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1196
19
Diffractive (Lensless) Imaging
John C.H. Spence
1 Introduction
Diffractive (or lensless) imaging refers to the use of theoretical methods
and computer algorithms to solve the phase problem for scattering by
a nonperiodic object. The name coherent X-ray diffractive imaging
(CXDI) is used in the X-ray community, which we could generalize to
CDI. Additional information about the object, such as the sign of the
scattering potential and the approximate boundary of the object, may
be combined with the measured scattered intensity to solve for the
phases of the scattered amplitudes. In this way, under conditions of
single scattering (and other approximations that often apply in optics
and in electron, X-ray, and neutron diffraction) it may be possible to
reconstruct a real-space image of an object by Fourier transform of the
complex scattering distribution, or Fraunhoffer diffraction pattern.
(Applications to Fresnel near-fi eld imaging are also possible. In this
geometry, resolution is, however, limited by detector pixel size, since
the magnifi cation is unity if lenses are not used.) In this review we will
not discuss the recently developed and powerful transport of intensity
method, which is also applicable to the near fi eld (Paganin and Nugent,
1998). By avoiding the need for a lens, the aberrations and resolution
limits introduced by lenses are thus avoided. Within the past decade
this process has been demonstrated experimentally for neutron, X-ray,
and electron scattering, so that the fi eld has reached an exciting point.
The electron work has produced atomic-resolution images, while
experiments with soft X-rays have fi nally produced three-dimensional
(tomographic) reconstructions. It now offers the real possibility of dif-
fraction-limited imaging with any radiation for which lenses do not
exist. Since each radiation interacts differently with matter, the method
can be expected to provide us with new information on matter in fi elds
as diverse as biology, materials science, and astronomy. Much current
interest focuses on tomographic imaging of whole cells, nanoparticles
and mesoporous materials, and on the imaging of proteins at near-
atomic resolution. At present, the inversion of even a two-dimensional
diffraction pattern is a lengthy process, and while three-dimensional
Chapter 19 Diffractive (Lensless) Imaging 1197
inversion may involve weeks of work for a 1024 × 1024 × 1024 voxel data
set (with much time devoted to data merging and preparation issues),
it is clear that automated processing will soon be possible. While it can
be shown rigorously that some of the algorithms described here do not
diverge, because of the nonconvex nature of the constraints involved
no formal proof of convergence or of the uniqueness of solutions exists.
Yet computational trials for simulated data with little noise practically
always converge rapidly to the known solution, while the inevitable
stagnation occurs in the presence of excessive noise. [The ability of
these algorithms to “climb out” of local minimum and fi nd a global
minimum in a data set with perhaps a million unknown parameters
(phases) is one of their most remarkable features.] The challenge now
is to perform experiments under conditions in which the experimental
parameters are suffi ciently well known that inversion of experimental
data performs as well as the simulations. In summary, over the past
decade many algorithms for lensless imaging have been published that
work well on simulated noisy data; the more diffi cult problem is exper-
imental data collection under the conditions for which the algorithms
converge. (This may require, for example, that the experimental scat-
tered intensity distribution possesses inversion symmetry, consistent
with the theoretical assumption of a two-dimensional hermitian poten-
tial or “real object.”)
2 History
The phase problem in optics has early origins: in a letter to Michelson,
Rayleigh comments that “the phase problem in interferometry is insol-
uble without a-priori information on the symmetry of the data” (Strutt,
1892). Several fi elds have contributed to the development of our current
working solution to this problem, including signal processing, wave-
front sensing, astronomy, electron microscopy, image processing, and
X-ray crystallography, in addition to applied mathematics. Thus a won-
derfully rich set of ideas has contributed, from communications theory
to diffraction physics and set theory. Since my background is in dif-
fraction physics, I must apologize to those workers in other areas (such
as acoustics) whose contributions to the vast literature on this subject
I have overlooked. In this review we will not discuss the crystallo-
graphic phase problem, except to note parallels (such as solvent fl at-
tening) with our noncrystallographic or “single particle” problem
(Carrozzini et al., 2004). [By the obvious method of periodic continua-
tion, it has been shown that the highly successful direct methods of
crystallography can be applied immediately to the continuous scatter-
ing from an isolated object, if the correct choice of sampling interval is
made (Spence et al., 2003b), however this may not be the most effi cient
solution.] In retrospect, we can now see that the key to a successful
solution was given in the crystallography literature as early as 1952. D.
