Mathematical Sciences: Collaborative Research on Iterative Methods for Image Restoration

Reichel 研究人员和他的同事进行图像处理的数值方法的合作研究。 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，两种不同的图像恢复方法：i）在代数恢复中，人们将问题视为线性的，可能是病态的方程组，ii）在随机恢复中，人们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。 这两种方法都产生线性方程组，其结构可用于图像恢复的有效算法的开发和分析。 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相当大;百万阶是常见的。 因此，即使采用有效的方法，计算工作也相当苛刻。 由于许多应用程序需要实时图像恢复，并行计算机的算法的发展是很重要的。 建议的研究重点是发展迭代方法，以及适合在多处理器上实现。 图像恢复的核心问题是估计原始图像，给出被噪声和模糊退化的图像的版本。 当图像是从地球上观察到的星星时，模糊可能是由于大气层造成的，而噪声可能是由于信号的传输造成的。 有几种不同的图像恢复方法。 它们都产生了大的方程组;一百万个方程是常见的。 方程组的结构取决于对图像和噪声的统计特性所做的假设。 该项目的目的是开发新的图像恢复算法。 这些算法的设计，使他们使用的方程组的结构，以减少计算工作。 它们也适合在并行计算机上使用。 后者是重要的，因为需要大量的计算。

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.