Evaluation of phase retrieval approaches in magnified X-ray phase nano computerized tomography applied to bone tissue

Boliang Yu; Loriane Weber; Alexandra Pacureanu; Max Langer; Cecile Olivier; Peter Cloetens; Françoise Peyrin

doi:10.1364/OE.26.011110

Optics Express
Vol. 26,
Issue 9,
pp. 11110-11124
(2018)
•https://doi.org/10.1364/OE.26.011110

Evaluation of phase retrieval approaches in magnified X-ray phase nano computerized tomography applied to bone tissue

Boliang Yu, Loriane Weber, Alexandra Pacureanu, Max Langer, Cecile Olivier, Peter Cloetens, and Françoise Peyrin

Open Access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Boliang Yu, Loriane Weber, Alexandra Pacureanu, Max Langer, Cecile Olivier, Peter Cloetens, and Françoise Peyrin, "Evaluation of phase retrieval approaches in magnified X-ray phase nano computerized tomography applied to bone tissue," Opt. Express 26, 11110-11124 (2018)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Check for updates

Related Topics
Table of Contents Category
- Image Processing
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: January 5, 2018
- Revised Manuscript: March 2, 2018
- Manuscript Accepted: March 5, 2018
- Published: April 16, 2018
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 13, Iss. 6

Abstract

X-ray phase contrast imaging offers higher sensitivity compared to conventional X-ray attenuation imaging and can be simply implemented by propagation when using a partially coherent synchrotron beam. We address the phase retrieval in in-line phase nano-CT using multiple propagation distances. We derive a method which extends Paganin’s single distance method and compare it to the contrast transfer function (CTF) approach in the case of a homogeneous object. The methods are applied to phase nano-CT data acquired at the voxel size of 30 nm (ID16A, ESRF, Grenoble, France). Our results show a gain in image quality in terms of the signal-to-noise ratio and spatial resolution when using four distances instead of one. The extended Paganin’s method followed by an iterative refinement step provides the best reconstructions while the homogeneous CTF method delivers quasi comparable results for our data, even without refinement step.

Full Article | PDF Article

More Like This

Soft tissue 3D imaging in the lab through optimised propagation-based phase contrast computed tomography

Berit Zeller-Plumhoff, Joshua L. Mead, Deck Tan, Tiina Roose, Geraldine F. Clough, Richard P. Boardman, and Philipp Schneider
Opt. Express 25(26) 33451-33468 (2017)

Fast algorithms for nonlinear and constrained phase retrieval in near-field X-ray holography based on Tikhonov regularization

Simon Huhn, Leon Merten Lohse, Jens Lucht, and Tim Salditt
Opt. Express 30(18) 32871-32886 (2022)

Comparison of quantitative multi-material phase-retrieval algorithms in propagation-based phase-contrast X-ray tomography

Ilian Häggmark, William Vågberg, Hans M. Hertz, and Anna Burvall
Opt. Express 25(26) 33543-33558 (2017)

Previous Article Next Article

Supplementary Material (8)

Name	Description
Visualization 1	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the homogeneous CTF method using 1 distance without iterative refinement
Visualization 2	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the homogeneous CTF method using 1 distance with 10 iterations' refinement
Visualization 3	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the homogeneous CTF method using 4 distances without iterative refinement
Visualization 4	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the homogeneous CTF method using 4 distances with 10 iterations' refinement
Visualization 5	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the extended Paganin’s method using 1 distance without iterative refinement
Visualization 6	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the extended Paganin’s method using 1 distance with 10 iterations' refinement
Visualization 7	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the extended Paganin’s method using 4 distances without iterative refinement
Visualization 8	100 slices in the middle of reconstructed volumes at 30 nm voxel size retrieved by the extended Paganin’s method using 4 distances with 10 iterations' refinement

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Fig. 1 Scheme of experimental setup of magnified X-ray phase nano-CT.

Z_{1}

Z_{2}

Z_{3}

and

Z_{4}

are 4 different focus-to-sample distances,

Z_{d}

the focus-to-detector distance.

