A Generalization of Polarization Formula and Its Application In

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ISSN: 1314-3344 Research Article

A Generalization of Polarization Formula and Its Application in Phase Re- trieval Zhitao Zhuang1* and Guochang Wu2 1College of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, P.R. China; 2Depart- ment of Mathematics, Henan University of Economics and Law Zhengzhou, P.R. China

ABSTRACT In this paper, some generalizations of the classical polarization formula are used to recover the relative phase in phase retrieval problem. Theoretically, in order to reconstruct any signal from its intensity measurements by polarization formula, the amount of needed measurements can be same as PhaseLift method. The numerical simulation also illustrates its good effect in (affine) phase retrieval with additive white Gaussian noised intensity Fourier measurements. Keywords: Polarization formula; Phase retrieval; Frame 2010 MS Classification 42C15

INTRODUCTION In this paper, the frame theory is used to obtain some polarization identities. At first, we briefly introduce some definitions and The aim of phase retrieval is to recover signal x from its intensity N notations. Let H denotes a separable Hilbert space with the inner measurements x,ϕi , i =1,..., N , where {ϕi}i=1 forms a d product 〈⋅ , ⋅〉 and J be a countable index set. frame of C . Since |< x,ϕi >|=|< αx,ϕi >| for any α ∈C Definition 1.1. A sequence { f j} j∈J of elements in H is a frame for H with|α |=1, the best reconstruction of x is up to a unimodular if there exists constants A, B > 0 such that constant. Phase retrieval arises in many areas of engineering and 2 ≤ < > 2 ≤ 2 ∈ applied physics, including X-ray crystallography [1-8], optics [9], A || f || ∑| f , f j | B || f || , f H. j∈J and computational biology [10-13]. In fact, it is di icult to solve The constants A, B are called lower and upper frame bounds the phase retrieval problem if one only knows the intensity for the frame. A frame is A-tight, if A=B. If A =B=1, it is called a ﬀ measurements. One way to overcome this issue is to collect more Parseval frame. prior information of the signal x [11]. Another way is to take more additional measurements. We only mention two different methods Frame theory not only provides an effective analysis method for of the second way. The PhaseLift algorithm is proposed by Candès signal processing, but also offers a reconstruction method. We et al. with the lift technique of semi-definite programming [6]. call {g j} j∈J a dual frame of { f j} j∈J if it is a frame for H and Polarization method using structured measurements is proposed by satisfying

Alexeev et al. [2]. Natural f = ∑ f , g j f j , f ∈ H. (1.1) j∈J nonconvex algorithms often work remarkably well in practice, The dual frame always exists but generally not unique. Since but lack clear theoretical explanations, therefore Sun et al. give a f {g j} j∈J is also a frame, the function has the expression geometric analysis of phase retrieval [14]. Recently, phase retrieval = { f j } f ∑ j∈J f , g j g j as well. For special case, if j∈J is an A-tight frame, in infinite dimensional space also attracts great attentions. 1 f f then {A j }j∈J is a dual frame. And if { j }j∈J is a Parseval frame, it is a Reconstruction of a bandlimited real-valued function f from dual frame of itself. unsigned intensity measurements is considered [8]. Unlike the Since the complex field C is closely related to the Euclidean finite dimensional case, phase retrieval in infinite dimensional 2 2 Hilbert space is never uniformly stable [4]. Therefore, Alaifari et al. space R , the frame theory in R can be rewritten with respect to proposed a new paradigm for stable phase retrieval by considering complex numbers. Explicitly, any complex number Z=X+IY can be the problem of reconstructing signal up to a phase factor that is considered as a bidimensional vector (x, y). Therefore, for any two ℜ(z z ) not global [1]. complex numbers z1 and z2, the real part 1 2 is an inner product

