EDIC RESEARCH PROPOSAL 1

Phase Retrieval in X-Ray Crystallography:the Sparse Way

Juri RanieriSchool of Computer and Communication Sciences, I&C, EPFL

Abstract—Phase loss problems occur in various scientific fields,particularly those involving optics. We focus our work on X–ray crystallography. The goal is to reconstruct a crystal fromthe magnitude of its Fourier transform, since the phases arelost during the measuring process. We analyze three papers togive the scientific foundations to our research proposal. The firsttwo deal with a mathematical model, reconstruction approachesand conditions for the unicity of the solution. This traditionalframework is based on two assumptions: bounded support andpositiveness of the crystals. We structure our research proposalon a new prior: sparsity. Therefore, the third paper describes aninnovative algorithm to recover sparse signals based on messagepassing. We conclude by outlining the research proposal, that isoriented towards the unicity of the reconstruction and efficientreconstruction algorithms.

Index Terms—X-Ray crystallography, phase retrieval, sparsity,unicity of the phase retrieval

I. INTRODUCTION

IN many real-world scenarios, it is often more convenient tomeasure the Fourier transform (FT) of a signal of interest

instead of the signal itself. Moreover, in many cases thephase of the FT is lost or irremediably distorted during the

Proposal submitted to committee: August 30th, 2011; Can-didacy exam date: September 6th, 2011; Candidacy examcommittee: Emre Telatar, Martin Vetterli, Gervais Chapuis.

This research plan has been approved:

Date: ————————————

Doctoral candidate: ————————————(name and signature)

Thesis director: ————————————(name and signature)

Thesis co-director: ————————————(if applicable) (name and signature)

Doct. prog. director:————————————(R. Urbanke) (signature)

EDIC-ru/05.05.2009

measuring process. The recovery of the phase is fundamentalto reconstruct the signal, and it is known as phase retrieval.

In what follows, we first set the notation and we statethe phase retrieval problem in a rigorous form. Particularly,we emphasize the role of sparsity in few applications, X–raycrystallography and channel estimation, and we show that thisproperty has been scarcely exploited in the past.

We consider a D-dimensional continuous real-valued signalf(x) : RD → R, where x is the position vector in thespatial domain. Observe that the exact parallel problem canbe discussed on a discrete domain using sequences, suchas f [n] : ZD → R. Unless otherwise stated, we use boldsymbols, such as x, for vectors and standard symbols, suchas x, for single elements and functions.

First, we define the FT of the signal as

F (u) =

∫RD

f(x) expj2π 〈u,x〉dx, (1)

where u is the position vector in the Fourier space, 〈·, ·〉 is theinner product between two vectors and capital letters indicatethe FT as in the rest of the paper. Clearly, the signal can bedirectly recovered by the inverse FT (IFT). Let us representthe FT in the polar form,

F (u) = |F (u)| exp jφ(u) , (2)

where φ(u) is the phase of the FT. Then, we define the phaseretrieval problem as follows: given the magnitude |F (u)| ofthe FT, recover the original signal f(x). Since the FT is abijective mapping, we can solve the problem by recovering thephase term φ(u). The IFT of |F (u)|2 is the autocorrelationfunction (ACF), defined as

a(x) =

∫RD

f(y)f∗(x+ y)dy. (3)

Therefore, we can equivalently state the problem as follows:given the ACF, recover the original signal f(x). Now, we candefine the more general form of the phase retrieval problem.

Problem 1. Phase Retrieval Given the magnitude |F (u)| ofthe FT or the ACF a(x) of a signal of interest f(x), recoverthe signal itself.

Two main questions arise from Problem 1:

• What are the properties that f(x) must satisfy to have aunique reconstruction?

• How can we recover the signal?

Remark 1. Some information on the signal f(x) is entirely

EDIC RESEARCH PROPOSAL 2

embedded into the phase of its FT and we cannot hope torecover this information from the magnitude. Namely, thefollowing transformations of f(x),

f(−x),−f(x), f(x− τ ), (4)

do not influence the magnitude of the FT. Hence, the time–reversal, the sign change and the absolute position cannot berecovered once the phase is lost.

It is then appropriate to define what unique means with thefollowing equivalence class,

f(x) ∼ g(x), if f(x) = ±g(k ± x), (5)

for any k ∈ RD. A reconstruction is unique if all the solutionsbelong to the same class of equivalence.

We focus our research proposal on X-ray crystallographythat is the primary method for determining the molecularconformations of molecules. The experiment is composed bythe following steps: the molecules are crystallized and theobtained crystals are put in front of a beam of X-rays, underdifferent angles. Each “projection” is a diffraction pattern and,mathematically speaking, is a slice of the three-dimensionalFT of the crystal. The diffraction patterns are recorded usingtraditional imaging techniques, such as CCDs, and only themagnitude can be acquired. Hence, we aim at the recovery ofthe spatial distribution of the crystal, called electron density,from the magnitude of the FT. The following remark explainswhy this phase retrieval problem is more complicated than inProblem 1.

