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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.


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

f ( x ) = ( k * u ¯ ) ( x ) + ω ( x ) , x Ω ,

where u ¯ ( x ) is an unknown clean image over a region Ω R 2 ,“ * " denotes the convolution operation, k ( x ) , n ( x ) and f ( x ) are real-valued functions from R 2 to R representing, respectively, convolution kernel, additive noise, and the blurry and noisy observation. Usually, theconvolution process neither absorbs nor generates optical energy, i.e., Ω k ( x ) d x = 1 , and the additive noise has zero mean.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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