.42310e+03 6.50968e-02 7.91021e-03
95 1.25000e-01 2.42341e+03 5.57374e-02 7.91021e-03
96 1.25000e-01 2.42366e+03 4.85352e-02 7.91021e-03
97 1.25000e-01 2.42388e+03 4.28481e-02 7.91021e-03
98 1.25000e-01 2.42407e+03 3.82393e-02 7.91021e-03
99 1.25000e-01 2.42423e+03 3.44296e-02 7.91021e-03
100 1.25000e-01 2.42437e+03 3.12197e-02 7.91021e-03
101 1.25000e-01 2.42689e+03 2.84783e-02 8.68707e-03
102 1.25000e-01 2.42745e+03 9.65193e-02 8.68707e-03
103 1.25000e-01 2.42791e+03 7.90365e-02 8.68707e-03
104 1.25000e-01 2.42829e+03 6.64621e-02 8.68707e-03
105 1.25000e-01 2.42860e+03 5.71290e-02 8.68707e-03
106 1.25000e-01 2.42887e+03 4.99854e-02 8.68707e-03
107 1.25000e-01 2.42909e+03 4.43675e-02 8.68707e-03
108 1.25000e-01 2.42928e+03 3.98289e-02 8.68707e-03
109 1.25000e-01 2.42945e+03 3.60829e-02 8.68707e-03
110 1.25000e-01 2.42959e+03 3.29215e-02 8.68707e-03
111 1.25000e-01 2.43216e+03 3.02073e-02 9.47051e-03
112 1.25000e-01 2.43274e+03 9.89313e-02 9.47051e-03
113 1.25000e-01 2.43321e+03 8.12606e-02 9.47051e-03
114 1.25000e-01 2.43361e+03 6.85909e-02 9.47051e-03
115 1.25000e-01 2.43393e+03 5.92209e-02 9.47051e-03
116 1.25000e-01 2.43420e+03 5.20683e-02 9.47051e-03
117 1.25000e-01 2.43443e+03 4.64455e-02 9.47051e-03
118 1.25000e-01 2.43463e+03 4.18961e-02 9.47051e-03
119 1.25000e-01 2.43480e+03 3.81351e-02 9.47051e-03
120 1.25000e-01 2.43494e+03 3.49568e-02 9.47051e-03
121 1.25000e-01 2.43754e+03 3.22196e-02 1.02485e-02
122 1.25000e-01 2.43813e+03 1.00318e-01 1.02485e-02
123 1.25000e-01 2.43862e+03 8.27781e-02 1.02485e-02
124 1.25000e-01 2.43902e+03 7.02139e-02 1.02485e-02
125 1.25000e-01 2.43935e+03 6.09267e-02 1.02485e-02
126 1.25000e-01 2.43963e+03 5.38352e-02 1.02485e-02
127 1.25000e-01 2.43986e+03 4.82562e-02 1.02485e-02
128 1.25000e-01 2.44006e+03 4.37412e-02 1.02485e-02
129 1.25000e-01 2.44023e+03 3.99959e-02 1.02485e-02
130 1.25000e-01 2.44038e+03 3.68180e-02 1.02485e-02
131 1.25000e-01 2.44302e+03 3.40751e-02 1.10259e-02
132 1.25000e-01 2.44362e+03 1.02179e-01 1.10259e-02
133 1.25000e-01 2.44412e+03 8.46594e-02 1.10259e-02
134 1.25000e-01 2.44453e+03 7.21285e-02 1.10259e-02
135 1.25000e-01 2.44487e+03 6.28762e-02 1.10259e-02
136 1.25000e-01 2.44516e+03 5.58117e-02 1.10259e-02
137 1.25000e-01 2.44540e+03 5.02531e-02 1.10259e-02
138 1.25000e-01 2.44560e+03 4.57472e-02 1.10259e-02
139 1.25000e-01 2.44577e+03 4.20006e-02 1.10259e-02
140 1.25000e-01 2.44592e+03 3.88105e-02 1.10259e-02
141 1.25000e-01 2.44856e+03 3.60418e-02 1.17898e-02
142 1.25000e-01 2.44917e+03 1.02472e-01 1.17898e-02
143 1.25000e-01 2.44968e+03 8.53659e-02 1.17898e-02
144 1.25000e-01 2.45009e+03 7.31150e-02 1.17898e-02
145 1.25000e-01 2.45044e+03 6.40482e-02 1.17898e-02
146 1.25000e-01 2.45073e+03 5.71030e-02 1.17898e-02
147 1.25000e-01 2.45097e+03 5.16177e-02 1.17898e-02
148 1.25000e-01 2.45118e+03 4.71545e-02 1.17898e-02
149 1.25000e-01 2.45135e+03 4.34283e-02 1.17898e-02
150 1.25000e-01 2.45150e+03 4.02401e-02 1.17898e-02
151 1.25000e-01 2.45420e+03 3.74608e-02 1.25622e-02
152 1.25000e-01 2.45484e+03 1.04930e-01 1.25622e-02
153 1.25000e-01 2.45536e+03 8.75396e-02 1.25622e-02