Remark 2. Given the periodicity of the crystal, we measurea set of samples |F [h]| of the magnitude of the FT. This set isnot sufficiently dense to reconstruct |F (u)| using Shannon’ssampling theorem.

However, there is some a-priori information about thecrystal that we can exploit: sparsity, positiveness and boundedsupport. In fact, we can assume the following model for theunit cell e(x) of the crystal f(x),

e(x) =

K∑n=1

cnen(x− xn), (6)

where en is the electron density of the n-th atom, that has anamplitude cn and is located at xn. We can now specify thephase retrieval problem for crystallography.

Problem 2. Phase Retrieval for Crystallography Given aset of samples |F [h]| of the magnitude of the FT of a unitcell e(x) that contains K atoms, is positive and has boundedsupport, estimate the locations and amplitudes of the atoms.

The first paper we analyze [1] gives a detailed problemstatement and a review of the reconstruction methods used forProblem 2.

As we previously mentioned, we are interested in the unicityof the solution of Problems 1, 2. Namely, we know that amagnitude |F (u)| can be assigned to an infinite number ofsignals f(x), by choosing a random phase φ(u). We alreadypointed out that there are many available priors on f(x)and, intuitively, these should reduce the number of possible

solutions. Therefore, we want to define the conditions on f(x)that guarantee a unique solution to the phase retrieval. Therehas been much work related to the question of the unicity,and the major part is oriented towards the reconstruction ofa discrete image from the magnitude of the FT. Formally, wehave the following problem.

Problem 3. Phase Retrieval for Images Let f [n] be adiscrete image of at least two dimensions, n ∈ ZD, D ≥ 2,positive and with bounded known support. Given the magni-tude |F [h]| of the image FT, estimate f [n].

The second paper [2] we chose to analyze is a rigorous studyof the unicity of the solution for Problem 3. In particular,Hayes obtains the following result: under the assumptionsgiven in Problem 3, the set of images f [n] which are notuniquely recoverable has measure zero. The results are derivedusing the theory of polynomials. A possible algorithm torecover signals from the magnitudes of the FT is also given,but it is does not perform well. Given Remark 2, this resultcannot be directly applied to Problem 2.

With this overview of the field, we would like to showthat there is room for new results regarding both the al-gorithmic and the unicity questions. Our research proposaltargets specifically the X-ray crystallography problem, and weaim to tackle its numerous challenges using new tools andtechniques derived from the theory of sparse signals. First,sparsity fits naturally in the application fields: crystals areusually sparse. Second, sparsity has already shown its powerin signal processing. In particular, the following problemgenerated many interesting results.

Problem 4. Compressed Sensing Let f be a vector with Knon-zero coefficients. Given a set of M N measurementsy = Af , where A is a M ×N matrix, recover f .

Among the vast literature, the authors of [3] present analgorithm to solve Problem 4, that is known as approximatemessage passing (AMP). The paper is remarkable for theperformance of the algorithm and for the analysis of therelative reconstruction bounds.

In what follows, we first discuss the three chosen papersin Section II. More precisely, we describe the mathematicalmodel of X-ray crystallography and we present some classicreconstruction techniques from [1]. Then, we sketch the proofof unicity of the solution of Problem 3 given in [2], underlyingthe main techniques used to prove this fundamental result.We conclude the survey describing the approximate mes-sage passing algorithm together with the relative performancebounds [3]. This material will serve as a basis for our researchproposal, given in Section III.

II. SURVEY OF THE SELECTED PAPERS

A. Phase Retrieval in Crystallography and Optics

The first paper [1] is a fairly recent review of the scientificadvances regarding the phase problem in two typical scenarios:images and X–ray crystallography. It attempts to outline theadvancements in the two fields, summarizing the main differ-ences and their implications. In particular, it shows the origin

EDIC RESEARCH PROPOSAL 3

Fig. 1: An example of a unit cell (a) and of a crystal (b).

and the details of the main difference, that has been introducedin Remark 2. We limit, for ease of exposition, the discussion toprimitive orthorhombic lattices in which the unit cell vectorsare mutually orthogonal and there is one molecule per unit cell,see Figure 1. Even if the author analyzes the reconstructiontechniques according to possible targets (small molecules,macromolecules and fiber diffraction), in this review, we coveronly the part related to small molecules for the followingreasons:

• for more complicated molecules we need particular ex-periments and the scenario drifts away from the onedescribed in Problem 2,

• many of the small molecules can be solved by directmethods. These algorithms rely on a prior similar tosparsity but less powerful: atomicity. Therefore, it isnatural to use sparsity to get better results.

We start outlining the mathematical model from the basicdefinition of the crystal to the derivation of its FT, and wecontinue describing the reconstruction of small molecules.

