Variable zoom X-ray computed tomography method for composite materials

Abstract

The variable zoom X-ray CT method significantly improves the resolution of structures with large in-plane dimensions, enabling the detection of complex structural damage due to low-velocity impacts, for example, in large, thin composite laminate panels. The variable zoom method involves emitting an X-ray beam from an X-ray source to project a region of interest (ROI) of a specimen within a field of view (FOV) onto a detector. The detector can scan the projection of the ROI while rotating the specimen around a rotation axis of a specimen stage and translating the specimen stage along an acquisition trajectory between the X-ray source and the detector. The acquisition trajectory specifies the source-to-object distance (SOD) between the X-ray source and the specimen stage rotation axis at each rotation angle of the specimen stage. A reconstruction computer reconstructs a three-dimensional volume of the specimen from the projections scanned by the detector.

Description

[Technical field]

[0001] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH This invention was made with Government support under Grant No. W911NF-17-2-0195 awarded by the Army Research Laboratory. The Government has certain rights in this invention. [Background technology]

[0002] The complexity of critical manufacturing defects and structural damage modes in polymer composites requires high-resolution nondestructive characterization capable of accurately measuring the location and geometry of such defects in three dimensions with sufficient objectivity for interpretation free of human error. X-ray computed tomography (CT) has demonstrated its unprecedented objectivity for nondestructive testing of polymer composite structures. However, high-resolution nondestructive characterization of small critical defects in composites with large in-plane dimensions poses a fundamental challenge for CT techniques.

[0003] Composite aircraft are beginning to dominate the commercial aircraft market with the production of the Boeing 787 and Airbus 350, and are expected to be upgrades to legacy systems. In the US Department of Defense application context, Lockheed Martin has received an order to manufacture over 3,000 F-35 aircraft. Similarly, the US Army and rotorcraft industry are facing the challenge of a future vertical take-off and landing aircraft aviation that will replace over 6,300 vertical take-off and landing aircraft [1]. Advanced fiber-reinforced polymer matrix composites are playing a major role in designing high performance and lightweight aircraft structures. However, uncertainties remain for the useful life of composite rotor and airframe structures due to the complexity of damage and failure mechanisms. In addition, the susceptibility of composites to manufacturing variations that may be precursors to structural damage imposes risks that require targeted structural diagnostics for risk mitigation [1-3].

[0004] To advance composites qualification and aircraft structural assurance, analysis must capture the complexity and variability of manufacturing of mission-critical flight components and structures. It is noteworthy that the fidelity of non-destructive testing (NDI) required to quantify even the smallest defects that impact structural performance is critical for structural diagnostics of composite parts [1]. However, the fidelity of NDI required to characterize such defects in manufactured composite parts is still insufficient. Small individual defects present at critical locations can dramatically affect the strength and durability of composite structures [2-3]. Challenges in composites include their sensitivity to manufacturing variability in the form of fiber waviness and voids at ply interfaces that reduce residual strength and fatigue behavior, as well as the complexity of structural damage modes and fatigue mechanisms [4-6]. Composite integrity assessment depends on accurate detection of structural defect locations and voids at ply interfaces, whose size reduces the residual strength and subsequent fatigue behavior, and leads to estimates of the complexity of structural or impact damage. Another challenge is the determination of the structural integrity of composite elements, which depends on the accurate detection of structural defect location and size, which leads to an estimation of the residual strength after fatigue or impact damage [7-8]. Due to the three-dimensional (3D) nature of critical defects and damage modes, high-resolution non-destructive methods capable of capturing the location and geometry / shape of individual defects that impact structural performance are required.

[0005] X-ray computed tomography (CT) has demonstrated its unprecedented objectivity for NDI of polymer composites [9-11]. However, the large in-plane dimensions relative to thickness of structures typical of composite aircraft pose a challenge in achieving sufficient resolution required to accurately identify critical defects and damage modes. Modern industrial microfocus X-ray CT tomography systems are based on cone-beam geometry, which uses a directional X-ray source that emits a collimated conical X-ray beam (e.g., 30 degrees wide) and projects the object onto a flat-panel detector, typically 8-16 in. (20-40 cm) in size. This setup allows the CT system to use a geometric magnification of the object depending on how close the object is located to the source

[12] .

[0006] Impact damage has been the target of many studies with X-ray CT methods due to the advantages of these studies in 3D imaging of damage topology. Reference

[13] used the objectivity of X-ray CT data at all sections to reconstruct layer-by-layer damage at curved laminate interfaces. To mitigate the issues of the large aspect ratio of impacting panels, such as artifacts and noise associated with incomplete data acquisition due to the nearly planar subject size, a variety of X-ray CT methods have been proposed, including dual-energy CT, synchrotron radiation sources, and computational laminography [14-15].

[0007] Despite the advantages of CT, high-resolution 3D reconstruction of small defects in objects with large in-plane dimensions remains a fundamental challenge for X-ray CT-based NDI. Current microfocus CT techniques are based on full-body scanning (currently 360° around the object, or at least 180° plus the cone beam angle

[12] ), which limits the applicability of this technique to small cross sections. Moreover, even objects that can be scanned with existing microCT equipment may not allow sufficient magnification of composite structures if the inspection object is placed too close to the X-ray tube due to resolution requirements such that a full-body scan cannot be completed.

[0008] 1A-1B illustrate the object size limitation. For example, FIG. 1A shows a test object 102 having a large in-plane dimension relative to the thickness. Rotation of the test object 102, shown by the dotted line, results in a collision between the test object 102 and the detector 104. Thus, the test object 102 can only rotate over a limited range of projection angles. Reconstructions using a limited range of projection angles less than 180° in today's available industrial systems quickly lose objectivity and often make errors during biased inspections, resulting in inconsistent 3D reconstructions associated with missing projections due to partial access. In another example, FIG. 1B shows a test object 106 placed too close to an x-ray tube 108. Rotation of the test object 106, shown by the dotted line, results in a collision between the test object 106 and the x-ray tube 108. Thus, the test object 106 may not be scanned at the desired magnification. Summary of the Invention

[0009] A first aspect of the present disclosure provides a variable zoom method for an X-ray computed tomography (CT) scanner. The method includes emitting an X-ray beam from an X-ray source to project a region of interest (ROI) of a specimen within a field of view (FOV) onto a detector. The method includes scanning the projection of the ROI of the specimen with the detector while rotating the specimen about a rotation axis of a specimen stage and translating the specimen stage along an acquisition trajectory between the X-ray source and the detector. The method includes reconstructing, by a reconstruction computer, a three-dimensional volume of the specimen from the projections scanned by the detector.

[0010] In some implementations of the first aspect of the present disclosure, the X-ray source and detector remain stationary while the specimen is rotated and translated.

[0011] In some implementations of the first aspect of the present disclosure, the ROI is projected onto a central area of ​​the detector.

[0012] In some implementations of the first aspect of the present disclosure, the acquisition trajectory identifies a source-to-object distance (SOD) between the X-ray source and the axis of rotation of the specimen stage at each rotation angle of the specimen stage.

[0013] In some implementations of the first aspect of the present disclosure, the acquisition trajectory translates the axis of rotation of the specimen stage along the center of the FOV.

[0014] In some implementations of the first aspect of the present disclosure, the initial SOD along the acquisition trajectory is SOD ROI And SOD ROI is the closest SOD whose ROI is completely within the FOV.

[0015] In some implementations of the first aspect of the present disclosure, the SOD is activated while the rotation angle of the specimen stage is less than a threshold angle. ROI is the closest SOD where the ROI remains in the FOV.

[0016] In some implementations of the first aspect of the present disclosure, the SOD at each rotation angle of the specimen stage is:

number

[0017] In some implementations of the first aspect of the present disclosure, while the rotation angle of the specimen stage is less than the threshold angle, SOD(θ)=SOD ROI It is.

[0018] In some implementations of the first aspect of the present disclosure, reconstructing the three-dimensional volume includes weighting a backprojection of the set of filtered radiographs with a weighting factor based on the SOD at each rotation angle of the specimen stage.

[0019] In some implementations of the first aspect of the present disclosure, the weighting coefficients include:

number

[0020] In some implementations of the first aspect of the present disclosure, reconstructing the three-dimensional volume further includes calculating a projection onto volume transform for each projection angle and SOD to generate a backprojection of the set of filtered radiographs. Reconstructing the three-dimensional volume further includes adding weighted backprojected pixel values ​​to voxels of the three-dimensional volume based on an interpolation method to generate a reconstruction of the three-dimensional volume.

[0021] In some implementations of the first aspect of the present disclosure, reconstructing the three-dimensional volume further includes computing a frequency-domain ramp filter, computing weighted and filtered radiographs based on the ramp filter, and applying a periodic-smooth decomposition to generate a set of filtered radiographs.

[0022] In some implementations of the first aspect of the present disclosure, computing the frequency domain ramp filter includes computing a one-dimensional direct Fourier transform for:

number

number

number

[0023] In some implementations of the first aspect of the present disclosure, when calculating the projection onto the volume transform, the projection coordinates are different for each projection angle as SOD(θ) is varied.

[0024] In some implementations of the first aspect of the present disclosure, calculating the projection onto the volume transformation includes: 3D coordinate transformation (x,y,z) T =R(θ)R V (t,s,r) T where (t,s,r) are the reconstructed volume coordinates, (x,y,z) are the projection coordinates, and R V is the volume transformation matrix, R θ is the sample rotation matrix.

[0025] In some implementations of the first aspect of the present disclosure, the interpolation method is a distance-driven approach or a separable footprint approach.

[0026] In some implementations of the first aspect of the present disclosure, the weighted and filtered radiograph is weighted to account for different ray lengths of the cone x-ray beam.

