Inpainting using Sparse Regularization¶
Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$
This numerical tour explores the use of sparse energies to regularize the image inpaiting problem.
In [2]:
using PyPlot
using NtToolBox
# using Autoreload
# arequire("NtToolBox")
Here we consider inpainting of damaged observation without noise.
Sparse Regularization¶
This tour consider measurements $y=\Phi f_0 + w$ where $\Phi$ is a masking operator and $w$ is an additive noise.
This tour is focused on using sparsity to recover an image from the measurements $y$. It considers a synthesis-based regularization, that compute a sparse set of coefficients $ (a_m^{\star})_m $ in a frame $\Psi = (\psi_m)_m$ that solves $$a^{\star} \in \text{argmin}_a \: \frac{1}{2}\|y-\Phi \Psi a\|^2 + \lambda J(a)$$
where $\lambda$ should be adapted to the noise level $\|w\|$. Since in this tour we consider damaged observation without noise, i.e. $w=0$, we use either a very small value of $\lambda$, or we decay its value through the iterations of the recovery process.
Here we use the notation $$\Psi a = \sum_m a_m \psi_m$$ to indicate the reconstruction operator, and $J(a)$ is the $\ell^1$ sparsity prior $$J(a)=\sum_m \|a_m\|.$$
Missing Pixels and Inpainting¶
Inpainting corresponds to filling holes in images. This corresponds to a linear ill posed inverse problem.
You might want to do first the numerical tour Variational image inpaiting that use Sobolev and TV priors to performs the inpainting.
First we load the image to be inpainted.
In [3]:
n = 128
f0 = load_image("NtToolBox/src/data/lena.png")
f0 = rescale(f0[256 - Base.div(n, 2) + 1 : 256 + Base.div(n, 2), 256 - Base.div(n, 2) + 1 : 256 + Base.div(n, 2)]);
Display it.
In [39]:
figure(figsize = (6, 6))
imageplot(f0, L"Image $f_0$")
Out[39]:
PyObject <matplotlib.text.Text object at 0x000000002566FA58>
Amount of removed pixels.
In [40]:
rho = .7;
Out[40]:
0.7
Then we construct a mask $\Omega$ made of random pixel locations.
In [41]:
Omega = zeros(n, n)
sel = randperm(n^2)
Omega[sel[1:Int(round(rho*n^2))]] = 1;
Out[41]:
1
The damaging operator put to zeros the pixel locations $x$ for which $\Omega(x)=1$
In [42]:
Phi = (f, Omega) -> f.*(1 - Omega);
Out[42]:
(::#21) (generic function with 1 method)
The damaged observations reads $y = \Phi f_0$.
In [43]:
y = Phi(f0, Omega);
Out[43]:
128×128 Array{Float64,2}:
0.0 0.0 0.0 0.596059 … 0.817734 0.0 0.0
0.571429 0.0 0.591133 0.610838 0.0 0.0 0.812808
0.581281 0.0 0.0 0.0 0.0 0.0 0.0
0.561576 0.0 0.0 0.586207 0.0 0.802956 0.0
0.0 0.0 0.522168 0.0 0.0 0.0 0.82266
0.0 0.0 0.0 0.0 … 0.0 0.802956 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.561576 0.0 0.0 0.0 0.82266 0.0
0.472906 0.0 0.0 0.394089 0.0 0.0 0.0
0.0 0.0 0.0 0.182266 0.0 0.0 0.0
0.0 0.0 0.0 0.0 … 0.0 0.0 0.832512
0.44335 0.0 0.0 0.0 0.0 0.812808 0.0
0.0 0.0 0.0 0.369458 0.807882 0.0 0.827586
⋮ ⋱ ⋮
0.0 0.0 0.0 0.472906 0.955665 0.932677 0.0
0.0 0.0 0.541872 0.0 0.0 0.935961 0.876847
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.497537 0.0 0.0 0.0 0.91133
0.0 0.507389 0.0 0.546798 … 0.0 0.0 0.906404
0.0 0.0 0.0 0.0 0.0 0.955665 0.0
0.0 0.0 0.0 0.689655 0.0 0.0 0.0
0.0 0.487685 0.0 0.0 0.990148 0.0 0.916256
0.0 0.0 0.0 0.788177 1.0 0.0 0.916256
0.0 0.0 0.0 0.0 … 1.0 0.97537 0.91133
0.0 0.0 0.0 0.561576 0.0 0.0 0.916256
0.0 0.0 0.0 0.738916 0.945813 0.0 0.0
Display the observations.
In [44]:
figure(figsize = (6, 6))
imageplot(y, "Observations y")
Out[44]:
PyObject <matplotlib.text.Text object at 0x0000000025309A90>
Soft Thresholding in a Basis¶
The soft thresholding operator is at the heart of $\ell^1$ minimization schemes. It can be applied to coefficients $a$, or to an image $f$ in an ortho-basis.
The soft thresholding is a 1-D functional that shrinks the value of coefficients. $$ s_T(u)=\max(0,1-T/|u|)u $$
Define a shortcut for this soft thresholding 1-D functional.
In [45]:
SoftThresh = (x,T) -> x.*max( 0, 1 - T./max(abs(x), 1e-10) );
Out[45]:
(::#23) (generic function with 1 method)
Display a curve of the 1D soft thresholding.
In [46]:
x = linspace(-1, 1, 1000)
figure(figsize = (7, 5))
plot(x, SoftThresh(x, .5))
show()
Note that the function SoftThresh can also be applied to vector which defines an operator on coefficients: $$ S_T(a) = ( s_T(a_m) )_m. $$
In the next section, we use an orthogonal wavelet basis $\Psi$.
We set the parameters of the wavelet transform.
In [47]:
Jmax = log2(n) - 1
Jmin = (Jmax - 3);
Out[47]:
3.0
Shortcut for $\Psi$ and $\Psi^*$ in the orthogonal case.
In [48]:
Psi = a -> NtToolBox.perform_wavelet_transf(a, Jmin, -1, "9-7", 0, 0)
PsiS = f -> NtToolBox.perform_wavelet_transf(f, Jmin, +1, "9-7", 0, 0);
The soft thresholding opterator in the basis $\Psi$ is defined as $$S_T^\Psi(f) = \sum_m s_T( \langle f,\psi_m \rangle ) \psi_m $$
It thus corresponds to applying the transform $\Psi^*$, thresholding the coefficients using $S_T$ and then undoing the transform using $\Psi$. $$ S_T^\Psi(f) = \Psi \circ S_T \circ \Psi^*$$
In [49]:
SoftThreshPsi = (f, T) -> Psi(SoftThresh(PsiS(f), T));
This soft thresholding corresponds to a denoising operator.
