9.1 A class of fast algorithms for total variation image restoration

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

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.