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, PyPlot
import TestImages: GrayUse a standard reference image
n = 128
k = 4
n1, n2 = k*n, n
img = Float32.(testimage("moonsurface.png"))
img = img[1:2:end, 1:2:end];[33m[1m┌ [22m[39m[33m[1mWarning: [22m[39m"moonsurface.png" not found in `TestImages.remotefiles`. Load "moonsurface.tiff" instead.
[33m[1m└ [22m[39m[90m@ TestImages ~/.julia/packages/TestImages/1zo60/src/TestImages.jl:163[39mimshow(img, cmap="magma", vmin=-.5, vmax=.5);
Measurment operator
A = vcat([joRomberg(n, n; DDT=Float32, RDT=Float32) for i=1:k]...);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);
b += .01f0*randn(Float32, size(b));imshow(reshape(b, n1, n2), cmap="magma", vmin=-.5, vmax=.5, aspect=.5)
PyObject <matplotlib.image.AxesImage object at 0x15a768810>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=.5, antichatter=true, spg=true)SlimOptim.BregmanParams(2, 1.0e-8, 200, false, true, 0.5, true, UniformScaling{Bool}(true), SlimOptim.var"#35#38"{Int64}(Core.Box(SlimOptim.var"#34#37"{Float64}(0.5)), 1), nothing)sol = bregman(A, W, zeros(Float32, n*n), b, opt);Running linearized bregman...
Progress tolerance: 1.00e-08
[33m[1m┌ [22m[39m[33m[1mWarning: [22m[39mdeprecation warning: please put TD in options (BregmanParams) for version > 0.1.7; now overwritting TD in BregmanParams
[33m[1m└ [22m[39m[90m@ SlimOptim ~/.julia/dev/SlimOptim/src/bregman.jl:88[39m
Maximum number of iterations: 200
Anti-chatter correction: 1
Iteration Step Length L1-2 ||A*x - b||_2^2 λ
1 NaN 0.00000e+00 8.62052e+03 1.34803e-06
2 2.49999e-01 7.31164e-06 8.61953e+03 1.34803e-06
3 2.50000e-01 2.15437e+03 2.42224e+00 1.34803e-06
4 2.50122e-01 2.15439e+03 2.42225e+00 1.34803e-06
5 2.26376e-01 2.15439e+03 2.42225e+00 1.34803e-06
6 2.12187e-01 2.15439e+03 2.42225e+00 1.34803e-06
7 2.20533e-01 2.15439e+03 2.42225e+00 1.34803e-06
8 2.23766e-01 2.15439e+03 2.42225e+00 1.34803e-06
9 2.38927e-01 2.15439e+03 2.42225e+00 1.34803e-06
10 2.14379e-01 2.15439e+03 2.42225e+00 1.34803e-06
11 2.34470e-01 2.15439e+03 2.42225e+00 1.34803e-06
12 2.15027e-01 2.15439e+03 2.42225e+00 1.34803e-06
13 2.24045e-01 2.15439e+03 2.42225e+00 1.34803e-06
14 2.34628e-01 2.15439e+03 2.42225e+00 1.34803e-06
15 2.18051e-01 2.15439e+03 2.42225e+00 1.34803e-06
16 2.17515e-01 2.15439e+03 2.42225e+00 1.34803e-06
17 2.29519e-01 2.15439e+03 2.42225e+00 1.34803e-06
18 2.36955e-01 2.15439e+03 2.42225e+00 1.34803e-06
19 2.21663e-01 2.15439e+03 2.42225e+00 1.34803e-06
20 2.25470e-01 2.15439e+03 2.42225e+00 1.34803e-06
21 2.19846e-01 2.15439e+03 2.42225e+00 1.34803e-06
22 2.32800e-01 2.15439e+03 2.42225e+00 1.34803e-06
23 2.40217e-01 2.15439e+03 2.42225e+00 1.34803e-06
24 2.36370e-01 2.15439e+03 2.42225e+00 1.34803e-06
25 2.33961e-01 2.15439e+03 2.42225e+00 1.34803e-06
26 2.29990e-01 2.15439e+03 2.42225e+00 1.34803e-06
27 2.36148e-01 2.15439e+03 2.42225e+00 1.34803e-06
28 2.32906e-01 2.15439e+03 2.42225e+00 1.34803e-06
29 2.33554e-01 2.15439e+03 2.42225e+00 1.34803e-06
