Patent Application
20190310208
The present invention relates to X-ray tensor tomography.
X-ray tensor tomography (XTT) is an imaging modality for reconstructing three-dimensional (3D) scattering tensors from a sequence of two-dimensional (2D) dark-field projections obtained in a grating interferometer. Due to the use of anisotropic dark-field signal XTT allows to provide information about the orientations of micro-scale structures which cannot be resolved in traditional X-ray imaging techniques, such as absorption contrast X-ray tomography and differential phase contrast (DPC) X-ray tomography.
XTT allows for non-destructive testing and can be used, for example, to resolve the orientations of fibers in dense carbon fiber composites. These materials are critical to the safety, weight and performance of automobiles and airplanes. Another exemplary application of XTT is to resolve the structure in polymer materials manufactured using injection molding. In addition, XTT can be used to resolve the structures of dentinal tubules in human tooth which are not directly visible in traditional X-ray imaging and to resolve the orientations of the structures within bones which allows to study the strength mechanisms of bones.
An Example of a state of the art XTT-system 10 including an X-ray grating interferometer in a so called Talbot-Lau configuration is shown in
Such a system 10 can provide three types of projection images of the specimen: absorption contrast, differential phase contrast and dark-field contrast. All three contrast modes provide complimentary information about the inside structure of the specimen. The absorption contrast can provide information about the 3D distribution of the absorption coefficient, which may be related to the imaginary part of the refractive index, for example, and the DPC can provide information about the 3D distribution of the real part of the refractive index within a specimen. Both in X-ray absorption contrast computer tomography (CT) and in DPC CT scalar quantities, which are respectively related to the absorption coefficient and to the refractive index, are reconstructed by using reconstruction algorithms for a plurality of volume elements or voxels of the specimen.
For the case of XTT a plurality of tensor quantities are reconstructed for a plurality of associated voxels, wherein each tensor quantity corresponds to a scattering tensor providing information about the directional distribution of the scattering power at the respective voxel. The scattering tensors are reconstructed from a sequence of 2D X-ray dark-field projections. The contrast in the dark-field projection images is caused by ultra-small angle scattering of X-rays by small structures, which can have a size in the sub-micron and micron range, and thus encode information about the 3D position and the 3D orientation of the structures within the specimen. The reconstruction of full 3D scattering tensors for a specimen is known in the art and describes, for example, in the article “Constrained X-ray tensor tomography reconstruction”, J. Vogel et al., Opt. Express, vol. 23, no 12, pp. 15134-15151, 2015.
For the purpose of XTT reconstruction several dark-field projections are acquired for different orientations of the specimen with respect to the XTT-system 10, wherein for each orientation the first and second gratings 16, 18 are shifted with respect to each other into different shifting positions, for each of which a projection is acquired with the detector 20. The projections corresponding to the different shifting positions of the gratings are combined to calculate absorption, differential phase contrast and dark-field signal for each sample orientation. The orientation of the specimen (not shown) is controlled by using the specimen stage 20. In the XTT-system 10 the specimen stage 20 corresponds to a three-circle Eulerian Cradle which allows rotating the specimen
For the acquisitions the sample is positioned at a center of the Eulerian Cradle on an optical axis 30 of the MT-system 10, wherein the center is intersected by all three axes 24, 26, 28.
As shown in
The direction of the third axis of rotation 28 coincides with the y-direction, such that the third axis of rotation 28 is stationary with respect to the XTT-system 10 and always orthogonal to the second axis of rotation 26.
In case of ψ=0°, the second axis of rotation 26 coincides with the optical axis 30 of the XTT-system 10 (z-direction). In case of θ=0°, the direction of the first axis of rotation 24 (y′-direction) coincides with the y-direction.