Download Full Size | PDF

Fig. 2 Plots of the filters in the Fourier domain. (a) The filters of both the homogeneous CTF method and Paganin’s method using a single distance; (c) the filters of the homogeneous CTF method using 4 different propagation distances; (e) the filters of the extended Paganin’s method using 4 different propagation distances; (b), (d) and (f) zoom on the filters corresponding to (a), (c) and (e) respectively.

Download Full Size | PDF

Fig. 3 Minimum Intensity Projections of reconstructed volumes at 30 nm voxel size retrieved by the homogeneous CTF method. (a), (b), (c) phase retrieved without iterative refinement using 1, 2 and 4 distances respectively; (d), (e), (f) phase retrieved with 10 iterations’ refinement using 1, 2 and 4 distances respectively (see Visualization 1, Visualization 2, Visualization 3, and Visualization 4).

Download Full Size | PDF

Fig. 4 Minimum Intensity Projections of reconstructed volumes at 30 nm voxel size retrieved by the extended Paganin’s method. (a), (b), (c) phase retrieved without iterative refinement using 1, 2 and 4 distances respectively; (d), (e), (f) phase retrieved with 10 iterations’ refinement using 1, 2 and 4 distances respectively (see Visualization 5, Visualization 6, Visualization 7, and Visualization 8).

Download Full Size | PDF

Fig. 5 Quantitative evaluation of SNR and spatial resolution of the reconstructions at 30 nm voxel size for both homogeneous CTF and Paganin’s methods followed by a refinement with 0 iteration or 10 iterations, using 1, 2 or 4 distances. (a) SNR; (b) spatial resolution. Blue: 1 distance, Red: 2 distances, Green: 4 distances.

Download Full Size | PDF

Tables (2)

Table 1 Specific experimental parameters.

View Table | View all tables in this article

Table 2 Measurements of SNR and estimation of the spatial resolution in the reconstructed images for different phase retrieval methods.

View Table | View all tables in this article

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

n (x, y, z) = 1 - δ_{n} (x, y, z) + i β (x, y, z),

\begin{matrix} T (x) = a (x) \exp [i φ (x)] = \exp [- B (x) + i φ (x)] \\ B (x) = (2 π / λ) \int β (x, y, z) d z \\ φ (x) = - (2 π / λ) \int δ_{n} (x, y, z) d z \end{matrix},

\begin{matrix} I_{D_{k}} (x) = {| T (x) * P_{D_{k}} (x) |}^{2} \\ P_{D_{k}} (x) = \frac{1}{i λ D_{k}} \exp (i \frac{π}{λ D_{k}} {| x |}^{2}) \end{matrix},

φ (x) = \frac{1}{2} \cdot \frac{δ_{n}}{β} \cdot \ln (F^{- 1} {\frac{F (I_{D_{k}} / I_{I n c}) (f)}{1 + λ D_{k} π \frac{δ_{n}}{β} {‖ f ‖}^{2}}} (x)),

T (x) \approx 1 - B (x) + i φ (x) .

{\tilde{I}}_{D_{k}} (f) = δ_{D i r a c} (f) - 2 \cos (π λ D_{k} {‖ f ‖}^{2}) \tilde{B} (f) + 2 \sin (π λ D_{k} {‖ f ‖}^{2}) \tilde{φ} (f),

B (x) = \frac{2 π}{λ} \frac{1}{δ_{n} / β} \int δ_{n} (x, y, z) d z = - \frac{1}{δ_{n} / β} φ (x) .

\tilde{φ} (f) = \frac{1}{2} \cdot \frac{δ_{n}}{β} \cdot \frac{{\tilde{I}}_{D_{k}} (f) - δ_{D i r a c} (f)}{\cos (π λ D_{k} {‖ f ‖}^{2}) + \frac{δ_{n}}{β} \sin (π λ D_{k} {‖ f ‖}^{2})} .

\begin{matrix} \begin{matrix} {\tilde{I}}_{n o r m, k} (f) = F (\frac{I_{D_{k}}}{I_{I n c}}) (f), & {\tilde{H}}_{k} (f) = 1 + \frac{D_{k} δ_{n} λ π}{β} {‖ f ‖}^{2} \end{matrix} \\ {\tilde{T}}_{1} (f) = F {\exp (\frac{2 β}{δ_{n}} φ (x))} (f) \end{matrix} .