30 2.27688e-01 2.15439e+03 2.42225e+00 1.34803e-06
31 2.32639e-01 2.15439e+03 2.42225e+00 1.34803e-06
32 2.35499e-01 2.15439e+03 2.42225e+00 1.34803e-06
33 2.25534e-01 2.15439e+03 2.42225e+00 1.34803e-06
34 2.33337e-01 2.15439e+03 2.42225e+00 1.34803e-06
35 2.30884e-01 2.15439e+03 2.42225e+00 1.34803e-06
36 2.34967e-01 2.15439e+03 2.42225e+00 1.34803e-06
37 2.25144e-01 2.15439e+03 2.42225e+00 1.34803e-06
38 2.31633e-01 2.15439e+03 2.42225e+00 1.34803e-06
39 2.37660e-01 2.15439e+03 2.42225e+00 1.34803e-06
40 2.27150e-01 2.15439e+03 2.42225e+00 1.34803e-06
41 2.22052e-01 2.15439e+03 2.42225e+00 1.34803e-06
42 2.38896e-01 2.15439e+03 2.42225e+00 1.34803e-06
43 2.40635e-01 2.15439e+03 2.42225e+00 1.34803e-06
44 2.35120e-01 2.15439e+03 2.42225e+00 1.34803e-06
45 2.37571e-01 2.15439e+03 2.42225e+00 1.34803e-06
46 2.30785e-01 2.15439e+03 2.42225e+00 1.34803e-06
47 2.33857e-01 2.15439e+03 2.42225e+00 1.34803e-06
48 2.88152e-01 2.15439e+03 2.42225e+00 1.34803e-06
49 2.33997e-01 2.15439e+03 2.42225e+00 1.34803e-06
50 2.19139e-01 2.15439e+03 2.42225e+00 1.34803e-06
51 2.15152e-01 2.15439e+03 2.42225e+00 1.34803e-06
52 2.16717e-01 2.15439e+03 2.42225e+00 1.34803e-06
53 2.32711e-01 2.15439e+03 2.42225e+00 1.34803e-06
54 2.35338e-01 2.15439e+03 2.42225e+00 1.34803e-06
55 2.22194e-01 2.15439e+03 2.42225e+00 1.34803e-06
56 2.17572e-01 2.15439e+03 2.42225e+00 1.34803e-06
57 2.17238e-01 2.15439e+03 2.42225e+00 1.34803e-06
58 2.13686e-01 2.15439e+03 2.42225e+00 1.34803e-06
59 2.32201e-01 2.15439e+03 2.42225e+00 1.34803e-06
60 2.30792e-01 2.15439e+03 2.42225e+00 1.34803e-06
61 2.44029e-01 2.15439e+03 2.42225e+00 1.34803e-06
62 2.38545e-01 2.15439e+03 2.42225e+00 1.34803e-06
63 2.32890e-01 2.15439e+03 2.42225e+00 1.34803e-06
64 2.36892e-01 2.15439e+03 2.42225e+00 1.34803e-06
65 2.30349e-01 2.15439e+03 2.42225e+00 1.34803e-06
66 2.28569e-01 2.15439e+03 2.42225e+00 1.34803e-06
67 2.33111e-01 2.15439e+03 2.42225e+00 1.34803e-06
68 2.28576e-01 2.15439e+03 2.42225e+00 1.34803e-06
69 2.38222e-01 2.15439e+03 2.42225e+00 1.34803e-06
70 2.31610e-01 2.15439e+03 2.42225e+00 1.34803e-06
71 2.27397e-01 2.15439e+03 2.42225e+00 1.34803e-06
72 2.39834e-01 2.15439e+03 2.42225e+00 1.34803e-06
73 2.28993e-01 2.15439e+03 2.42225e+00 1.34803e-06
74 2.25153e-01 2.15439e+03 2.42225e+00 1.34803e-06
75 2.38148e-01 2.15439e+03 2.42225e+00 1.34803e-06
76 2.38356e-01 2.15439e+03 2.42225e+00 1.34803e-06
77 2.44641e-01 2.15439e+03 2.42225e+00 1.34803e-06
78 2.90175e-01 2.15439e+03 2.42225e+00 1.34803e-06
79 2.38272e-01 2.15439e+03 2.42225e+00 1.34803e-06
80 2.27486e-01 2.15439e+03 2.42225e+00 1.34803e-06
81 2.32778e-01 2.15439e+03 2.42225e+00 1.34803e-06
82 2.25604e-01 2.15439e+03 2.42225e+00 1.34803e-06
83 2.26262e-01 2.15439e+03 2.42225e+00 1.34803e-06
84 2.11050e-01 2.15439e+03 2.42225e+00 1.34803e-06
85 2.06730e-01 2.15439e+03 2.42225e+00 1.34803e-06
86 2.39487e-01 2.15439e+03 2.42225e+00 1.34803e-06
87 2.29575e-01 2.15439e+03 2.42225e+00 1.34803e-06
88 2.22524e-01 2.15439e+03 2.42225e+00 1.34803e-06
