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# Phase retrieval for the wavelet transform

by I. Waldspurger

#### GdR MOA 2015

December 03, 2015

## Transcript

1. ### Phase retrieval for the wavelet transform Ir` ene Waldspurger Paris,

´ Ecole normale sup´ erieure (Joint work with St´ ephane Mallat) December 3, 2015 Journ´ ees annuelles du GdR MOA 2015
2. ### Introduction 2 / 26 Phase retrieval problems Let V be

a complex vector space ; {Li }i∈I be a ﬁxed family of linear forms on V . A phase retrieval problem is of the form Reconstruct x ∈ V from {|Li (x)|}i∈I . where |.| is the usual complex modulus ; the unknown x can be any element of V . If u ∈ C, |u| = 1, x and ux are indistinguishable. → We consider the reconstruction up to a global phase.
3. ### Introduction 3 / 26 Three families of issues arise. Uniqueness

The element x may not be uniquely determined. ∀i, |Li (x)| = |Li (y)| ⇒ x = y ? Stability Reconstruction can be unstable under measurement noise. ∀i, |Li (x)| ≈ |Li (y)| ⇒ x ≈ y ? Algorithms It is diﬃcult to have simultaneously precision and reasonable complexity.
4. ### Introduction 4 / 26 Plan 1. Two categories of phase

retrieval problems Well-behaved case : random or “generic” measurements Deterministic “physical” measurements Fourier transform Wavelet transform 2. Main two classes of algorithms Iterative algorithms Convexiﬁed algorithms PhaseLift PhaseCut 3. Numerical results : which algorithm for which problem ?
5. ### Two categories of phase retrieval problems 5 / 26 Random

or “generic” measurements Reconstruct x ∈ V from {|Li (x)|}i∈I . Finite dimension : assume V = Cn, Li ∈ Cn. Choose the Li at random. → Uniqueness / stability results that hold with high probability / probability 1. Theorem If Card(I) ≥ 4n − 4, then the set of measurements for which there is no uniqueness has Lebesgue measure zero. [Balan, Casazza, and Edidin, 2006] [Conca, Edidin, Hering, and Vinzant, 2015] Random “generic” measurements → uniqueness with probability 1
6. ### Two categories of phase retrieval problems 6 / 26 Stronger

hypotheses on the measurements distribution. Case 1 : the Li ’s are i.i.d. Gaussian, and isotropic. Case 2 : masked Fourier transform. M1 , ..., MJ ∈ Cn i.i.d. Lk,j : x → Mj x[k] (j ≤ L, k = 0, ..., n − 1) Theorem Uniqueness and stability hold with high probability, provided that : Card(I) ≥ Cn (case 1) J ≥ C log n (case 2) [Cand` es and Li, 2014] [Cand` es, Li, and Soltanolkotabi, 2015; Gross, Krahmer, and Kueng, 2015]
7. ### Two categories of phase retrieval problems 7 / 26 Random

measurements Advantages : strong results (uniqueness, stability). Drawbacks : depending on the application, the measurements may or may not be possible to choose at random. But the questions of uniqueness and stability tend to be harder to analyze when the measurements are deterministic.
8. ### Two categories of phase retrieval problems 8 / 26 Most

well-known case : the Fourier transform Reconstruct f ∈ L2(Rd ) from |ˆ f | s.t. f is compactly-supported Many applications in physics. → Has been studied for around 60 years. This case can be studied with harmonic analysis techniques. [Akutowicz, 1956; Walther, 1963] [Barakat and Newsam, 1984] Unfortunately, If d = 1, no uniqueness. If d ≥ 2, uniqueness “almost surely” but stability issues.
9. ### Two categories of phase retrieval problems 9 / 26 Tools

used for the Fourier transform can be adapted to more sophisticated cases. Example : fractional Fourier transform [Jaming, 2014] → Uniqueness, but no stability. The wavelet transform [Mallat and Waldspurger, 2014] The modulus of the wavelet transform is a widely-used representation of audio signals (scalogram ≈ spectrogram). Studying the phase retrieval problem allows to better understand why this representation is so useful in audio processing, modify the scalogram of a signal, then reconstruct a modiﬁed signal.
10. ### Two categories of phase retrieval problems 10 / 26 Example

for an audio signal
11. ### Two categories of phase retrieval problems 11 / 26 We

consider Cauchy wavelets. Uniqueness Any function f ∈ L2(R, R) is uniquely determined by the modulus of its wavelet transform, up to trivial ambiguities. Stability There is no “strong stability” : |Wf | ≈ |Wg| ⇒ f ≈ g. There is a form of “local stability”.
12. ### Two categories of phase retrieval problems 12 / 26 Summary