{\tilde{I}}_{n o r m, k} (f) = {\tilde{T}}_{1} (f) {\tilde{H}}_{k} (f) .

\min ε = \sum_{k = 1}^{K} {| {\tilde{T}}_{1} (f) {\tilde{H}}_{k} (f) - {\tilde{I}}_{n o r m, k} (f) |}^{2} + α {| {\tilde{T}}_{1} (f) |}^{2} .

{\hat{\tilde{T}}}_{1} (f) = \frac{\frac{1}{K} \sum_{k = 1}^{K} {\tilde{H}}_{k} (f) \cdot {\tilde{I}}_{n o r m, k} (f)}{\frac{1}{K} \sum_{k = 1}^{K} {\tilde{H}}_{k} {(f)}^{2} + α} .

\hat{φ} (x) = \frac{1}{2} \cdot \frac{δ_{n}}{β} \cdot \ln (F^{- 1} {\frac{\frac{1}{K} \sum_{k = 1}^{K} {\tilde{H}}_{k} (f) \cdot {\tilde{I}}_{n o r m, k} (f)}{\frac{1}{K} \sum_{k = 1}^{K} {\tilde{H}}_{k} {(f)}^{2} + α}} (x)) .

{\tilde{G}}_{k} (f) = \cos (π λ D_{k} {‖ f ‖}^{2}) + \frac{δ_{n}}{β} \sin (π λ D_{k} {‖ f ‖}^{2}) .

{\tilde{I}}_{D_{k}} (f) = δ_{D i r a c} (f) + 2 {\tilde{G}}_{k} (f) \tilde{φ} (f) .

\hat{\tilde{φ}} (f) = \frac{1}{2} \cdot \frac{δ_{n}}{β} \cdot \frac{\frac{1}{K} \sum_{k = 1}^{K} {\tilde{G}}_{k} (f) \cdot ({\tilde{I}}_{D_{k}} (f) - δ_{D i r a c} (f))}{\frac{1}{K} \sum_{k = 1}^{K} {\tilde{G}}_{k} {(f)}^{2} + α} .

T_{1} (x) = \exp (\frac{2}{δ_{n} / β} φ (x)) \approx 1 + \frac{2}{δ_{n} / β} φ (x) .

{\tilde{T}}_{1} (f) \approx δ_{D i r a c} (f) + \frac{2}{δ_{n} / β} \tilde{φ} (f) .

δ_{D i r a c} (f) + \frac{2}{δ_{n} / β} \tilde{φ} (f) = \frac{{\tilde{I}}_{n o r m, k} (f)}{1 + λ D_{k} π \frac{δ_{n}}{β} {‖ f ‖}^{2}} .

\tilde{φ} (f) \approx \frac{1}{2} \cdot \frac{δ_{n}}{β} \cdot \frac{{\tilde{I}}_{n o r m, k} (f) - δ_{D i r a c} (f)}{1 + λ D_{k} π \frac{δ_{n}}{β} {‖ f ‖}^{2}} .

{\tilde{G}}_{k} (f) = \cos (π λ D_{k} {‖ f ‖}^{2}) + \frac{δ_{n}}{β} \sin (π λ D_{k} {‖ f ‖}^{2}) \approx 1 + λ D_{k} π \frac{δ_{n}}{β} {‖ f ‖}^{2} .

