Applications of Sparse Representation, Low Rank Approximation and Dictionary Learning to Image Processing, Pattern Recognition and Computer Vision

人类的视觉擅长在各种条件下检测人脸等物体：光照、遮挡和姿势。多年来，科学家和工程师一直在训练计算机模仿人类的视觉。其中一个主要问题是处理大量的图像数据。由于人脸或视频帧等图像可以存储在大矩阵或张量中，因此对矩阵和张量的大规模数据分析研究已成为一个热点。特别是，稀疏表示、低秩近似、字典学习、鲁棒主成分分析和多线性主成分分析等技术引起了人们的广泛关注，并成为图像/视频处理、模式识别和计算机视觉领域的强大工具。这些技术中的一个共同问题是需要有效的计算方法来优化相关的目标函数。一个有效的优化技术将导致计算成本的显著降低以及解决方案的精度的提高。此外，真实的数据集往往是不完整的，有许多维度或元素缺失。它们可能包含由于测量或通信错误而损坏的信息。因此，对来自现实问题的大型数据集的分析一直是最具挑战性的任务之一。例如，在生物识别中，光照、分辨率、方向、表情和遮挡等综合作用下的人脸识别一直是计算机视觉中的主要问题之一。本研究的目的是开发可应用于上述问题的模型和解决方法，并将研究常见的数学问题。具体来说，我们关注三个主要问题：(1)开发用于大规模图像数据集分析的通用模型和框架；(2)分析影响系统的不同因素，如训练输入数据的选择，以及所提出的技术的局限性；(3)新模式和新技术的实施、实验和测试。建议的工作可以对许多领域产生重大影响，包括图像/视频处理，计算机视觉，安全，生物识别，电信，遥感和生物信息学。我们工作的一个应用是面部年龄的变化，这对失踪儿童的调查很有用

Human vision is good at detecting objects such as human faces under various conditions: illumination, occlusions, and poses. Training the computer to mimic human vision has occupied scientists and engineers for many years. One of the major problems is dealing with very large amount of image data. Since images such as human faces or video frames can be stored in large matrices or tensors, the research in large scale data analysis for matrices and tensors has become a hot topic. In particular, techniques such as sparse representation, low rank approximation, dictionary learning, robust principal component analysis, and multi-linear principal component analysis have brought much attention and are powerful tools in image/video processing, pattern recognition and computer vision. A common problem among these techniques is the need for efficient computational methods for optimizing the related objective functions. An efficient optimization technique will lead to a significant reduction in computational costs as well as an improvement on accuracy of the solutions. Furthermore, real data sets are often incomplete with many dimensions or elements missing. They may contain corrupted information due to measurement or communication errors. Hence the analysis of large data sets coming from real-world problems has been one of the most challenging tasks. As an example, in biometrics, face recognition under a combined effect of illumination, resolution, orientation, expression, and occlusion has been one of the major problems in computer vision. The objective of this research is to develop models and solution methods that can be applied to the problems mentioned above and common mathematical problems will be investigated. Specifically we are concerned with three main issues: (1) Development of common models and frameworks for the analysis of large scale image datasets; (2) Analysis of different factors that affect the systems such as the choice of input data for training, and the limitation of the proposed techniques; and (3) Implementation, experimentation and testing of the new models and techniques. The proposed work can have significant impact on many areas including image/video processing, computer vision, security, biometrics, telecommunications, remote sensing, and bioinformatics. One of the applications of our work is on face age progression that can be useful for the investigation of missing children.**