Random measurements Well-posed problems (uniqueness and stability). “Physical” case Not all problems are well-posed. There are stability issues, although they may not be dramatic.
13. ### Main two classes of algorithms 13 / 26 Algorithms Iterative

methods Convexiﬁed algorithms PhaseLift PhaseCut            Comparison in terms of precision, complexity.
14. ### Main two classes of algorithms 14 / 26 Iterative methods

Principle Start with an initial estimation. Reﬁne the estimation step after step. Hope to converge towards a solution. Most well-known : alternate projections. [Gerchberg and Saxton, 1972; Fienup, 1982] Advantages : relatively fast, simple to implement. Main drawback : the problem is not convex, and the algorithm may converge to a local optimum instead of the correct solution.
15. ### Main two classes of algorithms 15 / 26 Choice of

the initialization When the measurements are random, and follow a known probability law, a good estimation of x can be found with a spectral method. → Wirtinger Flow [Cand` es, Li, and Soltanolkotabi, 2015] [Chen and Cand` es, 2015] Theorem If the measurements are i.i.d. Gaussian and isotropic, then a variant of Wirtinger Flow succeds with high probability, provided that Card I ≥ Cn, for some C > 0. Drawback : a priori limited to random cases.
16. ### Main two classes of algorithms 16 / 26 Convexiﬁed algorithms

Principle Minimization of a non-convex functional. Approximate as the minimization of a convex functional on a larger space. Hope that the solutions on the small and large spaces coincide. First example : PhaseLift. [Cand` es, Eldar, Strohmer, and Voroninski, 2011] [Chai, Moscoso, and Papanicolaou, 2011] Advantage : much more precise (theoretical guarantees in some cases). Drawback : high complexity.
17. ### Main two classes of algorithms 17 / 26 Principle of

PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i   § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i
18. ### Main two classes of algorithms 17 / 26 Principle of

PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i   § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i Not convex
19. ### Main two classes of algorithms 17 / 26 Principle of

PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i   § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i Approximation − − − − − − − − − − → min Tr X s.t. ∀i, Tr(L∗ i Li X) = b2 i X 0
20. ### Main two classes of algorithms 17 / 26 Principle of

PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i   § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i Approximation − − − − − − − − − − → min Tr X s.t. ∀i, Tr(L∗ i Li X) = b2 i X 0 (Convex)
21. ### Main two classes of algorithms 18 / 26 PhaseCut Similar

principle, but we focus on the reconstruction of {Li (x)}, instead of the direct reconstruction of x. Advantages In terms of precision, there is an equivalence between PhaseCut and a modiﬁed version of PhaseLift. → PhaseCut is “as precise” as PhaseLift. Diﬀerent complexity. → PhaseCut is more eﬃcient is some cases.
22. ### Main two classes of algorithms 19 / 26 PhaseCut ﬁnd

x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd y ∈ Cp s.t. ∀i, |yi | = bi and y ∈ Im(L) ﬁnd u ∈ Cp s.t. ∀i, ui ui = 1 and ||P(Im L)⊥ (b × u)||2 = 0 ⇐⇒ ﬁnd u ∈ Cp s.t. ∀i, |ui | = 1 and b × u ∈ Im(L) ﬁnd U = uu∗ ∈ Mp (C) s.t. diag(U) = 1 Tr(MU) = 0   § ¦ ¤ ¥ rank (U) = 1
23. ### Main two classes of algorithms 19 / 26 PhaseCut ﬁnd

x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd y ∈ Cp s.t. ∀i, |yi | = bi and y ∈ Im(L) ﬁnd u ∈ Cp s.t. ∀i, ui ui = 1 and ||P(Im L)⊥ (b × u)||2 = 0 ⇐⇒ ﬁnd u ∈ Cp s.t. ∀i, |ui | = 1 and b × u ∈ Im(L) ﬁnd U = uu∗ ∈ Mp (C) s.t. diag(U) = 1 Tr(MU) = 0   § ¦ ¤ ¥ rank (U) = 1 −→ minimize Tr(MU) s.t. diag(U) = 1 U 0
24. ### Main two classes of algorithms 20 / 26 Complexity PhaseLift

