Feb 26 Adaptive Block Compressive Sensing for Noisy Images

本篇文章是来自于一本书的某一章

讲述图像去噪，获得原图X的时候，引进一种新的方法，是可以在前期处理的过程中，引入Lipschitz去做逼近的，同时利用Taylor展式，可以把原有的目标函数变换成convex function并且带有一个penalty 项的样子 分成了两部分处理，一部分可以继续用梯度下降的方法，另一部分就单独留着再继续处理。

除此之外，在image 处理过程中，也和以前的方法不同，以前通常都是一个大的image，并且大多是square形状的，在本文的paper中，允许多样性的存在，

Compressive Sensing Methodology

For a noisy image, Compressive Sensing can be expressed as :
$y = \Phi\Psi s+ \omega$ $~~~~~~~~~~~~~~~~~~~$(1)
$\omega$ is a N-dimension noise signal
$\Psi$ is a N $\times$N orthogonal basis matrix
$\Phi$ is an M $\times$ N random measurement matrix (M < N)
signal s in (1) can be estimated from measurement y by solving the convex minimization problem as follows.

argmin $_x||\Phi x -y ||_2^2+\lambda||x||_1$ $~~~~~~~$(2)
(2) is a constrained minimization problem of a convex function.
We can solve this problem by gradient-based method, which generate a new sequence $x_k$ via:

$x_0\in \mathbb{R}^N,x_k=x_{k-1}-t_k\nabla g(x_{k-1})$
where g(x) is a convex function, and $t_k$ is step size.

For (2),Let us look at the objective function, it can be rewritten as F(x) = g(x) + f(x)
g(x) is a convex function, and f(x) = $\lambda||x||_1$

Then the Function g(x) can be approximated by a quadratic function

$g(x,x_k)=g(x_{k-1})+&lt;(x-x_{k-1}),\nabla g(x_{k-1})&gt;+\frac{1}{2t_k}||x-x_{k-1}||_2^2$
this function $t_k$ can be replaced by a constant 1/L which is related to the Lipschitz constant.

Combined with other papers I have read. Applying the same idea to the non-smooth $l_1$ norm regularized problem:

minF(x) = min {g(x) + $\lambda||x||_1$ }

which can lead to the following iterative scheme:
$x_k = argmin_x{g(x_{k-1})+&lt;(x-x_{k-1}),\nabla g(x_{k-1})&gt;+\frac{1}{2t_k}||x-x_{k-1}||_2^2+\lambda||x||_1}$

After the constant term is ignored, we get:

$x_k =argmin_x\left( \large \frac{1}{2t_k}||x-(x_{k-1}-t_k\nabla g(x_{k-1})||_2^2+\lambda||x||_1\large \right)$

According to the Lipschitz gradient:
$||\nabla g(x) -\nabla g(y)||\leq L||x-y||$ for all x & y,

we know that when x is close to y, $\frac{||\nabla g(x) -\nabla g(y)||}{||x-y||}$ is the approximation of $g\prime\prime(x)$ at point x.

So our model and function can be approximated by:
$F(x)=g(x_{k-1})+&lt;(x-x_{k-1}),\nabla g(x_{k-1})&gt;+\frac{L}{2}||x-x_{k-1}||_2^2 + f(x)$

$x_k =argmin_x\left( \large \frac{L}{2}||x-(x_{k-1}-\frac{1}{L}\nabla g(x_{k-1})||_2^2+\lambda||x||_1\large \right)$

Or equivalently:
$x_k =argmin_x\left( \large \frac{L}{2}||x-d_k||_2^2+\lambda||x||_1\large \right)$
We recall the equation before, we know that:
$y = \Phi\Psi s+ \omega$

g(x)= $||\Phi x -y ||_2 ^2=||\Phi \Psi s -y ||_2 ^2$

$d_k = x_{k-1}-\frac{1}{L}\nabla g(x_{k-1})$

$d_k=x_{k-1}-\frac{1}{L}(\Phi\Psi^T)^T(\Phi\Psi^Tx_{k-1}-y)$

$\frac{1}{L}$ is the step size

The adaptive Block CS with sparsity

$l_0$ = #{j, $c_j$=0}
$l_\varepsilon^0$ = #{j, $c_j\leq\varepsilon$}

THE END of notes.