Phase retrieval of finite Blaschke projection

• Published in: Mathematical methods in the applied sciences Vol. 43; no. 15; pp. 9090 - 9101
• Main Authors: Li, Youfa, Zhou, Chunxu
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.10.2020
• Subjects: Algorithms; Blaschke product measurement; Computer simulation; Hardy space; instantaneous frequency; Mathematical models; Phase retrieval; recursive reconstruction; uniqueness
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.6603

Phase retrieval by Fourier measurements is a classical application in coherent diffraction imaging, and the modified Blaschke products (MBPs) are the generalization of linear Fourier atoms. Motivated by this, we investigate the phase retrieval modeled as to reconstruct P(f)=∑k=0∞⟨f,B{a0,a1,…,ak}⟩B{a0,a1,…,ak} by the intensity measurements {|⟨f,Bk1⟩|,|⟨f,Bk2⟩|,|⟨f,Bk3⟩|:k≥1}, where f lies in Hardy space ℋ2(D) such that f(a0)=0, B{a0,a1,…,ak} and Bki are all the finite MBPs. We establish the condition on Bki such that P(f) can be determined, up to a unimodular scalar, by the above measurements. A byproduct of our result is that the instantaneous frequency of the target can be exactly reconstructed by the above intensity measurements. Moreover, a recursive algorithm for the phase retrieval is established. Numerical simulations are conducted to verify our result.