Research Article | Open Access
Almost Phase Retrieval with Few Intensity Measurements
In signal processing, the phases of the measurements are often unknown. To recover a signal of length , Balan, Casazza, and Edidin showed that it suffices to know at least intensity measurements. We give another characterization of frames which give phase retrieval for almost all signals with only intensity measurements. We provide a method to test if a frame has this property or not. With our method, we can construct tight frames or modify any frames such that they give almost phase retrieval. Numerical results show that frames with this property can recover signals with high probability.
Given a set of vectors ( or ), the phase retrieval problem is to recover a signal from which is called intensity measurements. Phase retrieval is a key step in speech processing , X-ray crystallography , and so forth. Since frames are redundant systems of vectors in a Hilbert space, it is possible to reconstruct signals with suitable frames.
First, we introduce the definition of phase retrieval.
Definition 1 (see [3, Definition 2.4]). One calls that a set of vectors in gives phase retrieval if for all satisfying , for all , there exists a constant with such that .
In fact, this definition is equivalent to the following.
Definition 2. One calls that a set of vectors in gives phase retrieval if for all satisfying where and , one has .
Let be an matrix of rank . If the set of columns of gives phase retrieval, we also say that the matrix gives phase retrieval.
Balan et al.  studied the properties of such frames with minimal measurements. For the real frames, they gave the following result.
Proposition 3 (see [4, Theorem 2.2]). Let be a frame for . If , then a generic frame gives phase retrieval.
Here a generic frame means a member of some nonempty Zariski open subset of the set of all -element frames in . We refer to [5, Section 2.2] for details.
For the complex frame, a similar result also holds.
Proposition 4 (see [4, Theorem 2.2]). Let be a frame for . If , then a generic frame gives phase retrieval.
For , is necessary. But for , Bandeira et al. conjectured the following.
Conjecture 5 (see ). Let be a frame for , . Then the following holds.(i)If , then does not give phase retrieval.(ii)If and is a generic frame, then gives phase retrieval.
Recently, people have studied more properties of such frames. Ohlsson and Eldar  studied the minimal measurements of sparse signals. If sparse signals contain noise, Eldar and Mendelson  also studied the minimal measurements and gave a recovery algorithm. By convex programming, Candès, Strohmer, and Vladislav gave a method which is called phaselift to recover a signal from its magnitude measurements. Then Gross et al.  and Alexeev et al.  developed other methods with phaselift. Chen et al.  estimated the error of phaselift. Schniter and Rangan  used compressed sensing to recover sparse signals via generalized approximate message passing.
If we do not require that gives phase retrieval, for all , we may need fewer measurements. Fickus et al.  studied when gives phase retrieval for almost all vectors of . In this paper, we first introduce some preliminary results of frames which give phase retrieval. Then, we show our characterization and some properties of such frames. We give a method to test if a frame gives almost phase retrieval. Finally, we provide some examples to illustrate our results.
2. Main Results
First, we introduce some preliminary results.
A set of finitely many vectors in is called a frame if there are two positive constants such that, for every , A frame is said to be tight if and Parseval if . If the right-hand side of (2) holds, it is said to be a Bessel sequence. Define the analysis and synthesis operators of a frame by respectively.
We call the frame coefficients of . It can be shown that is invertible on and, for each , We call that is the frame operator and the canonical dual frame for . It is easy to see that is the frame operator of . Generally, if satisfies for every , we call that is a dual frame for . For more details on frame theory and applications, we refer the reader to [14, 15] and so forth.
Fickus et al. gave the following definition.
Definition 6 (see [13, Definition 7]). A set of vectors in gives almost phase retrieval if it gives phase retrieval for almost every .
Next, we study real frames which give almost phase retrieval. Characterizing the complex frames which give almost phase retrieval remains an open problem.
Denote the th row and the th column of a matrix by and , respectively. They gave a characterization of such frames.
Proposition 7 (see [13, Theorem 12]). Let be an matrix on . Suppose each column of is nonzero. Then gives almost phase retrieval if and only if is of and, for each nonempty proper subset , where and are the matrices consisting of and , respectively.