Sayre in “Some implications of a theorem due to Shannon” (Sayre,
1952) appears to have been the fi rst to consider the relationship between
Shannon’s sampling theorem and Bragg’s law. He pointed out that for
1198 J.C.H. Spence
centrosymmetric crystals, where only the sign of structure factors F
g
is
unknown, these could be determined if intensities |F|
2
g
were observ-
able at half-integral values of g, since a change of sign requires a zero
crossing of the molecular scattering factor there. By Shannon’s theorem,
these fractional and integral order intensities are needed and suffi cient
to completely defi ne the autocorrelation function of the molecule.
Sayre’s fractional orders could thus be generated only by a molecule
that fi lled half the cell, leaving the remaining “half” empty. [The Pat-
terson function or autocorrelation function of an inorganic, continu-
ously bonded crystal will differ from that of the basis (one “molecule”),
since the autocorrelation of the molecule is twice as large as the unit
cell. These concepts will be clarifi ed in the next section.] Sayre con-
cludes that a solution to the phase problem for centric crystals could
as well be obtained from measurements of the intensity of half-order
refl ections as from study of the Bragg intensities. [The paper was
prompted by the observation of nonintegral refl ections (following
hydration) in experimental patterns from hemoglobin by Perutz and
others, and a talk by Gay.] Since diffraction from an isolated nonperi-
odic object is a continuous function of scattering angle it provides
immediate access to these “fractional orders.” Thus a solution in prin-
ciple to the phase problem for real, centric single particles has existed
in the literature since 1952.
Sayre’s paper had no impact in the image processing or signal pro-
cessing communities, and we will not trace here its important infl uence
on the development of density modifi cation techniques in protein crys-
tallography. (Here the molecule is often surrounded by a water jacket
of comparable volume to the molecule, effectively doubling the cell size
and possibly giving rise to “half-integral” refl ections.) For imaging, the
next important development was work by Gerchberg and Saxton (G–S)
(Gerchberg and Saxton, 1971), which posed the following question: if
the intensity of a general complex two-dimensional image and its cor-
responding diffraction pattern intensity is known, can the complex
image (and diffraction pattern) be reconstructed? (By a complex image
we refer here to the sample exit-face wavefunction, as discussed in the
next section.) These data are available in a modern electron micro-
scope. By solving the resulting quadratic equations (equal in number,
as they comment, to the number of unknown phases) using an iterative
algorithm, they showed successful inversion for one-dimensional data.
It has often been noted that these Fourier equations, one for each pixel
in the detector, may not be independent, are nonlinear if a sign con-
straint is used, and are corrupted by noise. Dependence in the equa-
tions may be introduced by choice of support shape. Other approaches
have been explored for solving these nonlinear Fourier equations, but
the G–S paper established the use of iterations between real and recip-
rocal space, with known information imposed repeatedly in each
domain. G–S do not establish uniqueness or convergence in general.
The connection with Sayre’s paper was not made; however, in retro-
spect we see that the sampling interval used by G–S for their analysis
of a nonperiodic object necessarily corresponded to the Shannon half-
integral orders of Sayre’s paper. Modern work is focused on three-
Chapter 19 Diffractive (Lensless) Imaging 1199
dimensional data in which only the diffraction pattern intensity from
a real or complex object is measured. However, for a pure phase object,
the modulus of the exit face wavefunction is known a priori to be unity
everywhere, greatly reducing the number of unknowns in the G–S
analysis. (This unit modulus constraint is, unfortunately, nonconvex.)
For a real object (or, equivalently, a weak phase object) the diffraction
pattern has Hermitian symmetry (Friedel’s law), resulting in a further
reduction in unknowns. Following the Gerchberg and Saxton paper
(and a second paper, Gerchberg and Saxton, 1972), it was still necessary
to determine the minimum information needed about a real object to
solve the phase problem if the diffracted intensity was given. The role
of the object boundary (defi ning a “support”) and the need to sample
the continuous scattering fi nely enough to satisfy Shannon’s theorem
(treating the autocorrelation function as the “bandlimit”) were all rec-
ognized at this time.
The autocorrelation of a two-dimensional object is twice as large as
the object in any direction, so Shannon’s theorem requires sampling at
half the Bragg angle for a periodically continued object. Bates referred
to this as “oversampling,” a term still in use. (Since it is in fact optimum
sampling of the diffracted intensity, we avoid that term here.) The
problem was vigorously attacked by authors such as Bates, Fiddy,
Fienup, Gonslaves, and Papoulis in the early nineteen eighties, and the
use of lensless imaging based on these ideas using X-ray scattering was
advocated (Sayre, 1980). The early series of papers by Bates and co-
workers are especially signifi cant, and can be traced through Bates and
McDonnell (1989), while Fiddy provided a new approach based on
analysis of zeros in the diffraction pattern (e.g., Liao et al., 1997, and
earlier work). It was shown, for example, that the set of all bandlimited
functions having a given set of real zero crossings is convex, and that,
in the absence of noise, the data might be factorizable. By 1982 a useful
working solution had been obtained (Fienup, 1982) for real two-
dimensional images by the addition of a feedback feature to the G–S
error-reduction (ER) algorithm. An important realization at about
this time was that the “landscape” (a map of error metric in the N-
dimensional search space) for the phase problem was not rugged, and
usually consisted of a single shallow global minimum with a few small
bumps; the essential diffi culty is not caused by the number of local false
minima, but rather by the large number of directions in which search
is possible, in a space containing one dimension for each image pixel.
This Fienup algorithm has come to be known as the hybrid input–output
or HIO algorithm. Feedback greatly speeds up convergence and
improves the ability of the algorithm to escape from local minima,
however, most modern work is based on a combination of the ER and
HIO algorithms, since only the ER algorithm supplies a reliable error
metric. Fienup showed that the ER algorithm converges, in the sense
that the error monotonically decreases (Fienup, 1997). Real images
could then be reconstructed from Fraunhoffer diffraction pattern inten-
sity data, provided that the sign of the scattering potential was known
together with an approximate estimate of the object boundary (support).
Important work on the uniqueness problem for the HIO algorithm was
1200 J.C.H. Spence
published by Bruck and Sodin (1979) and Baraket and Newsam (1984)
who showed that ambiguous solutions are obtained with these con-
straints only on “pathologically rare” occasions. The aim of this work is
to show that only one image function (or images equivalent to this)
satisfi es the sign, support, and Fourier modulus constraints.
The HIO algorithm, which we give below, is the basis for most suc-
cessful modern work, and has led to the development of many variants
and much detailed mathematical analysis. The book by Stark (1987)
provided an excellent overview of the subject up to 1987, and can be
strongly recommended to students, together with an important review
article by Millane (1990), which unifi ed the crystallographic and single-
particle approaches and so brought the fi eld to the attention of a much
larger audience. [The striking resemblance of the HIO algorithm to the
independently developed solvent fl attening methods of X-ray crystal-
lography can be traced through Wang (1985).] The water-jacket around
a protein in a crystal plays the same role as the “zero-density” region
outside the support of our single particles.
The HIO algorithm and its variants have been highly successful for
real objects, however, rather less attention has been paid to the more
diffi cult problem of complex image reconstruction. Except for special
cases, it is found that a very precise knowledge of the support is
needed, and that in one and two dimensions this support should be
disjoint (separated into two parts) (Fienup, 1987). Iterative inversion
schemes fail for one-dimensional real data that are not disjoint, and
work better in higher dimensions.
The third more recent period of work since about 1990, during which
the fi eld has grown rapidly, is discussed in the next sections. During
that time a major effort was made to apply these methods to soft X-ray
transmission data at the Brookhaven synchrotron (Sayre et al., 1998;
Miao et al., 1998). Our understanding of the success of the HIO algo-
rithm has increased considerably in that period due to the contribution
of mathematicians using powerful methods based on convex set theory;
Bregman projections and constraint theory were fi rst described as
early as 1982 (Youla and Webb, 1982) but not taken up until recently
(for an excellent review, see Bauschke et al., 2002). Questions of unique-
ness in dimensions greater than two have been considered, where the
problem is overdetermined (Millane, 1995), and it is generally found
that the iterative algorithms work better. In addition, a number of
developments of the HIO algorithm have been published (Bauschke
et al., 2003; Elser, 2003); the standard symbols used by mathematicians
for the theory of projections has been adopted to describe the problem,
and a new algorithm, which dispenses with any need for knowledge
of the object support [the “shrinkwrap” algorithm (Marchesini et al.,
2003b)], has been published. The problem of lost information within
the synchrotron beam-stop has been addressed by reconstructing the
diffuse scattering around a Bragg refl ection from a nanocrystal
(Robinson et al., 2001). In 2001 the fi rst of a biannual series of confer-
ences on the noncrystallographic phase problem was held at Lawrence
Berkeley Laboratory (Spence et al., 2001), with a second in Cairns,
Australia. The latest, held at Porquelles in France in 2005, can be found
Chapter 19 Diffractive (Lensless) Imaging 1201
at http://www.esrf.fr/NewsAndEvents/Events/Non-Crys20-06-05/.
An entirely different approach to the solution of the phase problem,
based on the transport of intensity equations, has also been fully devel-
oped during this period and applied to optical, X-ray, and electron
imaging (Paganin and Nugent, 1998).
These theoretical developments, often based on computer simula-
tions, have been supported by far fewer demonstrations of inversions
from experimental data. Only experimental results can truly give con-
fi dence in theory, and early results were somewhat disappointing. An
early application of the HIO algorithm to experimental optical speckle
data can be found in Cederquist et al. (1988), and applications to images
formed with laser light can be found in Weierstall et al. (2001). In the
X-ray imaging community the work at Brookhaven fi nally paid off in
1999, when images of lithographed characters were reconstructed from
soft X-ray diffraction patterns (Miao et al., 1999). More recently, an
atomic-resolution image of a single double-walled carbon nanotube
has been reconstructed from its electron microdiffraction pattern (Zuo
et al., 2003). These successes have led to rapid growth of the subject in
recent years.
3 Objects, Images, and Diffraction Patterns:
Validit y Domains of Approximations
The terms object, image, diffraction pattern, exit-face wavefunction, real and
complex object, and transmission function have sometimes been confused
in the literature and are defi ned here for clarity. We defi ne the object
by its ground-state charge density ρ(R) (which diffracts X-rays) and by
the corresponding electrostatic potential V(R) (which diffracts elec-
trons). V(R) is related to the density ρ(R) by Poisson’s equation. As a
result of inelastic processes, both may be complex, and could be referred
to generally as the “complex optical potential,” but in this chapter
unless otherwise stated we take them to be real. Here R is a three-
dimensional vector while r will be two dimensional. We defi ne the
exit-face wavefunction Ψ(r) across the downstream face of a thin slab of
sample in the transmission diffraction geometry with transmission func-
tion T(r) by
Ψ(r) = T(r) Ψ
0
(r)
where Ψ
0
(r) is the wavefi eld incident on the sample. [For perfectly
coherent radiation from a point source, collimated by a lens, Ψ
0
(r) is
approximately a plane-wave.]
Transmission functions for X-ray and electron diffraction are derived
below in terms of the wanted object properties ρ(R) or V(R). We defi ne
the image as any magnifi ed, resolution-limited or aberrated copy of
Ψ
0
(r), formed by a lens. Unfortunately this term is now widely used in
the CDI literature as a synonym for either object or exit-face wavefunction.
This use of “image” to refer to an exit-face wavefunction is now fi rmly
established and will be continued here; however, the more important
distinction between object and exit-face wavefunction is likely to