154 1.25000e-01 2.45579e+03 7.51056e-02 1.25622e-02
155 1.25000e-01 2.45614e+03 6.59137e-02 1.25622e-02
156 1.25000e-01 2.45644e+03 5.88758e-02 1.25622e-02
157 1.25000e-01 2.45669e+03 5.33115e-02 1.25622e-02
158 1.25000e-01 2.45690e+03 4.87786e-02 1.25622e-02
159 1.25000e-01 2.45708e+03 4.49885e-02 1.25622e-02
160 1.25000e-01 2.45724e+03 4.17442e-02 1.25622e-02
161 1.25000e-01 2.45994e+03 3.89159e-02 1.33246e-02
162 1.25000e-01 2.46059e+03 1.05203e-01 1.33246e-02
163 1.25000e-01 2.46112e+03 8.80553e-02 1.33246e-02
164 1.25000e-01 2.46156e+03 7.57770e-02 1.33246e-02
165 1.25000e-01 2.46192e+03 6.66882e-02 1.33246e-02
166 1.25000e-01 2.46222e+03 5.97201e-02 1.33246e-02
167 1.25000e-01 2.46247e+03 5.42062e-02 1.33246e-02
168 1.25000e-01 2.46269e+03 4.97058e-02 1.33246e-02
169 1.25000e-01 2.46287e+03 4.59313e-02 1.33246e-02
170 1.25000e-01 2.46303e+03 4.26935e-02 1.33246e-02
171 1.25000e-01 2.46575e+03 3.98715e-02 1.40819e-02
172 1.25000e-01 2.46641e+03 1.05433e-01 1.40819e-02
173 1.25000e-01 2.46695e+03 8.84902e-02 1.40819e-02
174 1.25000e-01 2.46739e+03 7.63617e-02 1.40819e-02
175 1.25000e-01 2.46776e+03 6.73878e-02 1.40819e-02
176 1.25000e-01 2.46806e+03 6.05122e-02 1.40819e-02
177 1.25000e-01 2.46832e+03 5.50697e-02 1.40819e-02
178 1.25000e-01 2.46854e+03 5.06265e-02 1.40819e-02
179 1.25000e-01 2.46872e+03 4.69040e-02 1.40819e-02
180 1.25000e-01 2.46888e+03 4.37125e-02 1.40819e-02
181 1.25000e-01 2.47164e+03 4.09215e-02 1.48376e-02
182 1.25000e-01 2.47231e+03 1.06274e-01 1.48376e-02
183 1.25000e-01 2.47286e+03 8.94202e-02 1.48376e-02
184 1.25000e-01 2.47331e+03 7.73723e-02 1.48376e-02
185 1.25000e-01 2.47368e+03 6.84566e-02 1.48376e-02
186 1.25000e-01 2.47400e+03 6.16259e-02 1.48376e-02
187 1.25000e-01 2.47426e+03 5.62199e-02 1.48376e-02
188 1.25000e-01 2.47448e+03 5.18038e-02 1.48376e-02
189 1.25000e-01 2.47466e+03 4.80991e-02 1.48376e-02
190 1.25000e-01 2.47483e+03 4.49140e-02 1.48376e-02
191 1.25000e-01 2.47755e+03 4.21247e-02 1.55749e-02
192 1.25000e-01 2.47823e+03 1.05078e-01 1.55749e-02
193 1.25000e-01 2.47878e+03 8.88411e-02 1.55749e-02
194 1.25000e-01 2.47923e+03 7.72112e-02 1.55749e-02
195 1.25000e-01 2.47960e+03 6.85885e-02 1.55749e-02
196 1.25000e-01 2.47991e+03 6.19594e-02 1.55749e-02
197 1.25000e-01 2.48017e+03 5.66954e-02 1.55749e-02
198 1.25000e-01 2.48039e+03 5.23860e-02 1.55749e-02
199 1.25000e-01 2.48058e+03 4.87545e-02 1.55749e-02
200 1.25000e-01 2.48074e+03 4.56219e-02 1.55749e-02Plot result and convergence
figure(figsize=(8,8));
subplot(221)
imshow(img, cmap="magma", vmin=-.5, vmax=.5)
title("True")
subplot(222)
imshow(reshape(b, n1, n2), cmap="magma", vmin=-.5, vmax=.5, aspect=.5)
title("Measurment")
subplot(223)
imshow(reshape(sol.x, n, n), cmap="magma", vmin=-.5, vmax=.5)
title("recovered")
subplot(224)
imshow(img - reshape(sol.x, n, n), cmap="seismic", vmin=-.5e-2, vmax=.5e-2)
title("Difference x100")
tight_layout()
figure()
subplot(121)
plot(sol.ϕ_trace/sol.ϕ_trace[1], label="Objective")
legend()
subplot(122)
plot(sol.r_trace/sol.r_trace[1], label="||A*x -b||_2^2")
legend()
PyObject <matplotlib.legend.Legend object at 0x14aef2d60>