Mathematical Sciences: Iterative Methods for Image Processing

这一职业发展奖支持在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。计划将其应用于图像处理和医学成像。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。将开展开发和分析用于计算估计图像的新的快速算法的工作。图像恢复主要有两种不同的方法，(I)代数恢复，被视为线性的，可能是病态的方程组，和(Ii)随机恢复，其中人们考虑受随机干扰的估计向量的问题。这两种方法的数值方法都将得到发展。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。计算工作，即使使用有效的方法，也是要求很高的。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。在这个项目上所做的工作主要集中在迭代方法的开发上，这些方法将有助于在多处理器上实现。

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.