Signal Processing - September 2016 - 158

lecture notes
Yonina C. Eldar, Nathaniel Hammen, and Dustin G. Mixon

recent advances in Phase retrieval

n many applications in science and
engineering, one is given the modulus
squared of the Fourier transform of an
unknown signal and then tasked with
solving the corresponding inverse problem, known as phase retrieval. Solutions to this problem have led to some
noteworthy accomplishments, such as
identifying the double helix structure of
DNA from diffraction patterns, as well
as characterizing aberrations in the Hubble Space Telescope from point spread
functions. Recently, phase retrieval has
found interesting connections with algebraic geometry, low-rank matrix recovery, and compressed sensing. These
connections, together with various new
imaging techniques developed in optics,
have spurred a surge of research into the
theory, algorithms, and applications of
phase retrieval. In this lecture note, we
outline these recent connections and
highlight some of the main results in
contemporary phase retrieval.

Relevance
Phase retrieval finds applications in areas
such as optics, X-ray crystallography,
astronomical imaging, speech processing, and computational biology. Every
application of this inverse problem
encounters several fundamental questions: How do we reconstruct the desired
signal from the measurements? To what
extent is the reconstruction unique and/
Digital Object Identifier 10.1109/MSP.2016.2565061
Date of publication: 2 September 2016

158

or stable? Can we develop new measurement devices that allow for robust signal
recovery? These questions form the
basis for the notes presented below.

Example 1

Suppose that x = (2, i, 0, 0) . Then
Fx 2 = (5, 9, 5, 1) . Let z be any of
(2i, -1, 0, 0), (0, 2, i, 0), or (0, 0, - i, 2) .
These are trivial ambiguities for which
Prerequisites
Fz 2 = (5, 9, 5, 1) . However, trivial
ambiguities
are not the only soluWe assume the reader has a basic
tions. For example, z = ( 5 + 1/2, i,
understanding of linear algebra, optimization, and probability. Some familiari5 - 1/2, - i) is not a trivial ambiguity with compressed sensing is helpful
ty, but still Fz 2 = (5, 9, 5, 1) .
but not necessary.
To obtain a well-posed problem, we
must acquire additional information.
Historically, this has been accomplished
Problem statement
by imposing structure on the signal. For
In standard applications of phase retrieval,
example, one might assume that x 0 is
we receive measurements of the modulus
squared of the Fourier transform of an
real and has compact support. In fact,
unknown signal x 0, y = Fx 0 2, where
this uniquely determines almost every
the magnitude is taken component-wise.
x 0 up to trivial ambiguities when taking
(For simplicity, we
a two-dimensional
model x 0 as a vector
(2-D) Fourier transPhase retrieval finds
form [or three-di in C n so that the Fouapplications in areas
mensional (3-D) or
rier operator F may
such as optics, X-ray
higher] [5]. However,
be represented by a
crystallography,
unfortunately, in 1-D
matrix.) Observe that,
astronomical imaging,
there is no uniquewithout additional
speech processing, and
ness even if the signal
information, this insupport is bounded.
verse problem is tercomputational biology.
Furthermore, even in
ribly ill posed. For
settings in which uniqueness is guaranexample, if y is the vector of all ones,
teed, there is no known general algothen x 0 can be any standard basis element
rithm to find the unknown signal from
multiplied by an arbitrary phase factor. In
its Fourier magnitude. A more recent
general, the set of solutions can have as
class of signal structures used in phase
many as n real degrees of freedom.
retrieval relies on sparsity [4]. A vector
Examples of these degrees of freedom
is defined to be k-sparse if it has at
include the so-called trivial ambiguities:
iz
2
2
most k nonzero entries. Sparsity priors
Fz = Fx if z = e x for some
have been used extensively in many
z ! [0, 2r), if z is a translation of x, or
fields of engineering and statistics and
if z is the conjugate reversal of x.
IEEE SIgnal ProcESSIng MagazInE

September 2016

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