CCF (CIF): Small: Recursive Reconstruction of Sparse Signal Sequences

稀疏信号序列的递归重建本研究的重点是递归算法，用于从少量“非相干”线性投影中因果地重建（近似）稀疏信号的时间序列。该算法将用于实时动态磁共振成像（MRI）介入放射学应用，如图像引导手术或功能MRI。由于“相对较慢的图像采集”（大数据采集时间和/或较慢的图像重建算法），MRI目前无法用于此类实时应用。其他潜在的应用包括太阳成像的动态断层扫描或实时单像素视频成像。自从最近引入压缩感知（CS）以来，上述问题的静态版本已经得到了深入的研究。但现有的动态问题算法大多只是利用CS来一次联合重构整个时间序列。这是一个批量解决方案，具有非常高的复杂性。另一种选择-每次CS (simpleCS) -需要更多的测量。本研究首次开发和分析了用于信号序列重建的递归算法，该算法具有与简单CS相同的复杂性，但(a)使用比简单CS所需的无噪声测量少得多的方法实现精确重建；(b)当使用有噪声的测量时，特别是当测量数量较少时，可证明比简单的CS实现更小的重建误差；和(c)可以证明是随时间稳定的（重构误差仍然有界）。更少的测量意味着减少MRI扫描时间，而递归重建意味着实时成像成为可能。通过利用稀疏模式随时间缓慢变化的事实，该问题被描述为具有部分已知支持的压缩感知之一。CS和顺序CS被纳入研究生/本科课程和适当级别的高级设计中。

Recursive Reconstruction of Sparse Signal SequencesThis research focuses on recursive algorithms for causally reconstructing a time sequence of (approximately) sparse signals from a small number of ``incoherent" linear projections. The algorithms will be useful for real-time dynamic magnetic resonance imaging (MRI) in interventional radiology applications such as image-guided surgery or in functional-MRI. MRI is currently not usable for such real-time applications due to its "relatively slow image acquisition" (large data acquisition times and/or slow image reconstruction algorithms). Other potential applications include dynamic tomography for solar imaging or real-time single-pixel video imaging.Since the recent introduction of compressive sensing (CS), the static version of the above problem has been thoroughly studied. But most existing algorithms for the dynamic problem just use CS to jointly reconstruct the entire time sequence in one go. This is a batch solution and has very high complexity. The alternative - CS at each time (simpleCS) - requires many more measurements. This research is the first to develop and analyze recursive algorithms for signal sequence reconstruction, which have the same complexity as simple CS, but which (a) achieve exact reconstruction using much fewer noise-free measurements than those needed by simple CS; (b) achieve provably smaller reconstruction error than simple CS, when using noisy measurements, especially when the number of measurements is small; and (c) are provably stable over time (reconstruction error remains bounded). Fewer measurements means reduced scan times for MRI, while recursive reconstruction means real-time imaging is possible. By exploiting the fact that sparsity patterns change slowly over time, the problem is formulated as one of compressive sensing with partially known support.CS and sequential CS are incorporated into the graduate/undergraduate curriculum and into senior-design at appropriate levels.