# A generalization for optimized phase retrieval algorithms.

**ABSTRACT** In this work, we demonstrate an improved method for iterative phase retrieval with application to coherent diffractive imaging. By introducing additional operations inside the support term of existing iterated projection algorithms, we demonstrate improved convergence speed, higher success rate and, in some cases, improved reconstruction quality. New algorithms take a particularly simple form with the introduction of a generalized projection-based reflector. Numerical simulations verify that these new algorithms surpass the current standards without adding complexity to the reconstruction process. Thus the introduction of this new class of algorithms offers a new array of methods for efficiently deconvolving intricate data.

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**ABSTRACT:**Many imaging techniques provide measurements proportional to Fourier magnitudes of an object, from which one attempts to form an image. One such technique is intensity interferometry which measures the squared Fourier modulus. Intensity interferometry is a synthetic aperture approach known to obtain high spatial resolution information, and is effectively insensitive to degradations from atmospheric turbulence. These benefits are offset by an intrinsically low signal-to-noise (SNR) ratio. Forward models have been theoretically shown to have best performance for many imaging approaches. On the other hand, phase retrieval is designed to reconstruct an image from Fourier-plane magnitudes and object-plane constraints. So it's natural to ask, "How well does phase retrieval perform compared to forward models in cases of interest?" Image reconstructions are presented for both techniques in the presence of significant noise. Preliminary conclusions are presented for attainable resolution vs. DC SNR.Proceedings of SPIE - The International Society for Optical Engineering 09/2013; · 0.20 Impact Factor -
##### Article: Tabletop coherent diffractive imaging of extended objects in transmission and reflection geometry

Matthew D. Seaberg, Bosheng Zhang, Daniel E. Adams, Dennis F. Gardner, Henry C. Kapteyn, Margaret M. Murnane[Show abstract] [Hide abstract]

**ABSTRACT:**Recent breakthroughs in high harmonic generation have extended the reach of bright tabletop coherent light sources from a previous limit of ≍100 eV in the extreme ultraviolet (EUV) all the way beyond 1 keV in the soft X-ray region. Due to its intrinsically short pulse duration and spatial coherence, this light source can be used to probe the fastest physical processes at the femtosecond timescale, with nanometer-scale spatial resolution using a technique called coherent diffractive imaging (CDI). CDI is an aberration-free technique that replaces image-forming optics with a computer phase retrieval algorithm, which recovers the phase of a measured diffraction amplitude. This technique typically requires the sample of interest to be isolated; however, it is possible to loosen this constraint by imposing isolation on the illumination. Here we extend previous tabletop results, in which we demonstrated the ability to image a test object with 22 nm resolution using 13 nm light [3], to imaging of more complex samples using the keyhole CDI technique adapted to our source. We have recently demonstrated the ability to image extended objects in a transmission geometry with ≍100 nm resolution. Finally, we have taken preliminary CDI measurements of extended nanosystems in reflection geometry. We expect that this capability will soon allow us to image dynamic processes in nanosystems at the femtosecond and nanometer scale.Proceedings of SPIE - The International Society for Optical Engineering 09/2013; · 0.20 Impact Factor - SourceAvailable from: opticsinfobase.org[Show abstract] [Hide abstract]

**ABSTRACT:**Coherent X-ray diffraction imaging (CXDI) of the displacement field and strain distribution of nanostructures in kinematic far-field conditions requires solving a set of non-linear and non-local equations. One approach to solving these equations, which utilizes only the object's geometry and the intensity distribution in the vicinity of a Bragg peak as a priori knowledge, is the HIO+ER-algorithm. Despite its success for a number of applications, reconstruction in the case of highly strained nanostructures is likely to fail. To overcome the algorithm's current limitations, we propose the HIOO<sub>R</sub>M+ER<sup>M</sup>-algorithm which allows taking advantage of additional a priori knowledge of the local scattering magnitude and remedies HIO+ER's stagnation by incorporation of randomized overrelaxation at the same time. This approach achieves significant improvements in CXDI data analysis at high strains and greatly reduces sensitivity to the reconstruction's initial guess. These benefits are demonstrated in a systematic numerical study for a periodic array of strained silicon nanowires. Finally, appropriate treatment of reciprocal space points below noise level is investigated.Optics Express 11/2013; 21(23):27734-49. · 3.53 Impact Factor

Page 1

A generalization for optimized phase

retrieval algorithms

Daniel E. Adams,1∗Leigh S. Martin,1Matthew D. Seaberg, Dennis F.