*Corresponding author: Zhitao Zhuang, College of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China Received May 03, 2019; Accepted July 22, 2019; Published July 29, 2019 Citation: Zhuang Z, Wu G (2019) A Generalization of Polarization Formula and Its Application in Phase Retrieval. Mathematica Eterna. 9: 101. 10.35248/1314-3344.19.09.101. Copyright: © 2019 Zhuang Z, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Mathematica Eterna, Vol. 9 Iss. 1 No: 101 1 Zhuang Z, et al. OPEN ACCESS Freely available online of the vectors with respect to them. Without confusion, we call the One important thing is whether an equal norm tight frame exists. collection {z } is a frame for C if its corresponding collection The answer is positive and explicit constructions are given in finite i i∈J 2 of vectors is a frame for R . And the frame reconstruction formula frames: theory and applications [7] and reference therein. In the can be rewritten as rest of this section, we consider the frame that consists of Nth roots z = ℜ(z~z )z = ℜ z~z ~z , z ∈C, of unity. ∑ k k ∑ ( k ) k (1.2) k∈J k∈J Lemma 2.2. Let kT for N ≥ where {z˜ } is the dual frame of {z } . Some polarization xN = (cos(2kp / N), sin(2kp / N )) k k∈J k k∈ J − 2 3, then k N 1 is an equal norm tight frame for with frame identities and examples are given in Section 2. We discuss the {ξN }k=0 R − applications of polarization identities in (a ine) phase retrieval bound N/2 and satisfy N 1ξ k = ∑k=0 N 0. problem in Section 3. 2 ﬀ Proof. For any f = (x, y)∈ R2 with norm f = x2 + y2 , there exists x y Polarization Formulas an angle such that cos(θ ) = , sin(θ ) = . Then we have x2 + y2 x2 + y2

In this section, we show some polarization identities that are N −1 2 N −1 kθ 2 2 deduced from frame theory in C. The classical polarization identity ∑ f ,ξ N = f ∑ cos(θ )cos(2kπ / N) − sin(θ )sin(2kπ / N ) k=0 k=0 in functional analysis becomes a special case. N −1 2 = f ∑cos2 (θ + 2kπ / N) Lemma 2.1. Suppose {ηk }k∈J is a frame for C with dual frame k=0 ~ N −1 {η } . Then 2  N 1  k k∈J = f  + ∑cos(2θ + 4kπ / N) 2 2 = 2z z = η~ | z +η z |2 − η~ (| z |2 + |η |2| z |2 ).  k 0  1 2 ∑ k 1 k 2 ∑ k 1 k 2 N 2 k∈J k∈J = f , 2 Proof. We expand the modulus: N −1 2 2 2 2 where the equation cos(2θ + 4kπ / N) = 0 is used in the last +η = + η + ℜ η ∑k=0 | z1 k z2 | | z1 | | k | | z2 | 2 (z1 z2 k ). equation. Taking summation over k and applying reconstruction formula By the frame properties we have (1.2) to the expansion, we get the desired result. 2 N −1 = ξ k ξ k ∀∈ 2 Above lemma can be generalized to any Hilbert space to get a f ∑ f , N N , R . N k=0 polarization identity with similar proof. As mentioned before, there is an equivalent formula corresponding Theorem 2.1. Suppose is a frame for C with dual frame to complex field C. Explicitly, for any complex number z, we have {ηk }k∈J {η~ } . Then for any two elements f, g in a Hilbert space H, we {η~ } k k∈J k k∈J (2.4) N −1 have 2 −k k z = ∑ℜ(zζ N )ζ N , ~ 2 ~ 2 2 2 N k=0 2 f, g = ηk || f +ηk g || − ηk (|| f || + |ηk | || g || ). ∑ ∑ π k∈J k∈J 2 i / N k where ζ N = e . Taking ηk = ζ N in Lemma 2.1 and Theorem 2.1, By imposing some constraints to the frames, we can get some we get the following two corollaries. compact results. For instance, if is an equal norm frame {ηk }k∈J 2πi / N ∈ Corollary 2.1. Take ζ N = e for N ≥ 3. Then for any z1, z2 ∈C, that is |ηk | = c for some constant c > 0 and all k J, then we have − 1 N 1 2 = ζ k + ζ k ~ 2 2 2 2 ~ z1z2 ∑ z1 z2 . (2.5) 2 f, g = η || f +η g || −(|| f || +c || g || ) η . N k=0 ∑ k k ∑ k π k∈J k∈J 2 i / N Corollary 2.2. Let H be a complex Hilbert space and ζ N = e for η~ = Furthermore, if we require∑ k 0 , then N ≥ 3. Then for any f, g ∈ H, k∈J N −1 = η~ +η 2 1 k k 2 2 f , g ∑ k || f k g || . (2.3) f , g = ∑ζ N f + i g . (2.6) k∈J N k=0

