# 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
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
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
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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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