Primal-Dual Proximal Splitting

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This tour explores a primal-dual proximal splitting algorithm, with application to imaging problems.

In [2]:

Convex Optimization with a Primal-Dual Scheme

In this tour we use the primal-dual algorithm detailed in:

Antonin Chambolle and Thomas Pock A First-order primal-dual algorithm for convex problems with application to imaging, Journal of Mathematical Imaging and Vision, Volume 40, Number 1 (2011), 120-145

One should note that there exist many other primal-dual schemes.

We consider general optimization problems of the form $$ \umin{f} F(K(f)) + G(f) $$ where $F$ and $G$ are convex functions and $K : f \mapsto K(f)$ is a linear operator.

For the primal-dual algorithm to be applicable, one should be able to compute the proximal mapping of $F$ and $G$, defined as: $$ \text{Prox}_{\gamma F}(x) = \uargmin{y} \frac{1}{2}\norm{x-y}^2 + \ga F(y) $$ (the same definition applies also for $G$).

The algorithm reads: $$ g_{k+1} = \text{Prox}_{\sigma F^*}( g_k + \sigma K(\tilde f_k) $$ $$ f_{k+1} = \text{Prox}_{\tau G}( f_k-\tau K^*(g_k) ) $$ $$ \tilde f_{k+1} = f_{k+1} + \theta (f_{k+1} - f_k) $$

The dual functional is defined as $$ F^*(y) = \umax{x} \dotp{x}{y}-F(x). $$ Note that being able to compute the proximal mapping of $F$ is equivalent to being able to compute the proximal mapping of $F^*$, thanks to Moreau's identity: $$ x = \text{Prox}_{\tau F^*}(x) + \tau \text{Prox}_{F/\tau}(x/\tau) $$

It can be shown that in the case $\theta=1$, if $\sigma \tau \norm{K}^2<1$, then $f_k$ converges to a minimizer of the original minimization of $F(K(f)) + G(f)$.

More general primal-dual schemes have been developped, see for instance

L. Condat, A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optimization Theory and Applications, 2013, in press.

Inpainting Problem

We consider a linear imaging operator $\Phi : f \mapsto \Phi(f)$ that maps high resolution images to low dimensional observations. Here we consider a pixel masking operator, that is diagonal over the spacial domain.

Load an image.

In [3]:
name = 'lena';
n = 256;
f0 = load_image(name);
f0 = rescale(crop(f0,n));

Display it.

In [4]:

We consider here the inpainting problem. This simply corresponds to a masking operator.

Load a random mask $\La$.

In [5]:
rho = .8;
Lambda = rand(n,n)>rho;

Masking operator $ \Phi $.

In [6]:
Phi = @(f)f.*Lambda;

Compute the observations $y=\Phi f_0$.

In [7]:
y = Phi(f0);

Display it.

In [8]:

Total Variation Regularization under Constraints

We want to solve the noiseless inverse problem $y=\Phi f$ using a total variation regularization: $$ \umin{ y=\Phi f } \norm{\nabla f}_1 $$

This can be recasted as the minimization of $F(K(f)) + G(f)$ by introducing $$ G(f)=i_H(f), \quad F(u)=\norm{u}_1 \qandq K=\nabla, $$ where $H = \enscond{x}{\Phi(x)=y}$ is an affine space, and $i_H$ is the indicator function $$ i_H(x) = \choice{ 0 \qifq x \in H, \\ +\infty \qifq x \notin H. } $$

Shorcut for the operators.

In [9]:
K  = @(f)grad(f);
KS = @(u)-div(u);

Shortcut for the TV norm.

In [10]:
Amplitude = @(u)sqrt(sum(u.^2,3));
F = @(u)sum(sum(Amplitude(u)));

The proximal operator of the vectorial $\ell^1$ norm reads $$ \text{Prox}_{\lambda F}(u) = \max\pa{0,1-\frac{\la}{\norm{u_k}}} u_k $$

In [11]:
ProxF = @(u,lambda)max(0,1-lambda./repmat(Amplitude(u), [1 1 2])).*u;

Display the thresholding on the vertical component of the vector.

In [17]:
t = -linspace(-2,2, 201);
[Y,X] = meshgrid(t,t);
U = cat(3,Y,X);
V = ProxF(U,1);
% 3D display
colormap jet(256);
shading interp;

For any 1-homogeneous convex functional, the dual function is the indicator of a convex set. For the $\ell^1$ norm, it is the indicator of the $\ell^\infty$ ball $$ F^* = i_{\norm{\cdot}_\infty \leq 1} \qwhereq \norm{u}_\infty = \umax{i} \norm{u_i}. $$

The proximal operator of the dual function is hence a projector (and it does not depend on $\sigma$ ) $$ \text{Prox}_{\sigma F^*}(u) = \text{Proj}_{\norm{\cdot}_\infty \leq 1}(u). $$

A simple way to compute the proximal operator of the dual function $F^*$, we make use of Moreau's identity: $$ x = \text{Prox}_{\tau F^*}(x) + \tau \text{Prox}_{F/\tau}(x/\tau) $$

In [15]:
ProxFS = @(y,sigma)y-sigma*ProxF(y/sigma,1/sigma);

Display this dual proximal on the vertical component of the vector.

In [16]:
V = ProxFS(U,1);
% display
colormap jet(256);
camlight; shading interp;

The proximal operator of $G = i_H$ is the projector on $H$. In our case, since $\Phi$ is a diagonal so that the projection is simple to compute $$ \text{Prox}_{\tau G}(f) = \text{Proj}_{H}(f) = f + \Phi(y - \Phi(f)) $$

In [15]:
ProxG = @(f,tau)f + Phi(y - Phi(f));

Primal-dual Total Variation Regularization Algorithm

Now we can apply the primal dual scheme to the TV regularization problem.

We set parameters for the algorithm. Note that in our case, $L=\norm{K}^2=8$. One should has $L \sigma \tau < 1$.

In [16]:
L = 8;
sigma = 10;
tau = .9/(L*sigma);
theta = 1;

Initialization, here |f| stands for the current iterate $f_k$, |g| for $g_k$ and |f1| for $\tilde f_k$.

In [17]:
f = y;
g = K(y)*0;
f1 = f;

Example of one iterations.

In [18]:
fold = f;
g = ProxFS( g+sigma*K(f1), sigma);
f = ProxG(  f-tau*KS(g), tau);
f1 = f + theta * (f-fold);

Exercise 1

Implement the primal-dual algorithm. Monitor the evolution of the TV energy $F(K(f_k))$ during the iterations. Note that one always has $ f_k \in H $ so that the iterates satisfies the constraints.

In [19]:
In [20]:
%% Insert your code here.

Display inpainted image.

In [21]:

Exercise 2

Use the primal dual scheme to perform regularization in the presence of noise $$ \umin{\norm{y-\Phi(f)} \leq \epsilon} \norm{\nabla f}_1. $$

In [22]:
In [23]:
%% Insert your code here.

Inpainting Large Missing Regions

It is possible to consider a more challening problem of inpainting large missing regions.

To emphasis the effect of the TV functional, we use a simple geometric image.

In [24]:
n = 64;
name = 'square';
f0 = load_image(name,n);

We remove the central part of the image.

In [25]:
a = 4;
Lambda = ones(n);
Lambda(end/2-a:end/2+a,:) = 0;
Phi = @(f)f.*Lambda;


In [26]:
imageplot(f0, 'Original', 1,2,1);
imageplot(Phi(f0), 'Damaged', 1,2,2);

Exercise 3

Display the evolution of the inpainting process.

In [27]:
In [28]:
%% Insert your code here.
In [28]: