X-ray absorption tomography employing a conical shell beam

: We demonstrate depth-resolved absorption imaging by scanning an object through a conical shell of X-rays. We measure ring shaped projections and apply tomosynthesis to extract optical sections at different axial focal plane positions. Three-dimensional objects have been imaged to validate our theoretical treatment. The novel principle of our method is scalable with respect to both scan size and X-ray energy. A driver for this work is to complement previously reported methods concerning the measurement of diffracted X-rays for structural analysis. The prospect of employing conical shell beams to combine both absorption and diffraction modalities would provide enhanced analytical utility and has many potential applications in security screening, process control and diagnostic imaging.


Introduction
Three-dimensional imaging of opaque objects using X-rays is exploited extensively in medicine, industrial inspection and security screening.In recent years, tomosynthesis has undergone a resurgence of interest in diagnostic imaging applications such as breast cancer screening [1] and the detection of pulmonary nodules [2].More advanced approaches have investigated coded apertures and multiple X-ray sources for compressive tomosynthesis to reduce radiation exposure [3].The tomographic approach enables the reconstruction and visualisation of volumetric information from a series of slice images.The tomographic principle involves summing projections from different relative locations to effect a blurring out of structures either side of a plane of interest.In general, computed axial tomography (CAT) employs in-plane one-dimensional projections, while classical tomography or tomosynthesis [4] employs two-dimensional projections over a limited angular range.In particular, the shape of flat objects can produce highly attenuated projections in directions at or near the object plane.Under such conditions, the two-dimensional point projections employed typically in tomosynthesis can provide a more complete sampling of flat objects in comparison to CAT, which in contrast requires a 360° rotation about the normal to the X-ray beam [5].
Microlaminography [6,7] has been demonstrated by employing polycapillary optics to focus radiation from laboratory X-ray sources or synchrotrons into micro-spots to form an inspection volume.For example, lamellae structures found in integrated circuits placed within the focal spot can be imaged in a similar way to classical tomography but without the requirement for sample or source rotation [6].Recent advances in the design and construction of X-ray sources [8] promises rapid switching to replace mechanical scanning with switched source alternatives.
Many applications would benefit from material composition or structural information attributed directly to spatially distributed components within an inspection volume.Recent advances in coded aperture coherent scatter tomography [9] have demonstrated the use of a fan beam of interrogating X-rays to ultimately, exploit object motion to produce a threedimensional materials discriminating imaging system.Although, significant challenges in terms of spatially complex objects and cluttered scenes, which are prevalent in real-world applications, require further investigation and development.An important and challenging example is encountered in the security screening of air passenger luggage, which routinely employs X-ray absorption imaging to help detect shape-based threats such as guns, knives and sharps.In contrast, threats including narcotics and a wide range of plastic, liquid and homemade explosives (HMEs) present highly shape-variant threats that also require material specific information for enhanced false-alarm resolution [10][11][12].Given that coherently scattered X-rays occur naturally during the radiographic process and that X-ray diffraction is the 'gold standard' for conducting materials structural discrimination it is a very attractive proposition to combine both methods to reduce potentially false-alarm rates.We have previously reported; see Section 2.1, the benefits of depth-resolved materials characterization employing X-ray diffraction tomography along with other conical shell or annular beam techniques.If depth resolved absorption and diffraction imaging [12,13] were combined successfully in a compact and cost-effective technology, then it could be deployed to increase the throughput of carry-on and checked luggage at international travel hubs.This technological development would be a major and disruptive advance for material specific security screening applications.
To complement earlier annular beam work we present a study of the absorption contrast signals propagating along the direction of the primary beam.We demonstrate our method by scanning a multi-layered object through an annular beam of X-rays.The beam projections have been collected from a corresponding ring shaped region of interest on the input window of an X-ray image intensifier optically coupled to a CCD camera.Pixels recorded at fixed circumferential positions along the ROI are composited to form a sequence of oblique views during a two axis translational scan.This image sequence encodes the position of inspected object features along the normal to the scan direction as depth dependent circular parallax.The application of tomosynthesis enables the extraction of optical sections at corresponding axial focal plane positions.This bright field capability is highly promising as it supports and informs the earlier dark-field work in the ongoing development of a new material specific high-energy X-ray imaging modality.
Our paper is organised as follows.Section 2 presents the methods including the theory background, our new imaging technique and describes the experiment conditions.Section 3 presents our experiment results and discussion.Section 4 summarizes our conclusions, discusses the implications of our findings and the future direction of the work.