Gardner, Henry C. Kapteyn, and Margaret M. Murnane

JILA, University of Colorado Boulder, Colorado 80309-0440, USA

1These authors contributed equally to this work

∗daniel.e.adams@gmail.com

Abstract:

phase retrieval with application to coherent diffractive imaging. By intro-

ducing additional operations inside the support term of existing iterated

projection algorithms, we demonstrate improved convergence speed, higher

success rate and, in some cases, improved reconstruction quality. New algo-

rithms take a particularly simple form with the introduction of a generalized

projection-based reflector. Numerical simulations verify that these new

algorithms surpass the current standards without adding complexity to the

reconstruction process. Thus the introduction of this new class of algorithms

offers a new array of methods for efficiently deconvolving intricate data.

In this work, we demonstrate an improved method for iterative

© 2012 Optical Society of America

OCIS codes: (100.5070) Phase retrieval; (110.1758) Computational imaging; (110.7440) X-

ray imaging; (170.0180) Microscopy.

References and links

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“Multiscale gigapixel photography.” Nature 486, 386–389 (2012).

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11. S. Marchesini, “Invited article: a [corrected] unified evaluation of iterative projection algorithms for phase re-

trieval.” Rev. Sci. Instrum. 78, 011301 (2007).

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1.Introduction

The last decade has seen significant advances in x-ray sources. Third generation synchrotrons,

fourth generation x-ray free electron lasers (XFELs) and high harmonic generation (HHG)

sources are currently paving the way for powerful new characterization techniques with un-

paralleled temporal and spatial resolution [1–3]. EUV light and higher energy soft x-rays are

ideal probes for nano- and atomic-scale systems due to their inherent elemental contrast and

ability to penetrate thick samples using novel imaging techniques. The last decade has also

seen great progress in coherent diffraction imaging (CDI) as an alternative to more traditional

zone-plate or mirror based imaging techniques [4]. In CDI, a plane wave illuminates an isolated

specimen (i.e. the entire finite sample is illuminated by the wave) and the diffracted light is de-

tected in the far-field or so-called Fraunhofer zone. In this case the diffracted light represents

the scaled, complex Fourier transform of the scattering electronic potential. In practice how-

ever, only the diffracted intensity can be measured, resulting in the loss of phase information.

The goal of iterated phasing algorithms is to solve for the missing phases, ultimately revealing

details about the diffracting electronic potential. Recent work using CDI has explored phase

contrast inspection of biological systems [5–7] strain fields in nano-particles [8] and demon-

strated that high numerical aperture imaging with near-wavelength resolution is possible [9].

These diverse applications of CDI show great promise for widespread biological and industrial

imaging. Moreover, the necessity for computational phase retrieval casts CDI into a broader

class of inverse problems, which are interesting to the mathematical community as well.

However, a drawback of CDI is that computation time for image reconstruction is a sub-

stantial bottleneck, especially considering that most common imaging modalities offer direct

imaging or involve non-iterative post processing. Some illustrative examples include applica-

tions in the semiconductor industry, which will require the imaging of massive areas, and the

emerging use of use of gigapixel detectors [10]. Numerous algorithms have been proposed with

the intent of accelerating the reconstruction process [11]. Some succeed in reducing the number

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of iterations necessary, but occasionally at the expense of computational simplicity, resulting in

a slower algorithm [12]. Keeping in mind the importance of convergence speed, we use physical

intuition in this paper to build on the concept of iterated, generalized projections that quickly

push an initial random guess toward convergence without introducing additional steps. We pro-

vide an outline for deriving augmented fixed point equations and hence new iterative phasing

algorithms, which we call generalized interior feedback (GIF). These algorithms have applica-

tion in CDI, and are motivated by the convex analysis employed by Luke [13] and the linear

systems approach used by Fienup [14,15]. We also show through extensive numerical studies

that GIF algorithms consistently outperform unmodified algorithms in terms of convergence

rate, and even offer improved reconstruction quality in some cases.

