MSPA-MCS: High-Dimensional, Nonparametric Density Estimation for the Analysis of Images and Shapes

技术工具、计算资源和数字媒体的激增正在安全、娱乐、科学和医学领域开辟新的机遇。 我们正在自动获取大型测量数据库，这些数据库涵盖了从分子到细胞、组织、生物体、生态系统以及恒星和星系的物理尺度。 然而，这些数据的获取速度超过了我们分析它们的能力，许多应用程序的障碍不是由于缺乏数据，而是由于理解所有数据的艰巨任务。该项目解决了利用数据本身的实例构建此类数据集的统计模型的问题。 研究人员将从数学和计算的角度研究与在高维空间中构建统计模型相关的技术问题。 本研究将集中在这些方法的应用，在多维图像分析的问题，包括去噪，分割和图像合成。 该项目将开发用于分析大型数据集的新算法。 这些算法对数据的统计结构做了非常弱的假设，而是通过大量的例子来学习统计数据，使用来自非常高维空间的原始测量。 该项目结合了来自概率论，应用数学和计算机科学的一小组研究人员，同时解决了与高维概率密度函数估计有关的几个重要的基本问题，与这些估计相关的计算挑战，以及这些想法在科学和医学中的应用。 在概率论的工作解决了关于概率密度函数在非常高维空间的基本问题。 应用数学的工作是开发通过低维或稀疏表示来近似这些密度的方法。 计算机科学研究正在研究实现问题以及这些方法在需要更好的数据分析算法的各种持续问题中的应用。

A proliferation of technical instruments, computational resources, and digital media are opening up new opportunities in security, entertainment, science and medicine. We are automatically acquiring large databases of measurements that span physical scales from the molecule, to the cell, tissue, organism, ecosystem, and beyond to stars and galaxies. However, the rate of acquisition of these data exceeds our ability to analyze them, and many applications are stymied not by the lack of data but by the daunting task of understanding it all. This project addresses the problem of constructing statistical models of such data sets using examples from the data itself. The researchers will study the technical problems associated with constructing statistical modeling in high-dimensional spaces from mathematical and computational points of view. This research will focus on applications of these methods to problems in multidimensional image analysis, including denoising, segmentation, and image synthesis. This project will develop new algorithms for analyzing large sets of data. These algorithms make very weak assumptions about the statistical structure of data, and instead learn the statistics through large sets of examples, using raw measurements from very high dimensional spaces. The project combines a small team of researchers from probability theory, applied mathematics, and computer science to simultaneously address several important, fundamental questions pertaining to estimates of high-dimensional probability density functions, the computational challenges associated with these estimations, and the applications of these ideas to ongoing problems in science and medicine. The work in probability theory addresses basic questions about probability density functions in very high dimensional spaces. The applied mathematics work is developing methods for approximating these densities through lower-dimensional or sparse representations. The computer science research is examining implementation issues and the application of these methods to a wide range of ongoing problems that demand better algorithms for data analysis.