In [50]:
figure(figsize = (6, 6))
imageplot(clamP(SoftThreshPsi(f0, 0.1)))
Inpainting using Orthogonal Wavelet Sparsity¶
If $\Psi$ is an orthogonal basis, a change of variable shows that the synthesis prior is also an analysis prior, that reads $$f^{\star} \in \text{argmin}_f \: E(f) = \frac{1}{2}\|y-\Phi f\|^2 + \lambda \sum_m \|\langle f,\psi_m \rangle\|. $$
To solve this non-smooth optimization problem, one can use forward-backward splitting, also known as iterative soft thresholding.
It computes a series of images $f^{(\ell)}$ defined as $$ f^{(\ell+1)} = S_{\tau\lambda}^{\Psi}( f^{(\ell)} - \tau \Phi^{*} (\Phi f^{(\ell)} - y) ) $$
Set up the value of the threshold.
In [52]:
lambd = .03;
In our setting, we have $ \Phi^* = \Phi $ which is an operator of norm
For $f^{(\ell)}$ to converge to a solution of the problem, the gradient step size should be chosen as $$\tau < \frac{2}{\|\Phi^* \Phi\|} = 2$$
In the following we use: $$\tau = 1$$
Since we use $ \tau=1 $ and $ \Phi = \Phi^* = \text{diag}(1-\Omega) $, the gradient descent step is a projection on the inpainting constraint $$ C = \{ f \backslash \forall \Omega(x)=0, f(x)=y(x) \} $$ One thus has $$ f - \tau \Phi^{*} (\Phi f - y) = \text{Proj}_C(f) $$
For the sake of simplicity, we define a shortcut for this projection operator.
In [53]:
ProjC = (f, Omega) -> Omega.*f + (1 - Omega).*y;
Each iteration of the forward-backward (iterative thresholding) algorithm thus reads: $$ f^{(\ell+1)} = S_{\lambda}^\Psi( \text{Proj}_C(f^{(\ell)}) ). $$
Initialize the iterations.
In [54]:
fSpars = y;
First step: gradient descent.
In [55]:
fSpars = ProjC(fSpars, Omega);
Second step: denoise the solution by thresholding.
In [56]:
fSpars = SoftThreshPsi(fSpars, lambd);
Exercise 1
Perform the iterative soft thresholding. Monitor the decay of the energy $E$ you are minimizing.
In [57]:
include("NtSolutions/inverse_5_inpainting_sparsity/exo1.jl")
In [21]:
## Insert your code here.
Display the result.
In [58]:
figure(figsize = (6, 6))
imageplot(clamP(fSpars))
Exercise 2
Since there is no noise, one should in theory take $\lambda \rightarrow 0$. To do this, decay the value of $\lambda$ through the iterations.
In [59]:
include("NtSolutions/inverse_5_inpainting_sparsity/exo2.jl")
Out[59]:
PyObject <matplotlib.text.Text object at 0x0000000024F89518>
In [24]:
## Insert your code here.
Inpainting using Translation Invariant Wavelet Sparsity¶
Orthogonal sparsity performs a poor regularization because of the lack of translation invariance. This regularization is enhanced by considering $\Psi$ as a redundant tight frame of translation invariant wavelets.
One thus looks for optimal coefficients $a^\star$ that solves $$a^{\star} \in \text{argmin}_a \: E(a) = \frac{1}{2}\|y-\Phi \Psi a\|^2 + \lambda J(a)$$
Important: The operator $\Psi^*$ is the forward translation invariant wavelet transform. It computes the inner product with the unit norm wavelet atoms: $$ (\Psi^* f)_m = \langle f,\psi_m \rangle \quad \text{with} \quad \|\psi_m\|=1. $$
The reconstruction operator $\Xi$ satisfies $ \Xi \Psi^* f = f $, and is the pseudo inverse of the analysis operator $ \Xi = (\Psi^*)^+ $.
For our algorithm, we will need to use $\Psi$ and not $\Xi$. Lukily, for the wavelet transform, one has $$ \Xi = \Psi \text{diag(U)} f $$ where $U_m$ account for the redundancy of the scale of the atom $\psi_m$.
Compute the scaling factor (inverse of the redundancy).
In [68]:
J = Jmax - Jmin + 1;
u = reshape([4^(-J); 4.^(-floor(J+2/3:-1/3:1))], (1, 1, 13))
U = repeat(u, inner = [n, n, 1])
Out[68]:
128×128×13 Array{Float64,3}:
[:, :, 1] =
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
⋮ ⋱ ⋮
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
[:, :, 2] =
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
⋮ ⋱ ⋮
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
[:, :, 3] =
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
⋮ ⋱ ⋮
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 … 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
0.00390625 0.00390625 0.00390625 0.00390625 0.00390625 0.00390625
...
[:, :, 11] =
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
⋮ ⋮ ⋱ ⋮
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
[:, :, 12] =
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
⋮ ⋮ ⋱ ⋮
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
[:, :, 13] =
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
⋮ ⋮ ⋱ ⋮
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 … 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
Choose a value of the regularization parameter.
In [70]:
lambd = .01;
Out[70]:
0.01
Shortcut for the wavelet transform and the reconstruction.
Important: Scilab users have to create files |Xi.m|, |PsiS.m| and |Psi.m| to implement this function.
In [71]:
Xi = a -> NtToolBox.perform_wavelet_transf(a, Jmin, -1, "9-7", 0, 1)
PsiS = f -> NtToolBox.perform_wavelet_transf(f, Jmin, + 1, "9-7", 0, 1)
Psi = a -> Xi(a./U);
The forward-backward algorithm now compute a series of wavelet coefficients $a^{(\ell)}$ computed as $$a^{(\ell+1)} = S_{\tau\lambda}( a^{(\ell)} + \Psi^*\Phi( y - \Phi\Psi a^{(\ell)} ) ). $$
The soft thresholding is defined as: $$\forall m, \quad S_T(a)_m = \max(0, 1-T/\|a_m\|)a_m. $$
The step size should satisfy: $$\tau < \frac{2}{\|\Psi\Phi \|} \leq 2 \min( u ). $$
In [72]:
tau = 1.9*minimum(u);
Out[72]:
0.007421875
Initialize the wavelet coefficients with those of the previous reconstruction.