[0027] In some implementations of the first aspect of the present disclosure, calculating the weighted and filtered radiograph and applying the cyclically smooth decomposition includes calculating: S θ (x,y k )=[P' θ (x,y k )*h(x)]=p x IFFT{FFTP' θ (x,y k ) ZP ·FFTh[np x ] shift}, and

number

number

[0028] In some implementations of the first aspect of the present disclosure, adding the weighted backprojected pixel values ​​to the voxels of the three-dimensional volume based on the interpolation method includes calculating:

number

number

[0029] These and other features will be more clearly understood from the following detailed description taken in conjunction with the accompanying drawings and claims. [Brief description of the drawings]

[0030] For a more complete understanding of the present disclosure, reference is now made to the following brief description, taken in conjunction with the accompanying drawings and detailed description, in which like reference characters represent like parts.

[0031] [Figure 1A] FIG. 1 is a simplified diagram of a CT system having a test piece that is too large for a full rotation. [Figure 1B] FIG. 1 is a simplified diagram of a CT system having a specimen placed too close to the x-ray tube for the selected magnification. [Figure 2A] FIG. 1 is a simplified schematic diagram of a top view of a CT system showing steps of an acquisition trajectory that includes both rotation and translation of the test piece for implementing some embodiments of the present disclosure. [Figure 2B] FIG. 1 is a simplified schematic diagram of a top view of a CT system showing steps of an acquisition trajectory that includes both rotation and translation of the test piece for implementing some embodiments of the present disclosure. [Figure 2C] FIG. 1 is a simplified schematic diagram of a top view of a CT system showing steps of an acquisition trajectory that includes both rotation and translation of the test piece for implementing some embodiments of the present disclosure. [Figure 2D] FIG. 1 is a simplified schematic diagram of a top view of a CT system showing steps of an acquisition trajectory that includes both rotation and translation of the test piece for implementing some embodiments of the present disclosure. [Figure 2E] FIG. 1 is a simplified schematic diagram of a top view of a CT system showing steps of an acquisition trajectory that includes both rotation and translation of the test piece for implementing some embodiments of the present disclosure. [Figure 2F] FIG. 1 is a simplified schematic diagram of a top view of a CT system showing steps of an acquisition trajectory that includes both rotation and translation of the test piece for implementing some embodiments of the present disclosure. [Diagram 3] FIG. 1 shows radiometric plots of source-to-object distance (SOD) for conventional, variable zoom, and limited angle scan trajectories. [Figure 4] FIG. 1 is a system diagram of a variable zoom CT system. [Diagram 5] 4A-4D are radial diagrams of SOD for conventional, variable zoom, and limited angle scanning trajectories according to a first embodiment. [Figure 6A] 3D reconstructed volumes of carbon / epoxy for conventional, limited angle, and variable zoom techniques, respectively. [Figure 6B] 3D reconstructed volumes of carbon / epoxy for conventional, limited angle, and variable zoom techniques, respectively. [Figure 6C] 3D reconstructed volumes of carbon / epoxy for conventional, limited angle, and variable zoom techniques, respectively. [Figure 7A] 13A-13C are reconstructed slices in the through-thickness direction for conventional scanning technique, variable zoom technique, and limited angle acquisition. [Figure 7B] 13A-13C are reconstructed slices in the through-thickness direction for conventional scanning technique, variable zoom technique, and limited angle acquisition. [Figure 7C] 13A-13C are reconstructed slices in the through-thickness direction for conventional scanning technique, variable zoom technique, and limited angle acquisition. [Figure 7D] 13A-13C are reconstructed slices in the through-thickness direction for conventional scanning technique, variable zoom technique, and limited angle acquisition. [Figure 7E] 13A-13C are reconstructed slices in the through-thickness direction for conventional scanning technique, variable zoom technique, and limited angle acquisition. [Figure 7F] 13A-13C are reconstructed slices in the through-thickness direction for conventional scanning technique, variable zoom technique, and limited angle acquisition. [Figure 8A] 13A-13C are reconstructed slices across the thickness for conventional scan, variable zoom, and limited angle techniques. [Figure 8B] 13A-13C are reconstructed slices across the thickness for conventional scan, variable zoom, and limited angle techniques. [Figure 8C] 13A-13C are reconstructed slices across the thickness for conventional scan, variable zoom, and limited angle techniques. [Figure 9] FIG. 1 is a system diagram of a variable zoom CT system. [Figure 10] 1A-1C are radial diagrams of schematic trajectories of the SOD for conventional, variable zoom, and limited angle scanning trajectories. [Figure 11A] 13 illustrates 3D reconstructed volumes of a hybrid composite laminate panel obtained by conventional and variable zoom scanning trajectories. [Figure 11B] 13 illustrates 3D reconstructed volumes of a hybrid composite laminate panel obtained by conventional and variable zoom scanning trajectories. [Figure 12A] Reconstructed slices across and through the thickness for conventional scanning, variable zoom, and limited angle acquisition. [Figure 12B] Reconstructed slices across and through the thickness for conventional scanning, variable zoom, and limited angle acquisition. [Figure 12C] Reconstructed slices across and through the thickness for conventional scanning, variable zoom, and limited angle acquisition. [Figure 12D] Reconstructed slices across and through the thickness for conventional scanning, variable zoom, and limited angle acquisition. [Figure 12E] Reconstructed slices across and through the thickness for conventional scanning, variable zoom, and limited angle acquisition. [Figure 12F] Reconstructed slices across and through the thickness for conventional scanning, variable zoom, and limited angle acquisition. [Figure 13A] 13. Reconstructed slices across and through the thickness for variable zoom with and without weighting. [Figure 13B] 13. Reconstructed slices across and through the thickness for variable zoom with and without weighting. [Figure 13C] 13. Reconstructed slices across and through the thickness for variable zoom with and without weighting. [Figure 13D] 13. Reconstructed slices across and through the thickness for variable zoom with and without weighting. [Figure 14] 1 illustrates an exemplary computer system suitable for implementing certain embodiments of the present disclosure. [Figure 15] Normalized intensity profiles across the defect boundaries are shown. [Figure 16] 13 shows an estimate of boundary sharpness based on line profiles. [Figure 17] 1 shows the phantom boundary unsharpness and normalized variance as a function of window size. [Figure 18] 1 shows the perforation boundary unsharpness and normalized variance as a function of window size. [Figure 19A] 13 shows a slice along a plane with variable zoom. [Figure 19B] 13 shows a slice along a plane with variable zoom. [Figure 19C] 1 shows a planar slice of the conventional method. [Figure 19D] 1 shows a planar slice of the conventional method. [Figure 20A] 4A and 4B show slices through the thickness for the variable zoom method and the conventional method, respectively. [Figure 20B] 4A and 4B show slices through the thickness for the variable zoom method and the conventional method, respectively. [Figure 21] 1 shows the boundary blur and normalized variance of the impact defect. [Figure 22A] 4A and 4B show slices in the plane and through thickness directions, respectively, of the variable zoom technique. [Figure 22B] 4A and 4B show slices in the plane and through thickness, respectively, of the conventional method. [Figure 22C] 4A and 4B show slices in the plane and through thickness directions, respectively, of the variable zoom technique. [Figure 22D] 4A and 4B show slices in the plane and through thickness, respectively, of the conventional method. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0032] Although exemplary implementations of one or more embodiments are illustrated below, it should be understood at the outset that the disclosed system and method may be implemented using any number of techniques, whether currently known or in existence. The present disclosure is not limited in any way to the exemplary implementations, drawings, and techniques illustrated below, but may be modified within the scope of the appended claims together with the full scope of their equivalents. The use of the term "and / or" indicates that any option or any combination of the listed options may be used. For example, "A, B, and / or C" means "A", or "B", or "C", or "A and B", or "A and C", or "B and C", or "A and B and C".

[0033] Throughout this disclosure, the following references are referenced in superscript: For example, document 23 may be identified from the reference list by placing the number 23 in parentheses

[23] .

[0034] Attempts to overcome the size limitations of X-ray CT examinations can be grouped into two main categories: local tomography and limited-angle tomography. Local tomography deals with scenarios where the specimen does not always remain in the field of view (FOV) of the detector during scanning. This group of methods incorporates experimental techniques as well as improved analytical and iterative algorithms to overcome the presence of artifacts in the reconstructed volume due to truncated projections, which are considered in a detailed overview

[13] . For example, the zoom-in technique has shown promising results for parallel beam X-ray geometries. It is based on a combination of full-angle acquisition of high-resolution data within the FOV and low-resolution data at all angles across the specimen

[25] . An experimental adaptive zooming technique has also been proposed

[26] , using a non-conventional scanning trajectory that keeps the specimen region of interest (ROI) entirely within the FOV.

[0035] An alternative technique to local tomography is limited-angle tomography. In some cases, the specimen width may be large enough that it is impossible to fit the entire specimen inside the X-ray CT system housing. Limited-angle scanning techniques are generally not as accurate as full-angle CT, but may present a potential solution. For example, computed laminography, in which the X-ray source and detector move synchronously to irradiate the specimen at an oblique angle, has been proposed as a promising NDT technique for objects with large width-to-thickness aspect ratios

[17] . The advantages and limitations of computed laminography for the inspection of composite materials are discussed in papers

[22] ,

[23] . Another possible technique refers to the case where projections are acquired from limited projection angles. This leads to an ill-posed problem with a large number of artifacts that degrade the resulting reconstruction quality. Despite the fact that limited-angle tomography helps to overcome the geometric constraints associated with large in-plane dimensions, current techniques are unable to provide the same quality of reconstructed volumes as local and full-angle tomography.

