Mathematical Sciences: Collaborative Research on Iterative Methods for Image Restoration

研究人员Reichel和他的同事对图像处理的数值方法进行了合作研究。该项目的目的是开发和分析新的快速算法，用于从受到噪声和模糊影响的退化图像中计算原始图像的估计。根据不同的观点，有两种哲学上不同的方法来处理图像恢复结果：i)在代数恢复中，一种是将问题视为一个线性的、可能是病态的方程组；ii)在随机恢复中，一种是将其视为在随机干扰下估计向量的问题。该项目为这两种方法开发了数值方法。无论采用哪种方法，都会产生线性方程组，其结构可用于开发和分析有效的图像恢复算法。该结构取决于对图像和噪声所做的假设。这些系统中的矩阵通常非常大；通常是一百万的数量级。因此，即使使用有效的方法，计算工作也是相当繁重的。由于许多应用都需要实时的图像恢复，因此并行计算机算法的开发是非常重要的。建议的研究集中于开发适合于在多处理器上实现的迭代方法。图像恢复的中心问题是在给定被噪声和模糊退化的图像的版本的情况下对原始图像的估计。当这张图像是从地球上观察到的一颗恒星时，模糊可能是由于大气造成的，而噪声可能源于信号的传输。有几种不同的图像恢复方法可用。它们都产生了大的方程系统；一百万个方程是常见的。方程系统有一个结构，它取决于对图像所做的假设和噪声的统计特性。该项目的目的是开发新的图像恢复算法。这些算法被设计成使用方程系统的结构来减少计算工作量。此外，它们还适合在并行计算机上使用。后者很重要，因为所需的计算量很大。

Reichel The investigator and his colleague undertake collaborative studies of numerical methods for image processing. The aim of the project is to develop and analyze new fast algorithms for computing an estimate of an original image from a degraded image that has been subjected to noise and blur. Depending on the point of view, two philosophically different approaches to image restoration result: i) in algebraic restoration one views the problem as a linear, possibly ill-conditioned, system of equations, and ii) in stochastic restoration one regards it as a problem of estimating a vector under random disturbances. The project develops numerical methods for both approaches. Either approach yields linear systems of equations that have a structure that can be used in the development and analysis of efficient algorithms for image restoration. The structure depends on the assumptions made on the image and the noise. The matrices in these systems are typically quite large; orders of a million are common. The computational work, even with efficient methods, is therefore quite demanding. Because many applications require real-time image restoration, the development of algorithms for parallel computers is important. The proposed research focuses on the development of iterative methods that lend themselves well to implementation on multiprocessors. The central problem of image restoration is the estimation of the original image, given a version of the image degraded by noise and blur. When the image is of a star observed from earth, the blur may be due to the atmosphere and the noise can stem from transmission of the signal. Several different approaches to image restoration are available. They all give rise to large systems of equations; a million equations are common. The system of equations has a structure, which depends on the assumptions made on the image and on the statistical properties of the noise. The purpose of the project is to develop new algorithms for the restoration of images. These algorithms are designed so that they use the structure of the system of equations in order to reduce the computational work. They are moreover suitable for use on parallel computers. The latter is important due to the large amount of computations required.