2. Fixed point equations and generalized interior feedback

In CDI, the phase retrieval problem was originally solved using iterated projections onto con-

straint subspaces, motivated by the pioneering work of Gerchberg and Saxton [16, 17]. The

diffraction amplitudes m serve as one constraint and if some element u ∈ L : ZN→ R satisfies

this constraint then we say u∈M ≡{v∈L :|Fv|=m}, where F is the discrete Fourier trans-

form and L is the Hilbert space of square integrable functions. Upon discretely sampling the

scatteredlightataratehigherthantheNyquistfrequency,wemayassumethattheelectronicpo-

tential is isolated. The latter condition indicates that the electronic potential u (or in some cases

the exit surface wave leaving the specimen) is supported on some set D i.e. supp(u) ⊂ D ⊂ ZN.

Further, we write CD as the compliment of D and 1CDwhich takes on a value of 0 on D and 1

elsewhere so that S={u∈L :u·1CD=0}, where we have defined ZNas the sample domain, R

is the reals and supp(u) is the support of u. If we also require that u is non-negative i.e. u → u∗,

then we require that R→R+, which leads to the definition S+={u∗∈L :u∗(x)≥0 ∀x∈ZN}.

Even with these constraints, the phase retrieval problem in CDI was intractable until Fienup

introduced the hybrid input-output (HIO) algorithm [15], which was sufficiently fast to en-

able real-world applications. Historically, Bauschke showed a relationship between Dykstra’s

algorithm and the Basic Input-Output of Fienup and Fienup’s Hybrid Input-Ouput algorithm

and the Douglas-Rachford algorithm [18]. Elser showed that the HIO is a specific instance of

the difference map, a general class of iterative projection algorithms. These algorithms gain

their speed-up over previous methods by searching off of the constraint subspaces. Specifi-

cally, Bauschke showed that the difference map moves past near intersections of the subspaces

in search of global minima [19]. Although this helps prevent stagnation near local minima, it

means that the HIO may diverge from the global minimum if the constraint subspaces do not

intersect. This behavior often becomes prominent in phase retrieval when noise is present in the

diffraction pattern.

The relaxed averaged alternating reflections (RAAR) algorithm [13] also demonstrates rapid

convergence and avoidance of local minima, making it practical for phase retrieval. In con-

trast, however, it can be shown that RAAR has fixed points that are related to points in one

constraint subspace nearest the other [20]. This means that RAAR searches for the best ap-

proximate solution to the phase problem when an exact solution does not exist. This behavior

is highly advantageous when no exact solution to the phase retrieval problem exists. Whereas

HIO (and when the non-negativity constraint is correctly implemented, the Hybrid Projection

Reflection (HPR) algorithm [20]) only have fixed points near intersections, RAAR explores

near approaches in search of a solution that optimizes the amplitude and support constraints.

2.1. Mathematical preliminaries

We present the problem of phase retrieval as a general feasibility problem, that is:

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find

u ∈ S∩M

(1)

Phase retrieval algorithms proceed by iteratively applying a projection-based operator to a ran-

dom initial guess u, which ultimately drives the current iterate toward a fixed point. We define

the projection PM(u), u ∈ L onto the set M:

?

m(ξ)

ˆ v is one of the multivalued Fourier domain projections and m is the measured Fourier modu-

lus. Additionally we define the projection onto the set D ⊂ ZNwith D being compact on RN

PS(u), u ∈ L as:

?

In the case where we insist the function u → u∗∈ R+and S → S+, the conditional statement is

modified to read max{0,u(x)} if x ∈ D (the otherwise case is left unchanged).

As the name suggests, RAAR is based on reflection operators, which are defined RC=2PC−

I. We now introduce the generalized reflector of an arbitrary projector, PCas:

PM(u) = F−1ˆ v

ˆ v(ξ) =

m(ξ)Fu(ξ)

|Fu(ξ)|

if

Fu(ξ) ?= 0

otherwise,

(2)

(∀x ∈ ZN)(PS(u))(x) =

u(x)

0

ifx ∈ D

otherwise.

(3)

Rγ

C≡ (1+γ)PC−γI

(4)

It is worth noting some properties of generalized reflector Rγ

For values of γ < 1, the generalized reflector is the relaxation parameter of Levi and Stark [21],

Cthat are summarized in Fig. 1.