The electron density of a crystal, that is the periodic spatialdistribution of charge given by the atoms, can be written as

f(x) = e(x) ∗ l(x) = e(x) ∗

[∑i∈Z3

δ(x− i⊗L)

], (7)

where e(x) is the electron density of a single unit cell, l(x)is the lattice of the crystal, ∗ is the convolution and ⊗ isthe element-wise multiplication. The unit cell is 3-dimensionaland we define its dimensions as L = [L1, L2, L3]. A graphicalexample of a unit cell and a crystal is given in Figure 1. Wediscard the knowledge of possible symmetries of the unit cell,to keep the notation simple and as general as possible. Weconsider the definition of the unit cell e(x) given in (6). Notethat we are more interested in the reconstruction of the singlecell e(x), than in the whole periodic crystal f(x), since theformer is just a repetition of the latter.

By the Bragg’s law, the diffraction pattern of an objectimpacted by X-ray is equivalent to its FT [4]. Taking the FTof (7) and using the convolution property, we get

F (u) = E(u)L(u) = E(u)∑j∈Z3

δ(u− j ⊗ L

), (8)

where E(u) and L(u) are the FTs of e(x) and l(x),respectively. Observe that we have a lattice also in the

Fig. 2: A typical example of diffraction pattern, where is itpossible to observe the sampling effect. Namely, the black dotsare the samples |F [h]|. Image taken from http://www.chem.ucla.edu/harding/IGOC/D/diffraction pattern.html

Fourier domain, that is called reciprocal lattice, and hasdimensions equal to the inverse of the direct lattice, that isL =

[1L1, 1L2, 1L3

]. Therefore, the FT of the crystal is the

sampling of the FT of the unit cell on the reciprocal lattice.The sampling grid is defined as h = j ⊗ L =

[j1L1, j2L2

, j3L3

]and, using (7) and (8), the samples are given as

F [h] =

∫V

f(x) exp−j2π 〈x,h〉dx, (9)

where we used a simple notation for the integration over the3D domain defined by the volume V of the unit cell. Anexample of real diffraction pattern where it is possible to noticethe sampling is given in Figure 2Using (6), we can rewrite (9)as

F [h] =

K∑n=1

fn[h] exp−j2π 〈xn,h〉, (10)

where fn[h] is the sampled FT of the n-th atom when it ispositioned at the origin (known as the scattering factor inliterature).

The whole electron density is therefore given by the inversediscrete FT (IDFT)

f(x) =1

V

∑h

F [h] exp j2π 〈x,h〉 . (11)

Unfortunately, we cannot use directly (11) because we do notmeasure F [h], but |F [h]|. This sampling of the FT preventsthe use of the uniqueness argument given for phase retrieval ofimages [2]. The theorem assumes the knowledge of the wholefunction |F (u)| but we can measure only a set of samples|F [h]|. We could try to recover the ACF taking the IDFT of|F [h]|2, but aliasing problems arise. In fact, the support ofthe ACF of e(x) is twice the support of e(x), leading to anundersampling factor of 2 for each dimension.

Even if unicity cannot be proved using the theory given in[2], many techniques have been developed to solve the phaseproblem in the last 50 years. These techniques have evolvedindependently from the optics community since the scenariocan be modified by the experimental setup to obtain more

EDIC RESEARCH PROPOSAL 4

information. These methods also evolved according to thecharacteristics of the crystal of interest: small molecules, largemolecules and fiber diffractions. The difference between smalland large molecules is the availability of high resolution X-ray data. For small molecules, the resolution can be sufficientlyhigh to resolve all the different atoms. In this case, the electrondensity satisfies the atomicity property, defined as follows,

Definition 1. Atomicity An electron density f(x) satisfies theatomicity property when the support of the electron densitiesof the single atoms are well separated.

In other words, the electron densities en(x) are sufficientlynarrow to be separated from each other.

In what follows, we present the main methods used torecover small molecules presented in [1]: heavy atoms methodand direct method. Even if most of the small molecules arenow reconstructed using the latter, we will also describe theformer since it contains some interesting insights regardingour ongoing research work.

1) Heavy Atom Method: The heavy-atom method attemptsto recover a molecule that contains a heavy atom (eithernaturally or one that is artificially introduced), i.e., an atomthat has a much larger atomic number than the other atoms inthe molecule. The ACF is dominated by peaks correspondingto interatomic distances between the heavy atoms. The positionof the heavy atom in the unit is are determined by inspectionof the ACF. The phase due to the heavy atom alone arethen calculated and combined with the measured amplitudesto calculate an approximate electron-density map. The heavyatom is then removed from the ACF and this usually allowsadditional atoms to be located. The process is repeated withatoms being removed at each iteration until all have beenlocated. This technique has many common points with ourongoing research on peeling algorithms [5].

2) Direct Methods: These methods are commonly usedto recover the majority of small molecules. The name isderived from the fact that it is possible to recover the structuredirectly from the magnitude of the FT, exploiting atomicity.Direct methods are based on a collection of different re-lationships regarding the magnitudes that are used togetherand/or subsequently to obtain the crystal’s structure. The firstsignificant relationship is due to Karle and Hauptman [6], whowon the Nobel prize in 1958. Namely, they showed that thenon-negativity of the electron density implies the followinginequality,∣∣∣∣∣∣∣∣∣∣

F [0] F [−h1] · · · F [−hn]

F [h1] F [0]. . .