Theory background
We have previously reported an alternate X-ray diffraction technique employing conical shell or annular interrogating X-ray beams, termed focal construct technology (FCT).This approach detects the presence of Bragg maxima from polycrystalline samples in the form of high intensity focal spots along the symmetry axis of the interrogating beam.This work encompassed angular dispersive methods realized by either translating the detector [14,15] or the sample [16] along the symmetry axis of the beam; the latter method employed pseudomonochromatic high energy X-rays (W-Kα) obtained via balance filtering [17].We have also demonstrated an energy dispersive approach [18] in which both the sample and detector remain stationary to enable the recovery of high-energy polychromatic focal spots from a centrally positioned point detector.These methods deal favorably with non-ideal samples such as those exhibiting large grain size, preferred orientation and liquid samples [19], which exhibit only short-range order.Separate independent studies using low X-ray energies and mono-capillaries [20] have also demonstrated the validity of the FCT method.In addition, we have demonstrated depth resolved material specific imaging by analysing caustics [21] in the diffracted X-ray intensity.Our prior work with annular beams has dealt with the analysis of coherently scattered X-rays collected from within the central dark field encompassed by the beam.This paper describes a complementary bright field absorption FCT approach.

New imaging technique
Our imaging technique employs translational scanning of an object through a conical shell beam of X-rays, see Fig. 1.Successive overlapping annular projections are measured along the x and y axes during the scan.A value of intensity on the discretised annular absorption profile ( ) is shown in Fig. 2. Intensity values measured at a fixed polar coordinate position ( ) , r γ on each annular projection A may be composited to form an oblique projection C .Thus a succession of different oblique views D may be recovered from a matrix of coplanar annular projections B , as illustrated in Fig. 3.A complete description of the pixel registration is given by , , , , Where γ Δ is the increment in angular separation between neighbouring elements v composing the annular sensor.The total number of samples or pixels composing each oblique image equates to the product , NM which is determined by the total number of discrete annular projections.Although the total number of oblique images is determined by 2 V π γ = Δ , this in practice is limited by the native resolution of the annular sensor.The conversion from a pixel separation along the x-axis or along the y-axis into linear distances is given by m x δ and , n y δ respectively.Where n and m are the total number of pixels under consideration and, , x y δ δ and are the step distances, respectively.The production of depth dependent circular parallax is recorded by the oblique projection series .D The diagram in Fig. 4 shows a focal plane position in object space together with the associated pseudo X-ray beam topology that equivalently applies when radially shifted oblique projections are considered.It can be appreciated from rotational symmetry that the magnitude of the parallax ( ) (4) requires employing shift-and-add tomosynthesis to recover the corresponding optical section.However, before presenting the digital tomosynthesis equation, Eq. ( 6), it will be helpful to consider the linear shift distance (5) It can be appreciated from this relationship, in combination with Fig. 4, that the inspection volume is bounded by two hypothetical focal planes; one is coincident with the point source plane ( ) while the other is coincident with the detection plane ( ) These conditions also bound the digitally reconstructed parameters.A digital optical section T requires that the oblique projections D be shifted along their respective radial directions via two orthogonal axial components before being summed and averaged according to where the digital shift indices j and k are given by ( ) In a practical system the value of parameters; , , , , , r L x y φ δ δ γ Δ are all known and fixed for a given scan.It can be appreciated from Eqs. ( 7) and ( 8) that a one-to-one registration or common index ( , ) m n for all pixel positions composing the oblique projections occurs at 0.