[0036] Extensive research efforts have been undertaken to overcome the size limitations of X-ray CT. Published work encompasses the development of unconventional trajectories of radiographic acquisition combined with novel reconstruction algorithms. One of the obvious solutions is the acquisition of projections with a limited angular (<180°) range. However, the lack of projections in this acquisition leads to a large number of artifacts that degrade the reconstruction quality, especially in the through-thickness direction

[16] . Computed laminography (CL) is a well-known one-sided technique for the identification of planar defects in flat objects such as electronic boards, where the X-ray source and detector move synchronously to irradiate the specimen at an oblique angle

[17] . CL has been shown to result in smaller artifacts compared to limited-angle tomography

[18] . Synchrotron radiation-based CL implementations offer more ways to improve the quality of defect detection

[19] , but cabinet system case studies have also been carried out

[20] . Various CL trajectory and reconstruction methods, especially for composite NDT, have been pursued in the literature [22–23]. Despite improvements that overcome the geometric constraints associated with large in-plane dimensions, CL cannot provide the same quality of out-of-plane defect detection as full-angle (360°) CT [21–23].

[0037] Another group of methods, local tomography or region of interest (ROI) tomography, incorporates experimental techniques as well as improved algorithms to overcome the presence of artifacts due to features outside the ROI present in the projections, which impair the reconstruction quality even for full-angle CT

[24] . For example, a zoom-in technique based on a combination of full-angle acquisition of high-resolution data within the field of view (FOV) and full-angle low-resolution data for the entire specimen demonstrated accurate ROI reconstruction

[25] . An experimental adaptive zooming technique using a non-conventional scanning trajectory that keeps the specimen ROI entirely within the FOV has been shown to result in improved spatial resolution

[26] . References [25-26] proposed radiographic acquisition trajectories similar to the trajectory proposed in this work, but none of them aimed at or demonstrated improved quality of through-thickness reconstruction for laminated composites. Interlaminar failures remain a major challenge in laminated composites, and therefore it is important to detect through-thickness associated and damage.

[0038] Extensive exploration efforts have been conducted to overcome the size limitations of X-ray CT. Pioneering work attempting to reconstruct small-scale regions of interest (ROI) in the human body appeared in the medical field, with later extensions to industrial CT. Penssel et al. investigated a modified 360° trajectory for ROI inspection of partially accessible subjects. The non-conventional scanning trajectory was driven by the size and shape of the ROI, while continuously displacing the specimen as a function of the rotation angle. Findings showed that the proposed trajectory could achieve high-fidelity scanning of regions of interest on simulated 2D phantoms. Dabravolski et al. used an acquisition trajectory that follows the convex hull of the specimen. The proposed adaptive zooming technique showed excellent reconstruction quality for artifactual data. Maass et al. tested several novel techniques that combine data from low- and high-resolution data to improve quality reconstruction. Kyrieleis et al. show that projection extension is suitable for reasonable approximation of regions of interest where high resolution is not required.

[0039] Computed laminography (CL) is an alternative technique for large objects that cannot be accommodated in an X-ray CT system. This technique allows partial access to the test specimen by irradiating the object at an oblique angle. CL has been shown to result in smaller artifacts compared to limited angle tomography. Despite improvements that overcome the geometric constraints associated with large in-plane dimensions, CL cannot provide the same quality of out-of-plane defect detection as full-angle (360°) CT.

[0040] Provided herein is a variable zoom X-ray CT method that can overcome the limitations of large width-to-thickness aspect ratios in cabinet X-ray CT systems and allow additional flexibility to achieve high resolution for structures with large in-plane dimensions. The variable zoom method eliminates the need to destroy the inspection item by cutting out small sections that allow accurate inspection of composite structures at sufficient magnification. Detection of complex structural damage, including accurate detection of features through thickness, due to low-velocity impacts in large and thin composite laminate panels that fall within the worst-case scenario of conventional CT methods illustrates the accuracy of the variable zoom method. In addition to the variable zoom CT scanning approach, a novel convolution backprojection reconstruction method is disclosed herein to achieve accurate high-resolution results. The disclosed technique includes two aspects: a non-conventional radiograph acquisition trajectory, and an industry-standard Feldkamp-Davis-Kress (FDK) reconstruction method that includes weighting of radiographs based on the distance from the panel to the X-ray source and allows for higher quality 3D reconstruction. Analysis of the reconstruction quality of CT images produced by the variable zoom technique, including measurements of dimension and sharpness, is performed on artificial 3D phantoms and on CT scans of objects with features of known dimensions. Specialized radiographic acquisition is also implemented in a Shimadzu inspeXio SMX-225CT FPD HR industrial microfocus X-ray CT system. The capabilities of the method are demonstrated for the detection of complex structural damage due to low-velocity impacts, including accurate detection of through-thickness features in large and thin composite panels.

[0041] 2A-2F are simplified schematic diagrams of top views of a CT system 200 illustrating steps of a variable zoom acquisition trajectory including both rotation and translation of a specimen 202 for implementing some embodiments of the present disclosure. The CT system 200 includes an X-ray source 204 and an X-ray detector 206. The source 204 may emit a cone-shaped X-ray beam to project the specimen 202 within a field of view (FOV) 208 onto the detector 206. The X-ray beam of the source 204 may be a collimated cone X-ray beam (e.g., 30 degrees wide). In one example, the source 204 includes a 225 kV X-ray tube with a focal spot size of 4 m. Other X-ray sources are contemplated by the present disclosure. In one example, the detector 206 is a 16-bit (65,536 intensity values) flat panel detector made of cesium iodide (CsI) scintillator material and has an operating size of 417 mm. The detector 206 can be used in two acquisition modes: high speed (1000x1000 pixels, 0.417mm pixel size) and high definition (3000x3000 pixels, 0.139mm pixel size). Other x-ray detectors are contemplated by this disclosure. In one embodiment, the source detector distance (SDD) may be 800mm and scanning was performed at 1 frame per second with 3 frame averaging. Other SSDs, frame rates, and set sizes of averaged frames are contemplated by this disclosure. In various embodiments, the CT system 200 is implemented with a Shimadzu inspeXio SMX-225CT FPD HR industrial microfocus x-ray CT system. The Shimadzu CT system hardware is capable of using non-conventional scan trajectories. The Shimadzu scanning software was modified to accept a custom definition of the object trajectory, i.e., the location of the center of rotation at each rotation angle, and was provided with reconstruction software with calibrated detector offsets at each location. The scanning parameters of the pre-impregnated continuous fiber reinforced polymer composite panels subjected to low velocity impact are summarized in Table 1. [Table 1]

[0042] The CT system 200 also includes a specimen stage (not shown) configured to fixably position a region of interest (ROI) 210 of the specimen 202 within the FOV 208. The specimen stage rotates the specimen 202 about an axis of rotation and translates the specimen 202 within the FOV 208 between the source 204 and the detector 206 along an acquisition trajectory. In various implementations, the acquisition trajectory keeps the axis of rotation at the center of the FOV 208. In some implementations, the acquisition trajectory may move the axis of rotation through other areas of the FOV 208. Thus, the ROI 210 remains projected onto a central area of ​​the detector 206 while the specimen 202 translates between the source 204 and the detector 206. In some implementations, the ROI 210 may be projected onto a non-central area of ​​the detector 206. The specimen stage may include a clamp for releasably clamping the specimen 202 to the specimen stage. The clamp is positioned on the specimen stage such that the central axis of the clamp is parallel to and coincident with the axis of rotation of the specimen stage. The specimen 202 is clamped to the clamp at the ROI 210.

[0043] A controller (not shown) of the CT system 200 controls the rotational and translational motion of the specimen stage during scanning of the specimen 202, and reconstructs the captured specimen using calibrated detector offsets at each location along the acquisition trajectory. The translational motion or acquisition trajectory of the specimen stage identifies the location of the specimen stage's axis of rotation (e.g., distance from the source 204 along the center of the FOV 208) at each rotation angle. The distance of the specimen stage's axis of rotation from the source 204, also known as the source-to-object distance (SOD), is used to estimate the maximum magnification factor and, consequently, the optimal spatial resolution. As the specimen rotates during a CT scan, it also translates following a sinusoidal path:

number

[0044] 3 is a radial diagram of schematic trajectories of SOD for conventional, variable zoom, and limited angle scanning trajectories. The radial coordinate corresponds to the SOD, whereas the angular coordinate defines the angle of the scan trajectory (e.g., the angle of the specimen stage). The SOD for the conventional trajectory remains constant during the scan, as shown by the first line 302. Meanwhile, the SOD for the variable zoom acquisition trajectory varies according to a sinusoidal path, as defined in Equation 1, and is shown by the "peanut" shaped second line 304.

[0045] In the example shown in FIG. 3, a schematic trajectory is provided for a specimen width of 401 mm and negligible thickness and offset. In one example, the minimum distance required to capture a desired ROI within the detector FOV within |θ|<15° is SOD ROI = 81 mm. The SOD for conventional CT geometry is 265 mm since the specimen must rotate 360° without hitting the source 204, while maintaining a safe offset S0 from the source 204. The SOD(θ) for the variable zoom trajectories ranges from 81 to 265 mm. The limited angle tomography (LAT) scan trajectories, also used for comparison, are a subset of the variable zoom trajectories, as shown by the thick third line 306, with SOD(θ) = SOD for |θ| < 15°. ROI The reason for such a small LAT angle range is that this is the largest angular span that can be achieved for the selected maximum magnification without the specimen interfering with the X-ray source.

[0046] In one embodiment, a Shimadzu microfocus X-ray CT cabinet system inspeXio SMX-225CT FPD HR is utilized for all X-ray CT scans. The system has a 225 kV X-ray tube capable of reaching a focal spot size of 4 μm. The system uses a high-resolution 16-bit flat panel X-ray detector with a size of 417×417 mm that can acquire radiographs with up to 3000×3000 pixels. The X-ray tube to flat panel detector distance SDD was set to 800 mm.

[0047] Reconstruction method Reconstruction of three-dimensional (3D) volumes from X-ray cone-beam projection radiographs represents a photon transmission tomography problem that has been widely studied since the 1970s. A historical review of explorations in this field can be found in

[12] . The analytical reconstruction method developed for variable zoom scanning trajectories disclosed herein is related to the industry standard Feldkamp-Davis-Kress (FDK) algorithm

[30] , the disclosure of which is incorporated herein by reference in its entirety. The analytical reconstruction method is superior in terms of performance and is used in the majority of commercial medical or industrial X-ray CT systems.