Here we give another characterization which shows more properties of the almost phase retrieval. Let be a subset of . Denote For a matrix , denote Denote the range and null space of an operator or matrix by and , respectively. Our first main result is the following.
Theorem 8. Let be a frame for with analysis operator . Suppose that the matrix satisfies Then gives almost phase retrieval if and only if holds for every nonempty proper subset .
Proof. First we prove the necessity. We assume that there exists a nonempty proper subset which does not satisfy (10). That is,
Since , we have
Hence for every
But for and . Hence does not give almost phase retrieval.
Sufficiency. Let be a nonempty proper subset of . Then is a proper subspace of . Let be the union of all such subspaces and . Since is the union of finitely many proper subspaces of , its Lebesgue measure in is . And for every , . That means gives phase retrieval on . Hence gives almost phase retrieval.
Remark 9. In Theorem 8, we only need .
Lemma 10. Let be an matrix on . Suppose that its columns are nonzero. Denote by the space spanned by rows of . If the rows of are orthogonal basis of , then, for every nonempty proper subset , holds if and only if, for every , ,
Proof. First we prove the necessity. We assume that (16) holds. Then each row of is a linear combination of rows of . Therefore there exists such that
where is an matrix on . Since the rows of form an orthonormal basis of , the rows of form also an orthonormal basis of . Hence is an orthogonal matrix. Hence, for every , ,
Sufficiency. Denote by and the matrices consisting of and , respectively. Let and be the spaces spanned by columns of and , respectively. By (17), we have Since each column of is nonzero, and are proper subspaces of . Therefore, there exists an matrix on such that, for every and , Then we have Hence we get (16).
Lemma 10 shows that if the matrix can not be split into two orthogonal parts, then the corresponding frame (see Theorem 8) gives almost phase retrieval. Here we need the fact that the rows of are orthogonal. If the columns of contain a natural basis, the result still holds.
Lemma 11. Let be an matrix on . Suppose that the columns of contain a natural basis of . For every nonempty proper subset , holds if and only if, for every , ,
Proof. For every nonempty proper subset , denote by and the matrices consisting of and , respectively. Suppose that contains and contains . Then, holds if and only if each column of is a linear combination of and each column of is a linear combination of , which is equivalent to and . Therefore,
Next we show that we can choose a suitable matrix to test if gives almost phase retrieval.
Theorem 12. Let be an matrix on . Each column of is nonzero. Denote by the space spanned by rows of . Let the matrix consist of some linearly independent vectors in . If gives almost phase retrieval on , then gives almost phase retrieval on . Moreover, if rows of the matrix on span , then also gives almost phase retrieval on .
Before proving Theorem 12, we introduce a result on the Naimark complement of Parseval frames.
Proposition 13 (Naimark’s Theorem [14, Theorem 1.9]). Let be a frame for with analysis operator . Let be the standard basis of , and let be the orthogonal projection onto . Then the following conditions are equivalent.(i) is a Parseval frame for .(ii)For all , we have .(iii)There exist such that is an orthonormal basis of .Moreover, if (iii) holds, then is a Parseval frame for .
We call that is a Naimark complement of .
Proof of Theorem 12. Take an matrix whose rows are orthogonal basis of . Then the columns of form a Parseval frame of . Since and gives almost phase retrieval, by Theorem 8, for any nonempty proper subset , we have By Lemma 10, there exist some and such that Let be a Naimark complement of . Then the rows of belong to . Since we have By Lemma 10, Since , gives almost phase retrieval on , thanks to Theorem 8.
From the above results, we give a method to test if an matrix gives almost phase retrieval on or not.
Denote by and the matrices consisting of the first columns and the last columns of , respectively. Without loss of generality, we assume that is of rank . And has the following form: Then gives almost phase retrieval if and only if does. Without loss of generality, we assume that each column of is nonzero; otherwise we can drop the columns which are zero. Denote where is the th element of . Next we do the following steps.(1)Set . If , end. Otherwise, set If , end. Otherwise, go to step .(2)Find a such that has two nonzero elements and satisfying and . If there exists no such column of , end. Otherwise, go to step .(3)If , end. Otherwise, set If , end. Otherwise, go to step .