......

.... . .

...F [hn] F [hn−1] · · · F [0]

∣∣∣∣∣∣∣∣∣∣≥ 0, (12)

that is valid for any determinant built on a sequence ofcoefficients F [h]. This relationship is extremely importantbecause it is the origin of many used inequalities. In particular,we describe the two principal ones, called sigma one andsigma two relationships. The former is obtained from the third

order determinant as

F [h]2 ≤ (1 + F [2h])/2. (13)

The latter is again derived from the third order determinant butholds only for FT coefficients with large magnitude. It showsthat

F [h] ≈ F [k]F [h− k]/F [0], (14)

which implies that the phases are

φ[h] ≈ φ[k] + φ[h− k]. (15)

Both inequality can be used to determine the phase of one FTcoefficient from partial knowledge of the FT.

Sayre [7] was the first one to actively exploit the atomicity.Namely, he showed that the FT samples of a crystal made ofidentical resolvable atoms satisfy the following equality,

F [h] = θ[h]∑k

F [k]F [h− k], (16)

where the sum is over all values of k and θ[h] is a parameterfunction of the known scattering factors fn[h]. Even if itis valid only for identical atoms, it is often approximatelysatisfied by crystals with non-homogeneous atoms. Observethat when the magnitudes of the FT are unusually large, (16)is dominated by few coefficients as stated by the sigma tworelationship (14).

Atomicity is introduced also by treating the atomic positionsas random variables. This is one of the main points whichmade direct methods powerful. The first step is to sharpenthe FT coefficients with a normalization factor given by thescattering patterns fn[h],

E[h] = F [h]

/(K∑n=1

fn[h]2

)1/2

. (17)

This procedure allows the peaks to be more visible.From (16), we can derive

tanφ[h] =

∑k |E[k]E[h− k]| sin(φ[k] + φ[h− k])∑k |E[k]E[h− k]| cos(φ[k] + φ[h− k])

, (18)

where the sum is restricted to the indices k such that both|E[k]| and |E[h− k]| are large. This relationship is called thetangent formula and has a higher probability of being correctthan (15).

Phase determination by direct methods uses all the describedrelationships in the following order:

1) the magnitudes are normalized using (17),2) the set with the largest |E[h]| is selected,3) a first set of phases is determined using (13),4) missing phases are estimated using (14) and (18),5) phases are refined by iterative use of (18).

If not enough phases are known to start phase determina-tion, symbols are assigned to the missing phases, permittingrelationships between phases to be carried through beforethe actual values are determined. This is known as symbolicaddition and was important in making direct methods practical.

Electron densities computed in this way show approximatelya recognizable distribution of atoms. Many refinements and

EDIC RESEARCH PROPOSAL 5

new techniques have been incorporated to improve directmethods. They include the use of higher-order determinants,neighborhood principle and further applications of the Sayreequation [9]. Recently, a new algorithm, called charge flipping,has been introduced [10] and follows a completely differentphilosophy. It randomly assigns the phase to the diffractionpattern and alternately projects on the spatial and on theFourier domain to constrain the solution.

B. The Reconstruction of a Multidimensional Sequence fromthe Phase or Magnitude of Its Fourier Transform

The second paper we review [2] is a theoretical milestone inthe study of the recovery of multidimensional sequences frompartial Fourier data. In particular, it answers the question ofunicity of the reconstruction from the phase or the magnitudeof the FT. Hayes derives conditions for which this unicity canbe proven and proposes algorithms to solve both problems.We underline the fact that many algorithms are presentedto solve practically the magnitude retrieval problem. On theother hand, only one algorithm is presented for phase retrieval.Unfortunately, it works only when the initial guess of the phase(usually done randomly) is sufficiently good.

Given the scope of this review as a foundation of ourresearch proposal, we focus on the theorems concerning theunicity of the solution of Problem 3.

1) Mathematical Toolbox: The results are based on thestudy of the z-transforms of the images. Since the z-transformis a polynomial, let us start with some basic definitionsregarding polynomials in two or more variables.

A polynomial in the D variables z = [z1, z2, . . . , zD] is afunction of the form

p(z) =∑· · ·∑

k1+···+kD≤N

c(k1, . . . , kD)zk11 · · · zkDD , (19)

where k1, . . . , kD are non-negative integers and c(k1, . . . , kD)are arbitrary numbers which are called coefficients of thepolynomial. Each term q(z) of the sum in (19) is called amonomial and has a degree equal to d(q) = k1 + · · ·+ kD. Ifall the coefficients of a polynomial p(z) belong to a particularfield F , we say that p(z) is a polynomial over F . The set ofall polynomials of D variables over F is denoted by F(z).

Definition 2. Reducible and Irreducible Polynomials Apolynomial p ∈ F(z) with d(p) > 0 is reducible over Fif there are two polynomials p1, p2 ∈ F(z) with d(p1) > 0and d(p2) > 0 such that p(z) = p1(z)p2(z).