In [119]:
a = U.*PsiS(fSpars) ;
Out[119]:
128×128×13 Array{Float64,3}:
[:, :, 1] =
0.0323998 0.0323408 0.032198 … 0.0497627 0.0497896 0.0498026
0.0323222 0.0322629 0.0321195 0.049747 0.0497735 0.0497863
0.0320984 0.0320383 0.0318918 0.0497036 0.0497298 0.0497425
0.031718 0.0316577 0.0315087 0.0496433 0.0496698 0.0496825
0.0311659 0.0311074 0.0309599 0.0495708 0.0495982 0.0496111
0.0304433 0.0303877 0.0302443 … 0.0494936 0.0495219 0.0495349
0.0295714 0.0295197 0.0293834 0.0494165 0.0494451 0.049458
0.0285937 0.0285468 0.0284203 0.0493344 0.0493626 0.0493751
0.0275173 0.0274758 0.0273607 0.0492568 0.0492838 0.0492956
0.026379 0.0263437 0.0262417 0.0491961 0.0492221 0.0492333
0.0252068 0.0251783 0.0250907 … 0.0491663 0.0491926 0.0492036
0.0239974 0.0239752 0.0239017 0.0491732 0.0492007 0.0492117
0.0227982 0.022781 0.0227206 0.0492064 0.0492362 0.049248
⋮ ⋱ ⋮
0.0382111 0.0381686 0.0379823 0.0554137 0.0557831 0.0559001
0.0386472 0.0385807 0.0383335 0.0555067 0.0558895 0.0560103
0.0389726 0.0388838 0.0385819 0.0555916 0.0559873 0.0561121
0.039167 0.0390565 0.038703 0.0556713 0.056079 0.0562073
0.0392543 0.0391239 0.0387253 … 0.0557556 0.0561754 0.0563073
0.0392432 0.039097 0.0386641 0.0558498 0.0562819 0.0564178
0.0391019 0.0389432 0.0384843 0.055957 0.056401 0.0565408
0.038889 0.0387209 0.038244 0.0560627 0.0565176 0.0566608
0.0387639 0.038588 0.0380964 0.0561424 0.0566062 0.0567523
0.0387139 0.0385291 0.0380195 … 0.056187 0.0566565 0.0568045
0.0387146 0.0385229 0.0379989 0.0562012 0.056674 0.056823
0.0387365 0.0385424 0.038014 0.0562015 0.0566757 0.0568252
[:, :, 2] =
0.00517647 0.00517974 0.00520866 … 0.000453717 0.000450113
0.00494234 0.00494263 0.00496545 0.000460674 0.000457591
0.00430781 0.00430343 0.00431392 0.000493755 0.000492376
0.003369 0.00336223 0.00336226 0.000513809 0.000514249
0.002139 0.00212951 0.00211789 0.000498846 0.000500436
0.000783836 0.00077358 0.000752523 … 0.00046975 0.000472512
-0.000455605 -0.000464663 -0.000491254 0.000428269 0.000432135
-0.00151366 -0.00152437 -0.00155787 0.000335391 0.000340167
-0.00234035 -0.00235136 -0.00238571 0.000150589 0.000155272
-0.00277925 -0.00278589 -0.0028116 -8.79292e-5 -8.46004e-5
-0.00278339 -0.00278594 -0.00280389 … -0.000359 -0.000358327
-0.00249213 -0.00249172 -0.00250609 -0.00060884 -0.000610867
-0.00190337 -0.00189785 -0.00190492 -0.000821571 -0.000825059
⋮ ⋱
0.00361317 0.0035157 0.00327191 -0.000156927 -0.000157751
0.00311397 0.00302163 0.00280189 7.32275e-5 7.75564e-5
0.00262674 0.00254535 0.00235944 0.000221212 0.000229091
0.00191046 0.00184993 0.00170897 0.000311088 0.000320092
0.00101829 0.000980576 0.00088745 … 0.000377979 0.00038726
3.51231e-5 2.00672e-5 -2.05707e-5 0.00041864 0.00042811
-0.00106653 -0.00106246 -0.00105575 0.000405405 0.000414133
-0.00205521 -0.00204404 -0.00200543 0.000322731 0.000330442
-0.00278134 -0.00276769 -0.00270948 0.000213383 0.000219765
-0.00340196 -0.00338264 -0.00330966 … 0.000131161 0.000135772
-0.00381386 -0.00379104 -0.00371005 9.1912e-5 9.55647e-5
-0.00393484 -0.00391203 -0.00382998 8.23326e-5 8.59554e-5
[:, :, 3] =
0.00138346 0.00134712 0.00132677 … 0.000294635 0.000295266
0.00135317 0.00131994 0.00130722 0.000291986 0.000292514
0.00126775 0.00123879 0.00123711 0.00029099 0.000291185
0.00117381 0.00114881 0.00115346 0.000299619 0.000300178
0.00112772 0.00110504 0.00110927 0.000316047 0.00031857
0.00113961 0.00111067 0.0010943 … 0.000339363 0.000343832
0.00117624 0.00113606 0.00108741 0.000364889 0.000369789
0.00121848 0.00116595 0.00108587 0.000383485 0.000387823
0.0012667 0.00120006 0.00108392 0.000396432 0.000398699
0.00129647 0.0012172 0.00106779 0.000404625 0.000404162
0.00130226 0.0012131 0.00103741 … 0.0004126 0.000409924
0.00125916 0.00116201 0.000964115 0.00042561 0.000420912
0.00111489 0.00101673 0.000814625 0.000437259 0.000431794
⋮ ⋱
-0.00701353 -0.00645471 -0.00488546 0.00423701 0.00419635
-0.0061419 -0.00556284 -0.00394824 0.00463457 0.00461425
-0.00523841 -0.00464047 -0.00298138 0.00500965 0.00500903
-0.00434972 -0.00373374 -0.00203747 0.00536678 0.00538489
-0.00349784 -0.00286825 -0.00114521 … 0.00571198 0.00574873
-0.00268225 -0.00204489 -0.00030208 0.0060428 0.00609817
-0.00197506 -0.00133154 0.000422803 0.00636174 0.00643585
-0.00138652 -0.000738818 0.00102683 0.00664149 0.00673154
-0.000848183 -0.000201675 0.0015706 0.00686179 0.00696506
-0.000418623 0.000221834 0.00199185 … 0.00701477 0.00712807
-0.000118745 0.000514737 0.00227896 0.00709912 0.00721828
8.54528e-6 0.000639047 0.00239924 0.00712595 0.00724738
...
