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2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance

M.A. Beltran, D.M. Paganin, K. Uesugi, and M.J. Kitchen

Open Access Open Access

Abstract

A method of tomographic phase retrieval is developed for multi-material objects whose components each have a distinct complex refractive index. The phase-retrieval algorithm, based on the Transport-of-Intensity equation, utilizes propagation-based X-ray phase contrast images acquired at a single defocus distance for each tomographic projection. The method requires a priori knowledge of the complex refractive index for each material present in the sample, together with the total projected thickness of the object at each orientation. The requirement of only a single defocus distance per projection simplifies the experimental setup and imposes no additional dose compared to conventional tomography. The algorithm was implemented using phase contrast data acquired at the SPring-8 Synchrotron facility in Japan. The three-dimensional (3D) complex refractive index distribution of a multi-material test object was quantitatively reconstructed using a single X-ray phase-contrast image per projection. The technique is robust in the presence of noise, compared to conventional absorption based tomography.

©2010 Optical Society of America

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Figures (5)
Fig. 1
Fig. 1 Experimental setup showing the imaging geometry and coordinate system, for propagation-based phase contrast tomography of a multi-material object from a single propagation-based phase contrast image per projection.

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Fig. 2
Fig. 2 Top view of the object, used as a test object to demonstrate interface-specific phase-retrieval tomography, using a single propagation-based X-ray phase contrast image per projection.

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Fig. 3
Fig. 3 Projection images of the multi-material test sample. (a) Absorption contrast image. The numbered regions correspond to the different interfaces in the object (Air/PMMA (Green), Al/PMMA (Red) and PTFE/PMMA (Blue)). (b) Phase contrast image acquired at d=1 m. (c) Phase retrieved image of (b) using Eq. (2). (d) and (e) Phase retrieved images of (b) using Eq. (9) with appropriate values of δ and µ for the Al/PMMA and PTFE/PMMA interface, respectively (see Table 1). Line profiles from the centre of the images on the left column are shown in (f), (g), (h), (i) and (j), respectively.

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Fig. 4
Fig. 4 Tomographic reconstruction of the multi-material test sample using (a) conventional absorption contrast images reconstructed via FBP; (b) phase contrast images reconstructed by applying FBP to raw PCI data. (c) Phase contrast images reconstructed using Eq. (2) to calculate T 1 ( r ⊥ , θ ) for each projection ( j = 1 ), followed by FBP. (d) and (e) are reconstructions using same procedure as (c), after calculating T 2 ( r ⊥ , θ ) and T 3 ( r ⊥ , θ ) with Eq. (9) for both Al and PTFE for each projection ( j = 2 and j = 3 ). (f), (g), (h), (i) and (j) respectively, show line profiles from left-to-right across the centre of the images in (a), (b), (c), (d) and (e). Numbered regions correspond to different interfaces in the object (Air/PMMA (Green), Al/PMMA (Red) and PTFE/PMMA (Blue)).

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Fig. 5
Fig. 5 (a) Spliced tomographic reconstruction. (b) Line profile of the different interfaces in (a).

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Tables (1)

Tables Icon

Table 1 Values of δ and µ at 24 keV for materials used to construct the test object in Fig. 2. Values obtained from http://henke.lbl.gov/optical_constants/ (accessed Nov. 4, 2009).

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Equations (13)

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∇ ⊥ ⋅ [ I ( r ⊥ , z ) ∇ ⊥ φ ( r ⊥ , z ) ] = − k ∂ z I ( r ⊥ , z ) .
T 1 ( r ⊥ ) = − 1 μ 1 log e ( F − 1 { 1 ( d δ 1 / μ 1 ) k ⊥ 2 + 1 F [ I ( r ⊥ , z = d ) I 0 ] } ) .
A ( r ⊥ ) = T j ( r ⊥ ) + T 1 ( r ⊥ ) .
I ( r ⊥ , z = 0 ) = I 0 exp [ − μ j T j ( r ⊥ ) − μ 1 T 1 ( r ⊥ ) ] ,
φ ( r ⊥ , z = 0 ) = − k [ δ j T j ( r ⊥ ) + δ 1 T 1 ( r ⊥ ) ] .
( δ j − δ 1 ) I 0 exp [ − μ 1 A ( r ⊥ ) ] ∇ ⊥ ⋅ { exp [ − ( μ j − μ 1 ) T j ( r ⊥ ) ] ∇ ⊥ T j ( r ⊥ ) } = ∂ z I ( r ⊥ , z ) .
∇ ⊥ ⋅ { exp [ − ( μ j − μ 1 ) T j ( r ⊥ ) ] ∇ ⊥ T j ( r ⊥ ) } = − 1 μ j − μ 1 ∇ ⊥ 2 exp [ − ( μ j − μ 1 ) T j ( r ⊥ ) ] ,
[ − d ( δ j − δ 1 ) ( μ j − μ 1 ) ∇ ⊥ 2 + 1 ] exp [ − ( μ j − μ 1 ) T j ( r ⊥ ) ] = I ( r ⊥ , z = d ) I 0 exp [ − μ 1 A ( r ⊥ ) ] .
T j ( r ⊥ ) = − 1 μ j − μ 1 log e ( F − 1 { 1 [ d ( δ j − δ 1 ) / ( μ j − μ 1 ) ] k ⊥ 2 + 1 F [ I ( r ⊥ , z = d ) I 0 exp [ − μ 1 A ( r ⊥ ) ] ] } ) .
1 α k ⊥ 2 + 1 = 1 ε k ⊥ 2 + 1 × ε k ⊥ 2 + 1 α k ⊥ 2 + 1 .
Δ x   Δ k ≥ 1 ,
Δ x ≥ α .
3 α ≥ Δ x ≥ 5 α .
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