Notice that the reducibility depends on the field F . Forexample p(z) = 1 + z2 is reducible over C but not over R.

Two important theorems are necessary to understand themain results described in the paper: one concerns the existenceand the unicity of the factorization of a polynomial, while theother deals with the unicity of a polynomial in terms of itsvalues over a finite set of points.

Theorem 1. Any polynomial p ∈ F(z) of nonzero degreecan always be factored into a product of polynomials whichare irreducible in F . Furthermore, if p(z) has two different

factorizations:

p(z) = q1(z) . . . qk(z) = r1(z) . . . rl(z),

then k = l and the factors may be ordered in such a way thatqi(z) = cirj(z) and ci ∈ F \ 0.

Theorem 1 says that the factorization of a polynomialalways exists and is unique once we have F .

For the second theorem, we need to define the concept oflattice. Let assume we have D sets of points Pk, one for eachdimension. Then the lattice L(P ) is the Cartesian productof these D sets of points. Now, we can state the followingtheorem, that defines when a polynomial is unique in terms ofa set of its samples.

Theorem 2. Suppose that p1, p2 ∈ F(z) are polynomials ofdegree at most Nk in the variable zk. If for each k, Pk is aset of Nk distinct numbers in the field F , and if

p1(z) = p2(z) ∀z ∈ L(P ), then

p1(z) = p2(z) ∀z.

We can now define the image as a D-dimensional sequencef [n] = f(n1, n2, . . . , nM ) and its z-transform F (z) as

F (z) =∑

n∈ZD

f [n]z−n, (20)

where z−n = zn11 zn2

2 · · · znD

D . We skip the necessary con-sideration about the convergence of F (z), that are availablein the original paper [2]. We consider images with finiteand bounded support and we assume for convenience thatf [n] = 0, ∀ 0 < n < N 1. The region of the support isdenoted as R(N) and P(n) denotes the set containing all thereal D-dimensional sequences that are supported on R(N).

Let f ∈ P(n) be an image. Its z-transform F (z) is apolynomial in z−1 and it can be uniquely factorized accordingto Theorem 1 as

F (z) = αzn0

p∏k=1

Fk(z), (21)

where α ∈ R, n0 ∈ ND and Fk(z) are non-trivial irreduciblepolynomials in z−1. We define also the time-reversed form of(21) and when the z-trasform is symmetric.

Definition 3. Let f ∈ P(n) be an image with z-transformF (z) of degree N in z−1 with no trivial factors. Then, thetime reversed z-transform is defined as

F (z) = z−NF (z−1). (22)

Moreover, the z-transform is said to be symmetric if, for somevector k ∈ ND,

F (z) = ±zkF (z−1). (23)

2) Uniqueness of the Solution to Problem 3: In this section,we summarize the result of [2] that is most relevant to us: acondition for the unicity of the solution to the D-dimensional

1We define the inequality between vectors as an element-wise inequality:if m < n, then we have mi < ni ∀i.

EDIC RESEARCH PROPOSAL 6

phase problem. First, the author shows why this problem isconsidered difficult. Then, the author counts how many ofthese equivalence classes (5) have the same magnitude ofthe FT under the assumptions of Problem 3. We concludehighlighting the necessary conditions to have just one class,leading to the desired unicity.

Let f ∈ P(n) be an image for which |F (u)| is known forevery u. Using the discrete version of (24) together with (21)and Definition 3, we obtain

A(z) = α2zNp∏k=1

F (z)F (z). (24)

Recovering F (z) from (24) is not trivial: the vector n0 andthe sign of α are lost. Moreover, it is not possible to determinewhether the factor Fk(z) or Fk(z) is a factor of F (z). Thisambiguity is not surprising: it is the D-dimensional extensionof the spectral factorization problem [11]. In 1D, we willhave a combinatorial number of equivalence classes havingthe same magnitude of the FT because p is equal to the lengthof the signal. Therefore, it requires a combinatorial effort tosearch among them. For the multidimensional case, Hayescounts the number of these classes with the following theorem.

Theorem 3. [2] Let f ∈ P(n) be an image with F (z) as z-transform. If g ∈ P(n) and |G(u)| = |F (u)| for all u, thenG(z) must be of the form

G(z) = ±αzm∏k∈I

Fk(z)∏k/∈I

Fk(z), (25)

where I is a subset of the integers between 1 and p.

The theorem is originated by the following intuition: theonly way to generate g[n] ∼ f [n] is to convolve g[n] with anall-pass filter h[n]. According to Theorem 3, to have f [n] ∼g[n], we need the following conditions to be satisfied:• G(z) must have the same number of nontrivial irreducible

factors as F (z).• the only way to generate G(z) from F (z) is to replace

one or more nontrivial factors Fk(z) with Fk(z).• if the terms Fk(z) are symmetric, then the swap withFk(z) may only change the sign of F (z).