[:, :, 11] =
0.00917278 0.0059813 -0.0003906 … 2.6135e-5 -0.00727329
-0.00578015 -0.00345774 -9.64634e-5 2.88868e-6 0.00466319
0.000226373 -0.00015325 0.000920243 -0.00055009 -0.000929758
0.00289008 0.00186457 4.47425e-5 1.84094e-6 -0.00195206
-0.0011004 -0.000295352 -0.000969456 0.00134247 0.00332507
-3.05964e-6 0.000455492 9.06466e-6 … -2.87702e-6 0.00015365
-0.00648117 -0.00686412 0.00171912 -0.0038552 -0.00532766
0.012836 0.0127208 5.0436e-6 0.00529647 0.00647929
-0.00939992 -0.0143707 -0.00260384 -0.00197791 -0.00421301
2.57869e-6 0.0121628 -9.07698e-6 -1.95516e-6 5.58853e-5
0.00194427 -0.00793574 0.00106706 … -0.000226475 0.0101109
-1.61525e-5 0.00201233 1.54474e-6 -0.000125637 -0.0156778
-0.000253978 0.00230396 -0.000617237 -0.00123861 0.00256596
⋮ ⋱
-0.0113686 -0.0107581 -0.00820473 0.00462062 0.00193607
-1.5835e-6 0.000135531 0.000844468 0.000351036 -0.0010109
0.00553721 0.0118431 0.0120688 -0.0017466 0.00297146
1.89804e-5 -0.0135594 -0.0155274 -6.6132e-6 -0.00105747
-0.00669351 0.00859285 0.00832474 … -0.000579034 -0.00308679
1.24352e-5 -0.00573448 -1.0406e-7 0.00178989 0.00218958
0.0247506 0.00643042 -0.000939542 -0.00130609 0.00198111
-0.0406369 -0.00350874 -7.5799e-6 -0.000372674 -0.00223376
0.024568 -0.00146143 -0.00114894 0.00132469 0.00165565
9.05832e-7 -4.31222e-6 -3.18369e-6 … -1.46323e-5 -0.00127304
-0.00402011 0.0061899 0.000796783 -0.000598393 -0.00490415
-5.99336e-7 -0.00909471 4.52905e-6 7.73119e-8 0.00990787
[:, :, 12] =
-0.00449085 0.00396476 -0.00232793 … -0.004166 0.000106307
-0.00319138 0.00245479 -0.00263145 -0.00319149 0.00150597
0.000276887 -4.09888e-7 -0.0035307 -5.42174e-5 1.72207e-6
-0.000669645 0.00224556 -0.00571916 0.000406822 -0.000740761
0.00156905 8.3065e-7 -0.00376568 -0.00287511 0.00385917
0.0121829 -0.0107244 0.00396029 … -0.00337479 0.0040742
0.00143818 3.64192e-6 -0.00286217 -0.000834918 -3.2223e-6
-0.0150026 0.0160502 -0.0109862 0.00254461 -0.00426762
-0.00333856 1.73205e-5 0.0149371 0.000350724 -8.21148e-6
0.0063219 -0.0110108 0.0278823 -0.0072361 0.0127171
0.00264213 1.18644e-6 -0.000154913 … -0.00335475 0.00419147
0.00187368 0.00315936 -0.0111698 0.00271088 -0.0100405
0.00291421 1.52903e-6 -0.00382745 -0.00441138 0.00492219
⋮ ⋱
-0.00155628 -2.30734e-6 0.00878927 0.0111443 -0.0242021
-0.00695112 -0.000924208 0.0135361 0.00185189 -0.0106929
-0.00444753 1.7005e-7 0.00510469 -0.0030131 -0.000350499
-0.00283263 -4.38726e-5 0.00161316 -0.000740089 -0.00451854
-0.00464792 -3.13702e-5 0.0019732 … 0.00214606 -0.00986261
0.00208197 0.000606016 -0.0084714 -0.00184894 -0.00279289
-0.0057928 -2.36751e-6 0.00556668 -0.00336604 -0.000118007
-0.0150847 -0.000946847 0.0154178 0.00298061 -0.0110905
0.00436221 -7.43884e-6 -0.0181692 0.00729842 -0.0202426
0.011324 0.000659877 -0.0245522 … 0.00633017 -0.0198908
0.000485084 -1.67179e-7 -0.0091768 0.00173709 -0.00808405
-0.00203203 -0.000379899 -0.00998926 -0.00163171 0.0013418
[:, :, 13] =
0.00355613 -0.00148139 0.00109446 … 0.00567658 -0.011787
-0.00390518 0.00225513 -0.000823849 -0.00585691 0.00992344
0.00291568 -0.00260679 0.00092917 0.00384719 -0.00452704
0.000769272 -4.32777e-5 -0.000172785 0.000322157 -0.0015609
-0.00507854 0.0045675 -0.00264505 -0.00148378 0.0028184
-0.000486914 -2.54141e-6 0.000993943 … 0.000106665 -2.74196e-5
0.0223541 -0.0205873 0.0111644 -0.000694964 -1.35017e-5
-0.037605 0.0342995 -0.0187872 0.000727461 -0.000130751
0.0284017 -0.0205045 -0.00207004 0.00220342 -0.00411383
-0.0131143 -1.62847e-6 0.0304887 6.58912e-5 -3.36512e-6
0.00633852 0.00410443 -0.025955 … -0.011318 0.0199959
-0.00288937 3.68533e-6 0.00357207 0.0198394 -0.0331738
-0.00139255 -0.000999686 0.00773926 -0.0161178 0.0205128
⋮ ⋱
0.00140687 -0.00146856 0.00183502 -0.00371426 0.00227529
-0.000101363 -2.76017e-6 -0.000649082 0.00185308 -0.00307042
-0.00647417 0.000397433 0.00787183 -0.00345536 0.00705918
0.0127634 2.95358e-5 -0.0174405 5.61066e-5 -0.000509055
-0.0132811 -0.000300215 0.019811 … 0.00292933 -0.0052473
0.00418413 1.09733e-5 -0.00906009 0.000416578 -0.000188827
0.0167908 0.00120511 -0.0217332 -0.00517419 0.00816235
-0.0329837 -0.00200856 0.0501814 0.00375625 -0.00551317
0.0230317 0.00119328 -0.0384689 -0.000278776 0.000643716
-4.47471e-6 8.36188e-7 -8.85988e-6 … -0.000206552 -0.000759992
-0.0103651 -0.000191526 0.0237438 0.00282315 -0.00670209
0.0106138 -5.11262e-7 -0.0282697 -0.00561311 0.014263
Gradient descent.