Even more, if {ηk }k∈J is an A-tight frame, then the expression is Taking N = 3 in Corollary 2.1, we get the polarization identity not related to dual frame in form. Explicitly, we have stated [2]. The classical polarization formula in functional analysis

1 2 is a special case of Corollary 2.2 with N = 4, that is f , g = ηk || f +ηk g || . ∑ 3 2 2A ∈ 1 k k k J f , g = ∑i f + i g . (2.7) All the above polarization identities can be generalized to 4 k=0 sesquilinear form. We only show the last one in the following. Consequently, the above corollaries can be viewed as a generalization of the canonical polarization formula. Theorem 2.2. Let W be a complex vector space, S a sesquilinear form on W, and q the quadratic form generated by S. Suppose The frames consisted of Nth roots of unity play an important role η = η = 0. is an A-tight equal norm frame for W with ∑ ∈ k 0. in above discussion. Since it’s not very e icient in phase retrieval ∑k∈J k k J Then we have when N is large, reduction of elements in frame is needed. In fact, 1 three elements of the Nth roots are enoughﬀ for recovering the S( f , g) = η q( f +η g). ∑ k k inner product. Let be any three different integers in the 2A k∈J k1,k2 ,k3 This theorem is easy to prove. From the theorem of Jordan and 2πik / N 3 set {0,1,..., N −1}. Then  must form a frame for C. If there Von Neumann, we know that a norm on a vector space is generated {e }=1 exists a dual frame η~ 3 satisfying by an inner product if and only if the parallelogram is satisfied. If {  }=1 this is so then the inner product is given by (2.7). In fact, one can 3 η~ = (2.8) prove that the inner product is also given by any frame in Theorem ∑  0, 2.1. =0

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Then we get a reconstruction formula Table 2: Three frames and their dual frames. 3 − π = ℜ 2 ik  / N η~ ∀ ∈ Three frames Dual frames f ∑ (e f )  , f C. (2.9) =  1 ~ 13 3 23 In fact, if we writeη = x + iy  , since the reconstruction 1, −+ i 1,+ ii formula holds if and only if it holds for f =1 and f=i by linearity of 22 33 innerproduct, the equations (2.8) and (2.9) are equivalent to the 13 3 23 1, −− i 1,−−ii matrix equation XA = B, where 22 33

 cos(2πk1 / N) sin(2πk1 / N) 1  x1 x2 x3  ,   , 1 0 0 . X =   = π π B =   1313 33 y y y A cos(2 k2 / N) sin(2 k2 / N) 1 0 1 0 −+ii, −− −+1ii ,1 −−  1 2 3      22 22 33 cos(2πk3 / N) sin(2πk3 / N) 1 If A is invertible, the matrix equation has a unique solution and the corresponding dual frame satisfy (2.8) and (2.9). By simple which are listed in Table 2. As a result, we can get three polarization computation, we have identities by Theorem 2.1. The first one is given by

2 2 2 2  (k − k )π   (k − k )π   (k − k )π  f , g = 1 + 3 i f + g + 3 i f + − 1 + 3 i g − 2 + 2 3i f + g . det A = 4sin 3 2 sin 3 1 sin 2 1 . (2 6 ) 3 ( 2 2 ) ( )( )  N   N   N  The others can be written out similarly. Since k1, k2, k3 are different from each other in the set {0,1,..., N −1}. the determinant of matrix A is not zero. In Example 2.2, the expression of polarization identity is not likely to become easier than Example 2.1. However, in phase retrieval, Combing the above discussion, we have the following conclusion: 2 2 since the original intensity measurements f , g are known