All gratings 14, 16, 18 are pairwise parallel and therefore orthogonal to the optical axis 30. Each of the gratings 14, 16, 18 has grating lines, wherein the grating lines of each of the gratings 14, 16, 18 extend in the same direction orthogonally to the optical axis 30. A sensitivity direction 32 of the XTT-system to, which corresponds to the direction in which a phase shift of the X-rays is measured, is parallel to the planes of the gratings 14, 16, 18 and orthogonal to the grating lines. In the XTT-system 10 of
The acquisition scheme M proposed in the above mentioned article by Malecki et al. is given by:
M={(ψ,θ,ϕ);ωϵ[0°,45°,90°,135°],
θϵ[0°,45°,90°,135°],
ϕϵ[−36,67°, −36°, . . . ,36,67°]}
resulting in 1776 different acquisition poses or orientations of the specimen, wherein for each orientation the first grating 16 is scanned or shifted with respect to the second grating 18 in eight steps over one grating period in the sensitivity direction, such that 14208 projections are acquired in total.
This acquisition with the above described acquisition geometry of the XTT-system 10 of
However, the space requirements of the acquisition geometry of the XTT-system 10 of
The problem to be solved by the present invention is to provide a more robust and reliable XTT-system being capable for reconstructing sets of full 3D scattering tensors with reduced acquisition time, reduced cost, reduced complexity and with lower space requirements. This problem is solved by claim 1. Preferred embodiments are defined in the dependent claims.
The present invention relates to an X-ray tensor tomography (XTT) system, comprising
Different from XTT-systems which are known from prior art, such as the XTT-system shown in
The inventors made the surprising discovery that it is possible to reconstruct full 3D scattering tensors by using only a single first axis of rotation for orienting the specimen, instead of three axes of rotation, if the first axis of rotation lies in a plane which is tilted by an angle ψ, 0°<ψ≤90, with respect to the planes of the gratings and by an angle β, 0°<β<90°, with respect to a plane being orthogonal to the direction of the beam path. Reconstruction of “full 3D scattering tensors” means that any scattering direction and any structure orientation within the specimen can be reconstructed, such that the reconstruction is not limited to certain directions only, e.g. only to such directions which lie in a common plane. In other words, the directions provided by the reconstructed scattering tensors and indicating a local orientation of a structure may span a 3D space and not just a plane or a line. A full 3D reconstruction of scattering tensors allows detecting structures having any orientation.
The XTT-system of the present invention is based on this discovery and uses only one or two axes of rotation. This significantly reduces the complexity of the system and makes the system more robust and reliable. Further, no additional free space is required for moving components in order to rotate the specimen about an additional axis of rotation, such that components of the XTT-system around the specimen can be arranged closer together. Therefore, XTT-system of the present invention can be smaller, more compact, cheaper and less complex.
Preferably, the source is arranged for providing the beam along an optical axis of the XTT-system, wherein the second grating lines and the first grating lines are orthogonal to the optical axis. In this case, the angels ψ and β designate the same fixed angle, such that the first axis of rotation lies in a plane which is tilted by the fixed angle ψ with respect to the planes of the gratings.
In one or more embodiments the shifting direction is orthogonal to the grating lines. In other embodiments the shifting direction is not orthogonal to the grating lines but comprises a component which is orthogonal to the grating lines.
In a preferred embodiment of the XTT-system the fixed angle ψ is in a range 10°<ψ<70°, preferably 30°<ψ<50°. These ranges allow for a good compromise between the quality of the reconstructed scattering tensors and the corresponding amount of poses of the specimen in which projections are acquired.
In one or more embodiments, the stage or the MT-system is designed to rotate the specimen about a second axis of rotation. This allows to acquire projections for a larger number of different orientations of the specimen and therefore for a more precise and finer reconstruction. Compared to embodiments in which only one axis of rotation is used, these embodiments are more complex.
In the before mentioned embodiments, the second axis of rotation is preferably orthogonal to the first axis of rotation. This allows covering large range of specimen orientations and allows for an efficient orienting of the specimen.
In the before mentioned embodiments, the second axis of rotation can be stationary with respect to the source and the detector and during a rotation about the second axis of rotation by an angle Δθ the first axis of rotation gets rotated by the same angle Δθ within a plane being orthogonal to the second axis of rotation.
In alternative embodiments the stage and the XTT-system is designed to rotate the specimen only about the first axis of rotation, wherein the first axis of rotation is tilted with respect to the direction being parallel to the grating planes and orthogonal to the grating lines by an angle α, with 0°<α<180°, preferably with 20°<α<70°, more preferably with 35°<α<55°. These embodiments allow for a particular compact and robust XTT-system.