Finally, the author states that the number of equivalenceclasses with magnitude |F (u)| is at most 2p

′−1, where p′ is thenumber of non-symmetric irreducible factors of F (z). Thus,we can state the unicity as follows.

Theorem 4. Let f ∈ P(n) be an image with the followingz-transform,

F (z) = W (z)

p′′∏k=1

Fk(z), (26)

where W (z) is irreducible and Fk(z) are irreducible andsymmetric. If g ∈ P(n) with |G(u)| = |F (u)| for all u,then g[n] ∼ f [n].

We conclude with a theorem that allows us to extend theresults of Theorem 4 to the case when we only know thediscrete FT (DFT) of f [n]. Hayes uses Theorem 2, to statethe following result.

Theorem 5. Let f, g ∈ P(n) two images with support R(N)and let Ωk a set of Mk distinct complex number on the unitcircle, with Mk ≥ 2Nk − 1 for every k. If the conditions onf [n] defined in Theorem 4 are satisfied and if |F (u)| = |G(u)|on the lattice Ω, then f [n] ∼ g[n].

Note that, if we pick for every Ωk a set of equidistantpoints, then we can specialize Theorem 5 to the case wherewe measure the magnitude of the DFT of f [n] and g[n].

The important point of [2] is the following one: for imagesthat have two or more dimensions, the constraint that F (z)is irreducible and non-symmetric is not particularly strict. Infact, it may be shown that within the set of the polynomialsin D > 1 variables, the subset of reducible polynomialshas measure zero [12]. Therefore, the result of the paper isextremely strong: when a sufficiently dense set of samples ofthe magnitude of the FT is known, the solution of Problem 3is unique almost surely.

C. Message-Passing Algorithms for Compressed Sensing

Compressed sensing, defined in Problem 4, aims at recon-structing a vector f of length N from a set of M Nmeasurements by exploiting the structure of the signal [13].The signal can be recovered using different algorithms, withdifferent trade-offs between computational complexity and theachievable undersampling factor. In the paper we review here[3], the authors present an algorithm called AMP and theyshow that it reaches the best known undersampling factor,while keeping a low computational cost.

In this section, we first describe the two classic algorithmsused in the field: linear programming (LP) and iterativethresholding (IT) [14]. Then, we introduce the approximatemessage passing (AMP) algorithm [3] and sketch the theoret-ical performance bounds of the algorithm obtained via a stateevolution (SE) formalism based on the mean squared error(MSE). Note that the paper analyzes three slightly differentconcepts of sparsity. Here, we consider only the case of K-sparse signals, that is ‖f‖0 = K, since it is the prior weconsider for our research proposal.

Let A be an M × N matrix, then we have to solve thefollowing optimization problem:

minf

‖f‖0

s.t. y = Af , (27)

where f ∈ RN and y ∈ RM . This is an NP-hard problem andit can be approximately solved instead by solving

minf

‖f‖1

s.t. y = Af . (28)

For the solution of (28) to be equal to the one of (27), weneed to satisfy some particular conditions on the matrix A. Inparticular, the following ones are some well-known sufficientconditions:

• the entries of A are independent and identically dis-tributed normal N (0, 1/M).

EDIC RESEARCH PROPOSAL 7

• the entries of A are independent and identically dis-tributed Bernoulli random variables.

• the matrix A is a randomly sampled Fourier operator.

We also need to collect a sufficient number of measurementsM w.r.t. to the number of non-zero coefficients K. Thissufficient number of measurements is studied by taking a large-system limit, where N → ∞. We define the undersamplingfactor as M/N → δ and the sparsity as K/N → ρ. Then, thesparsity-undersampling behavior is studied as a function of(δ, ρ). The domain (δ, ρ) ∈ (0, 1)2 has two phases, a successphase where reconstruction typically occurs and a failurephase where exact reconstruction typically fails. Many effortshave been directed to determine the function that defines themaximum ρ that is in the success phase, as a function of δ. Forthe `1 minimization problem, this function has been obtainedusing arguments from combinatorial geometry in [15], and wecall it ρCG(δ).

Even if this particular convex optimization problem is wellunderstood, the algorithms that solve it can be extremely slowwhen the number of variables increases, such as it does forimages. Therefore, many researchers have abandoned LP infavor of fast iterative methods. One of the classic examplesare IT algorithms, that are typically designed as

f t+1 = ηt(A∗zt + f t),

zt = y −Af t, (29)

where ηt(·) is a scalar threshold function (applied componentwise), f t ∈ RN is the current estimate of f and zt ∈ RM isthe current residual. These schemes are extremely fast and areappealing for their performance. On the other hand, IT cannotachieve the undersampling factor of LP.

The authors of [3] present a significant improvement to (29),called AMP. It achieves the same reconstruction performanceof LP while keeping the computation complexity almost aslow as IT. AMP proceeds iteratively according to

f t+1 = ηt(A∗zt + f t),

zt = y −Af t +1

δzt−1

⟨η′t−1(A∗zt−1 + f t−1)

⟩, (30)

where 〈u〉 =

N∑i=1

ui/N and η′t−1(u) = ∂∂uηt−1(u).