Theorem 2.3. Suppose k1,k2 ,k3 are three different integers from generally, only two additional measurements are needed in order ~ 3 each other in the set {0,1,..., N −1}. Then, the collection {η } forms to recover relative phase. ∥ ∥ ∥ ∥ 3  =1 a frame of C with a dual frame{η~ } satisfying (2.8) and (2.9).  =1 APPLICATIONS TO PHASE RETRIEVAL Example 2.1. The Nth roots of unity are 1, i, 1, i when N = 4. By Theorem 2.3, we can find four frames of C and their corresponding dual In this section, we apply the polarization identities to phase retrieval frames, which are listed in Table 1. As a result, we− can −get four polarization problem. The key ingredient is to add new measurements in order identities by formula (2.3). The first one is given by to gain the relative phase.

1− i 2 i 2 1− i 2 By leveraging the ideas of Alexeev et al. [2] and Bandeira et al. f , g = f + g + f + ig − f − g . 4 2 4 [3], we can implement polarization algorithms in phase retrieval The others can be written out similarly. problem with the help of polarization identities that are given in Section 2. Suppose the intensity measurements 2 are known, If we continue to reduce the number of elements in a frame to two, x,ϕi then the new frame should be a basis of C and have no redundancy. the key point to recover signal x with polarization method is to 2 compute the relative phase using , where η is a frame Therefore, the restriction η~ = is no longer hold. However, x,ϕi +ηkϕ j k ∑ =  0  1 of C. If x,ϕ ≠ 0 and , the relative phase between we still can recover the innerproduct by the following theorem i x,ϕ j ≠ 0 whose proof is similar to Theorem 2.3. x,ϕi and x,ϕ j is defined by

−1 Theorem 2.4. Taking out two numbers k1,k2 from the set  x,ϕ  x,ϕ x,ϕ x,ϕ ρ =  i  j = i j  N  i, j . {0,1,..., N −1} with k ≠ k if N is odd and  x,ϕ  x,ϕ x,ϕ x,ϕ 1 2 k1 ≠ k2 mod   i  j i j  2  2pik12 /N 2pik /N Since can be considered as an innerproduct in C, if N is even, then we have that the set {e ,e } forms a x,ϕi x,ϕ j f ≠ 2 it can be computed by Lemma 2.1. If x, i 0 for everyi ∈ J , we frame with dual frame h = x + iy , which is given by { ll l}l=1 can get the relative phase ρi,i+1 of two adjacent points, or relative phase ρ of two points with ϕ is fixed. Then the signal x can  x1 x2  1  sin(2πik 2 / N) − sin(2πik 1 / N) i0 ,i+1 i   =  . be recovered up to a global phase factor. In graph theory terms, if  y1 y2  sin(2π (k2 − k1) / N) − cos(2πik 2 / N) cos(2πik 1 / N)  the graph with vertex ϕ and edges is a circle or star, then we i x 1 3 1 3 can recover x . However, in general situation, there is a mask ϕ Example 2.2. The Nth roots of unity are 1, − 2 − 2 i , − 2 − 2 i i when N=3. By Theorem 2.4, we get three frames and their dual frames, orthogonal to the signal x, i.e., x,ϕι = 0 . Therefore the relative phases can’t propagate across this vertex. The authors of Alexeev et al. [2] propose to design full spark frame and expander graph to Table 1: Four frames and their dual frames. overcome this shortage. Four frames Dual frames In this section, we focus on phase retrieval simulations with 11 1, i, -1 (1−ii) ,,( −− 1 i) masked Fourier measurements, which are obtained by measuring 22 the Fourier power spectrum of signals with adding mask. In order 11 to have a high probability recovery, the mask is chosen randomly 1, i, -i 1,1,1(−+ii) ( −−) 22 with Gaussian distribution.

