# 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.

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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
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production place Price product
Benedict
Criticism of elasticity
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Gyamfi
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some one who not able to find a job
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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)
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uses of microeconomics
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this is a situation whereby a nation's outcome or available resources are not enough to the people thereby causing scarcity
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prices of Quality demanded is equal to Quality supplied
it's quantity demand and quantity supplied that's called equilibrium
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no
NABUBOLO
they deal With prices
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explain different types of elasticity
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anything else?
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Thomas, microeconomics is the study of how consumers, workers, and firms interact to generate outcomes in specific markets
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I Mean on the same curve..
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Difference between extinct and extici spicies
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