Note that the thresholding function depends always on both tand the problem setting. In the case of interest, we consider athresholding operator ηt(u) = η(u;λσt) defined as,

η(u;λσt) =

(ui − λσt) if ui ≥ λσt,(ui + λσt) if ui ≤ −λσt,0 otherwise,

∀i (31)

where σ2t =

⟨E

(f t − f)2⟩

is the MSE of the currentestimate of f .

If the MSE of the estimate f t is reduced significantly atevery iteration, the algorithm will converge; therefore a studyof the evolution of the MSE can lead to a characterizationof the performance of the algorithm. Namely, the authors

consider the evolution of the MSE of (29), defined as

σ2t+1 = Ψ(σ2

t ),

Ψ(σ2t ) = E

(η

(Θ +

σt√δZ;λσt

)−Θ

)2, (32)

where Θ is the distribution of the entries of f and the expecta-tion is with respect to the independent variables Z ∼ N (0, 1)and Θ.

In what follows, we summarize the main results of thepaper [3]. Given the recursive definition of the MSE map (32),we can define the two phases of the algorithm as follows:if Ψ(σ2

t ) < σ2t−1 for all σ2

t ∈ (0,EX2], then the SEconverges to zero as t → ∞ and the algorithm converges tothe proper solution. On the other hand, if Ψ(σ2

t ) ≥ σ2t−1 the

algorithm does not converge. Moreover, let ρSE(δ, λ, F ) bethe obtained phase transition, then the authors prove that thisfunction does not depend on the distribution F of the valuesof f . Subsequently, they show that this phase transition hasthe following expression,

ρSE(δ, λ) = maxz≥0

1− 2/δ((1 + z2)Φ(−z)− zΦ(z))

1 + z2 − 2((1 + z2)Φ(−z)− zΦ(z))

.

(33)

If the parameter λ is optimally chosen, then

supλρSE(ρ, λ) = ρCG(ρ),

up to numerical precision. This is equivalent to claim that theSE describes the reconstruction performance of LP.

Thus, the authors defined a novel technique to study thephase transition of AMP derived from the simpler IT frame-work, that is numerically equivalent to the one derived for LP.Now, it is easy to show that this phase transition for IT doesnot hold. In fact, the performance of IT is generally worse thanthe one of LP. Surprisingly, the authors show in their secondclaim that this formalism correctly predicts the phase transitionof AMP and other statistical properties of the algorithm, suchas the number of non-zeros, false alarms and missed detectionsin the estimation f t at a particular iteration t. Moreover, theyempirically show that if ρ < ρSE(δ, λ) then AMP convergesexponentially fast to the correct solution,

σ2t ≤ σ2

0 exp−bt, (34)

for some b > 0 and for all t.To conclude, we would like to underline the fundamental

points given in [3]: the trade-offs between computational com-plexity and achievable undersampling factor, the possibility ofrecovering a signal from a small number of linear measure-ments, exploiting the structure of the signals, with a provablyefficient algorithm that converges to the exact solution expo-nentially fast. Unfortunately, this theoretical framework cannotbe applied directly to the problem of phase retrieval becausethe ACF is not a linear function of the signal. Our researchproposal, that is outlined in the next and concluding section,shows how to possibly improve the theoretical and practicalknowledge of the well defined open questions regarding theX-ray crystallography problem [1], the unicity of its solution

EDIC RESEARCH PROPOSAL 8

[2] and the reconstruction algorithms [3].

III. RESEARCH PROPOSAL

The phase retrieval problems we discussed in the introduc-tion start from the same type of data: a partial informationregarding the FT. What changes are the assumptions: whatwe measure, how we measure it and how the signal isstructured. We focus on X–ray crystallography [1], and wetarget unanswered questions: unicity of the solution of thephase retrieval and efficient algorithms to solve it. Theseproblems have been tackled in the last 50 years using manytechniques, models and strategies. We claim that sparsity canbe a good leverage to improve the results available in literature:sparsity fits naturally in the problem and a similar–but lesspowerful–property, atomicity, has been already used in thepast with outstanding results, [1]. Other prior information onthe signal, such as bounded support, allowed to prove unicity[2] for Problem 3. Therefore, we expect sparsity to restrictsignificantly the number of the solutions of Problem 2. On theother hand, it is also clear that sparsity is not the El Dorado.The electron densities of the single atoms en(x) might be toodifferent to prevent from modeling the crystal using the `0norm. In other words, even if a crystal is clearly sparse, wecould need other mathematical tools to define its structure.

Sparsity has also been also key to reconstruct optimally anundersampled signal and allowed to derive fast algorithmswith proven performance. For example, AMP [3] has thebest known performance to solve Problem 4 and it is amongthe fastest algorithms. We also know when and how AMPconverges to the exact solution. Unfortunately, as we havealready mentioned in Section II-C, it cannot be applied directlyto phase retrieval problems because the ACF is a set of non-linear measurements of the signal.

