In this paper, we address the problem of computing the dimension of data space in phase retrieval problem.
Starting from the quadratic formulation of the phase retrieval, the analysis is performed in two steps. First, we
exploit the lifting technique to obtain a linear representation of the data. Later, we evaluate the dimension of data
space by computing analytically the number of relevant singular values of the linear operator that represents the data.
The study is done with reference to a 2D scalar geometry consisting of an electric current strip whose square amplitude of the electric radiated field is observed on a twodimensional extended domain in Fresnel zone

Raffaele Moretta

Rocco Pierri

English

arXiv

Archivio Istituzionale della Ricerca - Università degli Studi della Campania "Luigi Vanvitelli"

URSI GASS 2021, Rome, Italy, 28 August - 4 September 2021An evaluation of the data space dimension in phase retrieval: results in Fresnel zoneRocco Pierri(1), Raffaele Moretta(1)(1) Dipartimento di Ingegneria, Universita’ della Campania "L. Vanvitelli", Aversa, 81031, Italia.AbstractIn this paper, we address the problem of computing the di-mension of data space in phase retrieval problem.Starting from the quadratic formulation of the phase re-trieval, the analysis is performed in two steps. First, weexploit the lifting technique to obtain a linear representa-tion of the data. Later, we evaluate the dimension of dataspace by computing analytically the number of relevant sin-gular values of the linear operator that represents the data.The study is done with reference to a 2D scalar geome-try consisting of an electric current strip whose square am-plitude of the electric radiated field is observed on a two-dimensional extended domain in Fresnel zone.1 IntroductionPhase retrieval techniques find applications in all the con-texts where phase information is not available. In electro-magnetism, they arise in antenna or array diagnostics, inthe reconstruction of the far-field pattern from near-zonedata (phaseless near-field far-field techniques) [1, 2], and ininverse scattering problem.From the mathematical point of view, the lack of phase in-formation makes the problem non-linear and this compli-cates the task of finding a solution. Over the years, differ-ent numerical procedures to address the problem have beenproposed; some of them exploit the amplitude formulation,instead, others are based on the square amplitude formula-tion. The latter consists in retrieving the unknown functionf ∈ X from the quadratic model below|T f |2 = |g|2 (1)where T : f ∈ X −→ g ∈ Y .The most common techniques to tackle the problem exploita least-square minimization. However, since the cost func-tional to be minimized is not quadratic, trap points may oc-cur. The latter may avoid reaching the actual solution of theproblem even if the uniqueness conditions are satisfied.To overcome this drawback, the lifting technique can beused [3]. The latter, starting from the quadratic formulationin (1), exploits a redefinition of unknown space to recastthe phase retrieval problem as a linear one. Despite this,the new unknown function belongs to a functional spacewhose dimensions are the square of those of the originalunknowns space. Consequently, for large-scale problemsthe lifting approach is not feasible and the phase retrievalproblem must be necessarily addressed by recurring to non-convex formulations.In this framework, avoiding trap points is the main task.From this point of view, the least-squares minimizationbased on the square amplitude formulation brings to a costfunctional that is smoother that obtained by consideringamplitude formulation. Furthermore, the quadratic formu-lation of phase retrieval problem allows a deep analysis ofthe genesis of local minima and allows finding strategiesto “cure” them. In particular, it has been shown that if theratio between the dimension of data space (M) and the di-mension of unknowns space (N) is high enough, no trappoints appear in the functional to minimize [4, 5, 6]. Fromthis discussion, it is evident that the dimension of data spaceplays a key role in phase retrieval via quadratic