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A Generalization of Polarization Formula and Its Application In atica E em ter OPEN ACCESS Freely available online th n a a M Mathematica Eterna 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 ~ . Then for any two elements f, g in a Hilbert space H, we {η~ } {ηk }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.
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