We foresee two paths according to the model we assumefor the crystal.Continuous time model: we consider the model described inSection II-A, where each atom of the unit cell is defined bythe location xn, the amplitude cn and the electron densityen(x). We measure the samples of the FT magnitude |F [h]|.We argue that the annihilation method used for signals with afinite rate of innovation [16] can lead to significant resultsThismodel is closer to reality but presents more difficulties. Thismodel is closer to reality but presents more difficulties, themain one being the fact that we cannot estimate the ACF fromthe samples without degrading the reconstruction performance.In fact, the sparsity of the ACF is approximately K2 and thusthe algorithms that rely on sparsity are less efficient. In ourtechnical report, [17], we describe the first results along withthe main encountered difficulties.Discrete time model: we consider the crystal as a discreteimage and the measured diffraction pattern as the DFT ofthe image. This model is a simplification of the real-worldscenario, and a careful evaluation of its accuracy is needed.However, it seems that it is possible to obtain stronger the-oretical and practical results. A first example of the possibleattainable results is given in [18], where the authors proposea condition for the unicity and a reconstruction algorithm.We extend these results in [17], where we describe a convex

relaxation of the reconstruction algorithm, together with asimplification of the proof of the unicity condition. The convexrelaxation has given promising initial results and its theoreticalstudy, given the convexity, can be predicted to be easier thanthe study of the iterative algorithm proposed in [18].

Even if we focused this research proposal on X–ray crys-tallography, we would like to point out that the same analysiscan be done for the channel estimation problem. In particular,let us assume we measure the output of an unknown channelh(x) with a white noise input. We would like to know h(x) toimprove the communications over that channel. We can easilyestimate |H(ua)|2 from the output, but the phase retrieval isas usual harder. Even in this scenario, we have the prior ofsparsity and the number of elements is generally smaller than20. Again, we are interested in a condition for the unicity ofthe solution and in an efficient algorithm that can solve theproblem for the largest possible class of sparse signals. Wecan use a discrete or a continuous time model, depending onwether we can assume that the location of the taps can bediscretized or not.

In summary, the signals of interest–crystals and channels–are sparse. The phase retrieval problem is fundamental torecover these signals. The main open questions are about theunicity of the solution and the existence of efficient algorithmwith provable performance. We aim at obtaining a strongsolution to these two problems by exploiting the sparsity,a property that has never been considered before in phaseretrieval and fits naturally the problem.

REFERENCES

[1] R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt.Soc. Am. A, 1990.

[2] M. Hayes, “The reconstruction of a multidimensional sequence fromthe phase or magnitude of its Fourier transform,” IEEE Trans. Acoust.Speech Signal Process., 1982.

[3] D. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithmsfor compressed sensing,” PNAS, 2009.

[4] G. Rhodes, Crystallography made crystal clear. Academic Press, 2006.[5] J. Ranieri, A. Chebira, Y. M. Lu, and M. Vetterli, “Frame reconstruction

from magnitude: a spectral factorization approach,” LCAV - School ofCommunciations and Computer Science - EPFL, Tech. Rep., 2011.

[6] J. Karle and H. Hauptman, “The phases and magnitudes of the structurefactors,” Acta Crystallogr., 1950.

[7] D. Sayre, “The squaring method: a new method for phase determina-tion,” Acta Crystallogr., 1952.

[8] W. Cochran, “Relations between the phases of structure factors,” ActaCrystallogr., 1955.

[9] T. Debaerdemaeker, C. Tate, and M. M. Woolfson, “On the applicationof phase relationships to complex structures. XXIV. The Sayre tangentformula,” Acta Crystallogr., Sect. A: Found. Crystallogr., 1985.

[10] G. Oszlanyi and A. Suto, “Ab initio structure solution by chargeflipping,” Acta Crystallogr., Sect. A: Found. Crystallogr., 2004.

[11] A. H. Sayed and T. Kailath, “A survey of spectral factorization methods,”Numer. Linear Algebr., 2001.

[12] M. Hayes and J. McClellan, “Reducible polynomials in more than onevariable,” Proc. IEEE, 1982.

[13] “Special issue on compressive sensing,” IEEE J. Sel. Top. Sign. Proces.,2008.

[14] J. A. Tropp and S. Wright, “Computational methods for sparse solutionof linear inverse problems,” Proc. IEEE, 2010.

[15] D. Donoho, “Neighborliness of randomly projected simplices in highdimensions,” PNAS, 2005.

[16] M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rateof innovation,” IEEE Trans. Signal Process., 2002.

[17] J. Ranieri, A. Chebira, Y. M. Lu, and M. Vetterli, “Phase retrieval in X-ray crystallography: the sparse way,” LCAV - School of Communciationsand Computer Science - EPFL, Tech. Rep., 2011.

[18] Y. M. Lu and M. Vetterli, “Sparse spectral factorization: unicity andreconstruction algorithms,” in IEEE Trans. Acoust., Speech, and SignalProcess., 2011.