In many applications, signals are measured according to a linear process, but
the phases of these measurements are often unreliable or not available. To
reconstruct the signal, one must perform a process known as phase retrieval.
This paper focuses on completely determining signals with as few intensity
measurements as possible, and on efficient phase retrieval algorithms from such
measurements. For the case of complex M-dimensional signals, we construct a
measurement ensemble of size 4M-4 which yields injective intensity
measurements; this is conjectured to be the smallest such ensemble. For the
case of real signals, we devise a theory of "almost" injective intensity
measurements, and we characterize such ensembles. Later, we show that phase
retrieval from M+1 almost injective intensity measurements is NP-hard,
indicating that computationally efficient phase retrieval must come at the
price of measurement redundancy.Comment: 18 pages, 1 figur
