A Study On Phase Retrieval:Mathematical Model And Geometry Characterization Of Exact Recovery Condition

Phase is the intrinsic property of object,and records some important information.In reality,the detector can only record amplitude.Phase retrieval is the process of recon-structing phase from intensity measurements.The application fields of phase retrieval contains X-rays crystallography,optics,quantum mechanics,and signal processing.The research of phase retrieval mainly consists of two aspects:theory and algo-rithm.About algorithms,the development of phase retrieval has gone from Gerchberg-Saxton Algorithm to PhaseLift,which forms a mature system.About theory,the flour-ish of compressed sensing brings new research methods for phase retrieval.This paper studies the new problem phaseless compressed sensing.First,we establish the theoretical framework for the problem of phaseless com-pressed sensing with partially known signal support,which aims at generalizing the Null Space Property and the Strong Restricted Isometry Property from phase retrieval to partially sparse phase retrieval.We first introduce two exact recovery conditions,Par-tial Null Space Property?P-NSP?and the Partial Strong Restricted Isometry Property?P-SRIP?.We also prove that a random Gaussian matrix A?R^m×nsatisfies the P-SRIP with high probability when m=O?t?k-r?log?n-r/t?k-r???.Second,we research the the problem of phaseless compressed sensing with inter-ference.Aimed at the problem,we propose the exact recovery condition-Generalized Partial Strong Restricted Isometry Property?GP-SRIP?,which is the generalization of P-SRIP.Interestingly,the GP-SRIP constant is shaper than P-SRIP constant.Thus the new condition GP-SRIP outperforms previously one.Ultimately,we show that if a random Gaussian matrix A?R^m×nsatisfies m=O(tk log?n-|T|/tke?,the matrix satisfies GP-SRIP with the probability at least 1-?n-|T|/tke?^-tk.