# Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

@article{Bianchi2010PhaseRF, title={Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms}, author={Gabriele Bianchi and Richard J. Gardner and Markus Kiderlen}, journal={Journal of the American Mathematical Society}, year={2010}, volume={24}, pages={293-343} }

We propose strongly consistent algorithms for reconstructing the characteristic function 1K of an unknown convex body K in R n from possibly noisy measurements of the modulus of its Fourier transform c 1K. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K…

## 10 Citations

### On the Reconstruction of Planar Lattice-Convex Sets from the Covariogram

- MathematicsDiscret. Comput. Geom.
- 2012

A partial positive answer to the problem of the reconstruction of lattice-convex sets K from gK is provided by showing that for d=2 and under mild extra assumptions, gK determines K up to translations and reflections.

### The covariogram and Fourier–Laplace transform in ℂn

- Mathematics
- 2013

The covariogram gK of a convex body K in Rn is the function that associates to each x∈Rn the volume of the intersection of K with K+x . Determining K from the knowledge of gK is known as the…

### Covariograms Generated by Valuations

- Mathematics
- 2015

Let \phi be a real-valued valuation on the family of compact convex subsets of \mathbb{R}^n and let K be a convex body in \mathbb{R}^n. We introduce the \phi -covariogram g_{K,\phi} of K as the…

### Star discrepancy for new stratified random sampling I: optimal expected star discrepancy

- Mathematics
- 2022

We introduce a class of convex equivolume partitions. Expected star discrepancy results are compared for stratified samples under these partitions, including simple random samples. There are four…

### Covariogram of a cylinder

- MathematicsJournal of Contemporary Mathematical Analysis
- 2014

In this paper we establish relationships between the covariogram and the orientation-dependent chord length distribution function of a cylinder and those of its base. Also, we obtain explicit…

### Covariogram of a cylinder

- Mathematics
- 2014

In this paper we establish relationships between the covariogram and the orientation-dependent chord length distribution function of a cylinder and those of its base. Also, we obtain explicit…

### Expected $L_2-$discrepancy bound for a class of new stratified sampling models

- Mathematics
- 2022

. We introduce a class of convex equivolume partitions. Expected L 2 − discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate…

### Constructing the covariogram of a convex body for efficient infrared images restoration

- MathematicsInfrared Physics & Technology
- 2021

### Weighted Null Vector Initialization and its Application to Phase Retrieval

- Computer Science, MathematicsICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2020

A simple method, called weighted null vector initialization (WNI), which can be used to compute accurate initialization vectors for solving non-convex phase retrieval and outperforms the best spectral initializer, and null initializer.

### Efficient Algorithms for Ptychographic Phase Retrieval

- Mathematics, Physics
- 2014

The mathematical formulation of the ptychographic phase retrieval problem is examined, some of the existing methods for solving the inverse problem are analyzed, and a number of practical techniques are discussed that can improve the efficiency and robustness of numerical algorithms for solution.

## References

SHOWING 1-10 OF 58 REFERENCES

### Convergence of algorithms for reconstructing convex bodies and directional measures

- Mathematics
- 2006

We investigate algorithms for reconstructing a convex body K in R n from noisy measurements of its support function or its brightness function in k directions u 1 ,..., u k . The key idea of these…

### Reconstruction of Convex Bodies from Brightness Functions

- MathematicsDiscret. Comput. Geom.
- 2003

Abstract. Algorithms are given for reconstructing an approximation to an unknown convex body from finitely many values of its brightness function, the function giving the volumes of its projections…

### Shape Estimation from Support and Diameter Functions

- Computer ScienceJournal of Mathematical Imaging and Vision
- 2005

This work addresses the problem of reconstructing a planar shape from a finite number of noisy measurements of its support function or its diameter function and facilitates a systematic statistical analysis via the Cramér-Rao lower bound (CRLB), which provides a fundamental lower bound on the performance of estimation algorithms.

### Confirmation of Matheron's conjecture on the covariogram of a planar convex body

- Mathematics
- 2007

The covariogramgK of a convex bodyK in E d is the function which associates to each x2 E d the volume of the intersection ofK withKCx. In 1986 G. Matheron conjectured that for dD 2 the covariogramgK…

### THE DISTRIBUTION FUNCTION OF THE CONVOLUTION SQUARE OF A CONVEX SYMMETRIC BODY IN R n

- Mathematics

I n t r o d u c t i o n a n d n o t a t i o n s The start ing point of our investigation is a paper of K. Kiener [K]. Before we explain his results we have to introduce some notation. Let C be a…

### COMPUTATION OF THE PERIMETER OF MEASURABLE SETS VIA THEIR COVARIOGRAM. APPLICATIONS TO RANDOM SETS

- Mathematics
- 2011

The covariogram of a measurable set A ⊂ Rd is the function gA which to each y ∈ Rd associates the Lebesgue measure of A ∩ (y + A). This paper proves two formulas. The first equates the directional…

### Geometric Tomography

- Mathematics
- 1995

422 NOTICES OF THE AMS VOLUME 42, NUMBER 4 T omography, from the Greek τóμoς, a slice, is by now an established and active area of mathematics. The word is usually associated with computerized…

### Estimation of mean particle volume using the set covariance function

- MathematicsAdvances in Applied Probability
- 2003

Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard…