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1222 J.C.H. Spence

zone-plate monochromator can be used (Howells et al., 2002). For CDI

using electron beams, the coherence requirements are easily met for

the nanostructures of most interest. We note that the drive for higher

resolution, for a ﬁ xed number of object pixels, reduces the demand on

coherence. The number of pixels, especially in tomography, is likely to

be limited by computer processing power to less than 1024

in the

medium-term future, as discussed below.

It is important to devise a consistent deﬁ nition of resolution in CDI.

Each spatial frequency u = d

−1

= θ/λ in the object diffracts energy at

scattering angle θ into the far ﬁ eld. Consider a square area detector

used for diffractive imaging for which the largest scattering angle into

the midpoint of the side of the detector (not the corner) is θ

max

. Then,

in the small angle approximation, this angle deﬁ nes a cutoff in the

“transfer function” (see below) at u

max

= θ

max

/λ, and the full period of

the corresponding ﬁ nest periodicity in the object that can be recon-

structed is d

min

= λ/θ

max

= 1/u

max

. This value of d

min

has frequently been

quoted as the resolution limit in CXDI. However, while it is a most

important experimental parameter, it fails to consider the accuracy of

the phasing process, and is not equal to the corresponding Rayleigh

resolution limit. There are other considerations, which we now

discuss.

The Rayleigh resolution limit was intended for the imaging of binary

stars, which are incoherent point sources, unlike the phase contrast

usually important for CXDI. If we do nevertheless wish to apply the

Rayleigh condition to this situation, we may consider that the CXDI

area detector in the far ﬁ eld plays the role of a square aperture in the

back-focal plane of an ideal lens. The numerical aperture imposed on

the reconstruction is then θ

max

, and, if the reconstruction is perfect (no

errors in phasing), the image will be given by the ideal object charge

density convoluted with a sinc function amplitude (the impulse

response for linear imaging), whose full width at half maximum is

min

/2 = 0.5/u

max

. (The distance between ﬁ rst minima is d

min

. For a cir-

cular area detector the factor 0.5 becomes 0.61—the square detector

does slightly better because of contributions from the corners.) Thus

our adapted “Rayleigh resolution” is half the ﬁ nest spatial periodicity,

and is therefore equal to the sampling interval needed in real space for

linear phase-contrast imaging, to avoid loss of information. Two

samples are needed for every full periodicity in the object. (For an

incoherent imaging model the impulse response becomes a sinc

squared function, for which a sampling interval of d

min

/4 is needed,

since the autocorrelation of the transform of the sinc squared function

is a triangular function, doubling the bandwidth and resolution, as

pointed out in Rayleigh’s original paper.) For a lens-based system with

aberrations, the factor 0.5 depends on the aberrations of the lens, but

takes its minimum value for the diffraction-limited CXDI case.

There are two further considerations. For phase contrast, the ability

to distinguish adjacent small objects will depend on the phase shift

each introduces, and thus the resolution becomes a property of the

sample, not only of the instrument. It is then impossible to deﬁ ne reso-

lution in a meaningful sample-independent manner. In coherent optics,

Chapter 19 Diffractive (Lensless) Imaging 1223

this problem is partly addressed by introducing the concept of a lens

coherent transfer function (CTF). Such a function has been introduced

in a way that also tests the reliability of the phasing process in an

excellent recent proposal by V. Elser for a resolution deﬁ nition for

CXDI. The algorithm is repeatedly run to convergence, each time start-

ing with a different set of random phases. The results for image con-

trast are plotted as a function of spatial frequency, showing, if noise is

not too severe, a relatively smooth curve that falls to zero at some u

max

(If the phasing process fails, the average of these many runs will be

zero at each spatial frequency.) The resolution is then 0.5/u

max

if a

square detector is used. This appears to be the best current deﬁ nition

of resolution for CXDI; however, it ignores the dependence of resolu-

tion on sample properties for phase contrast.

For a known object, the faithfulness of the reconstruction may be

indicated by a cross-correlation function between the reconstructed

estimate and the known object, or crystallographic R-factor, as dis-

cussed in detail elsewhere (Spence et al., 2003a). For an unknown

object, the ER error metric ε deﬁ ned in Eq. (10) has been shown by

computational trials against known objects to vary monotonically with

a cross-correlation function (Fienup, 1997). The best resolution achieved

in CXDI is currently about 8 nm.

8 Computer Processing Demands

While noise-free data converge in a few iterations, several hundred

iterations of the HIO algorithm are typically needed for convergence

of good quality two-dimensional experimental data, or several thou-

sand if many data are missing in three dimensions. Computer process-

ing power can therefore impose serious limitations on CXI, especially

for tomography. Recall that for an area detector of linear pixel dimen-

sion N, the ratio of largest to smallest feature size is N/2. In many cases

we wish to obtain phase-contrast images of a real object, so that sample

thickness is limited by the need to avoid spatial variations in absorp-

tion, which would lead to the possible complications of complex object

restoration. (For CXDI, avoiding a beam energy near an inner-shell

ionization edge may therefore be important.) A second limit on thick-

ness may be set by the need to satisfy the weak-phase object condition,

or avoid extinction effects, both of which introduce multiple scattering

and a complex object. [The inversion of Eq. (1) for ∆n

is also referred

to as the phase unwrapping problem.] Fixing this limiting thickness

and N for a data cube then ﬁ xes the lowest resolution and largest ﬁ eld

of view possible.

For N = 1024, a double-precision complex N

array occupies 16 giga-

bytes, and three or four of these must be accessible directly in RAM

(plus a single-bit array for the support mask) to perform HIO iterations.

For the Mac G5, a single fast Fourier transform (FFT) on this array takes

about 1 min with the FFTW routine, although it is not possible to store

sufﬁ cient data in RAM at present to work with a single processor, and

hard disk transfers are prohibitively time consuming. Clusters of 16

1224 J.C.H. Spence

dual processor G5 machines with TCP gigabit ethernet or Inﬁ niband

appear to be ideal, when times of approximately 15 and 7 s per iteration,

respectively, are achieved for complex 1K × 1K × 1K arrays (Barty, 2005).

Many procedures may be used to reduce the total data processing time.

For example, working with some low-resolution 512

projections

selected from the three-dimensional data set can provide useful support

estimates using the two-dimensional shrinkwrap algorithm, before

attempting full three-dimensional Fourier iterations. Thus overnight

(or longer) computing times must be expected for tomographic CDI in

the near future. Once the three-dimensional data set has been phased

and transformed to real space, a variety of commercial tomographic

viewing programs can be used to present the data, making it possible

to “see inside” a nanostructure, such as a cell or metallic foam.

9 Summary

The past decade has been an exciting time for diffractive imaging. In it,

we have seen the ﬁ rst successful applications of the phasing algorithms

developed in the 1980s to experimental data, together with the develop-

ment of many new algorithms, ideas, and stimulating intellectual dis-

cussion at workshops and conferences. The interdisciplinary nature of

the subject has been striking and exciting. Experimental results have

appeared for phase-contrast imaging by neutrons (Nugent, 2003), elec-

trons (Weierstall et al., 2001), electrons at atomic resolution (Zuo et al.,

2003), and X-rays in both two and three dimensions (Miao et al., 1999,

2002; Chapman et al., 2006). Applications of the method have been

slower to develop, but already both in materials science and biology it

is clear that the power of imaging inside nanostructures (going beyond

the projection approximation) in three dimensions presents an exciting

prospect for the future. Preliminary applications of phase-contrast X-

ray imaging at lower resolution have included the imaging of nano-

structures within composite materials, mesoporous silicates and foams,

and the imaging of crack tips (Salvo et al., 2003). Bone is also of interest,

since it consists of 20-nm nanocrystals of hydroxyapatite in a labyrin-

thine structure. There is every reason to suppose that the pursuit by

tomographic CXDI imaging of these type of materials to higher resolu-

tion will be possible and of great interest to scientists. Using medium

energy X-rays, the imaging of much thicker material should be possible

than that studied by tomographic electron microscopy in materials

science (Midgely et al., 2001). For CDI by electron diffraction in materi-

als science the limiting problem is the method of sample support in the

microscope. One obvious solution is to use a thin crystal of known

structure (such as graphite) as the physical supporting membrane for

the nanostructure. Then the known atom positions of the graphite may

be used instead of the zero-density region in the HIO iterations. (This

allows a connection with the phasing method of fragment completion

in crystallography—the graphite atoms provide a kind of reference

structure in projection.) Experiments along these lines are in progress.

In biology, the situation will be clariﬁ ed only when careful studies

of radiation damage by spot-fading and other methods have been

Chapter 19 Diffractive (Lensless) Imaging 1225

completed. These are needed to determine the domain of applicability

of the CXDI method in comparison with other microscopies. Figure

19–6 shows a plot dose against resolution. The domain of applicability

of many microscopies is discussed on this ﬁ gure (and on plots of resolu-

tion against thickness) by Howells et al. (2005), showing the niche for

CXDI. Whole cell imaging has been pursued with both the zone-plate

X-ray microscope and by cryoelectron microscopy, where, using the

latter technique, a resolution of about 2 nm is possible in samples up to

100 n m thick or more. It seems likely that radiation damage will prevent

competitive performance by CXDI; however, the method may provide

useful images at perhaps 5 nm resolution in much thicker samples

using medium energy X-rays in an environment of vitreous ice. The

optimum choice of X-ray energy involves many issues, including the

variation of synchrotron undulator brightness with beam energy and

the variation in phase contrast with beam energy. While the coherent

ﬂ ux B available for a given synchrotron source brightness varies as λ

the required ﬂ uence (from the X-ray cross section) A scales as λ

−2

, so that

the recording time A/B varies as λ

−4

in a most unfavorable manner as

X-ray beam energy increases. The dose in Grays needed to scatter a

given number of photons into a voxel varies inversely as the fourth

power of the resolution in tomography. An analysis of the variation of

dose against resolution for several microscopies including CDI can be

found in Marchesini et al. (2003a). Here the statistical demands of good

imaging (based on the Rose equation) are compared with the maximum

tolerable dose for a given resolution for single-particle imaging.

In summary, the demand for higher resolution three-dimensional

noninvasive imaging with old and new radiation sources continues

unabated in both materials science and biology, and it seems clear

now that diffractive imaging will soon be able to make a decisive

contribution.

Spectacular 25 femtosecond single-shot images have just been

obtained at DESY by CXDI using 30 nm X-rays from a free-electron laser

with 90 nm spatial resolution (H. Chapman et al., Nature, 2006, in

press). It is clear that diffractive imaging will play a major role in future

time-resolved imaging efforts.

Acknowledgments. This chapter has summarized the work of many

groups and many of my collaborators, as indicated in the references.

I am particularly grateful for the help of Malcolm Howells, Uwe

Weierstall, and Anton Barty during the preparation of this review.

The work was supported by NSF, CBST, and IDBR award.

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(See page 1138)

1228

The Notion of Resolution

S. Van Aert, Arnold J. den Dekker, D. Van Dyck, and A. Van den Bos

1 Introduction

In microscopy, resolution has always been, and still is, an important issue.

Since it is not an unambiguously deﬁ ned physical quantity, it is interpreted

in many ways (den Dekker and van den Bos, 1997). The purpose of this

chapter is, on the one hand, to brieﬂ y review past and existing resolution

deﬁ nitions and methods, and, on the other hand, to present alternative

quantitative deﬁ nitions of resolution based on model ﬁ tting. Throughout

this chapter, emphasis will be placed on electron microscopy.

Using model ﬁ tting, the resolution will principally be discussed in

terms of the precision with which unknown quantities, atom positions

in particular, can be measured. It will be shown that a precision of the

order of 0.01 Å is in principle possible even with an electron microscope

that is not corrected for spherical and chromatic aberration. Once atom

positions can be measured with a precision of 0.01 Å they can be used

as input data for theoretical ab initio calculations (Muller, 1998, 1999;

Spence, 1999; Kisielowski et al., 2001b). Such calculations make it pos-

sible to calculate the properties of a material with a given structure. In

this way, the combination of precise experimental structure determina-

tion and theoretical calculations contributes to the understanding of

the properties–structure relation. A complete understanding of this

relation, combined with recent progress in building materials atom by

atom, will enable materials science to evolve toward materials design,

that is, from describing and understanding toward predicting materi-

als with interesting properties (Wada, 1996; Olson, 1997, 2000; Reed and

Tour, 2000; Browning et al., 2001).

The detailed outline of the sections in this chapter is as follows. In

Section 2, classical two-point resolution criteria are discussed. The

most widely known classical resolution criterion is that of Lord

Rayleigh. The Rayleigh criterion is derived from the assumption that

the human visual system needs a minimal contrast to discriminate two

points in its composite intensity distribution. Other classical criteria

can be seen as modiﬁ ed versions of Rayleigh’s criterion. The classical

criteria are expressed in terms of the width of the point spread function

Chapter 20 The Notion of Resolution 1229

of the imaging instrument. Section 3 discusses resolution in the spatial

frequency domain, addressing literature relating linear systems theory

to resolution. In this section, the diffraction limit to resolution and its

relation to the classical resolution criteria are introduced. Furthermore,

the notion of superresolution is discussed. It will be shown that to

attain superresolution, prior knowledge is required. In Section 4, deter-

ministic model-based resolution is considered. Here, prior knowledge

is taken into account in the form of a parametric model. However, the

images are supposed to be noise free. The unknown parameters, such

as the positions of projected atoms, are measured by means of model

ﬁ tting. Then, in Section 5, statistical model-based resolution is studied.

It is taken into account that the observations ﬂ uctuate about their

expectations due to the unavoidable presence of electron counting

noise in the images. The parameters are measured by means of model

ﬁ tting using parameter estimation methods. Hence, statistical model-

based resolution is discussed in terms of the precision with which the

parameters can be estimated. Finally, in Section 6, ultimate model-

based resolution is contemplated. It is shown that depending on the

observations, the solutions of the position estimates may be exactly

coinciding. Section 7 includes a discussion and conclusions.

2 Classical Two-Point Resolution

Two-poi nt resolution is a widely used criterion for the resolving capa-

bilities of an imaging system. It is deﬁ ned as the system’s ability to

resolve two point sources of equal brightness. Due to the ﬁ nite size of

the system’s optical components, a point source is not imaged as a point

but as the diffraction pattern of the system’s effective aperture. This

diffraction pattern therefore represents the system’s point spread func-

tion. In astronomical problems, two-point resolution is more than just

a resolution measure. It has direct practical signiﬁ cance, since in astro-

nomical problems many objects are effectively point sources.

2.1 Rayleigh Resolution

The most widely used criterion for two-point resolution is that of Lord

Rayleigh (Strutt, 1899). Rayleigh estimated the minimal resolvable dis-

tance between two points of equal brightness that are imaged by a

diffraction-limited imaging system. According to the Rayleigh crite-

rion, two point sources are just resolved if the central maximum of the

diffraction pattern generated by one point source coincides with the

ﬁ rst zero of the diffraction pattern generated by the second. This means

that Rayleigh’s resolution limit is given by the distance between the

central maximum and the ﬁ rst zero of the point spread function of the

imaging system. The criterion can be generalized to include point

spread functions that have no zero in the neighborhood of their central

maximum, by taking the resolution limit as the distance for which the

intensity at the central dip in the composite image is 81% of that at the

maxima on either side. This corresponds to the original Rayleigh limit

for a rectangular aperture.

1230 S. Van Aert et al.

For example, let us consider the Rayleigh limit, or alternatively, the

point resolution, of an imaging system with a (two-dimensional) Gauss-

ian point spread function of the following form:

ppr

(

)

(

)

=−













=−

πρ ρ πρ

exp exp

2ρ













(1)

where r is the absolute value of the two-dimensional vector r = (x y)

and ρ is the width of the Gaussian function. The superscript T denotes

transposition. According to Rayleigh, the point resolution ρ

, which is

the smallest distance at which two points can be resolved, is given by

the requirement that the value of the cross section of the composite

intensity distribution halfway between these two points is about 0.8

times the value at the maxima. Thus

exp .

−













(2)

from which it follows that

ρρ

≈ 22 (3)

2.2 Sparrow Resolution

Rayleigh’s choice of resolution limit is based on presumed resolving

abilities of the human visual system. Since Rayleigh’s days, several other

resolution criteria have been proposed that are similar to Rayleigh’s (for

a review, refer to den Dekker and van den Bos, 1997). A notable example

of such so-called classical criteria for two-point resolution is that of

Sparrow (1916). Compared to Rayleigh resolution, which is based on

presumed capabilities of the human visual system, the Sparrow resolu-

tion is based on a less subjective criterion that is valid for a hypothetical

perfect imaging instrument. It states that the smallest resolvable dis-

tance between two points is the point at which the minimum in the

composite image intensity distribution just disappears. When this deﬁ -

nition of resolution is applied to an imaging system with a Gaussian

point spread function, it follows from Eqs. (1) and (3) that,

≈

(4)

From the examples given in this section, it can be noted that classical two-

point resolution criteria are expressed in terms of the width of the point

spread function of the imaging instrument. A narrower point spread

function corresponds to an improved Rayleigh or Sparrow resolution.

3 Resolution in the Spatial Frequency Domain:

Diffraction Limit and Superresolution

3.1 The Diffraction Limit

In the previous section, classical resolution criteria have been dis-

cussed. An alternative resolution criterion is based on linear systems

Chapter 20 The Notion of Resolution 1231

theory. It is assumed that the imaging system is linear and shift invari-

ant. Coherent imaging systems are linear in complex amplitude and

incoherent imaging systems are linear in intensity. The characteristics

of a shift-invariant linear imaging system are deﬁ ned by its point

spread function, or, equivalently, by its transfer function, which is the

Fourier transform of the point spread function. Point spread functions

are more directly useful in the assessment of telescopes or spectro-

scopic instruments. Transfer functions, on the other hand, are more

often used in the case of microscopes and cameras. The amplitude

(coherent imaging) or intensity (incoherent imaging) distribution in

the image produced by a linear and shift-invariant system is the con-

volution of the amplitude (or intensity) distribution of the object and

the amplitude (or intensity) point spread function of the imaging

system. For the spatial frequency domain, the imaging system acts as

a ﬁ lter for spatial frequencies. Each spatial frequency is transferred

from the object to the image plane independently of all other frequen-

cies present; the corresponding amplitude (coherent imaging) or inten-

sity (incoherent imaging) is multiplied by the transfer function, or

frequency response function, of the system. Due to the ﬁ nite size of

the system’s aperture, transfer functions of (both coherent and incoher-

ent) imaging systems are band limited, i.e., they are equal to zero for

all frequencies above a certain cutoff frequency. Born and Wolf (1999)

discuss the more general class of partially coherent imaging systems

and show that these systems are also strictly band limited. Therefore,

independent of the degree of coherence, spectral components beyond

the cutoff frequency are not transferred by the imaging system.

For this reason, the cutoff frequency is called the diffraction limit to

resolution.

3.2 The Diffraction Limit and Its Relation to Rayleigh and

Sparrow Resolution

The diffraction limit, or equivalently, the cutoff frequency, is related to

the Rayleigh resolution. This can been seen as follows. It is well known

that a narrow function has a broad Fourier transform and vice versa.

Therefore a point spread function with a narrow main lobe corre-

sponds to a transfer function with a high cutoff frequency. In fact, the

product of the Rayleigh limit and the cutoff frequency is a constant

that is close or even equal to one. For example, this constant is equal

to 1 and 1.22 for incoherent imaging systems with a rectangular and a

circular aperture, respectively.

As mentioned above, transfer functions of practical imaging systems

are strictly band limited (due to the ﬁ nite size of the system’s aperture).

In this work, it will often be assumed that the point spread function of

the imaging systems under study can be described by a Gaussian func-

tion, such as (1). This assumption is made so as to keep calculations

simple. The transfer function of the imaging system of which the point

spread function is described by Eq. (1) is given by the two-dimensional

Fourier transform of Eq. (1):

P(g) = exp(−2π

) (5)