Z =
The separation between two neighboring focal plane positions, see Fig. 1, is obtained by differentiating Eq. ( 4): where p x δ δ = is the minimum detectable increment in parallax.

Experiment setup
We employed an industrial X-ray inspection system (X-tek Systems VTX 160) to provide computerised control of X-ray source/imaging parameters and scanning of the phantom in a radiological inspection chamber.A micro-focus generator with a 2 m μ spot size and a tungsten transmission target produced the X-rays.The accelerating voltage and current were 70 kV and 50 , A μ respectively.Annular projections with a half-opening angle 6.87  The X-ray absorption contrast produced by the steel material against the relatively low density plastic material provided the image forming signal for our experiments.Also, the coordinate positions of the targets were carefully measured and recorded prior to scanning and image reconstruction; see true coordinate values presented in Table 1.The phantom was raster scanned through 400 positions along the x and y axes in steps of 350 m. x y This setup provided sufficient sampling to resolve the minimum feature size of ≈ 1 mm on the steel shapes in the oblique projections and in the resultant optical sections.

Composite oblique projections
The phantom was placed within the inspection volume and was raster scanned along the xaxis and the y-axis according to the experiment conditions described in Section 2.3.The discrete annular projections A have been collected and stored as a series B ; the resultant composite oblique projections C have been reconstructed on discrete ( ) , r γ positions and stored as a sequence D in digital memory according to Eqs. ( 1)-( 3).An example of the composite images is shown in Fig. 6. = a zero parallax plane can be observed to translate through the sequence from 0 Z = to Z L = , respectively.These "stationary" planes are an analogue of the optical sections produced by the tomosynthesis analyzed in the following, Section 3.2.

Optical sections
A series of optical sections , T see Fig. 7, have been produced by applying shift-and-add tomosynthesis according to Eqs. ( 6)- (8).The linear characteristic shown in the graph of Fig. 8 is expected when employing parallel projections to reconstruct optical sections.This aspect is a unique feature of our "point source" technique, which is also highlighted by the constant (object space to image space) magnification observed between the different optical sections recorded, see Fig. 7.
These observations are counter to those expected from conventional or circular tomosynthesis i.e.where the magnification factor associated with each optical section decreases as the distance to the axial focal plane increases.A change in magnification has a concomitant effect upon the discretized sample size along the x-axis and y-axis.In addition, the separation between adjacent axial focal plane positions 2 .z z δ ∝ We did not observe any of these effects in the reconstructed , , x y z parameters recorded via annular beam tomosynthesis as supported and informed by the following analysis.
The linear regression line plotted in the graph of Fig. 8 has a gradient of 1.452 mm pixel , as predicted by Eq. ( 9).This result is consistent with the expression for differential parallax pixels, from consideration of Eq. ( 5).This digital shift value may be expressed as a distance with the aid of Eq. ( 9) as .Alternatively, the separation between optical sections along the z-axis was calculated from Eq. ( 4 The resultant error −1.3% represents <1.5 pixels error in the measurement of differential digital parallax across the full z-axis component of the phantom.This result is within the bounds of expected experimental error for digital measurements.Any other source of error such as the imperfect alignment of the source, detector and translation plane was assumed too small to be observed directly in our experiment results.Therefore, our theory accurately describes the reconstructed x, y and z parameters. The previously presented calibrated phantom, shown in Fig. 5, was designed to emphasise the details of the approach and support the analysis of the reconstructed image parameters.To complement this work we present the reconstruction results of a "real-life" sample featuring an irregular and complex morphology.The cylindrical phantom, see Fig. 9, comprises of an aggregate of objects to represent a section through a luggage item to emulate a security screening application.Again, a series of optical sections , T see Fig. 10, have been produced by applying shiftand-add tomosynthesis according to Eqs. ( 6)- (8).Fig. 10.Three optical sections of the cylindrical phantom: (left, nearest the source) a plastic Petri dish containing a disc of soap and a paperclip; (middle, around midway between the source/detector) elliptical cross-section of a plastic marker pen and the circular cross-section of the pressurised canister within a medical inhaler; (right, nearest the detector) printed circuit board (PCB) from a wristwatch.In addition, two fine gauge metal wires run the full length of the phantom.The wires can be observed as a twisted pair leading from a circular metal contact (source side) at the centre of the soap, and as two dots near the circular cross section of the pressurised canister, and finally in the rightmost image connecting to the PCB.The full series of raw optical sections at successive focal plane positions can be seen in (Visualization 2).