In [120]:
fTI = Psi(a)
a = a + tau.*PsiS(Phi(y-Phi(fTI, Omega), Omega));
Out[120]:
128×128×13 Array{Float64,3}:
[:, :, 1] =
0.0323998 0.0323408 0.032198 … 0.0497627 0.0497896 0.0498026
0.0323222 0.0322629 0.0321195 0.049747 0.0497735 0.0497863
0.0320984 0.0320383 0.0318918 0.0497036 0.0497298 0.0497425
0.031718 0.0316577 0.0315087 0.0496433 0.0496698 0.0496825
0.0311659 0.0311074 0.0309599 0.0495708 0.0495982 0.0496111
0.0304433 0.0303877 0.0302443 … 0.0494936 0.0495219 0.0495349
0.0295714 0.0295197 0.0293834 0.0494165 0.0494451 0.049458
0.0285937 0.0285468 0.0284203 0.0493344 0.0493626 0.0493751
0.0275173 0.0274758 0.0273607 0.0492568 0.0492838 0.0492956
0.026379 0.0263437 0.0262417 0.0491961 0.0492221 0.0492333
0.0252068 0.0251783 0.0250907 … 0.0491663 0.0491926 0.0492036
0.0239974 0.0239752 0.0239017 0.0491732 0.0492007 0.0492117
0.0227982 0.022781 0.0227206 0.0492064 0.0492362 0.049248
⋮ ⋱ ⋮
0.0382111 0.0381686 0.0379823 0.0554137 0.0557831 0.0559001
0.0386472 0.0385807 0.0383335 0.0555067 0.0558895 0.0560103
0.0389726 0.0388838 0.0385819 0.0555916 0.0559873 0.0561121
0.039167 0.0390565 0.038703 0.0556713 0.056079 0.0562073
0.0392543 0.0391239 0.0387253 … 0.0557556 0.0561754 0.0563073
0.0392432 0.039097 0.0386641 0.0558498 0.0562819 0.0564178
0.0391019 0.0389432 0.0384843 0.055957 0.056401 0.0565408
0.038889 0.0387209 0.038244 0.0560627 0.0565176 0.0566608
0.0387639 0.038588 0.0380964 0.0561424 0.0566062 0.0567523
0.0387139 0.0385291 0.0380195 … 0.056187 0.0566565 0.0568045
0.0387146 0.0385229 0.0379989 0.0562012 0.056674 0.056823
0.0387365 0.0385424 0.038014 0.0562015 0.0566757 0.0568252
[:, :, 2] =
0.00517647 0.00517974 0.00520866 … 0.000453717 0.000450113
0.00494234 0.00494263 0.00496545 0.000460674 0.000457591
0.00430781 0.00430343 0.00431392 0.000493755 0.000492376
0.003369 0.00336223 0.00336226 0.000513809 0.000514249
0.002139 0.00212951 0.00211789 0.000498846 0.000500436
0.000783836 0.00077358 0.000752523 … 0.00046975 0.000472512
-0.000455605 -0.000464663 -0.000491254 0.000428269 0.000432135
-0.00151366 -0.00152437 -0.00155787 0.000335391 0.000340167
-0.00234035 -0.00235136 -0.00238571 0.000150589 0.000155272
-0.00277925 -0.00278589 -0.0028116 -8.79292e-5 -8.46004e-5
-0.00278339 -0.00278594 -0.00280389 … -0.000359 -0.000358327
-0.00249213 -0.00249172 -0.00250609 -0.00060884 -0.000610867
-0.00190337 -0.00189785 -0.00190492 -0.000821571 -0.000825059
⋮ ⋱
0.00361317 0.0035157 0.00327191 -0.000156927 -0.000157751
0.00311397 0.00302163 0.00280189 7.32275e-5 7.75564e-5
0.00262674 0.00254535 0.00235944 0.000221212 0.000229091
0.00191046 0.00184993 0.00170897 0.000311088 0.000320092
0.00101829 0.000980576 0.00088745 … 0.000377979 0.00038726
3.51231e-5 2.00672e-5 -2.05707e-5 0.00041864 0.00042811
-0.00106653 -0.00106246 -0.00105575 0.000405405 0.000414133
-0.00205521 -0.00204404 -0.00200543 0.000322731 0.000330442
-0.00278134 -0.00276769 -0.00270948 0.000213383 0.000219765
-0.00340196 -0.00338264 -0.00330966 … 0.000131161 0.000135772
-0.00381386 -0.00379104 -0.00371005 9.1912e-5 9.55647e-5
-0.00393484 -0.00391203 -0.00382998 8.23326e-5 8.59554e-5
[:, :, 3] =
0.00138346 0.00134712 0.00132677 … 0.000294635 0.000295266
0.00135317 0.00131994 0.00130722 0.000291986 0.000292514
0.00126775 0.00123879 0.00123711 0.00029099 0.000291185
0.00117381 0.00114881 0.00115346 0.000299619 0.000300178
0.00112772 0.00110504 0.00110927 0.000316047 0.00031857
0.00113961 0.00111067 0.0010943 … 0.000339363 0.000343832
0.00117624 0.00113606 0.00108741 0.000364889 0.000369789
0.00121848 0.00116595 0.00108587 0.000383485 0.000387823
0.0012667 0.00120006 0.00108392 0.000396432 0.000398699
0.00129647 0.0012172 0.00106779 0.000404625 0.000404162
0.00130226 0.0012131 0.00103741 … 0.0004126 0.000409924
0.00125916 0.00116201 0.000964115 0.00042561 0.000420912
0.00111489 0.00101673 0.000814625 0.000437259 0.000431794
⋮ ⋱
-0.00701353 -0.00645471 -0.00488546 0.00423701 0.00419635
-0.0061419 -0.00556284 -0.00394824 0.00463457 0.00461425
-0.00523841 -0.00464047 -0.00298138 0.00500965 0.00500903
-0.00434972 -0.00373374 -0.00203747 0.00536678 0.00538489
-0.00349784 -0.00286825 -0.00114521 … 0.00571198 0.00574873
-0.00268225 -0.00204489 -0.00030208 0.0060428 0.00609817
-0.00197506 -0.00133154 0.000422803 0.00636174 0.00643585
-0.00138652 -0.000738818 0.00102683 0.00664149 0.00673154
-0.000848183 -0.000201675 0.0015706 0.00686179 0.00696506
-0.000418623 0.000221834 0.00199185 … 0.00701477 0.00712807
-0.000118745 0.000514737 0.00227896 0.00709912 0.00721828
8.54528e-6 0.000639047 0.00239924 0.00712595 0.00724738
...