C

RC

u

RC

u

RC

u

RC

u

u

Fig. 1. Generalized reflector, where C is the set around which the guess u is being reflected.

For γ < 1 the reflector is a relaxation parameter, γ = 1 recovers the standard reflector and

γ > 1 chooses the depth of reflection.

which appears in many iterative phase retrieval algorithms including the difference map. γ = 0

is simply a projection, γ = 1 produces the standard reflection RCas in [22,23] while γ > 1

essentially chooses the reflection depth.

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2.2.Generalized interior feedback

We now state the RAAR algorithm without further justification and develop the concept of

generalized interior feedback (GIF). RAAR can be written as a fixed point equation of the

form un+1= V (T ,βn) un, where, in general, T is a generic operator that can include any type

of averaging of reflectors and projectors. V (T ,βn) is an operator that includes some kind of

relaxation strategy through the use of the relaxation parameter βn, which is modified at the nth

iteration.UsingthisnotationweseethattheRAARalgorithm,usuallywrittenastheconditional

statement,

⎧

⎪

may be written as a fixed point algorithm

(∀x ∈ ZN)

un+1(x) =

⎪

⎩

⎨

(PM(un))(x)

if x ∈ D

βnun+(1−2βn)(PM(un))(x)

otherwise,

(5)

un+1= (1

2βn(RSRM+I)+(1−βn)PM)(un),

(6)

where T ≡1

the support reflector RS, Eq. (6) becomes,

2(RSRM+I) and the algorithm is relaxed by the term (1−βn)PM. Upon expanding

un+1= (1

2βn(2PS(RM)−RM+I)−(1+βn)PM)(un).

(7)

Fienup and others related iterative phase retrieval to control theory and motivated several de-

velopments with the concept of feedback. Most algorithms only include feedback terms outside

the support, which push object-domain amplitudes to zero. Here, we consider the addition of a

general feedback term of the form αE(PM,I) inside the support. In the case where the feasibil-

ity problem is consistent, we require the feedback term to vanish as the algorithm approaches a

fixed point. This leads us to a specific form of the feedback as αE(I−PM). Inserting this form

in the interior of the support in Eq. (7)

un+1= (1

2βn(2 PS(RM−αE(I−PM))−RM+I)−(1+βn)PM)(un),

(8)

and, for example, if we choose the especially simple form for E(I−PM) ≡ I −PMand collect

terms, we arrive at

un+1=1

2βn(RSRγ

M+I)un+1

2βn(Rγ

M−RM)un+(1−βnPM)un,

(9)

where we have defined γ ≡ α +1. It is now instructive to write Eq. (9) in the conditional form

preferred by the optics community. For any arbitrary, real signal z we can write the positive and

negative parts as z+= max{0,z} and z−= z−z+Returning to the more general form of GIF,

setting the scaling factor α to unity and enforcing positivity on u → u∗∈ R+Eq. (8) is,

un+1= (1D·[PM−β E(I−PM)]+

1CD·[βI+(I−2β)PM]+

1D·[RM−E(I−PM)]−)(un).

Then there are three cases to consider. If x ∈ D & RM≥ E(I −PM), then the right hand side

of Eq. (10) reduces to PM−βE(I −PM). If x ∈ D & RM≤ E(I −PM) then this term becomes

(10)

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βI +(I −2β)PM. Finally, if x ?∈ D then it reads βI +(I −2β)PM. Phrased in a conditional

statement Eq. (10) is (∀x ∈ ZN),

⎧

⎪

If we again choose a simple form for E(I−PM) = I−PM, (∀x ∈ ZN) we arrive at,

⎧

⎪

3.Numerical results

un+1(x) =

⎪

⎩

⎨

((PM(un)−β E(I−PM)un)(x)

if x ∈ D and

(RM−E(I−PM))un(x) ≥ 0

otherwise.

βnun(x)+(1−2βn)(PM(un))(x)

(11)

un+1(x) =

⎪

⎩

⎨

(Rβn

M(un))(x)

if x ∈ D and

(RM−I+PM)un(x) ≥ 0

otherwise.

