Basis extension and construction of tight frames

The notion of compression has received enormous attention in recent years because of its necessity in terms of the computational cost and other applicable features. But many times the notion expansion appears to be quite useful. Tight frames are quite useful in signal reconstruction, signal and image de-noising, compressed sensing because of the availability of a simple, explicit reconstruction formula. So in this paper, we discuss the extension of a basis by including some very sparse (at most two nonzero components) vectors so that the new frame becomes a tight frame. We do the basis extension in finite dimensional Hilbert spaces (both real and complex) to construct tight frames. We formulate constructive algorithms to do the aforementioned task. The algorithms guarantee us to produce tight frames with very less computational cost, and the new tight frames compensate for multiple erasures. The algorithms also do not disturb the vectors in the given basis. We also present one application of the aforementioned concept.‎‎