11 1 2πimm'/ N 1,+i −+ 1, ii − Let fm denote the complex sinusoid{ e } . Then the 1, -1, -i ( ) ( ) N m'∈Z N 22 discrete Fourier transform 2 2 is defined by F :  (Z N ) →  (Z N ) 11 i, -1, -i (1+−ii) , 1,( 1 −) (F* x) (m) = x,fm . 22

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Suppose the Fourier intensity measurements x, fm are known, much difference. However, for large N, we can still recover the additional measurements are needed to recover the relative phase. signal when erasures occur in intensity measurements. π Taking η = 2 i / N = ζ  = − then we have  e N ,  0,1,..., N 1, A ine phase retrieval introduced in Gao et al. [10] has an exact N −1 2 1   reconstruction but not up to a unimodular constant. Explicitly, we x, fi x, f j = ∑ζ N x, fi + ζ N f j . N =1 considerff recovering signal from the absolute values of the a ine Accordingly, we need additional vectors  for computing linear measurements N fi + ζ N f j ρ 2πim / N N −1 ff every relative phase i, j . Sincef = Ef , where E = diag{e } , i+1 i m=0 x,φ + b , j =1,...,m, one has j j T whereφ ∈ H , ∈ m and H = C or R . Let   j b =( b1m ,...,b) H f + ζ f = (I + ζ E)f . T m i N j N i b =( b ,...,b) ∈ Hm and b ∈ H , by vector augmentation, we If all intensity measurements , then we need N + 1 masks 1m x, fm ≠ 0 set − {I} + ζ  N 1 to recover x . By Theorem 2.3, one can reduce the  {I N E}=0 ~ T T ~ ~ ~ φ = φ T ~ = T and = φ φ number of total masks to four. When the intensity measurements j ( j ,bj ) , x (x ,1) A ( 1,..., m ). x, f are known, one can continue to reduce the amount of masks Then the measurements can be written as ~ ~ i x,φ j + bj = x,φ j . to three by Theorem 2.4. For instance, we can take the masks as in One can prove easily that (A,b) is a ine phase retrievable if A +η 2 ˜ Example 2.2, and then the total masks are{I}{I  E}=1. Comparing and A are both full spark. Generally, the polarization method is with the masks {I, I + Es, I + iEs} in Candes et al. [5], the same hardly used to high dimension data dueff to its high computation amount of masks are used. complexity. However, because of the exact reconstruction of a ine phase retrieval, this can be implemented by a ine phase retrieval in We demonstrate the performance of polarization method with one dimension iteratively. As a simulation, the polarization methodff different frames and their duals in phase retrieval problem. Given is used to reconstruct the cameraman image fromff its power spectrum a random signal, we add white Gaussian noise with different noise with SNR=20. This is implemented by reconstruction column by level to the intensity measurements such that the signal noise ratio column in one dimension. Finally, we get the reconstructed image (SNR) changes from 13 to 40 with step length 3. The effects of with SNR=17.9, which is illustrated in Figure 1. recovering the original signal with different frames are shown in 1 1 Table 3, where N,N4,3,N3,2 correspond to N-th roots of unity frames, CONCLUSION the first frame in Table 1 and Table 2 respectively. Observing Table By the advantage of frame theory, a class of generalization of polarization 3, we find that the polarization method also have denoising effects formula is given, which makes the classical polarization as a special due to the least square method that is used in the reconstruction case. It provides a strong support for recovering the relative phase process. For different frames, it seems that the effects have not in polarization method. Furthermore, the same amount of intensity