\hat{\tilde{φ}} (f) = \frac{1}{2} \cdot \frac{δ_{n}}{β} \cdot \frac{\frac{1}{K} \sum_{k = 1}^{K} [\cos (π λ D_{k} {‖ f ‖}^{2}) + \frac{δ_{n}}{β} \sin (π λ D_{k} {‖ f ‖}^{2})] \cdot ({\tilde{I}}_{D_{k}} (f) - δ_{D i r a c} (f))}{\frac{1}{K} \sum_{k = 1}^{K} {[\cos (π λ D_{k} {‖ f ‖}^{2}) + \frac{δ_{n}}{β} \sin (π λ D_{k} {‖ f ‖}^{2})]}^{2} + α},

\hat{\tilde{φ}} (f) = \frac{1}{2} \cdot \frac{δ_{n}}{β} \cdot \frac{\frac{1}{K} \sum_{k = 1}^{K} (1 + λ D_{k} π \frac{δ_{n}}{β} {‖ f ‖}^{2}) \cdot ({\tilde{I}}_{n o r m, k} (f) - δ_{D i r a c} (f))}{\frac{1}{K} \sum_{k = 1}^{K} {(1 + λ D_{k} π \frac{δ_{n}}{β} {‖ f ‖}^{2})}^{2} + α} .

M_{k} = Z_{d} / Z_{k} .

D_{k} = [Z_{k} \cdot (Z_{d} - Z_{k})] / Z_{d} .

F_{k} = A^{2} / λ D_{k},

r = \frac{π w}{4} \cdot \frac{1}{\sqrt{\log 2 \cdot \log (1 / a)}},

Focus-to-sample distances (mm)	Focus-to-detector distance (m)	Wavelength (nm)	Fresnel numbers for lacunae	Fresnel numbers for canaliculi	Voxel size (nm)
50.54; 52.70;	1.2648	0.0369	55.86; 53.66;	0.0223; 0.0215;	30.00; 31.29;
61.37; 79.37	1.2648	0.0369	46.41; 36.43	0.0186; 0.0146	36.43; 47.12

Distance number	Iteration	CTF		Paganin
Distance number	Iteration	SNR	Resolution (nm)	SNR	Resolution (nm)
1	0	5.44 ± 1.00	81.12 ± 4.84	7.89 ± 0.79	108.18 ± 8.14
1	10	4.73 ± 0.99	77.66 ± 6.57	5.29 ± 1.06	77.51 ± 4.84
2	0	6.27 ± 0.77	78.76 ± 3.45	8.29 ± 0.38	120.36 ± 9.60
2	10	5.41 ± 0.89	76.35 ± 3.19	6.55 ± 0.97	74.96 ± 4.53
4	0	7.13 ± 0.84	76.06 ± 6.02	8.32 ± 0.50	115.96 ± 8.39
4	10	7.11 ± 1.16	71.71 ± 2.62	8.33 ± 0.94	71.50 ± 3.21

Focus-to-sample distances (mm)	Focus-to-detector distance (m)	Wavelength (nm)	Fresnel numbers for lacunae	Fresnel numbers for canaliculi	Voxel size (nm)
50.54; 52.70;	1.2648	0.0369	55.86; 53.66;	0.0223; 0.0215;	30.00; 31.29;
61.37; 79.37	1.2648	0.0369	46.41; 36.43	0.0186; 0.0146	36.43; 47.12

Distance number	Iteration	CTF		Paganin
Distance number	Iteration	SNR	Resolution (nm)	SNR	Resolution (nm)
1	0	5.44 ± 1.00	81.12 ± 4.84	7.89 ± 0.79	108.18 ± 8.14
1	10	4.73 ± 0.99	77.66 ± 6.57	5.29 ± 1.06	77.51 ± 4.84
2	0	6.27 ± 0.77	78.76 ± 3.45	8.29 ± 0.38	120.36 ± 9.60
2	10	5.41 ± 0.89	76.35 ± 3.19	6.55 ± 0.97	74.96 ± 4.53
4	0	7.13 ± 0.84	76.06 ± 6.02	8.32 ± 0.50	115.96 ± 8.39
4	10	7.11 ± 1.16	71.71 ± 2.62	8.33 ± 0.94	71.50 ± 3.21

Abstract

Supplementary Material (8)

Cited By

Figures (5)

Tables (2)

Equations (27)

Optics Express