approach;hence, it is worth investigating how to evaluate it from ananalytical point of view.As shown in [3], the dimension of data space can be evalu-ated by counting the number of significant singular valuesof the lifting operator. However, to the best of our knowl-edge, analytical results concerning the singular values be-havior of lifting operator are not available in literature.For such reason, with reference to a 2D scalar geometry,we will provide a closed-form expression of the number ofsignificant singular values of the pertinent lifting operator.2 Geometry of the problem and preliminaryresultsIn this paper, we consider the 2D scalar geometry depictedin fig. 1 where the y-axis represents the axis of invariance.An electric current J(x) = J(x) îy supported on the set[−a,a] of the x-axis radiates within a homogeneousmedium with wavenumber β .The electric field E radiated by such strip source has onecomponent directed along the îy; hence, E(r,θ) =E(r,θ) îy.The square amplitude of the radiated electric field |E|2 isobserved in Fresnel zone on a two-dimensional domainthat extends along the polar coordinates (r,θ) on the set[rmin,rmax]× [−θmax,θmax].For the geometry at hand, the radiated electric field can byexpressed in the variables r and u = sin(θ) by the equationE(r,u) = T J(x) (2)where T is the linear integral operator that realizes the fol-lowing mappingT : J ∈ L2(SD)−→ E ∈ L4(OD) (3)Figure 1. Geometry of the problemwith L2(SD) denoting the space of square-integrable func-tions on the set SD = [−a,a], and L4(OD) indicating thespace of functions whose amplitude to the fourth power isintegrable on the set OD = [rmin,rmax]× [−umax,umax].Under the paraxial Fresnel approximation, the operator Tcan be explicitly written as in [7] in the formT J =1√β re− jβ r (1+12 u2)∫ +a−ae− jβ2r x2e jβux J(x)dx (4)3 The lifting operator and its singular valuesIn this section, first we provide a linear representation of|E(r,u)|2, i.e, the square amplitude of the electric field overthe observation domain OD. Later, we find the dimensionof data space by evaluating the number of significant singu-lar values of the linear operator which represents the data.To obtain a linear representation of |E|2, let us rewrite thequadratic model |E|2 = |T J|2 in the form below|E(r,u)|2 =(T J)(T J)∗==1β r∫∫De jβ2r (x2−x2) e− jβu(x−x) J(x)J∗(x)dxdx (5)with D = {(x,x) ∈ [−a,a]× [−a,a]}.From the last equation, it is evident that if we redefine theunknown space and we consider as unknown the functionF(x,x) = J(x)J∗(x) then the operator which links the un-known function F(x,x) with the data function |E(r,u)|2 islinear. Such operator is known in literature as lifting opera-tor and it is defined asA : F ∈ L2 (SD×SD) −→ |E|2 ∈ L+2 (OD) (6)whereAF =1β r∫∫De jβ2r (x2−x2) e− jβu(x−x) F(x,x)dxdx (7)A weighted adjoint operator A†w is given byA†w(·) =∫ rmaxrmin∫ +umax−umaxw(x,x)β re− jβ2r (x2−x2)e jβu(x−x) (·)dudr(8)where w(x,x) is a weight function, and (·) denotes the func-tion of the variables (r,u) on which the adjoint operatoracts.Naturally, the presence of the weight function changes thedynamics of the singular values of the lifting operator A.Despite this, we will show that the number of relevant sin-gular values remains unchanged. For such reason, we cantolerate the changes in the singular values behavior broughtby the weight function.To evaluate the number of significant singular values of thelifting operator, we will find the eigenvalues λm of the op-erator AA†w. The latter can be expressed asAA†w(·) =∫ rmaxrmin∫ +umax−umaxH(r,ro,u,uo)(·)dudr (9)whereH(r,ro,u,uo) = (10)1β 2rro∫∫Dw(x,x)e jβ2 (1ro−1r )(x2−x2)e− jβ (uo−u)(x−x)dxdxWith the aim to compute the integral (10) in a very simpleway, let us divide the integration domain D asD = D1∪D2where D1 = {(x,x) ∈ D : x 6= x}, D2 = {(x,x) ∈ D : x = x}.The set D2 is a null set with respect to the Lebesgue mea-sure; consequently, the kernel can be computed by perform-ing the integration only on D1.With the aim to evaluate H(r,ro,u,uo), let us perform thechange of variables X1 = x− xX2 =x2− x2rmax(11)which is injective and continuously differentiable on D1.By virtue of (11), the kernel of AA†w can