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

 Page 4 / 6
$\begin{array}{c}\hfill \underset{\mathbf{w},u}{min}\sum _{i}\parallel {\mathbf{w}}_{i}\parallel +\frac{\beta }{2}\sum _{i}\parallel {\mathbf{w}}_{i}-{D}_{i}{u\parallel }^{2}+\frac{\mu }{2}{\parallel Ku-f\parallel }^{2},\end{array}$

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

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

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By Rhodes By Robert Morris By Christine Zeelie By Brooke Delaney By OpenStax By Brooke Delaney By OpenStax By Alec Moffit By Michael Pitt By Rhodes