[0048] The algorithm described in this section follows the representation of filtered backprojection as a convolution backprojection method provided in Reference

[31] . In a first step, the x-ray radiogram P(x,y) is weighted to account for the different ray lengths in the cone beam, P θ '(x,y k ) and convolved with a ramp filter. The pixel coordinates are assumed to be scaled to the center of rotation, i.e., their coordinates (x,y k ) is divided by the scaling factor.

number

[0049] A step in this reconstruction algorithm is the filtering of the projections in the frequency domain. In the FDK algorithm, a 2D radiograph P taken at an angle θ is θ '(x,y k ), a 1D filtering is applied to each row of the image. In this work, we use the following discrete spatial sampling of a ramp filter

[31] :

number

number

number

[31] : [P' θ (x,y k )*h(x)]=p x IFFT{FFTP' θ (x,y k ) ZP ·FFTh[np x ] shift} Formula (3) where FFT / IFFT stands for 1D direct and inverse discrete Fourier transform for real inputs, and the filter h[np x The half-spaces of are swapped using the fftshift method

[32] and the radiographs are denoted by the subscript zp to avoid period artifacts.

number

[31]

[0050] In the final step, the backprojection of the set of filtered radiographs is vz This is performed over all projection angles with an additional weighting by:

number

number

[0051] where v(t,s,r) is the volume transformation matrix R V represents the reconstructed volume, R θis the matrix of specimen rotations. Note that due to the variable zoom method acquisition trajectory, the "projection" coordinates (x,y,z) are different for each projection angle according to the varying SOD(θ). The following weighting factors were found experimentally to significantly improve the quality of the reconstructed through-thickness sections:

number

[0052] Variable magnification presents another reconstruction difficulty not present in conventional CT reconstruction. Reconstruction of a volume larger than the initial ROI, which is only partially visible on the detector, leads to artifacts corresponding to changes in the effective size of the projections. These artifacts result from the assumption of data periodicity by the discrete Fourier transform. To eliminate these artifacts, we apply a periodic smooth decomposition

[33] to the weighted radiographs and use the periodic parts of the radiograph boundaries for filtering.

[0053] To avoid artifacts at high resolution, the reconstruction algorithm must accurately calculate the backprojection contribution to a voxel via any ray through the object. The separate application of equation (5) involves the interpolation of density values ​​from the detector grid to a rotated volume grid. It was found that "naive" methods based on the interpolation of projection values ​​or volume coordinates lead to low-quality reconstructions. In this work, we implemented two interpolation methods that were developed specifically for tomographic reconstruction: a distance-driven method

[34] or a separable footprint method

[35] . Both methods yield similar results for the presented test cases, and their disclosures are incorporated herein by reference in their entirety.

[0054] The following describes the high level steps in implementing the algorithm: 1. According to equation (3), the frequency domain FFTh[np x ]shift Calculate the ramp filter at 2. Projection / volume transformation R(θ) for each projection angle / SOD V Calculate 3. Using equations (2, 4), the weighted and filtered radiograph S θ (x,y k ) and apply a periodic smoothing decomposition

[33] (on the graphics card). 4. R(θ)R V Transform the volume coordinates using a transformation (on the graphics card), 5. Add the backprojected pixel values ​​to every voxel (on the graphics card) using an interpolation method [34-35], as shown in equation (5). 6. Repeat starting from step 2 for all projection angles / SODs.

[0055] Reconstruction of manufacturing variations or structural damage in composite structures typically requires high-resolution volumes with at least hundreds, preferably thousands, of points in each dimension. In the algorithms described above, backprojection is an infinitely time-consuming operation during reconstruction, and an efficient implementation of backprojection for large volumes must be based on highly parallel computations. In this work, we use standard implementations of FFT and IFFT available in modern graphics card hardware, as well as a custom backprojection implementation of Eq. (5) that takes advantage of vector processing by implementing complex, but identical, interpolation computations required for each volume point. In addition to voxel-wise parallelization, the graphics card processor offers multiple command streams that allow simultaneous execution of vectorized computations and memory transfers.

[0056] In various embodiments, a high performance reconstruction desktop computer is equipped with a CT system 200 (e.g., a Shimadzu InspeXio SMX-225CT FPT HD system)

[29] , the disclosure of which is incorporated herein by reference in its entirety. In one embodiment, the reconstruction computer has dual 2.3 GHz Intel Xeon E5-2650v3 processors and an NVIDIA Quadro M5000 graphics card with 2048 CUDA cores. As an example of computational performance, the reconstruction algorithm disclosed above performed the reconstruction of a 1329 x 1266 x 432 voxel volume (approximately 700 million variables) using 820 2000 x 2000 pixel radiographs (approximately 3.3 billion equations) in less than 7 minutes. In this case, the radiographs were cropped to 2000 x 2000 pixels to reduce the amount of projection data to the size required for reconstruction.

[0057] Referring to the pre-impregnated continuous fiber reinforced polymer composite panels listed in Table 1 above, each panel has a distinct material system typically encountered in aerospace applications. The first panel is an IM7-carbon / 8552-epoxy composite laminate and the second panel is a hybrid IM7-carbon and S2-glass / 913-epoxy composite laminate. Both panels were manufactured by Boeing using the prepreg (Hexcel) specification [27-28].

[0058] To evaluate the performance of the variable zoom technique for both material systems, the performance is compared to reconstructions based on a) limited angle (30°) CT with a magnification dictated by the size of the damaged area, and b) conventional full angle CT with a magnification dictated by the size of the panel. The 30° angular span is chosen to demonstrate the capabilities of conventional limited angle scanning with the best resolution.

[0059] FIG. 4 is a system diagram of a variable zoom CT system 400. The CT system 400 is substantially similar to the CT system 200 described above, with like numerals representing like parts. For example, the specimen 202, the source 204, the detector 206, and the ROI 210 are substantially the same as described above. As described above, the source 204 is configured to emit an X-ray beam along the FOV 208 to the detector 206. The CT system 400 also includes a specimen stage 402 that is substantially the same as the specimen stage described above with reference to the CT system 200. In particular, the specimen stage 402 rotates the specimen 202 about a rotation axis A and translates the specimen 202 within the FOV 208 between the source 204 and the detector 206 along an acquisition trajectory 408. In the illustrated embodiment, the specimen stage 202 rotates 45°. The specimen stage 402 may include a clamp 404 for releasably clamping the specimen 202 to the specimen stage 402. The clamp is positioned on the specimen stage such that a central axis B of the clamp 404 is parallel to and coincident with the axis of rotation A of the specimen stage 402. The CT system 400 includes a housing 406 that contains other components of the CT system 400. In various implementations, the housing 406 may provide radiation shielding to prevent radiation from the source 204 from entering the surrounding environment.

[0060] In the embodiment shown in Figure 4, the specimen 202 is a pre-impregnated continuous fiber reinforced polymer composite (prepreg) panel that has been subjected to low velocity impact damage. For example, the specimen may be a 401 mm carbon / epoxy composite laminate panel. The panel contains impact damage localized in the center of the laminate.

[0061] Panel specimens represent a challenge for conventional X-ray CT due to their large width-to-thickness aspect ratio. The large size of the panel prevents conventional CT scanning techniques from obtaining the desired spatial resolution in sensitive areas that typically have a size comparable to the panel thickness. Reliable attenuation of interlayer defects requires that the reconstructed voxels are several orders of magnitude smaller than the panel thickness. The variable zoom technique disclosed herein is compared to reconstructions based on conventional full-angle CT at minimum magnification, determined by the size of the panel, and limited-angle CT at maximum magnification. The variable zoom technique obtains high-resolution CT scans using acquisition trajectories with variable SOD. The trajectories proposed in this work acquire radiographs at higher geometric magnification by bringing the inspected object closer to the X-ray source.

[0062] The composite panels investigated represent a number of distinct material systems. In one example, the first panel is an IM7-carbon / 8552-epoxy laminate, while the second panel is a hybrid IM7-carbon and S2-glass / 913-epoxy composite laminate. Both panels were manufactured by Boeing using prepregger (Hexcel) specifications [27-28]. For both material systems, a comparison of CT reconstructions is performed for 1) a conventional CT acquisition trajectory using the reconstruction software provided with an InspeXio SMX-225CT FPD HR, 2) the variable zoom acquisition trajectory disclosed herein, and 3) a limited angle acquisition, both using the reconstruction algorithm described above. Line profile measurements across defect boundaries in the carbon / epoxy composite laminate are used to confirm the superior resolution of the variable zoom technique disclosed herein.

[0063] Carbon / Epoxy Composite Laminated Panel The first example illustrates the variable zoom technique for the inspection of an impact damaged area in a carbon / epoxy panel. The width and thickness of the panel are 401 mm and 3.5 mm, respectively, resulting in a width-to-thickness aspect ratio of 114.57. The impact damage is 4×4.5 mm, as illustrated in FIG. 2 The area is localized in ROI 210.

[0064] 5 is a radial diagram of schematic trajectories of SOD for conventional, variable zoom, and limited angle scanning trajectories. For the conventional scanning trajectory, SOD 502 remains constant and equals 265 mm. SOD 504 for the variable zoom trajectory varies from 81 mm to 265 mm using the sinusoidal path described in equation (1).

[0065] The conventional scanning trajectory achieves a 3x magnification on the specimen 202 corresponding to an optimal spatial resolution of 43 μm. Meanwhile, the variable zoom technique is implemented such that the ROI 210 of the specimen 202 is projected onto the detector 206 according to the trajectory outlined in equation (1). The variable zoom method achieves up to 10x magnification on the impacted area in the 401 mm carbon / epoxy specimen. The optimal spatial resolution may be 14 μm in this case.

