A uniqueness result for propagation-based phase contrast ...ip.math.uni-goettingen.de/data-smaretzke/slides_ifip_2014-12_smare… · Holotomography: Quantitative phase tomography

A uniqueness result for propagation-based phasecontrast imaging from a single measurement

IFIP TC7.4 Workshop on Inverse Problems and Imaging

Simon Maretzke

CRC 755 - Nanoscale Photonic Imaging

16/12/2014

S. Maretzke CRC 755 - Nanoscale Photonic Imaging

A uniqueness result for propagation-based phase contrast imaging from a single measurement

Physical Problem

Monochromatic EM-Waves: ∆Ψ + n2k2Ψ = 0, n = 1− δ + iβ

I Interaction: By geometrical optics → contact image

Ψ(·, 0) = 1 +[exp

(−ik

∫ 0

−L(δ − iβ) dz)− 1]

= 1 + h

I Fresnel Diffraction: Propagation of paraxial waves (no far-field)

Ψ(·, d) = D(F)d (Ψ(·, 0)) ∝ w (F)·F

(w (F) ·Ψ(·, 0)

), w (F) ∼ exp

(iξ2)

I Detected Intensities: I = |Ψ(·, d)|2 = |1 +D(F)d (h)|2



Inverse Problem

Forward Operator:

F : S ′c (Rn)→ C∞(Rn); F (h)(ξ) =∣∣∣exp

(−iξ2

)+ F(w (F) · h)(ξ)

∣∣∣2 (1)

⇒ Well-defined? Injective?

Inverse Problem: (Phase Retrieval in Phase Contrast Imaging)

Reconstruct the contact image h† ∈ S ′c (Rn) from intensities I † = F (h†).

Previous Results:

Contrast transfer function (linearize F for h ∈ L2(Rn) small):

F(F (h)− 1)(ξ) ∝ sin(χξ2

)F(=(h))(ξ) + cos

(χξ2

)F(<(h))(ξ)

Complex h unique from two measurements [Jonas and Louis, 2004]

Phase vortex [Nugent, 2007] → single intensity pattern insufficient?



Numerical Evidence

Phase Contrast Tomography:

3D-structure δ + iβ from contact images at different incident angles

Combined ansatz: Simultaneous phase retrieval and Radon inversion

Solve Ftot(δ + iβ) = I by iteratively regularized Gauss-Newton

δ: reconstruction β: reconstruction δ: exact β: exact

I Unique reconstruction of compact objects from a single measurement

I Tomographic correlations of contact images facilitate phase retrieval



Paley-Wiener-Schwartz Theorem

Theorem 1 (Paley-Wiener-Schwartz Theorem)

Let K ⊂ Rn compact, convex. Then, for any u ∈ S ′c (Rn), supp(u) ⊂ K ,u := F(u) defines an entire function in Cn and ∃C > 0,N ∈ N0 s.t.

|u(ξ)| ≤ C (1 + ‖ξ‖2)N exp

(supx∈K=(ξ) · x

)∀ ξ ∈ Cn (2)

Conversely, any entire function u, satisfying (2), is the complex extensionof the Fourier transform of such a distribution.

Correspondence:

S ′c (Rn)F←→ entire functions of order ≤ 1 (i.e. . exp(τ‖ξ‖))

I f := exp(−i(·)2

)+ F(w (F)h) entire of order 2 ⇒ F well-defined

I F (h) = f · f (·) =: f · f ∗ entire ⇒ uniquely determined by F (h)|U

I Uniqueness problem accessible by theory of entire functions!



Entire Functions in 1D

Setting:

f : C→ C entire, not identically zero

Complex zeros: Zf := {aj}j∈J ⊂ C \ {0}, J ⊂ NConvergence exponent: ρf := inf{ρ ≥ 0 :

∑j∈J |aj |−ρ <∞}

Rank: pf := min{p ∈ N0 :∑

j∈J |aj |−(p+1) <∞}

Theorem 1 (Hadamard’s factorization theorem)

Let f be entire of order λf <∞. Then pf ≤ λf and

f (ξ) = ξm exp(qf (ξ))∏j∈J

Epf

(ξ

aj

)∀ ξ ∈ C (3)

with m ∈ N0, deg(qf ) ≤ λf and En(z) = (1− z) exp(∑n

j=1z j

j

).

Conversely, for any sequence Zf , polynomial qf and m ∈ N0, (3) definesan entire funtion f such that λf = max{deg(qf ), ρf }



Phase Retrieval in 1D

Hadamard Factorization of |f |2

|f |2(ξ) = f · f ∗(ξ) = ξ2m exp(2<(qf )(ξ))∏j∈J

Epf

(ξ

aj

)· Epf

(ξ

aj

)I Quantification of the information obtained by measuring |f |2

I Uniqueness theory for (Fourier-)phase retrieval of compact signals[Akutowicz, 1956, Akutowicz, 1957, Walther, 1963]

Lemma 2

Let f , f : C→ C entire s.t. λf ≤ λf <∞, |f |2|U = |f |2|U for U ⊂ R open.Then there exist entire functions f1, f2 : C→ C of order ≤ λf such that

f = f1 · f2 and f = f1 · f ∗2 . (4)

Conversely, if f1 and f2 are entire of order λ, then f and f are entirefunctions of order ≤ λ satisfying |f |2|R = |f |2|R.



