1 of 1

Bi-Lipschitz Mapping on a Space of Orbits

Kathleen Fleming and Sankalp Yadav

Department of Arts and Sciences, University of North Carolina at Wilmington

Introduction

​

Phase retrieval is the idea of understanding the nature of an input based on the properties of the output. So, for instance, when we think of x-ray imaging or astronomical photo rendering, we observe two dimensional projections of three-dimensional entities.

​

Therefore, we must first produce results for a base case and then expand it into a general form to observe ubiquitously true results. In order to accomplish this task of retrieval, we start off by using the idea of bi-Lipschitz bounds.

​

A bi-Lipschitz bound bounds a functions’ outputs, or intensity measurements, by the distance between the original inputs scaled by a constant lower and upper bound.

With this knowledge, we can now list some objectives to explore within the mapping:

    • Create a stable algorithm that produces input and output dimensions
    • Assess an optimal number of samples in order to account for complexity of increasing dimensions
    • Retrieve properties regarding the input through phase retrieval from the output dimension
    • Produce and analyze graphs that express properties for a varying range of internal variables

​

​

 

Results

Conclusions

 

Future Work

 

 

Abstract

​

 

Acknowledgements

MAT 495 Seminar in Mathematics, Dr. Jameson Cahill, Dr. Daniel Guo

 

 

Steps

References Cited

Radu Balan, Pete Casazza, and Dan Edidin. “On signal reconstruction without phase”. In: Applied and Computational Harmonic Analysis 20.3 (2006), pp. 345–356.

​

Afonso S. Bandeira et al. “Saving phase: Injectivity and stability for phase retrieval”. In: Applied and Computational Harmonic Analysis 37.1 (2014), pp. 106-125

​

Simon Maretzke and Thorsten Hohage. “Stability Estimates for Linearized Near-Field Phase Retrieval in X-ray Phase Contrast Imaging”. In: SIAM Journal on Applied Mathematics 77.2 (2017), pp. 384–408

 

 

 

Figure 3. Simplistic example of a case of image-based phase retrieval for two inputs plotted onto the same dimensional output