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

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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.

## 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.

#### Questions & Answers

current economic plans (MDGS) needs
I don't know what is happening
surajkumar
What is economic
What is the importance of study economics
Wilma
Economic is the study of how humans make decisions in face of sacristy
Wilma
economics is the study of how humans makes decision in the face of scarcity
Kpienta
economics is the study of human behaviour when faced with difficult situation example when goods and services are scarcity.
Sydney
what is Economic
what is 4ps of economic?
production place Price product
Benedict
Criticism of elasticity
what is unemployment
ohk thanks
Gyamfi
why is unemployment rapid in the country
Gyamfi
I need more explanation
Odo
what is unemployment
not working
Bethel
some one who is willing qualified to work but can't find job
jackie
Abubakar
some one who is willing to work but can't find job
Hawa
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Brian
Hawa
unemployment refers to the ability for someone who is capable and willing to work but could not find a job..
Mnoko
some one who not able to find a job
Dennis
please what is the secret of learning?
thomas
What is stock market?
explain the various types of cost curve
Short-run average fixed cost (SRAFC) Short-run average total cost (SRAC or SRATC) Short-run average variable cost (AVC or SRAVC) Short-run fixed cost (FC or SRFC) Short-run marginal cost (SRMC) Short-run total cost (SRTC)
Romy
what's economic development and growth
what do you understand by Ceteris Paribus?
the external factor will remained constant, except the price
Hasib
everything being equal
Chenwi
explain the uses of microeconomics
uses of microeconomics
Nikita
Adam Smith's definition of economics
what is economic deficit
this is a situation whereby a nation's outcome or available resources are not enough to the people thereby causing scarcity
Ariel
prices of Quality demanded is equal to Quality supplied
it's quantity demand and quantity supplied that's called equilibrium
Romy
no
NABUBOLO
they deal With prices
NABUBOLO
define the elasticity
NABUBOLO
explain different types of elasticity
NABUBOLO
oops 😬 you are right you talk about quality I tell about quantity
Romy
elasticity is the measurement of the percentage change of one economic variable in response to a change in another
Romy
Cross Elasticity of Demand (XED) Income Elasticity of Demand (YED) Price Elasticity of Supply (PES)
Romy
anything else?
Romy
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Grace
Romy, what is microeconomic?
thomas
What is Economic please
Dauda
Thomas, microeconomics is the study of how consumers, workers, and firms interact to generate outcomes in specific markets
Kieran
Dauda, economics is the study of people and choices. it is on one side the study of wealth and on the more important side, a part of the study if man
Kieran
How does one analyze a market where both demand and supply shift?
That's equilibrium market
Ramon
but an equlibrum can appear twice on the same market... both in Movement along the Demand/supply curve of shift in the Curve
Gabriel
I Mean on the same curve..
Gabriel
how can consumer surplus be calculated
Franklyn
Difference between extinct and extici spicies
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