βnun(x)+(1−2βn)(PM(un))(x)

(12)

Heuristically, appendix A demonstrates which elements GIF-RAAR converges to if it con-

verges, and provides insight concerning the behavior of GIF-RAAR during phase retrieval.

However, the analysis found in appendix A does not bear on issues such as convergence rate

or behavior in a non-convex setting. To study these aspects, we explore the algorithm’s prop-

erties in the context of coherent diffraction imaging. In an ideal setting, the sets M and S+

intersect, and thus the fixed point of GIF-RAAR is the set of elements that exactly satisfy both

constraints. Real data sets contain noise, which eliminates exact intersections and dramatically

changes the nature of the problem.

We consider both noiseless and noisy data. In the latter case, white Gaussian noise is added

to the square of the Fourier amplitudes of the object to simulate detector read-out noise and

background light, which are usually the dominant contributions. Results on experimental data

are also given at the end of this section. Figure 2 shows the test image along with the support.

These algorithms may be used both with and without a non-negativity constraint. For objects

that contain a phase angle greater than π/2, the typical non-negativity constraint cannot be

used. Simulations consider both cases. When not using non-negativity, a phase profile shown

in Fig. 2 is added to the image and a tighter support is used to compensate for the missing

constraint.

Table 1 shows results from a series of reconstructions without non-negativity. Each case

consists of one of four different signal to noise ratios and one of three algorithms. Each case

is comprised of 100 individual runs starting with a random phase guess and then run for 3000

iterations. The translation of each reconstruction is determined by measuring the root mean

square difference between the object and the reconstruction during a pixel-by-pixel raster scan.

The real-space error is then the lowest error value given by the scan. A cutoff of 5% reciprocal-

space error defines the convergence condition. Runs that do not reach this cutoff are considered

unsuccessful and are not included in calculations of the convergence iteration or real-space

error.

In this setting, HIO surpasses RAAR and GIF-RAAR except in the presence of a large

amount of noise. However, it is important to realize that we have only considered the case of

β = 0.9, and that each algorithm behaves differently with respect to this parameter. We make

these comparisons because this choice of β is most common within the optics community re-

gardless of algorithm, but note that the most meaningful contrasts are between GIF algorithms

and their standard counterparts. In general, our research has consistently shown that RAAR

outperforms HPR (HIO with the correct non-negativity constraint) when non-negativity can be

used, but that HIO outperforms RAAR without this constraint. This important trend is reflected

in the following data.

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Fig. 2. (a) The object used for reconstruction. (b) Simulated diffraction pattern in the pres-

ence of noise raised to the quarter power for visibility. (c) The phase profile added to re-

constructions when the non-negativity constraint was omitted. (d) Support used for recon-

struction when the non-negativity constraint could be used. A similar but tighter support

(not shown) was used in simulations without non-negativity.

Notice in table 1 that although RAAR converges more quickly than GIF-RAAR, its real-

space error appears not to be noise limited, but is fixed at 20% regardless of the noise level.

On the contrary, GIF-RAAR consistently reaches a significantly lower real-space error, which

shows the expected correlation with noise level. GIF-RAAR also achieves a much higher suc-

cessrate,near100% forallcases.Wefound thatGIF-HIOperformed slightlybetterforα =0.9,

and thus used this value in reconstructions. GIF-HIO converges more quickly than HIO, but

with a consistently larger real-space error. With a signal-to-noise ratio of 11 (the smallest used),

GIF-HIO fails to converge after 3000 iterations.

RAAR was originally developed with the non-negativity constraint implemented. Table 2

shows improvement resulting from its use. GIF-RAAR surpasses unmodified RAAR in all cat-

egories. Once again, RAAR has a lower but still useful success rate. GIF-HPR also surpasses

its unmodified counterpart for higher signal-to-noise ratios, though in the presence of large

quantities of noise its convergence rate drops substantially. In this simulation, GIF algorithms

surpass their standard counterparts’ real-space error and convergence rate in all but this case.

The most dramatic improvement is the order of magnitude lower real-space error achieved by

GIF-RAAR over RAAR after the allotted number of iterations.

Reconstruction quality depends strongly on many details of the problem in intricate and sub-

tle ways. To ensure that the results above carry over to real-world applications, we present

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Table 1. Reconstruction quality, convergence speed and success rate without the use of the

non-negativity constraint.

