Phase retrieval concerns the solution to the phase problem

Phase problem

In physics the phase problem is the name given to the problem of loss of information concerning the phase that can occur when making a physical measurement. The name itself comes from the field of x-ray crystallography, where the phase problem has to be solved for the determination of a structure...

. Given a complex signal , of amplitude , and phase :

,

phase retrieval consists in finding the phase that for a measured amplitude satisfies a set of constraints.
For example if is the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 of , a typical constraint would be that is a bounded real and positive function.

Important applications of phase retrieval include X-ray crystallography

X-ray crystallography

X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...

 and transmission electron microscopy

Transmission electron microscopy

Transmission electron microscopy is a microscopy technique whereby a beam of electrons is transmitted through an ultra thin specimen, interacting with the specimen as it passes through...

.

See also

• Phase problem
  Phase problem
  In physics the phase problem is the name given to the problem of loss of information concerning the phase that can occur when making a physical measurement. The name itself comes from the field of x-ray crystallography, where the phase problem has to be solved for the determination of a structure...
  
  
• Crystallography
  Crystallography
  Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
  
  
• X-ray crystallography
  X-ray crystallography
  X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...