[0066] The scanning parameters of the first embodiment are listed in Table 1 above. In the first embodiment, an angular range of 210° (known as short scan) of variable zoom technique was used to reduce the acquisition time. The trajectory described in Eq. (1) was used in the angular range ∈ [−105°, 105°]. A fixed SOD of 81 mm was used in the range of ±15°. The limited angle reconstruction was accomplished by extracting the trajectory in the angular range ∈ [−15°, 15°] at the highest magnification. An optimal reconstruction voxel size of 43 μm is used for the conventional scanning method, and 14 μm is used for the variable zoom and limited angle scanning technique.

[0067] 6A-6C illustrate 3D reconstructed volumes of a carbon / epoxy panel obtained by three scanning trajectories. Although the resolution and sharpness of the reconstructed volume with the variable zoom technique is better than the other two, the three-dimensional view makes it difficult to distinguish individual features. Clearly, the limited angle reconstruction does not allow for the discrimination of the microstructure of the damaged area. The conventional scan shown in FIG. 6A shows insufficient resolution. The limited angle scan shown in FIG. 6B results in a very shallow depth of field and poor suitability for cross-sectional planes such as cross-sectional plane 602. The variable zoom technique shown in FIG. 6C produces adequate resolution in all three dimensions.

[0068] 7A-7F are reconstructed slices in a direction through the thickness for a conventional scanning technique, a variable zoom technique, and a limited angle acquisition. A quality analysis of the three-dimensional volume can be performed by examining slices at different positions through the thickness away from the damaged area shown in FIGS. 7A-7F. For example, FIGS. 7A-C are reconstructed slices of a first position through the thickness for a conventional scanning technique, a variable zoom technique, and a limited angle acquisition, respectively. FIGS. 7D-7F are reconstructed slices of a second position through the thickness for a conventional scanning technique, a variable zoom technique, and a limited angle acquisition, respectively.

[0069] The conventional acquisition mode provides images that lack sharpness, as shown in Figures 7A and 7D. Due to insufficient resolution, small-sized matrix damage cannot be distinguished. On the other hand, the variable zoom technique has the ability to improve spatial resolution. As a result, smaller-sized defects can be distinguished, as shown in Figures 7B and 7E. Moreover, the images obtained using the variable zoom technique appear to have better sharpness as observed by visual inspection. The slices obtained by the limited angle acquisition appear highly distorted in the through-thickness direction due to the lack of projection, as shown in Figures 7C and 7F.

[0070] 8A-8C are reconstructed slices across the thickness for conventional scan, variable zoom, and limited angle techniques. The same trends discussed above with slices in Figs. 7A-7F emerge by inspecting slices along the through thickness direction as shown in Figs. 8A-8C. Note that the variable zoom technique improves the sharpness and clarity of the ablation reconstruction shown in Fig. 8B in the through thickness direction. As shown in the limited angle acquisition example in Fig. 8C, the through thickness resolution is most compromised due to missing angular data, leading to detectable defects being obscured throughout a large thickness range for the 30° scan. Although the variable zoom acquisition trajectory defined in Eq. (1) was expected to improve the resolution of the planar slices shown in Figs. 7B and 7E, unexpectedly, the variable zoom technique also improved the through thickness slices shown in Fig. 8B. The apparently superior quality of the through thickness slices obtained by the variable zoom technique is attributed to the novel weighting scheme proposed in Eq. (6).

[0071] Hybrid Composite Laminate Panel In another embodiment, the variable zoom technique is used to characterize hybrid composite laminate panels. Hybrid composite laminate panel material systems impose additional challenges due to the large contrast variation between carbon and glass fibers. FIG. 9 is a system diagram of a variable zoom CT system 900. The CT system 900 is substantially similar to the CT systems 200, 400 described above, with like numerals representing like parts. For example, the specimen 202, source 204, detector 206, and ROI 210 are substantially the same as those described above. As described above, the source 204 is configured to emit an X-ray beam along the FOV 208 to the detector 206. The CT system 900 also includes a specimen stage 402 that is substantially the same as the specimen stage described above with reference to the CT system 200. In particular, the specimen stage 402 rotates the specimen 202 about a rotation axis A and translates the specimen 202 within the FOV 208 between the source 204 and the detector 206 along an acquisition trajectory 408. In the example shown, the specimen stage 202 rotates 45°. The specimen stage 402 may include a clamp 404 for releasably clamping the specimen 202 to the specimen stage 402. The clamp is positioned on the specimen stage such that a central axis B of the clamp 404 is parallel to and coincident with the axis of rotation A of the specimen stage 402. The CT system 900 includes a housing 406 that contains other components of the CT system 900. In various implementations, the housing 406 may provide radiation shielding to prevent radiation from the source 204 from entering the surrounding environment.

[0072] In the example shown in Figure 9, the specimen 202 is a hybrid composite laminate panel having a width of 152 mm and a thickness of 5.2 mm. In this case, the aspect ratio is 29.23. As illustrated in Figure 9, the impact damage is located at the center of the specimen 202 in the ROI 210 and measures approximately 3 x 3 mm. 2 It occupies an area of.

[0073] Figure 10 shows schematic trajectory radiograms of SOD for conventional, variable zoom and limited angle scanning trajectories. Following the example of Figure 9, the SOD 1002 for conventional acquisition mode is 100 mm. The SOD 1004 for the variable zoom trajectory ranges from 38 to 100 mm. Also, the results of limited angle acquisition are presented by extracting radiograms corresponding to the angular range ∈ [-15°, 15°] of maximum magnification, again with SOD 1006.

[0074] The relatively low width-to-thickness aspect ratio of the panel allows for the achievement of a magnification of 8x over conventional scanning. In contrast, the variable zoom technique was able to increase the magnification up to 20x. A summary of the CT scan parameters for specimen 202 of FIG. 9 is provided above in Table 1. It is important to note that in this example the variable zoom technique uses an angular range of 360° throughout the scan.

[0075] 11A-11B illustrate 3D reconstructed volumes of a hybrid composite laminate panel obtained by conventional and variable zoom scanning trajectories. It can be observed that the reconstructed volume of the variable zoom technique shown in FIG. 11B has superior quality compared to the conventional scan shown in FIG. 11A. Again, the apparently improved quality of the reconstructed slices through the thickness is not trivially expected and arises from the application of the projection weighting provided by equation (6).

[0076] 12A-12F are reconstructed slices across and through the thickness of the conventional scanning technique, the variable zoom technique, and the limited angle acquisition. FIG. 12A-12C show slices across the thickness of the conventional scanning technique, the variable zoom technique, and the limited angle acquisition. FIG. 12D-12F show slices through the thickness of the conventional scanning technique, the variable zoom technique, and the limited angle acquisition, respectively. Reconstruction is performed using a voxel size of 23 μm when projections are acquired by each of the methods under consideration. It is evident that the images obtained by the variable zoom acquisition trajectory have better sharpness as illustrated in FIG. 12B and 12E. Despite the relatively high magnification factor utilized in the conventional acquisition mode, the spatial resolution appears to be insufficient to provide the desired reconstruction quality. On the other hand, variable magnification through the scan allows the achievement of higher spatial resolution, thus leading to better reconstruction quality. In fact, all slices are subject to characteristic artifacts due to the presence of metal-coated fibers (X-ray tracers) detectable on plain radiographs. These artifacts are more pronounced in limited angle tomographic slices, leading to a degradation of imaging quality, as illustrated in Figures 12C and 12F.

[0077] 13A-13D are reconstructed slices through and across the thickness for a variable zoom with and without weighting. FIGS. 13A and 13B are reconstructed slices through and across the thickness for a variable zoom with weighting coefficients defined in Equation (6). In comparison, FIGS. 13B and 13D are reconstructed slices through and across the thickness for a variable zoom without weighting coefficients defined in Equation (6). While the reconstructed slices across the thickness in FIG. 13D show less noise, the reconstructed slices through the thickness in FIG. 13B show significant distortion features in the depth direction, most clearly illustrated by the star-shaped pattern from the high density tracer. FIG. 13D also shows distortions due to tracer obstructions in the section. In contrast, when using the weighting coefficients defined in Equation (6), these distortions from the high density tracer are not present in the reconstructed slices of FIGS. 13A and 13C.

[0078] Accuracy and Sharpness Measurements Disclosed herein is a novel X-ray computed tomography technique capable of improving the spatial resolution of inspection of composite laminates with large in-plane dimensions. The novel scanning technique utilizes a non-conventional scanning trajectory, where the specimen not only rotates but also translates toward the X-ray source as spatial dimensions permit, thus leading to greater resolution. The variable zoom method was demonstrated to outperform both conventional and limited angle scanning methods. Unlike conventional and limited angle techniques, the variable zoom CT produced outstanding resolution in all three dimensions.

[0079] The approach incorporates two proposals: a non-conventional trajectory of radiograph acquisition, and a novel reconstruction weighting scheme. The scanning trajectory involves simultaneous angular rotation and translation of the specimen towards the x-ray source as the specimen dimensions permit, thus leading to greater resolution. Each radiograph is scaled by a weighting factor proportional to the distance from the object to the x-ray source. The variable zoom method was demonstrated to outperform both conventional and limited angle scanning methods. Unlike the latter technique, the variable zoom method produced outstanding resolution in all three dimensions.

[0080] To validate the proposed method, a feasibility study was performed on two composite laminate panels manufactured from different composite material systems and subjected to low-velocity impact loading. Both panels had a relatively large thickness-to-width aspect ratio, a known challenge for conventional CT scanning techniques. Indeed, conventional CT was not able to achieve the spatial resolution required to clearly identify smaller impact-induced cracks and delaminations, while limited-angle tomography techniques showed a complete lack of interlaminar defect resolution. Variable zoom CT was able to reconstruct the volume with higher spatial resolution and therefore lead to better sharpness of the reconstructed slices. The improved sharpness of the reconstruction with the variable zoom method was especially notable for the through-thickness sections, leading to a significant improvement in interlaminar defect resolution. These results were consistent for both material systems under investigation. A quantitative analysis of the accuracy and sharpness of the measurements of the defect reconstruction was performed for machining-induced defects of known geometry and dimensions. The variable zoom technique provided accurate in-plane and out-of-plane measurements of defect dimensions and superior sharpness of the reconstructed sections based not only on visual inspection but also on analysis of the line profiles, compared to measurements obtained by mechanical tools.

