Abstract
One of the crucial and major issues in engineering, particularly in
signal processing, is signal reconstruction. Frames are flexible tools
that helps to reconstruct signals/vectors in a stable way. Frame theory
was introduced for Hilbert spaces by Duffin et al. in 1952. If it
happens that we only have the intensity measurements or the phaseless
measurements of the lost signal then reconstructing the original signal
becomes difficult. In such cases, phase retrieval sequences play
essential roles to reconstruct or regain the signal from its intensity
measurements or phaseless measurements. In 2006, Balan et al. introduced
phase retrievable frames for Hilbert spaces. Phase retrieval has
received significant attention in various fields, including image
processing and signal reconstruction. Similar to phase retrieval, norm
retrieval sequences help to regain the norm of the original signal. In
2015 Bahmanpour et al. introduced norm retrieval sequences for Hilbert
spaces. We provide a thorough overview of phase retrieval and norm
retrieval by vectors and subspaces in separable Hilbert space. We also
highlight and discuss the recent developments in phase retrieval and
norm retrieval of partially lost signals.