Phase retrieval of finite Blaschke projection. (24th June 2020)

• Record Type: Journal Article
• Authors: Li, Youfa; Zhou, Chunxu
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 15(2020)
• Page Start: 9090
• Page End: 9101
• Publication Date: 2020-06-24
• Subjects: Blaschke product measurement -- Hardy space -- instantaneous frequency -- phase retrieval -- recursive reconstruction -- uniqueness; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6603
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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 { a 0, a 1, …, a k } ⟩ B { a 0, a 1, …, a k } by the intensity measurements { | ⟨ f, B k 1 ⟩ |, | ⟨ f, B k 2 ⟩ |, | ⟨ f, B k 3 ⟩ | : k ≥ 1 }, where f lies in Hardy space ℋ 2 ( D ) such that f ( a 0 )=0, B { a 0, a 1, …, a k } and B k i are all the finite MBPs. We establish the condition on B k i 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.