# 9.1 A class of fast algorithms for total variation image restoration

 Page 1 / 6
This report summarizes work done as part of the Imaging and Optimization PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This module is based on the recent work of Junfeng Yang(jfyang2992@yahoo.com.cn) from Nanjing University and Wotao Yin, Yin Zhang, and Yilun Wang (wotao.yin, yzhang, yilun.wang@rice.edu) fromRice University. In image formation, the observed images are usually blurred by opticalinstruments and/or transfer medium and contaminated by noise, which makes image restoration a classical problem in image processing. Amongvarious variational deconvolution models, those based upon total variation (TV) are known to preserve edges and meanwhile removeunwanted fine details in an image and thus have attracted much research interests since the pioneer work by Rudin, Osher and Fatemi.However, TV based models are difficult to solve due to the nondifferentiability and the universal coupling of variables. In thismodule, we present, analyze and test a class of alternating minimization algorithms for reconstructing images from blurry andnoisy observations with TV-like regularization. This class of algorithms are applicable to both single- and multi-channel imageswith either Gaussian or impulsive noise, and permit cross-channel blurs when the underlying image has more than one channels. Numericalresults are given to demonstrate the effectiveness of the proposed algorithms.

## Introduction

In electrical engineering and computer science, image processing refers to any form of signal processing in which the input is animage and the output can be either an image or a set of parameters related to the image. Generally, image processing includes imageenhancement, restoration and reconstruction, edge and boundary detection, classification and segmentation, object recognition andidentification, compression and communication, etc. Among them, image restoration is a classical problem and is generally apreprocessing stage of higher level processing. In many applications, the measured images are degraded by blurs; e.g. theoptical system in a camera lens may be out of focus, so that the incoming light is smeared out, and in astronomical imaging theincoming light in the telescope has been slightly bent by turbulence in the atmosphere. In addition, images that occur in practicalapplications inevitably suffer from noise, which arise from numerous sources such as radiation scatter from the surface before the imageis sensed, electrical noise in the sensor or camera, transmission errors, and bit errors as the image is digitized, etc. In suchsituations, the image formation process is usually modeled by the following equation

$\begin{array}{c}\hfill f\left(x\right)=\left(k*\overline{u}\right)\left(x\right)+\omega \left(x\right),\phantom{\rule{1.em}{0ex}}x\in \Omega ,\end{array}$

where $\overline{u}\left(x\right)$ is an unknown clean image over a region $\Omega \subset {\mathbb{R}}^{2}$ ,“ $*$ " denotes the convolution operation, $k\left(x\right),n\left(x\right)$ and $f\left(x\right)$ are real-valued functions from ${\mathbb{R}}^{2}$ to $\mathbb{R}$ representing, respectively, convolution kernel, additive noise, and the blurry and noisy observation. Usually, theconvolution process neither absorbs nor generates optical energy, i.e., ${\int }_{\Omega }k\left(x\right)\mathrm{d}x=1$ , and the additive noise has zero mean.

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
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
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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!