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

“该奖项是根据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运算符来消除图像中的冗余，从而使数据稀疏化。对多分辨率的需求使得小波变换难以稀疏化局部不连续性，同时对旋转和平移保持不变而没有显著的冗余。基于小波的稀疏重建算法的能力有限，以占这种冗余限制了他们的表现在具有挑战性的实际应用。在这种情况下，有一个强烈的动机，开发多维图像稀疏化运营商是不变的平移和旋转，并在代表边缘主管。这项研究导致了多维信号恢复的几个领域的根本性进展。具体来说，研究人员应用该框架，以显着加快采集动态和光谱磁共振成像（MRI）data.The主要亮点的建议是梯度的泛化，以获得一个新的家庭的多维运营商。这些算子的使用导致更高阶的全变差（HD-TV）图像恢复方案，其（a）是旋转和平移不变的，（B）可以表示任意阶的多项式，以及（c）最小化振铃伪像。为了充分利用该框架在MRI应用中的强大功能，研究人员还开发了稳定有效的优化算法和计算方案，以获得强大的采样模式。拟议的研究项目涉及理论和计算的混合，专注于解决具有挑战性的实际问题。研究模块被调整为在多维图像数据处理中为研究生和本科生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.