In the before mentioned embodiments the first axis of rotation is preferably stationary with respect to the source and the detector.
In one or more of the before mentioned embodiments, the first and second gratings are preferably arranged in parallel planes and/or the first and second gratings are line gratings.
The source can comprise an X-ray source and a third grating with grating lines parallel to the first and second grating lines. The third grating allows providing an X-ray beam with at least partially coherent X-rays from an X-ray beam being less coherent.
The present invention further relates to a method for reconstructing scattering tensors for a specimen by using an XTT-system according to one or more of the mentioned embodiments, wherein the sequence of two-dimensional intensity distributions is acquired for a scanning process comprising
According to an embodiment the scanning process further comprises
Preferably, the first and/or second rotational positions are evenly distributed over the respective angle range.
Further details and advantages of the present invention will become apparent from the following description, in which preferred embodiments are described in detail with reference to the appended drawings, in which:
In the drawings same elements are designated with same reference numbers.
The XTT-system 34 comprises an X-ray source 12, a source grating 14, a first grating 16, a second grating 18, a specimen stage 36 and a detector 22.
Different from the specimen stage 20 of the XTT-system 10 of
Unlike the three-circle Eulerian Cradle 20
In XTT-systems according to other embodiments of the present invention, which are not shown, ψ can be equal to 90°. These embodiments can have a specimen stage with a cradle ring having holes in order not to block the beam or can have other specimen stages without a cradle ring with does not to block the beam for the configuration of ψ=90°.
In the XTT-system 34 the specimen is preferably positioned at a center of the cradle on the optical axis 30 of the XTT-system 34, wherein in the example of
In the XTT-system 34 of
For the embodiment of
In the XTT-system 34 of
The source grating 14 is parallel to the gratings 16, 18 and has grating lines extending parallel to the grating lines of the gratings 16, 18. The source grating 14 is used provide at least partially coherent X-ray beams along the optical axis 30, for cases in which the X-rays provided by the source 12 along the optical axis 30 are not sufficiently coherent. Other embodiments may use sources being capable to provide sufficiently coherent X-rays along the optical axis 30, such that these embodiments may not comprise a source grating 14.
In case of θ=0, the direction of the first axis of rotation 24 (y′-direction) coincides with the y-direction.
The XTT-system 134 comprises a source grating 114, first and a second gratings 116, 118 being pairwise parallel to each other but—different from the gratings 14, 16, 18 of the XTT-system 34—being tilted with respect to a plane 44 being orthogonal to the direction of the optical axis 30. The grating lines of the source grating 114, the first and second gratings 116, 118 extent parallel to the plane 44. Therefore, different from the XTT-system 34, in the XTT-system 134 the sensitivity direction 132, which corresponds to the direction in which a phase shift of the X-rays is measured and which is parallel to the planes of the gratings 116, 118 and orthogonal to the grating lines, is not orthogonal to the direction of the optical axis 30 at a location of the stage 36, as shown in
In the XTT-systems 34, 134 the sensitivity direction 32, 132 respectively corresponds to a shifting direction, in which the first and the second gratings 16, 18 or 116, 118 are shifted with respect to each Other when acquiring the 2D projections which are processed to get the dark-field signal.
In other embodiments, which are not shown, the shifting direction does not correspond the sensitivity direction but has a component in the sensitivity direction.
It is noted that both in the XTT-systems 34, 134—as well as in other preferred embodiments—for the measurements except for rotating the specimen by the stage and relatively shifting the gratings preferably no other movements are performed and no other corresponding degrees of freedom or parameter variations need to be introduced, such that the XTT-systems of the present invention can be quite robust and reliable.
In the XTT-system 134 the first axis of rotation 24 lies in a plane 38 for all orientations of the specimen during a measurement and can be rotated within this plane 38 by a rotation about the second axis of rotation 26, which extends in a direction being orthogonal to the plane 38, as shown in
Further, in the XTT-system 134, the plane 38 is tilted by an angle ψ with respect to the planes of the gratings 116, 118, wherein 0°<ψ<00°. The angle ψ corresponds to the angle between the direction being orthogonal to plane 38 (corresponding to the direction of axis 26 in
XTT-systems according to other embodiments of the present invention, which are not shown, comprise only one axis of rotation 24, which is preferably stationary with respect to the x,y,z coordinate system of the respective XTT-system.
