2013/05/27 by Jameson Cahill, Peter G. Casazza, Cahill, Jameson +5 · 1 citation
Engineering · Physics and Astronomy · #Advanced X-ray Imaging Techniques #Advancements in Photolithography Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Welding Techniques and Residual Stresses
paper · pdf · doi:10.48550/arxiv.1305.6226
openalex publication_date 2013/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with 2M-1 projections of arbitrary rank in \HHM. Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.