MeasureSNRHIO α = 0

Convergence

∞

76±4

Iteration400 74±4

81 73±4

11330±20

R-Space Error

∞

(24±6)×10−8

4000.0091±0.0001

810.0247±0.0003

110.138±0.002

Success Rate

∞

100%

400100%

81100%

11100%

HIO α = 0.9

49±3

49±2

50±2

> 3000

(4±2)×10−7

0.0135±0.0001

0.0327±0.0002

RAAR α = 0

210±50

150±30

170±40

180±40

0.20±0.01

0.20±0.01

0.20±0.01

0.20±0.01

67%

62%

62%

52%

RAAR α = 1

230±60

230±60

170±40

280±50

0.075±0.008

0.094±0.009

0.092±0.009

0.127±0.007

100%

100%

99%

97%

100%

100%

100%

0%

Table 2. Reconstruction quality, convergence speed and success rate with the use of the

non-negativity constraint.

MeasureSNRHPR α = 0

Convergence

∞

66±2

Iteration40067±2

8166±2

11 180±20

R-Space Error

∞

0.0521±0.0008

4000.0633±0.0006

810.0821±0.0007

110.158±0.002

Success Rate

∞

100%

400100%

81100%

11100%

HPR α = 0.9

47±1

46.9±0.9

47.4±0.8

1500±800

0.0372±0.0005

0.0489±0.0005

0.0660±0.0005

0.175±0.009

100%

100%

100%

3%

RAAR α = 0

90±10

100±30

88±8

180±30

0.076±0.007

0.091±0.009

0.10±0.01

0.111±0.006

77%

76%

77%

76%

RAAR α = 1

61±5

70±8

80±11

77±6

0.007±0.002

0.009±0.002

0.013±0.002

0.074±0.006

100%

100%

100%

100%

reconstructions on a set of experimental data. Figure 3 shows a representative diffraction pat-

tern and reconstruction using the GIF-RAAR algorithm. The object was a negative 1951 USAF

test target imaged using CDI, an optical laser (λ = 633nm) and the apertured illumination tech-

nique [24]. A significant amount of readout noise is present in the data. Panel (a) of Fig. 3 is a

diffraction pattern collected using the apertured illumination technique. Panel (b) shows a rep-

resentative GIF-RAAR reconstruction, averaging 1000 iterations after the algorithm converged.

Also shown in panel (b) is a dashed line indicating the border of the support used to reconstruct

the pattern and a photograph of the test pattern for comparison.

Panel (c) in Fig. 3 shows the relative error for both GIF-RAAR and RAAR highlighting the

plateau common in RAAR reconstructions. This particular comparison is motivated by the con-

sistentimprovement GIFseemstooffertoRAAR,basedonthedataintable2.Asinmostcases,

even a modest amount of averaging after the algorithm has converged significantly improves re-

constructed image quality. The overall reconstruction quality and relative reconstruction quality

between GIF-RAAR and RAAR depend heavily on the conditions of the problem. For example,

changing the support size dramatically changes the convergence rates. In this test, GIF-RAAR

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always converges faster than RAAR, at times by 500 iterations and under other conditions only

50. In some cases, RAAR achieves a slightly lower k-space error than GIF-RAAR. However,

the visual quality of the resulting reconstructions is identical.

Iteration

Relative Error

GIF-RAAR

RAAR

A

B

C

S

USAF 1951

Fig. 3. (a) Diffraction pattern data raised to the quarter power for visibility. (b) Recon-

structed object with dashed line showing the boundary of the support, S used for both GIF-

RAAR and RAAR. The inset shows a traditional microscope image of the test pattern for

comparison (Thorlabs R3L3S1N - Negative 1951 USAF Test Target, 3” x 3”). (c) Relative

error as a function of iteration number for both GIF-RAAR and RAAR respectively.

