Mathematical Sciences: Iterative Methods for Image Processing

该职业发展奖支持图像处理中产生的大型方程组的迭代方法领域的数学研究。 计划应用于图像处理和医学成像。 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。 工作将完成开发和分析新的快速算法的估计图像的计算。 两种不同的方法来图像恢复占主导地位，（i）代数恢复，被视为一个线性的，可能是病态的，方程组和（ii）随机恢复，其中一个方面的问题估计向量受到随机干扰。 两种方法的数值方法将被开发。 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。 计算工作，即使有有效的方法，是苛刻的。 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。 在这个项目上所做的工作集中在迭代方法的开发上，这些方法可以在多处理器上实现。

This career advancement award supports mathematical research in the field of iterative methods for large systems of equations arising in image processing. Application to image processing and medical imaging is planned. The central problem of image restoration is the estimation of the original image, given a version of the image degraded by noise and blur. Work will be done developing and analyzing new fast algorithms for the computation of the estimated image. Two different approaches to image restoration predominate, (i) algebraic restoration, viewed as a linear, possibly ill-conditioned, system of equations and (ii) stochastic restoration where one regards the problem of estimating vectors subject to random disturbances. Numerical methods for both approaches will be developed. 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. The computational work, even with efficient methods, is demanding. Since many applications require real-time image restoration, the development of algorithms for parallel computers is important. Work done on this project focuses on the development of iterative methods that lend themselves will to implementation on multiprocessors.