Table 3: Effects of different frames with different noise level measured by decibel. SNR 13 16 19 22 25 28 31 34 37 40 Frames N=3 17.1 20.2 21.9 25.0 26.4 29.0 32.7 37.0 39.9 40.8 N=4 17.5 20.0 23.7 24.8 29.6 30.2 30.8 33.9 40.1 41.9 N=5 17.5 19.5 22.5 26.4 26.8 29.9 31.3 35.4 40.0 43.1 N=6 18.1 18.7 23.0 24.0 27.4 31.0 34.6 36.4 38.5 43.8 N=7 17.3 19.9 23.2 25.0 27.3 30.9 33.3 36.8 38.3 41.0 N=8 17.5 20.0 21.4 24.5 28.8 31.2 32.6 36.1 40.0 39.6 N=9 15.8 20.5 22.9 23.2 28.3 28.2 33.0 37.3 40.0 39.9 N=10 17.8 20.3 21.5 27.7 26.0 32.0 33.4 36.7 40.2 42.5 1 N 4,3 19.6 22.8 23.6 23.5 27.6 28.3 30.2 38.2 38.6 41.3 1 N 3,2 17.1 19.1 23.6 24.8 27.3 30.2 33.2 33.2 37.8 41.5

Figure 1: The left is the original image, right is the reconstruction image from intensity measurements with SNR=20.

Mathematica Eterna, Vol. 9 Iss. 1 No: 101 4 Zhuang Z, et al. OPEN ACCESS Freely available online measurements are used as in PhaseLift method. The numerical 3. Bandeira AS, Chen Y, Mixon DG. Phase retrieval from power spectra simulations also demonstrate its good effect in (a ine) phase retrieval of masked signals. Information and Inference: a Journal of the IMA. of signal and image with Fourier measurements. 2014;3(2):83-102. ﬀ 4. Cahill J, Casazza P, Daubechies I. Phase retrieval in infinite-dimensional DECLARATIONS Hilbert spaces. Trans Am Math Soc, Series B. 2016;3(3):63-76. Acknowledgements 5. Candes EJ, Eldar YC, Strohmer T, Voroninski V. Phase retrieval via matrix completion. SIAM Rev Soc Ind Appl Math. 2015;57(2):225- The authors would like to thank the referees for their useful 251. comments and remarks. 6. Candes EJ, Strohmer T, Voroninski V. Phaselift: Exact and stable signal Availability of data and materials recovery from magnitude measurements via convex programming. Commun Pure Appl Math. 2013;66(8):1241-1274. The [cameraman.tif] data used to support the findings of this study are available from the corresponding author upon request. 7. Casazza PG, Kutyniok G. Finite frames: Theory and applications. Sprin Scie & Busi Med; 2012; Sep 14. Funding 8. Chen Y, Cheng C, Sun Q, Wang H. Phase retrieval of real-valued This study was partially supported by National Natural Science signals in a shift-invariant space. Appl Comput Harmon Anal. 2018. Foundation of China (Grant No.11601152). 9. Elser V. Phase retrieval by iterated projections. J Opt Soc Am A Opt Competing interests Image Sci Vis. 2003; 20(1): 40-55. 10. Gao B, Sun Q, Wang Y, Xu Z. Phase retrieval from the magnitudes of The authors declare that they have no competing interests. affine linear measurements. Adv Appl Math Mech. 2018; 93:121-141. Authors’ contributions 11. Gerchberg RW. A practical algorithm for the determination of phase from image and diffraction plane pictures. Optik. 1972;35:237-246. All authors contributed equally to this work. All authors read and approved the final manuscript. 12. Millane RP. Phase retrieval in crystallography and optics. J Opt Soc Am A Opt Image Sci Vis. 1990;7(3):394-411. REFERENCES 13. Stefik M. Inferring DNA structures from segmentation data. Arti 1. Alaifari R, Daubechies I, Grohs P, Thakur, G. Reconstructing real Intelli. 1978;11(1-2): 85-114. valued functions from unsigned coefficients with respect to wavelet 14. Sun J, Qu Q, Wright J. A geometric analysis of phase retrieval. Found and other frames. J Fourier Anal Appl, 2017;23(6);1–15. Comut Math. 2018;18(5):1131-1198. 2. Alexeev B, Bandeira AS, Fickus M, Mixon DG. Phase retrieval with polarization. SIAM J Imaging Sci, 2014;7(1):35-66.

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