4.Conclusion

Phase retrieval has proven to be a subtle but very important problem both physically and math-

ematically. Although practical implementations of phase retrieval have been available since the

1970’s, development of optimal algorithms continues to this day [13, 25]. Non-convex opti-

mization problems remain an advanced field of study within pure mathematics, and while some

algorithms have been well-characterized [26], the exact convergence properties of most algo-

rithms remain elusive. In a practical setting, phase retrieval of experimental data often relies on

the use of a combination of methods and empirical observation which are used to determine

optimal parameter values. The introduction of GIF-RAAR provides a tool which, in all of our

numerical simulations, outperforms the most commonly used methods while maintaining the

computational speed of iterative fast Fourier transform fixed point algorithms. Furthermore,

our simulations demonstrate the advantage of introducing feedback within the object domain

in general, suggesting that similar benefit may be gained from incorporation into other meth-

ods. Only a relatively simple internal feedback mechanism has been analyzed here, and other

types of terms may exhibit advantageous properties. Finally, we suggest that this internal feed-

back mechanism may provide a conduit relating RAAR type algorithms to a more generalized

difference map.

A. Appendix

By design, Eq. (12) strongly resembles RAAR. However, the internal feedback mechanism

modifies the update rule inside the support to be the generalized reflector instead of a projection.

The update rule for the so-called GIF-RAAR algorithm depends on the pointwise sign of the

reflector as in RAAR but additionally depends on the pointwise sign of the feedback term. The

question then arises; what, if anything, is fundamentally different about the new GIF-RAAR

algorithm? In the present section we examine the fixed points of the GIF-RAAR algorithm and

find them to be the same as those of RAAR in the convex case. Although a convex analysis only

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provides heuristics for the phase retrieval problem, which is non-convex, such analyses are in

general much simpler and have predicted the important properties of RAAR [13]. Non-convex

behavior of RAAR has been analyzed in reference [26].

The following proof is an extension of that given in ref. [20]. Let A and B be two closed

convex subsets of the Hilbert space L of square integrable functions, and replace to S+and M

respectively. PAand PBare projections onto these sets. L has an inner product ?·,·? and norm

||·||. Analogous to RAAR, we define the operator Tγ

Tγ

2(RARγ

∗ as

∗≡1

B+I)+1

2(Rγ

B−RB)

or in simplified form,

Tγ

∗= PARγ

A−PB+I,

(13)

GIF-RAAR is then the β-weighted average between Tγ

∗ and PB:

V (Tγ

∗,β) ≡ βTγ

∗+(1−β)PB

(14)

Let F be the set of points in B nearest A and let E be the set of points in A nearest B as

shown in Fig. 4. We now wish to show that the fixed points of GIF-RAAR correspond to those

of RAAR, which are given by:

u ∈ F −

β

1−βg

β

1−βg

(15)

or

u = PBu−

(16)

here, g is the gap vector, which is the shortest vector between A and B. g is rigorously defined

by considering the set D = B−A, which is the closure of the set of all vectors starting in A and

ending in B. g may be written as g = PD(0). Notice that even when the problem is ill posed so

that A and B have no intersection, RAAR still has well defined fixed points which are related

to the points in one set nearest the other. This feature makes RAAR advantageous over other

algorithms.

With the definition f = PBu, recall that PA(2f −u) = PA(2PBu−u) = PARBu. To show the

fixed points of GIF-RAAR, begin by rewriting Eq. (13) as

(∀u ∈ L)

u−Tγ

∗u = −PARγ

Bu+ f,

(17)

and for our choice of u

βTγ

∗u+(1−β)PBu = u,

(18)

which may be rewritten as

u−Tγ

∗u =(1−β)

β

(PBu−u) = −1−β

β

y.

(19)

where we have defined y = PBu−u. Using Eq. (17) we obtain

PARγ

Bu = f +1−β

β

y.

(20)

Because A is nonempty, closed and convex, ∀a ∈ A ?a−PAx,x−PAx? ≤ 0 for any x. Letting

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Fig. 4. Definition of relevant quantities for convex analysis, considering two sets in R2for

visualization. A and B are convex sets in L, E and F are the points in A and B closest to B

and A, respectively. The gap vector g is effectively the shortest distance between A and B.

x = Rγ

Bu, we can write

(∀a ∈ A)

?a−PARγ

Bu,Rγ

Bu−PARγ

Bu?≤ 0,

(21)

and using the previous results

?

(a− f)−1−β

β

y,−(γ +1−β

β

)y

?

≤ 0.