be recast asH(r,ro,u,uo) =1β 2rro(12)∫∫D̂1w(X1,X2)ej β2 (rmaxro −rmaxr )X2e− jβ (uo−u)X1∣∣∣∣ ∂ (x,x)∂ (X1,X2)∣∣∣∣dX1dX2where•∣∣∣ ∂ (x,x)∂ (X1,X2)∣∣∣ = − rmax2X1denotes the Jacobian determinantof the transformation,• D̂1 indicates the domain in which the original integra-tion domain D1 is mapped by (11).Note that, despite the Jacobian determinant is singular forX1 = 0, such point does not belong to the integration do-main D̂1. Hence, no singularity appears in the integral (12).Now, if we choosew(X1,X2) =(∣∣∣∣ ∂ (x,x)∂ (X1,X2)∣∣∣∣)−1 (13)Figure 2. Shape of D̂1 when a = 10λ , rmax = 100λ .we have thatH(r,ro,u,uo) =1β 2rro∫∫D̂1e jβ2 (rmaxro −rmaxr )X2e− jβ (uo−u)X1dX1 dX2 (14)According to equation (14), the integration should be doneon the set D̂1 which is sketched in fig. 2. However,since we want to recast the operator AA†w in a form whoseeigenvalues are known in closed-form, we will approxi-mate H(r,ro,u,uo) by integrating on the smaller domainthat encloses the set D̂1. The latter is made up by all thepoints (X1,X2) belonging to the rectangular set [−2a,2a]×[−a2/rmax, a2/rmax] except for the point (0,0). By per-forming the integration on such domain, we have thatH(r,ro,u,uo)≈8a3β 2rmax1rrosinc(βa22(1ro− 1r))sinc(2βa(uo−u))(15)Accordingly, the operator AA†w can be expressed asAA†w(·)≈8a3β 2rmax∫ rmaxrmin∫ +umax−umax1rrosinc(βa22(1ro− 1r))sinc(2βa(uo−u)) (·)dudr(16)From (16), it is evident that the operator AA†w becomes moresimilar to a convolution operator if we set s = rmax/r. Infact, by doing this, we have thatAA†w(·)≈8a3β 2r2max∫ rmaxrmin1∫ +umax−umaxsossinc(βa22rmax(so− s))sinc(2βa(uo−u))(·)duds(17)Despite the previous operator is only an approximation ofAA†w, the positive aspect is that its eigenvalues are known inclosed-form.It is worth noting that the operator (17) has been obtainedby exploiting two changes of variables. The idea of exploit-ing a change of variable in such a way that the kernel of theconsidered operator assumes a desired form has been ex-ploited also in other recent works [8].In order to compute the eigenvalues of (17), we must solvethe eigenvalue problemAA†wvm = λmvm (18)where vm represents the m− th eigenfunctions of AA†w.By fixing ṽm(s,u) = vm(s,u)/s the eigenvalue problemabove can be recast as8a3β 2r2max∫ rmaxrmin1∫ umax−umaxsinc(βa22rmax(so− s))sinc(2βa(uo−u))ṽm(s,u)duds = λmvm(so,uo)(19)The eigenvalues of (19) are known in closed-form. In fact,according to [9], they are given by the equationλm(AA†w) = λ(u)m1 λ(s)m2 (20)where λ (u)m1 and λ(s)m2 denote the eigenvalues of theSlepian-Pollak operators whose kernels are respectivelysinc(2βa(so− s)) and sinc(βa22rmax(so− s)).Since the sequences {λ (u)m1 } and {λ(s)m2 } are relevant respec-tively until to the indexesMu =4πβ aumax+1 Ms =βa22π(1rmin− 1rmax)+1, (21)it results that the eigenvalues of the problem (19) and, con-sequently, the eigenvalues of the operator (17) are signifi-cant until to the index M = Mu Ms.Let us remember that the kernel of the operator in (17) hasbeen obtained by integrating on the smaller rectangular thatencloses D̂1; for such reason, M is not exactly equal to thenumber of relevant eigenvalues of AA†w but it represents anupper bound.Until now, we have focused on the eigenvalues of AA†w;instead, the actual singular values of A are related to theeigenvalues of AA† by the equation σm(A) =√λm(AA†)where A† denotes the usual adjoint operator defined with-out the weight function. For such reason, we know only anapproximation of the singular values of A that is given bythe square root of the eigenvalues of AA†w.In the next section, by means of some simulations, we willcheck that the actual singular values of A and their approx-imated version become negligible at the same index. Thisverification allows to state that for the considered geome-try, the number of significant singular values of the liftingoperator or in other words the dimension of data space Msatisfies the inequalityM ≤Mu Ms (22)4 Numerical resultsIn this section, we check that actual singular values of Aand their approximations become negligible at the same in-dex. As test case, we consider the configuration in whicha = 10λ , umax = 0.5, rmin = 25λ (smax = 4), rmax = 100λ(smin = 1). With reference to such configuration, in fig. 3we have sketched the actual singular values of A and theirapproximated versions in dB. In particular, the blue, red andblack diagrams sketch respectivelyFigure 3. Singular values of A, and their approximated ver-sions in dB.