[:, :, 11] =
0.00917278 0.0059813 -0.0003906 … 2.6135e-5 -0.00727329
-0.00578015 -0.00345774 -9.64634e-5 2.88868e-6 0.00466319
0.000226373 -0.00015325 0.000920243 -0.00055009 -0.000929758
0.00289008 0.00186457 4.47425e-5 1.84094e-6 -0.00195206
-0.0011004 -0.000295352 -0.000969456 0.00134247 0.00332507
-3.05964e-6 0.000455492 9.06466e-6 … -2.87702e-6 0.00015365
-0.00648117 -0.00686412 0.00171912 -0.0038552 -0.00532766
0.012836 0.0127208 5.0436e-6 0.00529647 0.00647929
-0.00939992 -0.0143707 -0.00260384 -0.00197791 -0.00421301
2.57869e-6 0.0121628 -9.07698e-6 -1.95516e-6 5.58853e-5
0.00194427 -0.00793574 0.00106706 … -0.000226475 0.0101109
-1.61525e-5 0.00201233 1.54474e-6 -0.000125637 -0.0156778
-0.000253978 0.00230396 -0.000617237 -0.00123861 0.00256596
⋮ ⋱
-0.0113686 -0.0107581 -0.00820473 0.00462062 0.00193607
-1.5835e-6 0.000135531 0.000844468 0.000351036 -0.0010109
0.00553721 0.0118431 0.0120688 -0.0017466 0.00297146
1.89804e-5 -0.0135594 -0.0155274 -6.6132e-6 -0.00105747
-0.00669351 0.00859285 0.00832474 … -0.000579034 -0.00308679
1.24352e-5 -0.00573448 -1.0406e-7 0.00178989 0.00218958
0.0247506 0.00643042 -0.000939542 -0.00130609 0.00198111
-0.0406369 -0.00350874 -7.5799e-6 -0.000372674 -0.00223376
0.024568 -0.00146143 -0.00114894 0.00132469 0.00165565
9.05832e-7 -4.31222e-6 -3.18369e-6 … -1.46323e-5 -0.00127304
-0.00402011 0.0061899 0.000796783 -0.000598393 -0.00490415
-5.99336e-7 -0.00909471 4.52905e-6 7.73119e-8 0.00990787
[:, :, 12] =
-0.00449085 0.00396476 -0.00232793 … -0.004166 0.000106307
-0.00319138 0.00245479 -0.00263145 -0.00319149 0.00150597
0.000276887 -4.09888e-7 -0.0035307 -5.42174e-5 1.72207e-6
-0.000669645 0.00224556 -0.00571916 0.000406822 -0.000740761
0.00156905 8.3065e-7 -0.00376568 -0.00287511 0.00385917
0.0121829 -0.0107244 0.00396029 … -0.00337479 0.0040742
0.00143818 3.64192e-6 -0.00286217 -0.000834918 -3.2223e-6
-0.0150026 0.0160502 -0.0109862 0.00254461 -0.00426762
-0.00333856 1.73205e-5 0.0149371 0.000350724 -8.21148e-6
0.0063219 -0.0110108 0.0278823 -0.0072361 0.0127171
0.00264213 1.18644e-6 -0.000154913 … -0.00335475 0.00419147
0.00187368 0.00315936 -0.0111698 0.00271088 -0.0100405
0.00291421 1.52903e-6 -0.00382745 -0.00441138 0.00492219
⋮ ⋱
-0.00155628 -2.30734e-6 0.00878927 0.0111443 -0.0242021
-0.00695112 -0.000924208 0.0135361 0.00185189 -0.0106929
-0.00444753 1.7005e-7 0.00510469 -0.0030131 -0.000350499
-0.00283263 -4.38726e-5 0.00161316 -0.000740089 -0.00451854
-0.00464792 -3.13702e-5 0.0019732 … 0.00214606 -0.00986261
0.00208197 0.000606016 -0.0084714 -0.00184894 -0.00279289
-0.0057928 -2.36751e-6 0.00556668 -0.00336604 -0.000118007
-0.0150847 -0.000946847 0.0154178 0.00298061 -0.0110905
0.00436221 -7.43884e-6 -0.0181692 0.00729842 -0.0202426
0.011324 0.000659877 -0.0245522 … 0.00633017 -0.0198908
0.000485084 -1.67179e-7 -0.0091768 0.00173709 -0.00808405
-0.00203203 -0.000379899 -0.00998926 -0.00163171 0.0013418
[:, :, 13] =
0.00355613 -0.00148139 0.00109446 … 0.00567658 -0.011787
-0.00390518 0.00225513 -0.000823849 -0.00585691 0.00992344
0.00291568 -0.00260679 0.00092917 0.00384719 -0.00452704
0.000769272 -4.32777e-5 -0.000172785 0.000322157 -0.0015609
-0.00507854 0.0045675 -0.00264505 -0.00148378 0.0028184
-0.000486914 -2.54141e-6 0.000993943 … 0.000106665 -2.74196e-5
0.0223541 -0.0205873 0.0111644 -0.000694964 -1.35017e-5
-0.037605 0.0342995 -0.0187872 0.000727461 -0.000130751
0.0284017 -0.0205045 -0.00207004 0.00220342 -0.00411383
-0.0131143 -1.62847e-6 0.0304887 6.58912e-5 -3.36512e-6
0.00633852 0.00410443 -0.025955 … -0.011318 0.0199959
-0.00288937 3.68533e-6 0.00357207 0.0198394 -0.0331738
-0.00139255 -0.000999686 0.00773926 -0.0161178 0.0205128
⋮ ⋱
0.00140687 -0.00146856 0.00183502 -0.00371426 0.00227529
-0.000101363 -2.76017e-6 -0.000649082 0.00185308 -0.00307042
-0.00647417 0.000397433 0.00787183 -0.00345536 0.00705918
0.0127634 2.95358e-5 -0.0174405 5.61066e-5 -0.000509055
-0.0132811 -0.000300215 0.019811 … 0.00292933 -0.0052473
0.00418413 1.09733e-5 -0.00906009 0.000416578 -0.000188827
0.0167908 0.00120511 -0.0217332 -0.00517419 0.00816235
-0.0329837 -0.00200856 0.0501814 0.00375625 -0.00551317
0.0230317 0.00119328 -0.0384689 -0.000278776 0.000643716
-4.47471e-6 8.36188e-7 -8.85988e-6 … -0.000206552 -0.000759992
-0.0103651 -0.000191526 0.0237438 0.00282315 -0.00670209
0.0106138 -5.11262e-7 -0.0282697 -0.00561311 0.014263
Soft threshold.
In [121]:
a = SoftThresh(a, lambd*tau);
Out[121]:
128×128×13 Array{Float64,3}:
[:, :, 1] =
0.0323256 0.0322666 0.0321238 … 0.0496885 0.0497154 0.0497284
0.032248 0.0321887 0.0320453 0.0496728 0.0496992 0.0497121
0.0320242 0.0319641 0.0318176 0.0496294 0.0496555 0.0496682
0.0316437 0.0315835 0.0314344 0.0495691 0.0495955 0.0496082
0.0310917 0.0310332 0.0308857 0.0494966 0.049524 0.0495368
0.0303691 0.0303135 0.0301701 … 0.0494194 0.0494477 0.0494607
0.0294972 0.0294455 0.0293092 0.0493423 0.0493709 0.0493838
0.0285195 0.0284726 0.0283461 0.0492602 0.0492884 0.0493009
0.0274431 0.0274016 0.0272865 0.0491826 0.0492095 0.0492214
0.0263047 0.0262695 0.0261674 0.0491219 0.0491479 0.0491591