PhaseCut min Tr X s.t. ∀i, Tr(L∗ i Li X) = b2 i X 0 min Tr(MU) s.t. diag(U) = 1 U 0 X is of size n × n ; U is of size p × p, with p ∼ 4n. → U is larger than X. The constraint diag(U) = 1 is simpler. It makes PhaseCut and instance of Max-Cut. → Possible optimizations for PhaseCut.
25. ### Main two classes of algorithms 21 / 26 The complexity

depends on the algorithm used to minimize the functional. With an interior-point method, PhaseLift : O a2n4.5 log 1 ; PhaseCut : O a3.5n3.5 log 1 . 10 20 30 0 2 4 6 8 x 108 PhaseLift PhaseCut (a = p/n ≈ 4 ; is the precision.) Interior-point methods are adapted to problems where high precision is required.
26. ### Numerical results 22 / 26 Three family of measurements Fourier

transform. (No uniqueness.) Masked Fourier transform, with 4 masks. (Uniqueness and strong stability, with high probability.) Wavelet transform. (Uniqueness ; weak stability.) Signals of size 128. Four algorithms Alternate projections Wirtinger Flow PhaseLift PhaseCut
27. ### Numerical results 23 / 26 Percentage of exact reconstruction Fourier

Masked Fourier Wavelet transform Projections 2% 51% 0% Wirtinger 3% 93% 0% PhaseLift 2% 100% 74% PhaseCut 1% 100% 100% Analysis Random cases with strong stability Iterative algorithms are suﬃcient. Deterministic cases with stability issues Convexiﬁed algorithms are more precise. PhaseCut has a better complexity than PhaseLift.
28. ### Numerical results 24 / 26 Larger problems Reconstruction from the

modulus of the wavelet transform for audio signals : n ∼ 10000. The cost of convexiﬁed methods is prohibitive. Alternative Iterative algorithms that use the particular structure of the signal or the measurements.
29. ### Bibliography I 25 / 26 E. J. Akutowicz. On the

determination of the phase of a Fourier integral, I. Transactions of the American Mathematical Society, 83(1) :179–192, 1956. R. Balan, P. Casazza, and D. Edidin. On signal reconstruction without noisy phase. Applied and Computational Harmonic Analysis, 20 :345–356, 2006. R. Barakat and G. Newsam. Necessary conditions for a unique solution to two-dimensional phase recovery. Journal of Mathematical Physics, 25(11) : 3190–3193, 1984. E. J. Cand` es and X. Li. Solving quadratic equations via phaselift when there are about as many equations as unknowns. Foundations of Computational Mathematics, 14 (5) :1017–1026, 2014. E. J. Cand` es, Y. C. Eldar, T. Strohmer, and V. Voroninski. Phase retrieval via matrix completion. SIAM Journal on Imaging Sciences, 6(1) :199–225, 2011. E. J. Cand` es, X. Li, and M. Soltanolkotabi. Phase retrieval from coded diﬀraction patterns. Applied and Computational Harmonic Analysis, 39(2) :277–299, 2015. E. J. Cand` es, X. Li, and M. Soltanolkotabi. Phase retrieval via wirtinger ﬂow : theory and algorithms. IEEE Transactions of Information Theory, 61(4) :1985–2007, 2015. A. Chai, M. Moscoso, and G. Papanicolaou. Array imaging using intensity-only measurements. Inverse Problems, 27(1), 2011. Y. Chen and E. J. Cand` es. Solving random quadratic systems of equations is nearly as easy as solving linear systems. Preprint, 2015.
30. ### Bibliography II 26 / 26 A. Conca, D. Edidin, M.

Hering, and C. Vinzant. Algebraic characterization of injectivity in phase retrieval. Applied and Computational Harmonic Analysis, 32 (2) :346–356, 2015. J. R. Fienup. Phase retrieval algorithms : a comparison. Applied Optics, 21(15) : 2758–2769, 1982. R. Gerchberg and W. Saxton. A practical algorithm for the determination of phase from image and diﬀraction plane pictures. Optik, 35 :237–246, 1972. D. Gross, F. Krahmer, and R. Kueng. Improved recovery guarantees for phase retrieval from coded diﬀraction patterns. To appear in Applied and Computational Harmonic Analysis, 2015. P. Jaming. Uniqueness results in an extension of pauli’s phase retrieval problem. Applied and Computational Harmonic Analysis, 37 :413–441, 2014. S. Mallat and I. Waldspurger. Phase retrieval for the cauchy wavelet transform. to appear in the Journal of Fourier Analysis and Applications, 2014. A. Walther. The question of phase retrieval in optics. Optica Acta, 10(1) :41–49, 1963.