# 9.1 A class of fast algorithms for total variation image restoration  (Page 4/6)

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$\begin{array}{c}\hfill \underset{\mathbf{w},u}{min}\sum _{i}\parallel {\mathbf{w}}_{i}\parallel +\frac{\beta }{2}\sum _{i}\parallel {\mathbf{w}}_{i}-{D}_{i}{u\parallel }^{2}+\frac{\mu }{2}{\parallel Ku-f\parallel }^{2},\end{array}$

where $\beta \gg 0$ is a penalty parameter. It is well known that the solution of ( ) converges to that of ( ) as $\beta \to \infty$ . In the following, we concentrate on problem ( ).

## Basic algorithm

The benefit of ( ) is that while either one of the two variables $u$ and $\mathbf{w}$ is fixed, minimizing the objective function with respect to the other has a closed-form formula that we willspecify below. First, for a fixed $u$ , the first two terms in ( ) are separable with respect to ${\mathbf{w}}_{i}$ , and thus the minimization for $\mathbf{w}$ is equivalent to solving

$\begin{array}{c}\hfill \underset{{\mathbf{w}}_{i}}{min}\parallel {\mathbf{w}}_{i}\parallel +\frac{\beta }{2}{\parallel {\mathbf{w}}_{i}-{D}_{i}u\parallel }^{2},\phantom{\rule{1.em}{0ex}}i=1,2,...,{n}^{2}.\end{array}$

It is easy to verify that the unique solutions of ( ) are

$\begin{array}{c}\hfill {\mathbf{w}}_{i}=max\left\{\parallel ,{D}_{i},u\parallel -,\frac{1}{\beta },,,0\right\}\frac{{D}_{i}u}{\parallel {D}_{i}u\parallel },\phantom{\rule{1.em}{0ex}}i=1,...,{n}^{2},\end{array}$

where the convention $0·\left(0/0\right)=0$ is followed. On the other hand, for a fixed $\mathbf{w}$ , ( ) is quadratic in $u$ and the minimizer $u$ is given by the normal equations

$\begin{array}{c}\hfill \left(\sum _{i},{D}_{i}^{\top },{D}_{i},+,\frac{\mu }{\beta },{K}^{\top },K\right)u=\sum _{i}{D}_{i}^{\top }{\mathbf{w}}_{i}+\frac{\mu }{\beta }{K}^{\top }f.\end{array}$

By noting the relation between $D$ and ${D}_{i}$ and a reordering of variables, ( ) can be rewritten as

$\begin{array}{c}\hfill \left({D}^{\top },D,+,\frac{\mu }{\beta },{K}^{\top },K\right)u={D}^{\top }w+\frac{\mu }{\beta }{K}^{\top }f,\end{array}$

where

$\begin{array}{c}\hfill w\triangleq \left(\begin{array}{c}{w}_{1}\\ {w}_{2}\end{array}\right)\in {\mathbb{R}}^{2{n}^{2}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{w}_{j}\triangleq \left(\begin{array}{c}{\left({\mathbf{w}}_{1}\right)}_{j}\\ ⋮\\ {\left({\mathbf{w}}_{{n}^{2}}\right)}_{j}\end{array}\right),\phantom{\rule{0.277778em}{0ex}}j=1,2.\end{array}$

The normal equation ( ) can also be solved easily provided that proper boundary conditions are assumed on $u$ . Since both the finite difference operations and the convolution are notwell-defined on the boundary of $u$ , certain boundary assumptions are needed when solving ( ). Under the periodic boundary conditions for $u$ , i.e. the 2D image $u$ is treated as a periodic function in both horizontal and vertical directions, ${D}^{\left(1\right)}$ , ${D}^{\left(2\right)}$ and $K$ are all block circulant matrices with circulant blocks; see e.g. , . Therefore, the Hessianmatrix on the left-hand side of ( ) has a block circulant structure and thus can be diagonalized by the 2D discreteFourier transform $\mathbf{F}$ , see e.g. . Using the convolution theorem of Fourier transforms, the solution of( ) is given by

$\begin{array}{c}\hfill u={\mathbf{F}}^{-1}\left(\frac{\mathbf{F}\left({D}^{\top },w,+,\left(\mu /\beta \right),{K}^{\top },f\right)}{\mathrm{diag}\left(\mathcal{F},\left(,{D}^{\top },D,+,\left(\mu /\beta \right),{K}^{\top },K,\right)\right)}\right),\end{array}$

where the division is implemented by componentwise. Since all quantities but $w$ are constant for given $\beta$ , computing $u$ from ( ) involves merely the finite difference operation on $w$ and two FFTs (including one inverse FFT), once the constant quantities are computed.

Since minimizing the objective function in ( ) with respect to either $\mathbf{w}$ or $u$ is computationally inexpensive, we solve ( ) for a fixed $\beta$ by an alternating minimization scheme given below.

Algorithm :

• Input $f$ , $K$ and $\mu >0$ . Given $\beta >0$ and initialize $u=f$ .
• While“not converged”, Do
• Compute $\mathbf{w}$ according to ( ) for fixed $u$ .
• Compute $u$ according to ( ) for fixed $w$ (or equivalently $\mathbf{w}$ ).
• End Do

## Optimality conditions and convergence results

To present the convergence results of Algorithm "Basic Algorithm" for a fixed $\beta$ , we make the following weak assumption.

Assumption 1 $\mathcal{N}\left(K\right)\cap \mathcal{N}\left(D\right)=\left\{0\right\}$ , where $\mathcal{N}\left(·\right)$ represents the null space of a matrix.

Define

$\begin{array}{c}\hfill M={D}^{\top }D+\frac{\mu }{\beta }{K}^{\top }K\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}T=D{M}^{-1}{D}^{\top }.\end{array}$

Furthermore, we will make use of the following two index sets:

$\begin{array}{c}\hfill L=\left\{i\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}\parallel ,{D}_{i},{u}^{*},\parallel <,\frac{1}{\beta }\right\}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}E=\left\{1,...,{n}^{2}\right\}\setminus L.\end{array}$

Under Assumption 1, the proposed algorithm has the following convergence properties.

Theorem 1 For any fixed $\beta >0$ , the sequence $\left\{\left({w}^{k},{u}^{k}\right)\right\}$ generated by Algorithm "Basic Algorithm" from any starting point $\left({w}^{0},{u}^{0}\right)$ converges to a solution $\left({w}^{*},{u}^{*}\right)$ of ( ). Furthermore, the sequence satisfies

• $\parallel {w}_{E}^{k+1}-{w}_{E}^{*}\parallel \le \sqrt{\rho \left({\left({T}^{2}\right)}_{EE}\right)}\parallel {w}_{E}^{k}-{w}_{E}^{*}\parallel ;$
• $\parallel {u}^{k+1}-{u}^{*}{\parallel }_{M}\le \sqrt{\rho \left({T}_{EE}\right)}{\parallel {u}^{k}-{u}^{*}\parallel }_{M};$

what is the stm
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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scanning tunneling microscope
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what is Nano technology ?
write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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what king of growth are you checking .?
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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for teaching engĺish at school how nano technology help us
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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