0.0251326 0.0251041 0.0250165 … 0.0490921 0.0491183 0.0491293
0.0239232 0.0239009 0.0238275 0.049099 0.0491264 0.0491375
0.0227239 0.0227068 0.0226463 0.0491322 0.049162 0.0491738
⋮ ⋱ ⋮
0.0381369 0.0380944 0.0379081 0.0553394 0.0557089 0.0558259
0.038573 0.0385065 0.0382593 0.0554324 0.0558153 0.0559361
0.0388984 0.0388096 0.0385077 0.0555174 0.0559131 0.0560379
0.0390928 0.0389823 0.0386288 0.0555971 0.0560048 0.0561331
0.03918 0.0390497 0.0386511 … 0.0556814 0.0561011 0.0562331
0.039169 0.0390228 0.0385898 0.0557755 0.0562076 0.0563436
0.0390277 0.0388689 0.0384101 0.0558827 0.0563268 0.0564666
0.0388148 0.0386467 0.0381698 0.0559885 0.0564434 0.0565866
0.0386897 0.0385137 0.0380221 0.0560682 0.056532 0.0566781
0.0386397 0.0384549 0.0379453 … 0.0561127 0.0565823 0.0567303
0.0386404 0.0384486 0.0379247 0.0561269 0.0565998 0.0567488
0.0386623 0.0384681 0.0379398 0.0561273 0.0566014 0.056751
[:, :, 2] =
0.00510226 0.00510552 0.00513444 … 0.000379498 0.000375894
0.00486813 0.00486841 0.00489124 0.000386455 0.000383373
0.00423359 0.00422921 0.0042397 0.000419537 0.000418157
0.00329478 0.00328801 0.00328804 0.000439591 0.00044003
0.00206479 0.0020553 0.00204367 0.000424627 0.000426217
0.000709618 0.000699362 0.000678304 … 0.000395531 0.000398293
-0.000381386 -0.000390444 -0.000417035 0.000354051 0.000357917
-0.00143944 -0.00145015 -0.00148365 0.000261173 0.000265948
-0.00226614 -0.00227714 -0.00231149 7.63704e-5 8.10535e-5
-0.00270503 -0.00271167 -0.00273738 -1.37104e-5 -1.03817e-5
-0.00270917 -0.00271172 -0.00272967 … -0.000284781 -0.000284108
-0.00241791 -0.0024175 -0.00243187 -0.000534622 -0.000536648
-0.00182916 -0.00182363 -0.0018307 -0.000747352 -0.00075084
⋮ ⋱
0.00353895 0.00344148 0.00319769 -8.27083e-5 -8.35321e-5
0.00303975 0.00294742 0.00272767 0.0 3.33761e-6
0.00255252 0.00247113 0.00228522 0.000146993 0.000154873
0.00183624 0.00177572 0.00163476 0.000236869 0.000245873
0.000944073 0.000906358 0.000813231 … 0.000303761 0.000313041
0.0 0.0 -0.0 0.000344421 0.000353891
-0.000992316 -0.000988242 -0.000981535 0.000331186 0.000339915
-0.001981 -0.00196982 -0.00193121 0.000248512 0.000256223
-0.00270712 -0.00269348 -0.00263526 0.000139164 0.000145546
-0.00332774 -0.00330843 -0.00323545 … 5.69425e-5 6.15534e-5
-0.00373964 -0.00371682 -0.00363583 1.76932e-5 2.1346e-5
-0.00386062 -0.00383781 -0.00375576 8.11381e-6 1.17366e-5
[:, :, 3] =
0.00130924 0.00127291 0.00125255 … 0.000220417 0.000221047
0.00127895 0.00124572 0.001233 0.000217767 0.000218295
0.00119353 0.00116457 0.00116289 0.000216771 0.000216966
0.00109959 0.00107459 0.00107924 0.000225401 0.000225959
0.0010535 0.00103082 0.00103505 0.000241829 0.000244351
0.0010654 0.00103645 0.00102008 … 0.000265144 0.000269613
0.00110202 0.00106184 0.00101319 0.00029067 0.00029557
0.00114426 0.00109173 0.00101165 0.000309267 0.000313605
0.00119248 0.00112585 0.0010097 0.000322214 0.000324481
0.00122225 0.00114298 0.000993568 0.000330406 0.000329943
0.00122804 0.00113888 0.000963195 … 0.000338382 0.000335706
0.00118494 0.00108779 0.000889896 0.000351391 0.000346693
0.00104067 0.000942508 0.000740406 0.00036304 0.000357576
⋮ ⋱
-0.00693931 -0.00638049 -0.00481124 0.00416279 0.00412214
-0.00606768 -0.00548862 -0.00387402 0.00456035 0.00454003
-0.0051642 -0.00456625 -0.00290716 0.00493543 0.00493481
-0.00427551 -0.00365952 -0.00196325 0.00529256 0.00531067
-0.00342362 -0.00279403 -0.001071 … 0.00563776 0.00567451
-0.00260803 -0.00197067 -0.000227861 0.00596858 0.00602395
-0.00190084 -0.00125732 0.000348584 0.00628752 0.00636163
-0.0013123 -0.000664599 0.000952607 0.00656727 0.00665733
-0.000773965 -0.000127456 0.00149638 0.00678757 0.00689084
-0.000344404 0.000147616 0.00191763 … 0.00694055 0.00705385
-4.45263e-5 0.000440518 0.00220474 0.0070249 0.00714406
0.0 0.000564828 0.00232503 0.00705173 0.00717317
...
[:, :, 11] =
0.00909856 0.00590708 -0.000316382 … 0.0 -0.00719907
-0.00570593 -0.00338352 -2.22446e-5 0.0 0.00458897
0.000152154 -7.90309e-5 0.000846024 -0.000475871 -0.000855539
0.00281586 0.00179035 0.0 0.0 -0.00187784
-0.00102618 -0.000221133 -0.000895237 0.00126825 0.00325085
-0.0 0.000381273 0.0 … -0.0 7.94315e-5
-0.00640695 -0.0067899 0.0016449 -0.00378098 -0.00525344
0.0127618 0.0126466 0.0 0.00522225 0.00640507
-0.0093257 -0.0142965 -0.00252962 -0.0019037 -0.00413879
0.0 0.0120885 -0.0 -0.0 0.0
0.00187006 -0.00786152 0.000992838 … -0.000152256 0.0100367
-0.0 0.00193811 0.0 -5.14187e-5 -0.0156036
-0.00017976 0.00222974 -0.000543018 -0.00116439 0.00249174
⋮ ⋱
-0.0112943 -0.0106839 -0.00813051 0.0045464 0.00186185
-0.0 6.13123e-5 0.000770249 0.000276817 -0.000936678
0.00546299 0.0117689 0.0119946 -0.00167239 0.00289724
0.0 -0.0134852 -0.0154532 -0.0 -0.000983246
-0.00661929 0.00851863 0.00825052 … -0.000504815 -0.00301257
0.0 -0.00566026 -0.0 0.00171567 0.00211536
0.0246764 0.0063562 -0.000865324 -0.00123187 0.00190689
-0.0405626 -0.00343452 -0.0 -0.000298456 -0.00215954
0.0244938 -0.00138722 -0.00107472 0.00125047 0.00158143
0.0 -0.0 -0.0 … -0.0 -0.00119882
-0.00394589 0.00611568 0.000722564 -0.000524174 -0.00482993
