CAREER: Efficient Image Sparsifying Operators: Theory, Algorithms and Applications

这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。许多图像处理应用程序依赖于变换或操作符来消除图像中的冗余，从而使数据稀疏。对多分辨率的需求使得小波类变换很难稀疏局部不连续，同时在没有显著冗余的情况下对于旋转和平移是不变的。基于小波的稀疏重构算法处理这种冗余的能力有限，限制了它们在具有挑战性的实际应用中的性能。在这种背景下，人们迫切需要开发一种对平移和旋转不变并且能够表示边缘的多维图像稀疏算子。这项研究在多维信号恢复的几个领域取得了根本性的进展。具体地说，研究人员应用该框架显著加快了动态和光谱磁共振成像(MRI)数据的获取。这一提议的主要亮点是对梯度的推广，以获得一族新的多维算子。这些算子的使用导致了高次全变分(HD-TV)图像恢复方案，其(A)是旋转和平移不变的，(B)可以表示任意次数的多项式，以及(C)最小化振铃伪影。为了充分利用这一框架在磁共振成像应用中的强大功能，研究人员还开发了稳定和高效的优化算法和计算方案，以获得稳健的采样模式。拟议的研究项目涉及理论和计算的结合，侧重于解决具有挑战性的实际问题。研究模块经过调整，为研究生和本科生在处理多维图像数据方面提供直接的实践经验。

"This award is funded under the American Recovery and Reinvestment Act of 2009(Public Law 111-5)."Many image processing applications rely on a transform or an operator to eliminate the redundancies in images, thus sparsifying the data. The need for multi-resolution makes it difficult for wavelet-like transforms to sparsify local discontinuities, while being invariant to rotations and translations without significant redundancy. The limited ability of wavelet-based sparse reconstruction algorithms to account for this redundancy limits their performance in challenging practical applications. In this context, there is a strong motivation to develop multidimensional image sparsifying operators that are invariant to translations and rotations and is competent in representing edges. This research leads to fundamental advances in several areas of multidimensional signal recovery. Specifically, the investigators apply the framework to significantly accelerate the acquisition of dynamic and spectroscopic magnetic resonance imaging (MRI) data.The main highlight of this proposal is the generalization of the gradient to obtain a new family of multidimensional operators. The use of these operators results in higher degree total variation (HD-TV) image recovery schemes that (a) are rotation and translation invariant, (b) can represent polynomials of arbitrary degree, and (c) minimize ringing artifacts. To fully exploit the power of this framework in MRI applications, the investigators also develop stable and efficient optimization algorithms and a computational scheme to derive robust sampling patterns. The proposed research projects involves a mix of theory and computation, focused on solving challenging practical problems. The research modules are tuned to provide direct, hands-on experience for graduate and undergraduate students in the processing of multidimensional image data.