Fig. 1 .
Fig. 1.A conical shell beam is incident normally upon a flat detector.The X-ray source and detector form a rigid body during the scan i.e. each undergoes a relative in-plane translation with respect to the object under inspection.The resultant focal plane positions within the inspection volume are parallel with the detection plane.

Fig. 3 .
Fig. 3. (a) A sample position v at angle γ on an annular projection A ; part of a series of coplanar annular projections B collected at discrete intervals along the x,y axes.(b) Composite oblique projection C comprised of pixels , x y associated with a fixed angular position γ (constant r value); part of a series of oblique projections D collected at discrete angular intervals in γ (or annular sample positions v ).

Fig. 4 .P
Fig. 4. A focal position is shown by considering coincident annular beam rays originating from a continuum of different point source locations during a rectilinear scan.The separation between conjugate points yields depth dependent parallax ( ) r r P P − − where r P − and r P are linear distances measured with respect to reference locations at r ± , respectively.
in a given focal plane.The position is a function of the tomosynthesis half angle in the oblique image series D is related to parallax in object space by the following relationship ( Thales TH 9464 QX) 6" image intensifier with a CsI phosphor input screen.The separation between the input window of the intensifier and the point X-ray source was fixed, 443 , L mm = throughout the experiments.The intensifier was optically coupled to an AVT Manta GigE camera capturing 1280 x 1024 x 8 bit images.The corresponding annular ROI from each frame was exposed for 70 .ms To demonstrate the technique a multi-level phantom shown in Fig. 5 was fabricated to support a number of different thin 1 mm ≈ steel shapes together with a "NTU logo" in copper laminate 0.3 mm ≈ all arranged in arbitrary x,y positions upon radiotranslucent sheets.

Fig. 7 .
Fig. 7. Raw optical sections of the phantom at successive focal plane positions (Visualization 1).Out of plane object features appear increasingly blurred as a function of their axial separation from the focal plane position.

Table 1 .
Linear distances highlighted by the arrows in Fig. 7 are tabulated together with their corresponding true, and calculated values via Eq.(stated in Fig. 7, were computed from m x δ and n y δ , where m and n are the corresponding distances in pixels.Again, the nominal 1:1 scale of the optical sections T was expected due to the equal scan step size, and the focal plane position as predicted by the theoretical analysis.The true z-axis positions of the optical sections recorded in Fig. 7 are plotted against parallax, measured in pixels, in the graph of Fig. 8.

Fig. 8 .
Fig. 8. True z-axis positions of six different optical sections plotted against parallax measured in pixels.Linear extrapolation shows a nominal intercept at the origin as z-axis distances were measured with respect to the point X-ray source.
of Eq. (4), being constant and independent of the shift value applied for two arbitrary points a and b located within the inspection volume.For example, the total number of predicted focal plane positions sandwiched between the square shape (nearest the source) and the cross shape (nearest the detector) of the phantom may be calculated from the difference between their respective shift values 2

Fig. 9 .
Fig.9.Cylindrical phantom mounted on the translation table in the X-ray inspection system.An aggregate of discrete objects arranged along a ~38 cm component of the z-axis were supported by polystyrene foam.