A uniqueness result for propagation-based phase contrast 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
14
Embed
A uniqueness result for propagation-based phase contrast ...ip.math.uni-goettingen.de/data-smaretzke/slides_ifip_2014-12_smare… · Holotomography: Quantitative phase tomography
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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 }
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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.
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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!
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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.
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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.
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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.
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement
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!
S. Maretzke CRC 755 - Nanoscale Photonic Imaging
A uniqueness result for propagation-based phase contrast imaging from a single measurement