[0081] The variable zoom CT method developed in this study has important implications for NDI of composite materials and structures. On the material side, the method can address the need to achieve maximum resolution of the CT system without destroying the test article by cutting small pieces that can be placed close enough to the x-ray source for sufficient geometric magnification. There is a similar need for composite structures that have small defects that can fit into existing CT system housings but have three-dimensional significant features that are not discernible using conventional scanning techniques. The new method also provides additional flexibility toward enabling high-resolution CT for larger structures that are not currently suitable for microfocus CT systems.

[0082] It should be understood that the logical operations described herein with respect to the various figures may be implemented (1) as a series of computer-implemented operations or program modules (i.e., software) executing on a computing device (e.g., the computing device described in FIG. 14), (2) as interconnected machine logic circuits or circuit modules (i.e., hardware) within a computing device, and / or (3) as a combination of software and hardware in a computing device. Thus, the logical operations discussed herein are not limited to a specific combination of hardware and software. The implementation aspect may be selected depending on the performance and other requirements of the computing device. Thus, the logical operations described herein are variously referred to as operations, structural devices, operations, or modules. These operations, structural devices, operations, and modules may be implemented in software, firmware, special purpose digital logic, and any combination thereof. It should also be understood that more or fewer operations may be performed than shown in the figures and described herein. The operations may also be performed in a different order than described herein.

[0083] Referring to FIG. 14, an exemplary computing device 1900 is illustrated in which an embodiment of the present invention may be implemented. For example, each of the controller systems described herein may be implemented as a computing device such as computing device 1900. It should be understood that the exemplary computing device 1900 is only one example of a suitable computing environment in which an embodiment of the present invention may be implemented. If desired, the computing device 1900 may be a well-known computing system, including but not limited to a personal computer, a server, a handheld or laptop device, a multiprocessor system, a microprocessor-based system, a network personal computer (PC), a minicomputer, a mainframe computer, an embedded system, and / or a distributed computing environment including a plurality of any of the above systems or devices. In a distributed computing environment, various tasks may be performed on remote computing devices connected to a communications network or other data transmission medium. In a distributed computing environment, program modules, applications, and other data may be stored in local and / or remote computer storage media.

[0084] In one embodiment, the computing device 1900 may comprise two or more computers communicating with each other to perform tasks in a collaborative manner. For example, but not by way of limitation, an application may be partitioned in a manner that allows simultaneous and / or parallel processing of the application's instructions. Alternatively, data processed by the application may be partitioned in a manner that allows simultaneous and / or parallel processing of different portions of the data set by two or more computers. In one embodiment, virtualization software may be used by the computing device 1900 to provide the functionality of several servers that are not directly coupled to several computers of the computing device 1900. For example, the virtualization software may provide 20 virtual servers on four physical computers. In one embodiment, the functionality disclosed above may be provided by running the application and / or multiple applications in a cloud computing environment. Cloud computing may include providing computing services over a network connection using dynamically scalable computing resources. Cloud computing may be supported, at least in part, by the virtualization software. The cloud computing environment may be established by an enterprise and / or rented as needed from a third party provider. Some cloud computing environments may comprise cloud computing resources that are owned and operated by an enterprise, as well as cloud computing resources that are rented and / or leased from third-party providers.

[0085] In its most basic configuration, a computing device 1900 typically includes at least one processing unit 1920 and a system memory 1930. Depending on the exact configuration and type of computing device, the system memory 1930 may be volatile (such as random access memory (RAM)), non-volatile (such as read only memory (ROM), flash memory, etc.), or some combination of the two. This most basic configuration is illustrated in FIG. 14 by dashed line 1910. The processing unit 1920 may be a standard programmable processor that performs arithmetic and logical operations necessary for the operation of the computing device 1900. Although only one processing unit 1920 is shown, there may be multiple processors. Thus, while instructions are discussed as being executed by a processor, instructions may be executed by one or multiple processors simultaneously, serially, or otherwise. The computing device 1900 may also include a bus or other communication mechanism for communicating information between various components of the computing device 1900.

[0086] The computing device 1900 may have additional features / functionality. For example, the computing device 1900 may include additional storage, such as removable storage 1940 and non-removable storage 1950, including, but not limited to, magnetic or optical disks or tape. The computing device 1900 may also include a network connection 1980 that allows the device to communicate with other devices, such as over a communication path as described herein. The network connection 1980 may take the form of a modem, a modem bank, an Ethernet card, a Universal Serial Bus (USB) interface card, a serial interface, a token ring card, a Fiber Distributed Data Interface (FDDI) card, a Wireless Local Area Network (WLAN) card, a radio transceiver card, such as a Code Division Multiple Access (CDMA), Global System for Mobile Communications (GSM), Long Term Evolution (LTE), Worldwide Interoperability for Microwave Access (WiMAX), and / or other air interface protocol radio transceiver card, as well as other well-known network devices. The computing device 1900 may also have input devices 1970, such as a keyboard, keypad, switches, dials, a mouse, a trackball, a touch screen, a voice recognizer, a card reader, a paper tape reader, or other well-known input devices. Output devices 1960, such as a printer, a video monitor, a liquid crystal display (LCD), a touch screen display, a display, speakers, etc., may also be included. Additional devices may be connected to the bus to facilitate communication of data between components of the computing device 1900. All of these devices are well known in the art and need not be discussed at length here.

[0087] The processing unit 1920 may be configured to execute program code encoded on a tangible computer-readable medium. A tangible computer-readable medium refers to any medium that can provide data that causes the computing device 1900 (i.e., machine) to operate in a particular manner. A variety of computer-readable media may be utilized to provide instructions to the processing unit 1920 for execution. Examples of tangible computer-readable media may include, but are not limited to, volatile, non-volatile, removable and non-removable media implemented in any method or technology for storing information such as computer-readable instructions, data structures, program modules or other data. The system memory 1930, the removable storage 1940, and the non-removable storage 1950 are all examples of tangible computer storage media. Examples of tangible computer-readable recording media include, but are not limited to, integrated circuits (e.g., field programmable gate arrays or application specific ICs), hard disks, optical disks, magneto-optical disks, floppy disks, magnetic tape, holographic storage media, solid state devices, RAM, ROM, Electrically Erasable Program Read Only Memory (EEPROM), flash memory or other memory technologies, CD-ROMs, digital versatile disks (DVDs) or other optical storage, magnetic cassettes, magnetic tapes, magnetic disk storage, or other magnetic storage devices.

[0088] It is fundamental to electrical and software engineering to translate functionality that can be implemented by loading executable software into a computer into a hardware implementation by well-known design rules. The decision to implement a concept in software versus hardware is driven by considerations of design stability and number of units, rather than any issues involved in translating from the software domain to the hardware domain. In general, designs that continue to undergo frequent changes may be preferred to be implemented in software, since respinning a hardware implementation is more expensive than respinning a software design. In general, designs that are stable and will be produced in large quantities may be preferred to be implemented in hardware, e.g., application specific integrated circuits (ASICs), since hardware implementations may not be as expensive for large production runs as software implementations. Often, designs are developed and tested in software form, and can later be translated by well-known design rules into an equivalent hardware implementation in an application specific integrated circuit that hardwires the software's instructions. In the same way that a machine controlled by a new ASIC is a particular machine or device, a computer that has been programmed and / or loaded with executable instructions may similarly be considered a particular machine or device.

[0089] In an exemplary implementation, processing unit 1920 can execute program code stored in system memory 1930. For example, a bus can carry data to system memory 1930, from which processing unit 1920 receives and executes instructions. Data received by system memory 1930 can be stored on removable storage 1940 or non-removable storage 1950, as desired, either before or after execution by processing unit 1920.

[0090] It should be understood that the various techniques described herein may be implemented in connection with hardware or software, or combinations thereof, as appropriate. Thus, the methods and apparatus of the presently disclosed subject matter, or certain aspects or portions thereof, may take the form of program code (i.e., instructions) embodied in a tangible medium, such as a floppy disk, CD-ROM, hard drive, or any other machine-readable storage medium, where the program code, when loaded and executed on a machine, such as a computing device, makes the machine an apparatus for executing the presently disclosed subject matter. In the case of program code execution on a programmable computer, the computing device generally includes a processor, a storage medium readable by the processor (including volatile and non-volatile memory and / or storage elements), at least one input device, and at least one output device. One or more programs may implement or utilize the processes described in connection with the presently disclosed subject matter, for example, through the use of application programming interfaces (APIs), reusable controls, or the like. Such programs may be implemented in a high-level procedural or object-oriented programming language to communicate with a computer system. However, if desired, the programs may be implemented in assembly or machine language. In any case, the language may be a compiled or interpreted language, and combined with hardware implementations.

[0091] Embodiments of the methods and systems may be described herein with reference to block diagrams and flowchart illustrations of methods, systems, apparatuses, and computer program products. It will be understood that each block of the block diagrams and flowchart illustrations, and combinations of blocks in the block diagrams and flowchart illustrations, respectively, can be implemented by computer program instructions. These computer program instructions may be loaded onto a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce machine, whereby the instructions executing on the computer or other programmable data processing apparatus create means for implementing the functions specified in one or more of the flowchart blocks.

[0092] These computer program instructions may also be stored in a computer readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer readable memory produce an article of manufacture that includes computer readable instructions for implementing functions specified in one or more of the flowchart blocks. These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause the computer or other programmable apparatus to perform a series of operational steps to produce a computer-implemented process, such that the instructions executing on the computer or other programmable apparatus provide steps for implementing functions specified in one or more of the flowchart blocks.

