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的因果重建（近似）稀疏信号的时间序列从少量的“非相干”线性投影。该算法将是有用的实时动态磁共振成像（MRI）在介入放射学应用，如图像引导手术或功能MRI。MRI由于其“相对缓慢的图像采集”（大数据采集时间和/或缓慢的图像重建算法）而目前不可用于这种实时应用。其他潜在的应用包括太阳成像或实时单像素视频imaging.Since最近推出的压缩传感（CS）的动态层析成像，上述问题的静态版本已被彻底研究。但现有的动态问题的算法大多只是使用CS来一次性地联合重构整个时间序列。这是一个批处理解决方案，具有非常高的复杂性。替代方案-每次CS（simpleCS）-需要更多的测量。本研究首次提出并分析了信号序列重建的递归算法，该算法与简单CS算法具有相同的复杂度，但（a）与简单CS算法相比，该算法使用更少的无噪声测量来实现精确的重建;（B）当使用噪声测量时，尤其是当测量数量很小时，可以证明比简单CS算法实现更小的重建误差;以及（c）可证明随时间稳定（重构误差保持有界）。更少的测量意味着减少MRI的扫描时间，而递归重建意味着实时成像是可能的。通过利用稀疏模式随时间缓慢变化的事实，将问题公式化为部分已知支持的压缩感知问题之一。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.