Compressed sensing, also known as compressive sensing, is a technique for finding sparse solutions to underdetermined linear systems. In signal processing, compressed sensing is the process of acquiring and reconstructing a signal that is known to be sparse or compressible. The theory asserts that certain signals or images can be recovered from what was previously thought to be highly incomplete measurements. This report reviews this rapidly growing field and its application in structural health monitoring. The report also presents a theoretical problem of constructing the full-field strain similar to that traditionally obtained from digital image correlation using very few samples of discrete strain measurements.