89 2.17675e-01 2.15439e+03 2.42225e+00 1.34803e-06
90 2.31493e-01 2.15439e+03 2.42225e+00 1.34803e-06
91 2.16341e-01 2.15439e+03 2.42225e+00 1.34803e-06
92 2.24872e-01 2.15439e+03 2.42225e+00 1.34803e-06
93 2.16945e-01 2.15439e+03 2.42225e+00 1.34803e-06
94 2.24635e-01 2.15439e+03 2.42225e+00 1.34803e-06
95 2.23042e-01 2.15439e+03 2.42225e+00 1.34803e-06
96 2.34892e-01 2.15439e+03 2.42225e+00 1.34803e-06
97 2.27087e-01 2.15439e+03 2.42225e+00 1.34803e-06
98 2.17167e-01 2.15439e+03 2.42225e+00 1.34803e-06
99 2.25401e-01 2.15439e+03 2.42225e+00 1.34803e-06
100 2.34606e-01 2.15439e+03 2.42225e+00 1.34803e-06
101 2.38159e-01 2.15439e+03 2.42225e+00 1.34803e-06
102 2.22516e-01 2.15439e+03 2.42225e+00 1.34803e-06
103 2.44695e-01 2.15439e+03 2.42225e+00 1.34803e-06
104 2.39945e-01 2.15439e+03 2.42225e+00 1.34803e-06
105 2.27302e-01 2.15439e+03 2.42225e+00 1.34803e-06
106 2.22394e-01 2.15439e+03 2.42225e+00 1.34803e-06
107 2.11417e-01 2.15439e+03 2.42225e+00 1.34803e-06
108 2.68428e-01 2.15439e+03 2.42225e+00 1.34803e-06
109 2.32341e-01 2.15439e+03 2.42225e+00 1.34803e-06
110 2.22020e-01 2.15439e+03 2.42225e+00 1.34803e-06
111 2.20446e-01 2.15439e+03 2.42225e+00 1.34803e-06
112 2.35209e-01 2.15439e+03 2.42225e+00 1.34803e-06
113 2.31724e-01 2.15439e+03 2.42225e+00 1.34803e-06
114 2.14602e-01 2.15439e+03 2.42225e+00 1.34803e-06
115 2.30873e-01 2.15439e+03 2.42225e+00 1.34803e-06
116 2.36182e-01 2.15439e+03 2.42225e+00 1.34803e-06
117 2.17700e-01 2.15439e+03 2.42225e+00 1.34803e-06
118 2.31121e-01 2.15439e+03 2.42225e+00 1.34803e-06
119 2.26273e-01 2.15439e+03 2.42225e+00 1.34803e-06
120 2.28442e-01 2.15439e+03 2.42225e+00 1.34803e-06
121 2.36458e-01 2.15439e+03 2.42225e+00 1.34803e-06
122 2.19529e-01 2.15439e+03 2.42225e+00 1.34803e-06
123 2.18114e-01 2.15439e+03 2.42225e+00 1.34803e-06
124 2.23112e-01 2.15439e+03 2.42225e+00 1.34803e-06
125 2.30561e-01 2.15439e+03 2.42225e+00 1.34803e-06
126 2.31983e-01 2.15439e+03 2.42225e+00 1.34803e-06
127 2.39946e-01 2.15439e+03 2.42225e+00 1.34803e-06
128 2.30612e-01 2.15439e+03 2.42225e+00 1.34803e-06
129 2.28289e-01 2.15439e+03 2.42225e+00 1.34803e-06
130 2.20381e-01 2.15439e+03 2.42225e+00 1.34803e-06
131 2.19100e-01 2.15439e+03 2.42225e+00 1.34803e-06
132 2.28832e-01 2.15439e+03 2.42225e+00 1.34803e-06
133 2.01979e-01 2.15439e+03 2.42225e+00 1.34803e-06
134 2.56993e-01 2.15439e+03 2.42225e+00 1.34803e-06
135 2.35915e-01 2.15439e+03 2.42225e+00 1.34803e-06
136 2.39654e-01 2.15439e+03 2.42225e+00 1.34803e-06
137 2.35434e-01 2.15439e+03 2.42225e+00 1.34803e-06
138 2.43991e-01 2.15439e+03 2.42225e+00 1.34803e-06
139 2.42739e-01 2.15439e+03 2.42225e+00 1.34803e-06
140 2.39365e-01 2.15439e+03 2.42225e+00 1.34803e-06
141 2.38661e-01 2.15439e+03 2.42225e+00 1.34803e-06
142 2.10096e-01 2.15439e+03 2.42225e+00 1.34803e-06
143 2.23454e-01 2.15439e+03 2.42225e+00 1.34803e-06
144 2.27167e-01 2.15439e+03 2.42225e+00 1.34803e-06
145 2.37637e-01 2.15439e+03 2.42225e+00 1.34803e-06
146 2.15325e-01 2.15439e+03 2.42224e+00 1.34803e-06