Examples for these embodiments with only one stationary axis of rotation 24 correspond to the XTT-system 34 of
Other examples for these embodiments with only one axis of rotation 24 correspond to the XTT-system 134 of
In these embodiments with only one axis of rotation 24 the single first axis of rotation 24 is tilted with respect to the sensitivity direction of the respective XTT-system by an angle larger than 0° and smaller than or equal to 90° and with respect to the direction of the X-ray beam at the location of the specimen by an angle larger than 0° and smaller than 90°.
The inventors have found that with the tilting of the first and second axis of rotation 24, 26 and with the tilting of the first axis of rotation 24 of the before mentioned XTT-systems according to the invention, it is possible to scan over a set of different orientations, which is large enough to allow for full 3D reconstruction of X-ray scattering tensors, by using only two axes of rotation and even by using one single axis of rotation.
This surprising discovery is illustrated by
B={(ψ,θ,ϕ);ψϵ[0°,20°,40°],
θϵ[0°,11,25°, . . . ,90°],
ϕϵ[0°,3,6°, . . . ,360°]}.
The acquisition scheme B comprises 2700 different poses for the specimen (=3 (for ψ)×9 (for θ)×100 (for ϕ)), with 8 grating steps in shifting direction per pose and with 5 s exposure time per grating step in each pose.
The scattering tensors which have been reconstructed for corresponding voxels of the specimen are presented by corresponding ellipsoids, wherein for each ellipsoid the distance from the center of the ellipsoid to a point on the surface of the ellipsoid indicates the relative scattering intensity or strength for the corresponding direction at the voxel location for which scattering tensor is reconstructed. Accordingly, the direction of the smallest half axis of an ellipsoid representing a scattering tensor corresponds to the direction with the lowest relative scattering strength at the corresponding location. This direction with the lowest relative scattering strength indicates an orientation of a microstructure within the specimen and is further indicated in
The specimen for which the scattering tensors of
Compared to the scattering tensors of
Different from acquisition scheme B, which comprises 2700 different poses, uses of three different axes of rotation 24, 26, 28 and needs a measurement time of 30 hours, acquisition schemes C (
Acquisition scheme C defines a scan over the following poses.
C={(ψ,θ,ϕ);ψϵ[0°,20°,40°],
θϵ[0°,11,25°, . . . ,90°],
ϕϵ[0°,10,91°, . . . ,360°]}.
Acquisition scheme D defines a scan over the following poses.
D={(ψ,θ,ϕ);ψ=0°,
θϵ[0°,11,25°, . . . ,90°],
ϕϵ[0°,3,6°, . . . ,360°]}.
Acquisition scheme E defines a scan over the following poses.
E={(ψ,θ,ϕ);ψ=0°,
θϵ[0°,11,25°, . . . ,90°],
ϕϵ[0°,3,6°, . . . ,360°]}.
The reconstruction of
The reconstructions of
The inner structure of the specimen can still be determined from the reconstructions shown in
In
The inventors have found that it is possible to achieve a full 3D scattering tensor reconstruction with sufficient quality not only with two axes of rotation but also with one single axis of rotation, if the angles are chosen to be in the before mentioned ranges.
This is demonstrated by
F={(ψ,θ,ϕ);ψ=40°,
θ=56,25°,
ϕϵ[0°,3,78°, . . . ,288°]}.
The use of only one and only two axes of rotation allows significantly reducing the complexity of XTT-systems and to make XTT-system even more robust and compact. For example, in
Tensor Reconstruction:
In order to reconstruct a set of scattering tensors a so called forward model can be used, according to which the dark-field projection d(x) measured at the detector 22 for an acquisition pose x=(ψ, θ, ϕ) is given by:
wherein ζk(ri) denotes the scattering strengths at every voxel ri, i=1, . . . , I, for a number of sampling directions ϵk, k=1, . . . , K, and wherein w(ϵk, x) are the weight factors that determine how well the scattering direction ϵk is measured at the acquisition pose x.