[0093] Thus, the blocks of the block diagrams and flowchart illustrations support combinations of means for performing the specified functions, combinations of steps for performing the specified functions, and program instruction means for performing the specified functions. It will also be understood that each block of the block diagrams and flowchart illustrations, and combinations of blocks in the block diagrams and flowchart illustrations, can be implemented by a dedicated hardware-based computer system that performs the specified functions or steps, or a combination of dedicated hardware and computer instructions.

[0094] Phantom-Based Evaluation of Variable Zoom Techniques Definition of a phantom In this section, an artificial 3D phantom is used to determine the performance of the variable zoom technique. The phantom volume represents a rectangular 400 mm wide and 3.5 mm thick block with a cylindrical defect (diameter and height 0.5 mm) located in the center of the phantom. The dimensions of the phantom simulating a composite panel are presented below. A cone beam projection of the phantom is generated using the projector function and using the geometric parameters and the acquisition trajectory of the carbon / epoxy panel shown in Figure 3. To avoid "inverse crime", the projection of the phantom volume is generated using a voxel size smaller than the voxel size in the projected volume. Due to memory limitations, the entire volume is divided into two parts. The smaller inner part with the defect is projected with a voxel size of 2 μm and the larger outer part is projected with a voxel size of 10 μm. The resulting projections are combined and used as input for reconstruction based on the algorithm described above in the reconstruction method section.

[0095] Measuring defect sharpness Visual inspection reveals substantial differences in sharpness between variable zoom and conventional CT reconstructed slices. To quantitatively determine this difference, we adopted the variance estimation method proposed in Kraemer A, Kovacheva E, Lanza G, “Projection based evaluation of CT image quality in dimensional metrology,” Digit. Ind. Radiol. Comput. Tomogr., 2015, p. 1-10. For each point on the defect boundary, the variance of the intensity values ​​inside the support window indicates the local sharpness of the material defect boundary. The maximum variance among all measurements along the defect boundary is chosen to represent the sharpness.

[0096] While ideal for complex boundaries, variance values ​​are difficult to quantitatively interpret and compare. To alleviate this problem, we propose a relationship to estimate boundary unsharpness based on the measured variance. A relationship is established for one dimensional boundaries such that the variance of intensity values ​​along the line profile is related to the apparent unsharpness. Using an assumed intensity profile of length w across a boundary with unsharpness u, we calculate the variance of the line profile illustrated in Figure 15. Here, I m corresponds to the material strength, and I a corresponds to the defect / air strength.

[0097] The intensity of the profile shown in FIG. 15 can be expressed as a function of the profile coordinate:

number

[0098] The variance of the intensity of this profile can be calculated and further estimated as follows:

number

number

[0099] By substituting equation (7) into equation (8) and calculating the integral, it is possible to express the unsharpness u as a function of the variance:

number

number

[0100] Figure 16 compares the apparent unsharpness of a typical profile of a 500 μm cylindrical void in a 400 mm phantom for both reconstruction methods. The 16 pixel line profile is centered on a point on the material-air boundary defined above. Note that the material / air intensity is averaged over a large section away from the boundary. Due to reconstruction artifacts that increase pixel intensity on the material side and decrease pixel intensity on the air side, the apparent unsharpness is measured as the distance between the intersections of the sloping line with the average material / air level. Figure 16 shows the apparent unsharpness of the variable zoom. vz is the unsharpness of the conventional method. c This shows that it is almost three times smaller than the

[0101] FIG. 17 illustrates the normalized variance calculated from the discrete form of Eq. (8) and the unsharpness from Eq. (9) as a function of window size. A w×w pixel support window is used for each point on the hole contour to obtain the maximum variance of the slice. As in FIG. 16, to avoid an overestimation of the unsharpness due to boundary artifacts, the intensity values ​​within the window are fixed to the average material intensity from above and the average air intensity from below. This correction removes the effect of boundary artifacts, so that the measured profile follows that of FIG. 15 and Eq. (9) can be used to calculate the unsharpness. Measurements were performed on 20 sections along the hole depth. Error bars indicate standard deviation. As expected, the maximum normalized variance of the contour increases approximately linearly with window size, while the calculated boundary unsharpness stabilizes after an increase at small window sizes. Comparison with Fig. 16 confirms that the accuracy of estimating unsharpness based on variance as in Fig. 17 shows similar ratios of unsharpness values ​​for the two methods. In addition, it is noted that for all window sizes, the variable zoom technique consistently achieves superior sharpness of the reconstruction compared to the conventional method.

[0102] Measurement-Based Validation of Variable Zoom Techniques Measuring dimensions of composite panels Measurement-based validation was performed on a large aspect ratio carbon / epoxy composite panel shown in Figure 4, manufactured by Boeing using Hexcel prepreg. A 0.5 mm hole was drilled in the center of the panel and the hole diameter was confirmed with a Keyence Digital Microscope VHX-950F. The defect depth was 2.5 mm, which was estimated with a dial test indicator with 0.001 inch (25.4 μm) accuracy. Dimensional as well as unsharpness measurements were performed using the procedures outlined in the previous section.

[0103] CT scans of the panel were generated using an X-ray tube voltage of 180 kV, a 90 μA target current, and an acquisition rate of 1 frame / s. Seven hundred and twenty 3000 × 3000 radiographs were acquired with an angular range of 360 degrees. The magnification factor was homologous to 3 times for conventional CT (optimal voxel size of 46 μm) and varied from 3 times to 10 times for the variable zoom technique.

[0104] Figure 18 provides normalized variance measurements and unsharpness calculation results for 20 sections of drilled holes in a carbon / epoxy composite panel. It is clear that the variable zoom technique provides better sharpness than conventional CT. Both methods exhibit higher unsharpness values ​​compared to the phantom due to the presence of noise in the radiographs.

[0105] Application of the variable zoom technique to composite panels with impact damage Test specimen The variable zoom technique is demonstrated for X-ray CT scanning of pre-impregnated continuous fiber reinforced polymer composite panels that suffered low-velocity impact damage. These specimens represent a challenge for conventional X-ray CT due to their large width-to-thickness aspect ratio. The large size of the panels prevents conventional CT scanning techniques from obtaining the desired spatial resolution in the damage-prone areas, which typically have sizes comparable to the panel thickness. Reliable attenuation of interlaminar defects requires that the reconstructed voxels are orders of magnitude smaller than the panel thickness. [Table 2]

[0106] Each panel has a distinct material system typically encountered in aerospace applications. The first panel is an IM7-carbon / 8552-epoxy composite laminate and the second panel is a hybrid IM7-carbon and S2-glass / 913-epoxy composite laminate. Both panels were manufactured by Boeing using Hexcel prepregs. Table 2 details the CT scanning parameters used to evaluate and compare the variable zoom CT technique and the conventional CT technique.

[0107] Damage detection in carbon / epoxy composite laminates. The first example demonstrates the variable zoom technique for inspection of an impacted area in a carbon / epoxy panel. The width and thickness of the panel are 401 mm and 3.5 mm, respectively, resulting in a width-to-thickness aspect ratio of 114.57. The panel is fitted with an approximately 4×4.5 mm impact area located in the center of the panel. 2 The area sustained low-velocity impact damage.

[0108] Figure 3 illustrates the variation of SOD for conventional and variable zoom trajectories. For the conventional scanning trajectory, the SOD remains constant and is homogenous to 265 mm resulting in a 3x magnification corresponding to an optimal spatial resolution of 46 μm. The SOD for the variable zoom trajectory varies from 81 to 265 mm using a sinusoidal path (Eq. 1), which leads to a maximum of 10x magnification for the impacted area of ​​the 401 mm carbon / epoxy specimen. The reconstruction voxel is chosen for the highest magnification and is homogenous to 14 μm. The 210° angular range of the variable zoom technique (known as short scan) was used to reduce the acquisition time. The trajectory was defined by Eq. (1) with an angular range θ ∈ [−105°, 105°] and the SOD was set to 81 mm for |θ|<15°.

[0109] A qualitative analysis of the reconstruction quality with the variable zoom technique can be performed by examining slices at different positions through the thickness, away from the damaged surface area presented in Figs. 19A-19D. The conventional acquisition mode provides images that lack sharpness, which can be seen in Figs. 19C-19D. Due to insufficient resolution, smaller matrix cracks cannot be distinguished. On the other hand, the variable zoom technique improves the perceived spatial resolution and can distinguish small size defects, as shown in Figs. 19A-19B.

[0110] The same trend emerges by inspecting slices of different depths, as illustrated in Figs. 20A-30B. Note that the improvement in sharpness and clarity of the ablation reconstruction shown in Fig. 20A by the variable zoom technique is truly remarkable. The quality of the through-thickness slices is strongly affected by the lack of angular data, which leads to detectable defects appearing unclear throughout the thickness range. In the spirit of this observation, the zooming technique is expected to improve the resolution of planar slices, which are not necessarily thick slices, as shown in Figs. 19A-19D. The superior quality of the thick slices obtained by the variable zoom technique is due to the weighting provided by equation (6).

[0111] To quantify the sharpness difference between the two methods, we present an analysis of CT slices using the variance estimation method described above. The boundaries of the defect in both the plane and thickness directions were identified by using the contour method. Ten slices adjacent to the slice shown in Figures 19A-20B were used to determine local variance measurements for each point on the contour.

[0112] Due to the extreme complexity of defect boundaries, larger windows contain a larger percentage of boundary pixels than smaller windows, which leads to a slower variance growth and an increase in the unsharpness estimate for larger window sizes. Thus, the comparison of variance values ​​is more meaningful for complex defects. FIG. 21 shows normalized variance measurements for variable zoom and conventional CT reconstructions using the same voxels of 14 μm. Higher variance values ​​for both planar and thickness slices with variable zoom indicate a reconstructed defect with lower unsharpness. For example, the normalized variance Var n = 0.83 corresponds to an unsharpness of 2.6 for a window size w = 10, while the unsharpness for the conventional method is 8.8 pixels. Figure 21 clearly shows that the variable zoom technique provides sharper resolution defects than conventional CT.