-0.0 -0.00902049 0.0 0.0 0.00983365
[:, :, 12] =
-0.00441663 0.00389054 -0.00225371 … -0.00409178 3.20887e-5
-0.00311716 0.00238057 -0.00255723 -0.00311727 0.00143175
0.000202668 -0.0 -0.00345648 -0.0 0.0
-0.000595426 0.00217134 -0.00564495 0.000332604 -0.000666543
0.00149483 0.0 -0.00369146 -0.00280089 0.00378495
0.0121087 -0.0106502 0.00388607 … -0.00330057 0.00399998
0.00136396 0.0 -0.00278795 -0.000760699 -0.0
-0.0149284 0.015976 -0.010912 0.00247039 -0.0041934
-0.00326434 0.0 0.0148629 0.000276505 -0.0
0.00624769 -0.0109366 0.027808 -0.00716188 0.0126429
0.00256791 0.0 -8.06943e-5 … -0.00328053 0.00411725
0.00179946 0.00308514 -0.0110956 0.00263666 -0.00996628
0.00283999 0.0 -0.00375323 -0.00433716 0.00484797
⋮ ⋱
-0.00148207 -0.0 0.00871505 0.0110701 -0.0241279
-0.0068769 -0.000849989 0.0134619 0.00177767 -0.0106187
-0.00437331 0.0 0.00503047 -0.00293888 -0.00027628
-0.00275841 -0.0 0.00153894 -0.00066587 -0.00444432
-0.0045737 -0.0 0.00189899 … 0.00207184 -0.00978839
0.00200775 0.000531797 -0.00839718 -0.00177472 -0.00271867
-0.00571858 -0.0 0.00549247 -0.00329182 -4.37879e-5
-0.0150105 -0.000872628 0.0153435 0.00290639 -0.0110163
0.00428799 -0.0 -0.018095 0.0072242 -0.0201684
0.0112498 0.000585658 -0.024478 … 0.00625595 -0.0198166
0.000410866 -0.0 -0.00910258 0.00166287 -0.00800983
-0.00195781 -0.000305681 -0.00991504 -0.00155749 0.00126758
[:, :, 13] =
0.00348191 -0.00140717 0.00102024 … 0.00560236 -0.0117128
-0.00383096 0.00218091 -0.00074963 -0.0057827 0.00984923
0.00284146 -0.00253257 0.000854952 0.00377297 -0.00445282
0.000695053 -0.0 -9.85663e-5 0.000247939 -0.00148669
-0.00500433 0.00449328 -0.00257083 -0.00140956 0.00274418
-0.000412696 -0.0 0.000919725 … 3.24464e-5 -0.0
0.0222799 -0.0205131 0.0110902 -0.000620745 -0.0
-0.0375308 0.0342253 -0.018713 0.000653242 -5.65321e-5
0.0283275 -0.0204303 -0.00199582 0.0021292 -0.00403962
-0.0130401 -0.0 0.0304144 0.0 -0.0
0.0062643 0.00403021 -0.0258808 … -0.0112437 0.0199217
-0.00281515 0.0 0.00349786 0.0197651 -0.0330996
-0.00131833 -0.000925467 0.00766504 -0.0160436 0.0204386
⋮ ⋱
0.00133265 -0.00139434 0.0017608 -0.00364004 0.00220107
-2.71445e-5 -0.0 -0.000574863 0.00177886 -0.0029962
-0.00639995 0.000323214 0.00779761 -0.00338114 0.00698496
0.0126892 0.0 -0.0173663 0.0 -0.000434836
-0.0132069 -0.000225997 0.0197367 … 0.00285511 -0.00517308
0.00410991 0.0 -0.00898587 0.000342359 -0.000114609
0.0167165 0.00113089 -0.021659 -0.00509997 0.00808813
-0.0329095 -0.00193434 0.0501072 0.00368203 -0.00543895
0.0229575 0.00111906 -0.0383947 -0.000204558 0.000569497
-0.0 0.0 -0.0 … -0.000132334 -0.000685773
-0.0102909 -0.000117307 0.0236696 0.00274893 -0.00662787
0.0105395 -0.0 -0.0281955 -0.00553889 0.0141887
Exercise 3
Perform the iterative soft thresholding. Monitor the decay of the energy $E$.
In [122]:
include("NtSolutions/inverse_5_inpainting_sparsity/exo3.jl")
# niter = 1000
# a = U.*PsiS(fSpars)
# E = []
# for i in 1 : niter
# fTI = Psi(a)
# d = y - Phi(fTI, Omega)
# E = [E; 1/2.*vecnorm(d )^2 + lambd*sum(abs(a))]
# # step
# a = SoftThresh(a + tau.*PsiS(Phi(d, Omega)), lambd*tau)
# end
# figure(figsize = (7, 5))
# plot(E)
# show()
In [33]:
## Insert your code here.
Perform the reconstruction.
In [123]:
fTI = Psi(a);
Out[123]:
128×128 Array{Float64,2}:
0.5959 0.595353 0.595991 0.599185 … 0.807213 0.808388 0.808516
0.584907 0.588184 0.595149 0.599971 0.806676 0.808078 0.808128
0.57642 0.578371 0.584624 0.59133 0.805662 0.80676 0.80688
0.553808 0.557782 0.563702 0.575164 0.804513 0.804783 0.806422
0.526524 0.527638 0.531838 0.545507 0.802583 0.805117 0.810185
0.515926 0.507072 0.512286 0.511787 … 0.801578 0.802938 0.805109
0.535189 0.514696 0.511119 0.498153 0.797553 0.801766 0.801545
0.490022 0.548615 0.486463 0.4634 0.799584 0.809141 0.811728
0.476114 0.453588 0.431169 0.393877 0.798867 0.803229 0.803014
0.426983 0.410282 0.35747 0.199237 0.797629 0.803465 0.808521
0.416674 0.400481 0.384433 0.374821 … 0.796928 0.805276 0.816949
0.429307 0.413717 0.373537 0.33173 0.801037 0.811888 0.804466
0.385212 0.386148 0.384318 0.368313 0.803085 0.809197 0.817707
⋮ ⋱ ⋮
0.580924 0.568513 0.543235 0.486878 0.961058 0.931482 0.879598
0.545986 0.544897 0.540175 0.517487 0.967928 0.931906 0.889483
0.515031 0.523947 0.534806 0.539136 0.965063 0.933182 0.911034
0.493114 0.493515 0.509994 0.589143 0.96075 0.94113 0.917824
0.479228 0.499803 0.535113 0.551319 … 0.969464 0.946206 0.917224
0.477929 0.490436 0.529352 0.613429 0.963602 0.947622 0.925302
0.481471 0.490679 0.535707 0.686777 0.973385 0.94616 0.926498
0.476539 0.501362 0.550068 0.481006 0.987526 0.94981 0.921532
0.501373 0.517568 0.600884 0.776717 0.998177 0.962172 0.91704
0.530728 0.546292 0.586503 0.622514 … 0.994391 0.968419 0.916097
0.546093 0.563104 0.596076 0.570503 0.96796 0.954034 0.923728
0.540378 0.556386 0.599143 0.728456 0.9488 0.941611 0.930387
Display the result.
In [124]:
figure(figsize = (6, 6))
imageplot(clamP(fTI))
Exercise 4
Perform the iteration with a decaying value of $\lambda$
In [125]:
include("NtSolutions/inverse_5_inpainting_sparsity/exo4.jl")