The weight factor w(ϵk, x) is given by
w(ϵk,x)=(|ϵk×{circumflex over (t)}(x)|ϵk,ŝ(x)
)2, (2)
wherein {circumflex over (t)}(x) is the unit vector in the direction of the optical axis 30 or the direction of the beam path 42 at the location of the specimen, respectively, and wherein ŝ(x) is the unit vector in the sensitivity direction 32, 132 at the acquisition pose x. The sampling directions ϵk are stationary with respect to the x′,y′,z′ coordinate system of the specimen and do not change within the x′,y′,z′ coordinate when the pose x of the specimen is changed, while the directions {circumflex over (t)}(x) and ŝ(x) change within the x′,y′,z′ coordinate when the pose x of the specimen is changed.
By using the above equations, the scattering strengths ζk(ri) can be reconstructed at every voxel ri, i=1, . . . , I, for each sampling direction ϵk, k=1, . . . , K. Then, an ellipsoid fit can be performed on these K sampling directions ϵk to recover the scattering tensor at every voxel ri.
From the above equation for the weight factor w(ϵk, x) it can be seen that the dark-field projection d(x) measured at a certain acquisition pose x depends on the relative orientation of the specimen sampled by sampling directions ϵk with respect to the direction {circumflex over (t)}(x) of the optical axis 30 and with respect to the sensitivity direction ŝ(x).
The inventors have found that in the proposed arrangements the specimen can be scanned through a set of acquisition poses x which still allow for a correct full 3D reconstruction of the scattering tensors, although only two or even only a single axis of rotation is used. This allows for XTT-systems which are more compact and robust and less complex.
Theoretical Model for Predicting the Efficiency of an Acquisition Scheme
In the following a theoretical model is presented which has been developed by the inventors for predicting the efficiency of an acquisition scheme:
Let Σ and T be two sets of uniformly distributed points on a unit sphere:
Σ={σkϵ2,k=1,2, . . . ,N},
={τjϵ
2,j=1,2, . . . ,N},
wherein 2 is the unit sphere and N=10000. The mean vector mΣ of Σ is given by:
Let X be an acquisition scheme using n different acquisition poses xi:
X:={x
i:=(ψi,θi,ϕi);i=1, . . . ,n}.
The objective is to define a quantity of coverage(X,k) that represents how well the orientation σkϵΣ is measured by the acquisition scheme X. The proposed procedure for computing the Coverage(X,k) is outlined in the following:
X
k
:={xϵX;w(σk,x)>Tc}.
w(σk,x)=(|σk×{circumflex over (t)}(x)|σk,ŝ(x)
)2 (3)
wherein |⋅×⋅| denotes the magnitude of the cross product and ⋅,⋅
denotes the standard scalar product.
wherein ∥⋅∥ denotes the Euclidean length of a vector.
:={τϵ
:
,{circumflex over (t)}(x)
>cos(2.5°)∀xϵXk}.
Next, the unit vectors σk ϵΣ are scaled with the values of Coverage(X,k) for all values of k resulting in:
Γ(X):={γk:=σk·Coverage(X,k),k=1, . . . ,N},
and its mean:
The angular deviation of the normalized mean vector {circumflex over (m)}Γ(X) from the normalized mean vector {circumflex over (m)}Σ is a measure of the non-uniformity of the Coverage Sphere. Finally, the proposed Coverage Metric CM(X) is introduced as:
CM(X)ϵ[−1,1] can be used as a measure of the efficiency of the acquisition scheme X. Positive and higher values of CM imply a more comprehensive and uniform measurement of all orientations on the unit sphere. Negative values, in particular, indicate severe inhomogeneities on the Coverage Sphere.
In
Accordingly, the proposed concept of the Coverage Metric allows theoretically comparing the quality of different acquisition schemes with each other and to predict which scheme provides the best quality. By an additional comparison with actual experimental results (as shown in
While specific embodiments have been described in detail, it is not intended that the scope of protection is limited by the specific embodiments. The scope of protection is defined by the appended claims.
| Number | Date | Country | Kind |
|---|---|---|---|
| 16174203.6 | Jun 2016 | EP | regional |
| Filing Document | Filing Date | Country | Kind |
|---|---|---|---|
| PCT/EP2017/064440 | 6/13/2017 | WO | 00 |