[0113] Damage detection in hybrid composite laminates As a second example, the variable zoom technique was applied to characterize the impact damage of a hybrid composite laminate. This material system poses additional challenges due to the large contrast variation between carbon and glass fibers. The laminate has a width of 152 mm and a thickness of 5.2 mm, with an aspect ratio of 29.2 and a dimension of 3 × 3 mm. 2 of impact damage is in the center of the plate.

[0114] The scanning trajectory of this specimen is shown in Figure 3. The SOD for conventional acquisition is 100 mm, while the SOD for the variable zoom trajectory varies from 38 to 100 mm. The relatively low panel width-to-thickness ratio makes it possible to achieve a magnification of 8x, while the variable zoom technique increases the magnification up to 20x. The reconstruction uses an angular range of 360° and a reconstruction voxel of 21 μm for both techniques.

[0115] It is clear that the images obtained by the variable zoom technique show better sharpness as illustrated in Figs. 22A-22D. Despite the relatively high magnification factor utilized in the conventional acquisition mode, the spatial resolution appears to be insufficient to provide the desired reconstruction quality. On the other hand, the variable magnification allows the achievement of higher spatial resolution, which leads to better reconstruction quality. Again, note the excellent sharpness of the thickness section reconstructed by the variable zoom technique shown in Fig. 22C.

[0116] This work has demonstrated a novel X-ray computed tomography technique capable of improving the spatial resolution of non-destructive testing of plate-like objects with large in-plane dimensions relative to the thickness. For example, damage detection in large composite plates can be identified as an important application of the proposed technique. The method incorporates two parts: a non-conventional trajectory of radiograph acquisition and a novel reconstruction weighting. The scanning trajectory includes simultaneous angular rotation and translation of the specimen towards the X-ray source as the specimen dimensions allow. The technique increases the magnification during the scan and, consequently, the overall spatial resolution. The reconstruction is based on the FDK method and incorporates a radiograph weighting based on the distance from the panel to the X-ray source. The proposed method allows reconstructions with excellent quality, especially in the thickness direction.

[0117] Validation of the variable zoom technique for accurate dimensional measurements was demonstrated on phantom models and on CT scans of subjects with measurable features. The estimated dimensions of the drilled holes in the carbon / epoxy panel and the aluminum wires attached to the panel were consistent with those obtained by mechanical measurements. In addition, a quantitative assessment of the sharpness of the reconstructed details illustrated the improved resolution of defects with the variable zoom technique compared to conventional CT. Finally, a feasibility study was performed on two composite laminate panels fabricated from different composite material systems and subjected to low-velocity impact loading. Both panels had relatively large thickness-to-width aspect ratios, a known challenge for conventional CT scanning techniques. Indeed, while conventional CT methods were unable to achieve the spatial resolution required to clearly distinguish smaller impact-induced cracks and delaminations, the variable zoom technique was able to reconstruct volumes with higher spatial resolution leading to better sharpness of the reconstructed slices, including significantly improved defect resolution in the thickness direction. These results were consistent for both material systems under investigation.

[0118] The variable zoom CT method developed in this study has important implications for NDI of materials and structures. On the material side, the method can address the need to achieve maximum resolution of the CT system without destroying the test piece by cutting a small piece that can be placed close enough to the x-ray source for sufficient geometric magnification. There is a similar need for small defected structures that have three-dimensional significant features that can fit into existing CT system housings but are not recognizable using conventional scanning techniques. The new method also provides additional flexibility toward enabling high-resolution CT for larger structures that are not currently suitable for microfocus CT systems.

[0119] Although several embodiments have been provided in this disclosure, it should be understood that the disclosed systems and methods may be embodied in many other specific forms without departing from the spirit and scope of the disclosure. The present examples are to be considered as illustrative and not limiting, and the intention is not to be limited to the details given herein. For example, various elements or components may be combined or integrated into another system, or certain features may be omitted or not implemented.

[0120] Also, the techniques, systems, subsystems, and methods disclosed and illustrated individually or separately in various embodiments may be combined or integrated with other systems, modules, techniques, or methods without departing from the scope of the present disclosure. Other items shown or discussed as being directly coupled or in communication with each other may also be indirectly coupled or in communication, whether electrically, mechanically, or otherwise, through some interface, device, or intermediate component. Other examples of variations, substitutions, and alterations will be ascertainable by those skilled in the art and may be made without departing from the spirit and scope of the present disclosure.

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Claims

1. 1. A method for variable zoom in an X-ray computed tomography (CT) scanner, comprising: emitting an X-ray beam from an X-ray source to project a region of interest (ROI) of the specimen within a field of view (FOV) onto a detector; obtaining a projection of the ROI of the specimen at the detector while rotating the specimen about an axis of rotation of a specimen stage and translating the specimen stage along an acquisition trajectory between the X-ray source and the detector; reconstructing, by a reconstruction computer, a three-dimensional volume of the specimen from the projections scanned by the detector; the acquisition trajectory identifies a source-object distance (SOD) between the x-ray source and the axis of rotation of the specimen stage at each angle of rotation of the specimen stage; reconstructing the three-dimensional volume weighting a set of radiographs based on the SOD at each rotation angle of the specimen stage.

2. The method of claim 1 , wherein the x-ray source and the detector are stationary while the specimen is rotated and translated.

3. The method of claim 1 , wherein the ROI is projected onto a central area of ​​the detector.

4. The method of claim 1 , wherein the acquisition trajectory translates the axis of rotation of the specimen stage along a center of the FOV.

5. The initial SOD along the acquired trajectory is SOD ＲＯＩ and the SOD ＲＯＩ The method of claim 1 , wherein is the closest SOD whose ROI is entirely within the FOV.

6. While the rotation angle of the specimen stage is less than a threshold angle, the SOD ＲＯＩ The method of claim 5 , wherein the ROI is the closest SOD that remains within the FOV.

7. The SOD at each rotation angle of the specimen stage is: [0010] where θ is the rotation angle of the specimen stage, SOD(θ) is the SOD at each rotation angle of the specimen stage, and SOD ＲＯＩ is the initial SOD, and S ０ is the safety offset, and S Ｐ is the sample width, and T Ｐ The method of claim 6, wherein is the specimen thickness.

8. While the rotation angle of the specimen stage is less than the threshold angle, SOD(θ)=SOD ＲＯＩ The method according to claim 7, wherein

9. reconstructing the three-dimensional volume 2. The method of claim 1, comprising weighting a backprojection of a set of filtered radiographs with a weighting factor based on the SOD at each rotation angle of the specimen stage.

10. The weighting factors include: [0025] During the ceremony, w ｖｚ 10. The method of claim 9, wherein: θ is the weighting factor, SOD(θ) is the SOD at each angle of rotation of the specimen stage, and SDD is the source-detector distance.

11. reconstructing the three-dimensional volume calculating a projection onto a volume transform for each projection angle and the SOD to generate the backprojection of the set of filtered radiographs; 10. The method of claim 9, further comprising: adding weighted backprojected pixel values ​​to voxels of the three-dimensional volume based on an interpolation method to generate the reconstruction of the three-dimensional volume.

12. reconstructing the three-dimensional volume Computing a frequency domain ramp filter; 12. The method of claim 11, further comprising: computing weighted and filtered radiographs based on the ramp filter; and applying a cyclic smooth decomposition to generate the set of filtered radiographs.

13. Computing the ramp filter in the frequency domain includes computing a one dimensional direct Fourier transform for: [0030] In the formula, n is an integer. [0045] and p ｘ is the row pixel spacing, [0050] rounded up to the next power of two, n ｘ The method of claim 12 , wherein x is the number of pixels in a projected row.

14. The method of claim 11 , wherein when calculating the projection onto the volume transform, the projection coordinates are different for each projection angle as SOD(θ) is varied.

15. Calculating the projection onto the volume transformation 3D coordinate transformation (x, y, z) Ｔ = R(θ)R Ｖ (t, s, r) Ｔ where (t,s,r) are the reconstructed volume coordinates, (x,y,z) are the projection coordinates, and R Ｖ is the volume transformation matrix, and R θ The method of claim 14 , wherein x is a matrix of sample rotations.

16. The method of claim 11 , wherein the interpolation method is a distance-driven approach or a separable footprint approach.

17. 12. The method of claim 11, wherein the weighted and filtered radiograph is weighted to account for different ray lengths of a cone x-ray beam.

18. The computing the weighted and filtered radiograph and applying a cyclically smooth decomposition includes computing: S θ (x, y ｋ )=[P' θ (x, y ｋ ) * h(x)] = p ｘ IFFT {FFTP' θ (x, y ｋ ) ＺＰ FFTh [np ｘ ] ｓｈｉｆｔ }, and [006] where FFT is the one-dimensional direct Fourier transform, IFFT is the one-dimensional inverse discrete Fourier transform, and h[np ｘ ] ｓｈｉｆｔ is the swapped ramp filter h[np ｘ ] is a half-space of [0070] 18. The method of claim 17, wherein x is a zero padded radiograph to avoid periodic artifacts and p is the periodic smooth decomposition such that only periodic portions of the weighted radiograph boundary are used.

19. Adding the weighted backprojected pixel values ​​to voxels of the three-dimensional volume based on an interpolation method includes calculating: [0080] During the ceremony, [0090] v(t,s,r) is the reconstructed volume value, the sum is calculated for every coordinate triple (t,s,r), and the interpolated value v(t,s,r) is obtained using the interpolation method, w ｖｚ (θ) is the weighting coefficient, and S θ (xz ｄ , yz ｄ 12. The method of claim 11, wherein the image is a filtered radiograph.

Images