On conditions for uniqueness in sparse phase retrieval

The phase retrieval problem has a long history and is an important problem in many areas of optics. Theoretical understanding of phase retrieval is still limited and fundamental questions such as uniqueness and stability of the recovered solution are not yet fully understood. This paper provides several additions to the theoretical understanding of sparse phase retrieval. In particular we show that if the measurement ensemble can be chosen freely, as few as 4k - 1 phaseless measurements suffice to guarantee uniqueness of a k-sparse M-dimensional real solution. We also prove that 2(k² - k + 1) Fourier magnitude measurements are sufficient under rather general conditions.