After finitely many steps, say steps, we get and finally. There are two cases.(i)Consider . For every and , we have By Lemma 11 and Theorem 8, does not give almost phase retrieval and neither does .(ii)Consider . For any , there exist such that Therefore, there exists some nonempty proper subset satisfying (26). Hence gives almost phase retrieval, and so does .
Remark 14. Balan et al. [4, Theorem 2.9] proved that gives almost phase retrieval while has a column such that . By the above method, it is easy to see that gives almost phase retrieval. And we find more general matrices which give almost phase retrieval in some sense.
Remark 15. In our method, we need to compute the inverse of . But is necessary; the processes of transforming to and computing are conducted at the same time. Next it is very easy to compute . Therefore, the computation complexity is acceptable.
Here we give a method to test if a frame gives almost phase retrieval or not. With this method, we can construct frames which have this property. For the unit norm tight frames, Fickus et al.  gave a sufficient condition.
Proposition 16 (see [13, Theorem 13]). If and are relatively prime, then every unit norm tight frame gives almost phase retrieval.
Different from Proposition 16, for any , we can construct a tight frame which gives almost phase retrieval from the above methods.
First, we can construct an matrix of the form (34) such that it gives almost phase retrieval. For example, we can choose such that one of the last columns does not contain zero element. Second, by the Gram-Schmidt process or other methods, we get a matrix from such that the rows of are orthogonal. Third, let where is a constant. Then the columns of form a tight frame with frame bound by the following proposition.
Proposition 17 (see [16, Lemma 1.1]). The columns of an matrix are a tight frame with frame bound if and only if
If an matrix does not give almost phase retrieval, we know that from the above method. Then we can choose a column of , say , such that and . Let be a matrix satisfying where Then we can check that set with respect to is . Let . Then gives almost phase retrieval. On the other hand, the canonical dual frame is always used to reconstruct the original vector. Sun  gave the following stability result of canonical dual frame.
Proposition 18 (see [17, Theorem 4.1]). Let and and and be two pairs of canonical dual frames for . Denote the frame operator of and by and , respectively. Denote the frame bounds of and by and , respectively. Then(i)if is a Bessel sequence with an upper bound , then so is with an upper bound .(ii)If then
Since for every , for small enough, the matrix is a good approach of . Denote the frames consisting of the columns of and by and , respectively. From Proposition 18, we see that the canonical dual frame of is also a good approach of the canonical dual frame of . Hence, if does not give almost phase retrieval, we can get a new matrix which gives almost phase retrieval with a little change. And the process system is stable.
3. Numerical Results
In this section, we provide some examples to show the frames in Theorem 8 give almost phase retrieval.
Let be a frame for with analysis operator . For , we want to recover from . Based on Theorem 8, we use the following algorithm.(i)Take some matrix such that . For every , , , denote (ii)Compute . Set (iii)Set , where is a dual frame of . Then .
A practical method to construct the desired frame is using random vectors. Given and , let be a set of vectors in with entries generated with the Gaussian random variable. In our tests, any frame generated in this way satisfies (10). For simplicity, we take and .
We test our algorithm to recover signals from their intensity measurements with different noise levels. That is, the measurement is where is a noise with the norm . We perform trials. In each trial, we generate a set of new random vectors and solve the phase retrieval problem with 1000 random vectors with norm . Figures 1 and 2 show the results of the trials. We see that the result is perfect and noise free. Whenever , the performance is good. Observe that the aliasing occurs for signals in consisting of proper subspaces. The results are also acceptable with larger noises. This shows that the frames in Theorem 8 can give phase retrieval with high probability.
(a) Trial 1
(b) Trial 2
(c) Trial 3
(d) Trial 4
(e) Trial 5
(a) Trial 1
(b) Trial 2
(c) Trial 3
(d) Trial 4
(e) Trial 5
In this paper, we show that intensity measurements are enough to reconstruct almost all signals of length . With our characterization, we provide a method to test if a frame has this property. Numerical results show that we can use these frames to recover signals with high probability.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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