• the square root of the eigenvalues of the approximatedversion of AA†w provided by (17),• the square root of the eigenvalues of AA†w,• the square root of the eigenvalues of AA†.As can be seen from fig. 3, the square root of the eigenval-ues of the operator (17) exhibits a multi-step behavior andthey are relevant until to the index M = MuMs = 164.The multi-step behavior can be understood if we rememberthat the eigenvalues of such an operator are given by (20).Now, as shown in fig. 4, the sequence {λ (u)m1 } have a step-like behavior, instead, the sequence {λ (s)m2 } is not exactlystep-like. This automatically implies that the eigenvaluesof the operator (17) have a multi-step behavior also beforethe index M = 164 and, consequently, also their square root.However, our aim is to forecast the critical index at whichthe actual singular values of A become negligible. By ob-serving the behavior of the actual singular values (black di-agram in fig. 3), it is evident that the singular values beyondthe index M = 164 are surely negligible while those beforeare almost all significant if the noise level is not so high.This implies that the use of the weighted adjoint changesonly the dynamics of the singular values but not the criticalindex at which they become negligible. For such reason,we can state that M = MuMs is an upper bound for the di-mension of data space that is very close to its actual value.5 ConclusionIn this article, we have addressed the problem of evaluatingthe dimension of data space in phase retrieval. In partic-ular, with reference to a 2D geometry consisting of a stripcurrent observed on a two dimensional observation domain,we first have introduced a linear operator that represents thesquare amplitude of the radiated field. After, studying theFigure 4. Eigenvalues of the Slepian-Pollak opera-tors whose kernels are sinc(2βa(uo − u)) (left panel),sinc(βa22rmax(so− s))(right panel).singular values of such an operator, we have provided anupper bound for the dimension of data space which is verynear to its actual value.References[1] A. Paulus, J. Knapp, T. F. Eibert, “Phaseless Near-Field Far-Field Transformation Utilizing Combinations of Probe Sig-nals,” IEEE Trans. Antennas Propag., 65, 10, Oct. 2017, pp.5492-5502.[2] B. Fuchs, M. Mattes, S. Rondineau, L. Le Coq, “Phase-less Near-Field Antenna Measurements From Two SurfaceScans - Numerical and Experimental Investigations,” IEEETrans. Antennas Propag., 68, 3, March 2020, pp. 2315-2322.[3] R. Moretta, R. Pierri, “Performance of phase retrieval viaPhaselift and Quadratic Inversion in circular scanning case,”IEEE Trans. Antennas Propag., 67, 12, Dec. 2019, pp. 7528-7537.[4] F. Soldovieri, A. Liseno, G. D’Elia, R. Pierri, “Global con-vergence of phase retrieval by quadratic approach,” IEEETrans. Antennas Propag., 53, 10, Oct. 2005, pp. 3135-3141.[5] R. Pierri, R. Moretta, “On Data Increasing in Phase Re-trieval via Quadratic Inversion: Flattening Manifold and Lo-cal Minima,” IEEE Trans. Antennas Propag., 68, 12, Dec.2020, pp. 8104-8113.[6] J. Sun, Q. Qu, J. Wright, “A geometric analysis of phaseretrieval,” Foundations of Computational Mathematics, 18,5, 2018, pp. 1131-1198.[7] R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depthresolution for a strip source in the Fresnel zone,” J. Opt. Soc.Am. A, 18, 2, 2001, pp. 352-359.[8] M.A. Maisto, R. Pierri, R. Solimene, “Near-Field Warp-ing Sampling Scheme for Broad-Side Antenna Character-ization,” Electronics, 9, 6, 2020, pp. 1047.[9] D. Slepian, “Prolate spheroidal wave functions, Fourieranalysis and uncertainty - IV: Extensions to many dimen-sions; generalized prolate spheroidal functions,” The BellSystem Technical Journal,, 43, 6, Nov. 1964, pp. 3009-3057.

An evaluation of the data space dimension in phase retrieval: results in Fresnel zone

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