(22)

Rearranging gives the result

?−a+ f,y?+1−β

β

||y||2≤ 0.

(23)

On the other hand, since B is also nonempty, closed and convex, with f = PBu we have

∀b ∈ B,

?b− f,y? ≤ 0

and upon combining results

(24)

?b− f,y?+?−a+ f,y?+1−β

β

||y||2≤ 0,

(25)

simplifying and noting that β is positive definite and so is ||y||2we have

?b−a,y? ≤ −1−β

β

||y||2≤ 0.

(26)

Now, taking a sequence a0,a1,a2, in A and a sequence b0,b1,b2, in B such that gn= bn−

an→ g then we have

(∀n ∈ N)

?gn,y? ≤ −1−β

β

||y||2.

(27)

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Taking the limit and using the Cauchy-Schwartz inequality yields

||y|| ≤

β

1−β||g||.

(28)

Conversely u = (βTγ

and solving for y,

∗u+(1−β)PBu) = 0, is also y = β(PARγ

β

1−β(PARγ

Bu− f)+βy,

y =

Bu− f).

(29)

Taking the l2norm gives

||y|| =

β

1−β||PARγ

Bu− f||.

(30)

Using the previous result

||y|| =

β

1−β||PARγ

Bu− f|| ≥

β

1−β||g||

hence

||y|| =

β

1−β||g||,

(31)

and taking the limit gives y = −

chosen this u ∈ Fix{V (Tγ

β

1−βg, =⇒ u = f −

β

1−βg, where we have appropriately

∗,β)}. We may now write

u = F −

β

1−βg

β

1−βg, we show mutual containment of the sets

(32)

To show the equality Fix V (Tγ

F−

[20], and shows that if u is of the form given in Eq. (16), then it is fixed by GIF-RAAR, so that

V (Tγ

We use the concept of a cone, which is similar to a linear vector space, but is instead closed

only under multiplication by positive scalars. We define the normal cone map of a convex set A

as

?

/ 0

This may be thought of as the set of elements that, when added to some x ∈ A are projected

back to x by PA, i.e. x = PA(x+u) ⇔ u ∈ NA(x).

Let u = f −

−g ∈ ND(g). From the definitions of E, F and D, it follows that ND(g) = NB(f)∩(−NA(e)).

By closure of cones, for 0 ≤ β < 1, −

∗,β) = F −

β

1−βg and Fix V (Tγ

∗,β). The following proof is a close adaptation of that given in reference

∗,β)u = u.

NA: x ?→

{u : ∀a ∈ A, ?a−x,u? ≤ 0}

x ∈ A

otherwise

(33)

β

1−βg and e = f −g, where f = PBu. As g = PD(0) = PD(g−g), it follows that

β

1−βg ∈ ND(g) ⊂ NB(f). It follows that

PBu = PB

?

f −

γβ

1−βg

β

1−βg

?

= f

Rγ

Bu = f +

(34)

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As g ∈ NA(e) and f = e+g, it follows that

PARγ

Bu = PA

?

?

f +

γβ

1−βg

?

?

= PA

e+

1+

γβ

1−β

?

g

?

= e

(35)

So long as

?

1+

γβ

1−β

?

≥ 0. Acting V (Tγ

∗,β) on u, we find

V (Tγ

∗,β) = βTγ

= β(I−PB+PARγ

= β(u−g)+(1−β)f

= u

∗u+(1−β)PBu

B)u+(1−β)PBu

(36)

or u ∈ Fix V (Tγ

Acknowledgments

∗,β). Thus, the fixed points of GIF-RAAR coincide with those of RAAR.

We gratefully acknowledge fruitful conversations with Dr. D. Russell Luke, Institut f¨ ur Nu-

merische und Angewandte Mathematik Universit¨ at G¨ ottingen and Dr. Keith A. Kearnes, Uni-

versity of Colorado at Boulder. We would also like to extend our gratitude to Moritz Barkowski,

Technische Universitt Kaiserslautern. The authors also gratefully acknowledge support from a

National Security Science and Engineering Faculty Fellowship and from the National Science

Foundation Engineering Research Center in EUV Science and Technology.

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22 October 2012 / Vol. 20, No. 22 / OPTICS EXPRESS 24790