147 2.24937e-01 2.15439e+03 2.42225e+00 1.34803e-06
148 2.39013e-01 2.15439e+03 2.42225e+00 1.34803e-06
149 2.35509e-01 2.15439e+03 2.42225e+00 1.34803e-06
150 2.30825e-01 2.15439e+03 2.42225e+00 1.34803e-06
151 2.89046e-01 2.15439e+03 2.42225e+00 1.34803e-06
152 2.34444e-01 2.15439e+03 2.42225e+00 1.34803e-06
153 2.40667e-01 2.15439e+03 2.42225e+00 1.34803e-06
154 2.33535e-01 2.15439e+03 2.42225e+00 1.34803e-06
155 2.32912e-01 2.15439e+03 2.42225e+00 1.34803e-06
156 2.29976e-01 2.15439e+03 2.42225e+00 1.34803e-06
157 2.23140e-01 2.15439e+03 2.42225e+00 1.34803e-06
158 2.13382e-01 2.15439e+03 2.42225e+00 1.34803e-06
159 2.38296e-01 2.15439e+03 2.42225e+00 1.34803e-06
160 2.38884e-01 2.15439e+03 2.42225e+00 1.34803e-06
161 2.32889e-01 2.15439e+03 2.42225e+00 1.34803e-06
162 2.80621e-01 2.15439e+03 2.42225e+00 1.34803e-06
163 2.30690e-01 2.15439e+03 2.42225e+00 1.34803e-06
164 2.20397e-01 2.15439e+03 2.42225e+00 1.34803e-06
165 2.30224e-01 2.15439e+03 2.42225e+00 1.34803e-06
166 2.33585e-01 2.15439e+03 2.42225e+00 1.34803e-06
167 2.26235e-01 2.15439e+03 2.42225e+00 1.34803e-06
168 2.49580e-01 2.15439e+03 2.42225e+00 1.34803e-06
169 2.41928e-01 2.15439e+03 2.42225e+00 1.34803e-06
170 2.39097e-01 2.15439e+03 2.42225e+00 1.34803e-06
171 2.35508e-01 2.15439e+03 2.42225e+00 1.34803e-06
172 2.35760e-01 2.15439e+03 2.42224e+00 1.34803e-06
173 2.37985e-01 2.15439e+03 2.42225e+00 1.34803e-06
174 2.26707e-01 2.15439e+03 2.42225e+00 1.34803e-06
175 2.36418e-01 2.15439e+03 2.42225e+00 1.34803e-06
176 2.40366e-01 2.15439e+03 2.42225e+00 1.34803e-06
177 2.36853e-01 2.15439e+03 2.42225e+00 1.34803e-06
178 2.36730e-01 2.15439e+03 2.42225e+00 1.34803e-06
179 2.22309e-01 2.15439e+03 2.42225e+00 1.34803e-06
180 2.67462e-01 2.15439e+03 2.42225e+00 1.34803e-06
181 2.35026e-01 2.15439e+03 2.42225e+00 1.34803e-06
182 2.33264e-01 2.15439e+03 2.42225e+00 1.34803e-06
183 2.33983e-01 2.15439e+03 2.42225e+00 1.34803e-06
184 2.33984e-01 2.15439e+03 2.42225e+00 1.34803e-06
185 2.31955e-01 2.15439e+03 2.42225e+00 1.34803e-06
186 2.27195e-01 2.15439e+03 2.42225e+00 1.34803e-06
187 2.24366e-01 2.15439e+03 2.42225e+00 1.34803e-06
188 2.28888e-01 2.15439e+03 2.42225e+00 1.34803e-06
189 2.36564e-01 2.15439e+03 2.42225e+00 1.34803e-06
190 2.24446e-01 2.15439e+03 2.42225e+00 1.34803e-06
191 2.25318e-01 2.15439e+03 2.42225e+00 1.34803e-06
192 2.21183e-01 2.15439e+03 2.42225e+00 1.34803e-06
193 2.35291e-01 2.15439e+03 2.42225e+00 1.34803e-06
194 2.11666e-01 2.15439e+03 2.42225e+00 1.34803e-06
195 2.39179e-01 2.15439e+03 2.42225e+00 1.34803e-06
196 2.40375e-01 2.15439e+03 2.42225e+00 1.34803e-06
197 2.40929e-01 2.15439e+03 2.42225e+00 1.34803e-06
198 2.41813e-01 2.15439e+03 2.42225e+00 1.34803e-06
199 2.38086e-01 2.15439e+03 2.42225e+00 1.34803e-06
200 2.26569e-01 2.15439e+03 2.42225e+00 1.34803e-06Plot 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-1, vmax=.5e-1)
title("Difference x10")
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 0x15d263310>