Mathematical Sciences: Iterative Methods for Image Processing

该职业发展奖支持在图像处理中出现的大型方程组的迭代方法领域的数学研究。计划在图像处理和医学成像方面的应用。图像恢复的核心问题是给定一个被噪声和模糊退化的图像版本的原始图像的估计。我们将开发和分析新的快速算法来计算估计的图像。两种不同的图像恢复方法占主导地位，(i)代数恢复，被视为线性的，可能是病态的，方程系统；（ii）随机恢复，其中考虑到受随机干扰的矢量估计问题。将开发这两种方法的数值方法。这两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相当大。即使使用有效的方法，计算工作也要求很高。由于许多应用需要实时图像恢复，因此开发并行计算机的算法非常重要。在这个项目中所做的工作集中在迭代方法的开发上，这些方法使自己能够在多处理器上实现。

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.