Main Result

Theorem 2 (Uniqueness of phase contrast imaging for compact objects)

For w ∈ C∞(Rn) everywhere nonzero, α ∈ C \R and P0 ∈ S ′c (Rn) \ {0}define

F : S ′c (Rn)→ C∞(Rn); F (h) = |F(P0) exp(α(·)2) + F(w · h)|2 (5)

Then F is well-defined and injective. Moreover, any h ∈ S ′c (Rn) isuniquely determined by F (h)|U on an arbitrary open set U ⊂ Rn.

General Idea of the Proof:

X Well-definedness + unique extension F (h)|U 7→ F (h) by PWS-Thm

X 1D case: Show that the “factorization-construction” of alternatesolutions in Lemma 2 is incompatible with the structure of F

X Reduce case n > 1 to a family of 1D-problems



Proof of the Main Result I

Setting:

Let U ⊂ R open, h, h ∈ S ′c (R) s.t. F (h)|U = F (h)|U

Define f , f : C→ C by f (ξ) := F(P0)(ξ) exp(αξ2) + F(wh)(ξ)

Assume h 6= h

I f , f entire of order 2 satisfying |f |2|U = F (h)|U = F (h)|U = |f |2|UI By Lemma 2: ∃ order ≤ 2 entire functions f1, f2 : C→ C such that

f = f1 · f2 and f = f1 · f ∗2

I g := f1 · (f2 − f ∗2 ) = f − f = F(w · (h − h)) is entire of order ≤ 1and non-zero since h − h ∈ S ′c (R) \ {0}

I g has rank ≤ 1 by Theorem 1



Proof of the Main Result II

I Zeros of f1 contained in those of g = f1 · (f2 − f ∗2 )

I f1 of order ≤ 2 and rank ≤ 1Hadamard⇒ ∃ µ ∈ C, f0 of order ≤ 1

f1 = exp(µ(·)2) · f0 (6)

I Set γ := =(µ), f3 := exp(<(µ)(·)2) · f2 and substitute (6) into g :

g · f ∗0 = exp(iγ(·)2) · (f0 · f ∗0 ) · (f3 − f ∗3 ) = − exp(2iγ(·)2) · g∗ · f0

I rhs of order 2, lhs ≤ 1 [Boas, 2011, Ch. 3] ⇒ γ = 0

⇒ f = f0 · f3 and f = f0 · f ∗3

I Setting a := F(P0), b := F(w · h), e := exp(α(·)2), this implies

f ∗0 · (a · e + b) = f ∗0 · f = f0 · f ∗ = f0 · (a∗ · e∗ + b∗)

I e∗ = exp(α(·)2) 6= e ⇒ inconsistent lhs/rhs ⇒ Contradiction!



Conclusions

Physical Implications:

X Unique imaging of compact objects from a single diffraction pattern!

X Applicable to a large class of incident/background wave fields

X In general: Recovery of compactly perturbed paraxial wave fronts

X Relevant to QM by equivalence Schrodinger ∼ paraxial Helmholtz

Open Questions and Future Work:

Phase retrieval may be severely ill-posed → stability estimates?

Uniqueness robust under relaxation of approximations? Analogue inphaseless Helmholtz scattering? [?, ?]

Tailored regularization methods for numerical reconstructions

Maretzke, S. (2014).A uniqueness result for propagation-based phase contrast imagingfrom a single measurement. arXiv:1409.4794.



References I

Akutowicz, E. J. (1956).On the determination of the phase of a Fourier integral, i.Transactions of the American Mathematical Society, pages 179–192.

Akutowicz, E. J. (1957).On the determination of the phase of a Fourier integral, ii.Proceedings of the American Mathematical Society, 8(2):234–238.

Boas, R. P. (2011).Entire functions, volume 5.Academic Press.

Cloetens, P., Ludwig, W., Baruchel, J., Van Dyck, D., Van Landuyt,J., Guigay, J., and Schlenker, M. (1999).Holotomography: Quantitative phase tomography with micrometerresolution using hard synchrotron radiation X-rays.Applied Physics Letters, 75(19):2912–2914.



References II

Jonas, P. and Louis, A. (2004).Phase contrast tomography using holographic measurements.Inverse Problems, 20(1):75.

Nugent, K. A. (2007).X-ray noninterferometric phase imaging: a unified picture.JOSA A, 24(2):536–547.

Walther, A. (1963).The question of phase retrieval in optics.Journal of Modern Optics, 10(1):41–49.



Proof of Concept: Experimental Data Set

Intensity data δ: reconstructed slices δ: 3D contour plot

I Colloidal Crystal of 415nm polystyrene-beads

I Spherical shape and binary refractive index resolved

I β ∼ δ2500 ∼ 10−9 [Cloetens et al., 1999] → no absorption contrast!



Documents

A uniqueness result for propagation-based phase contrast ...ip.math.uni-goettingen.de/data-smaretzke/slides_ifip_2014-12_smare… · Holotomography: Quantitative phase tomography