Applications of frame theory to diffraction data

Reference No.	2026a037
Type/Category	Grant for Young Researchers and Students- Short-term Visiting Researcher
Title of Research Project	Applications of frame theory to diffraction data
Principal Investigator	Navneet Redhu（Research Scholar at the Department of Mathematics, Indian Institute of Technology Indore・Ph.D. course in Mathematics）
Research Period	May 5,2026. - May 15,2026.
Keyword(s) of Research Fields	Harmonic Analysis, Frame Theory, Generalized Sampling, Inverse Problems, Phase Retrieval, Mathematical Crystallography, Diffraction Data Analysis
Abstract for Research Report	This project aims to develop a rigorous and unified theoretical framework for applying generalized sampling reconstruction and frame theory to the analysis of discrete, incomplete, and noisy measurement data arising in crystallographic structure determination and materials science. A central challenge in this area is the stable and accurate recovery of structural information from diffraction patterns or partial Fourier measurements, particularly in the presence of missing data and the well-known phase problem, where only magnitude (quadratic) information is available. The primary objective of this research is to formulate appropriate measurement and reconstruction frames that are adapted to the structure of diffraction data. Special attention will be given to scenarios involving erasures, limited sampling, and non-linear measurement constraints. By leveraging tools from harmonic analysis and modern sampling theory, the project seeks to identify conditions under which stable reconstruction is possible, and to design reconstruction schemes that are robust with respect to perturbations and incomplete observations. A key aspect of the project is the integration of two complementary areas of expertise: the host researcher’s work on lattice determination and diffraction-based structure analysis, and the applicant’s background in frame theory, generalized sampling, and functional analytic methods. Building on prior discussions and preliminary insights, the research will systematically extend generalized sampling techniques to settings motivated by crystallographic applications. This includes the study of reconstruction guarantees, error bounds, and the development of mathematically justified algorithms. The expected outcomes of the project include: (i) a well-defined theoretical framework connecting frame-based sampling methods with crystallographic reconstruction problems; (ii) new stability results and error estimates for reconstruction from incomplete or phaseless data; and (iii) potential algorithmic approaches that can be further explored for practical implementation. In the longer term, this work aims to bridge abstract harmonic analysis with real-world inverse problems in materials science, contributing to both theoretical advancements and application-oriented methodologies.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Navneet Redhu（Research Scholar at the Department of Mathematics, Indian Institute of Technology Indore・Ph.D. course in Mathematics）