On Recovering the Best Rank-r Approximation from Few Entries

In this note, we investigate how well we can reconstruct the best rank-$r$ approximation of a large matrix from a small number of its entries. We show that even if a data matrix is of full rank and cannot be approximated well by a low-rank matrix, its best low-rank approximations may still be reliably computed or estimated from a small number of its entries. This is especially relevant from a statistical viewpoint: the best low-rank approximations to a data matrix are often of more interest than itself because they capture the more stable and oftentimes more reproducible properties of an otherwise complicated data-generating model. In particular, we investigate two agnostic approaches: the first is based on spectral truncation; and the second is a projected gradient descent based optimization procedure. We argue that, while the first approach is intuitive and reasonably effective, the latter has far superior performance in general. We show that the error depends on how close the matrix is to being of low rank. Both theoretical and numerical evidence is presented to demonstrate the effectiveness of the proposed approaches.

Automatic Identification of the End-Diastolic and End-Systolic Cardiac Frames from Invasive Coronary Angiography Videos

Automatic identification of proper image frames at the end-diastolic (ED) and end-systolic (ES) frames during the review of invasive coronary angiograms (ICA) is important to assess blood flow during a cardiac cycle, reconstruct the 3D arterial anatomy from bi-planar views, and generate the complementary fusion map with myocardial images. The current identification method primarily relies on visual interpretation, making it not only time-consuming but also less reproducible. In this paper, we propose a new method to automatically identify angiographic image frames associated with the ED and ES cardiac phases by using the trajectories of key vessel points (i.e. landmarks). More specifically, a detection algorithm is first used to detect the key points of coronary arteries, and then an optical flow method is employed to track the trajectories of the selected key points. The ED and ES frames are identified based on all these trajectories. Our method was tested with 62 ICA videos from two separate medical centers (22 and 9 patients in sites 1 and 2, respectively). Comparing consensus interpretations by two human expert readers, excellent agreement was achieved by the proposed algorithm: the agreement rates within a one-frame range were 92.99% and 92.73% for the automatic identification of the ED and ES image frames, respectively. In conclusion, the proposed automated method showed great potential for being an integral part of automated ICA image analysis.