Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames

作者：Alaifari Rima^*; Daubechies Ingrid; Grohs Philipp; Thakur Gaurav

来源：Journal of Fourier Analysis and Applications, 2017, 23(6): 1480-1494.

DOI：10.1007/s00041-016-9513-7

• Phase retrieval
• Semi-discrete frames
• Injectivity
• Stability
• Sampling

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions that constitutes a semi-discrete frame, we ask whether any real-valued function can be uniquely recovered from its unsigned convolutions . We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at , it suffices to know on suitably fine lattices to uniquely determine f (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of , , we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.

• 出版日期2017-12