A combination interferometric and morphological image processing approach to rapid quality assessment of additively manufactured cellular truss core components

Advanced manufacturing (AM) processes such as laser powder bed fusion (LPBF) are increasingly capable of fabricating components with useful and unprecedented mechanical properties by incorporating complex internal bracing structures. From the standpoint of quality control and assessment, however, internally complex assemblies present significant build-verification challenges. Here we propose a hybrid approach to the inspection involving the application of computer-aided speckle interferometry (CASI) and morphological image processing as a rapid, inexpensive, and facile method for AM quality control. The described methodology has low capital equipment costs, is full-field and non-contact, can be used in an industrial setting, and has very low requirements in terms of operator training and expertise. Consisting primarily of the combination of image processing software with a simple optical system of variable sensitivity, the method is shown to be effective for inspection of a titanium honeycomb component subjected to differential pressure. Results are compared to those achieved with computed tomography (CT), immersion ultrasound testing (UT), and optical holographic interferometry. Lastly, we propose several possible processing strategies for automated quality assessment based on this powerful hybrid approach.


Introduction
With the maturation of advanced manufacturing (AM) technology, it is now possible to manufacture lightweight components with internal truss supports that are highly optimized for mechanical performance. These designs, typically referred to as "cellular truss core structures," offer a lightweight alternative to solid components while simultaneously exhibiting greater strength than a fully hollow structure. In some cases, these structures have even shown superior mechanical and/or thermal properties compared to their solid counterparts for the desired application [1]. Using AM methods to make cellular core structures can further improve structures' mechanical strength by reducing the number of weld seams, thus limiting the number of fatigue initiation locations [2]. The advantages offered by these designs are attractive for many applications, notably the fabrication of strong, lightweight structures required by the aerospace industry [3,4].
Unlike traditional subtractive methods of manufacturing, AM technology enables the manufacture of cellular truss cores with almost arbitrary complexity such as honeycomb components with irregular prismatic cells and variational infill patterns, thereby allowing their mechanical responses to be optimized for a particular application [5][6][7][8][9][10]. This complexity places a large emphasis on post-build verification which, in turn, poses new challenges for non-destructive validation and testing, especially in a high-volume industrial setting where throughput (i.e., simple pass/fail acceptance testing) and automation may be overriding concerns. Devising practical methods to test for internal flaws in these components is therefore essential [11][12][13][14]. Classical methods of non-destructive evaluation such as X-ray computed tomography (CT) and ultrasound testing (UT) can be challenging to deploy in a high-volume environment. CT can be effective at identifying build flaws such as excess porosity and missing or incomplete internal structures but is less effective at identifying cracks or zero-volume flaws (i.e., complete but unbonded layers). Furthermore, X-ray CT scanning can be time-consuming and often requires a trade-off between component size and scan resolution. Perhaps most importantly, CT equipment is often expensive to deploy since it requires operation in bulky safety cabinets that can occupy manufacturing floor space, and skilled operators are necessary for proper operation and interpretation of results. UT is generally less expensive than X-ray CT, poses no safety hazard, and is very effective at localizing flaws such as cracks and debonds. In many cases, however, the inspection must be conducted within an immersion tank, requiring components to be surrounded by an acoustically conductive medium such as mineral oil, which may or may not be acceptable. Results can have high spatial resolution (~ micrometric), but the scans are often time-consuming, and the inspection tank must be able to physically accommodate (and manipulate) the component, which may be large, heavy, or both.
Here, we describe an inspection methodology that combines an optical technique originally conceived for experimental mechanics research, computer-aided speckle interferometry (CASI), with common image processing morphological filters. We apply this approach to the problem of rapid non-contact full-field inspection of thin-shelled cellular truss core structures. By leveraging modern highspeed computers, commodity machine vision cameras, and coherent illumination, we illustrate how the method can be automated. The technique shares characteristics with shearography [15,16] but benefits from key features that allow for simplified optical setup and data analysis. As with shearography, the approach described here hinges on the application of small magnitude stress to induce micron-level (or smaller) deflections in the surface of an illuminated component. Because the induced strain is small and elastic, the technique is considered to be non-destructive. The approach is useful for high-volume automated quality control and part validation. The driving assumption is the same as that which underlies shearography, namely, that the observation of unexpected non-uniformities in a deformation field is indicative of manufacturing anomalies. More specifically, a priori knowledge regarding the design of internal truss cores, in combination with the application of hydrostatic pressure, will manifest as a predictable array of features in the resulting deformation field image. An altered mechanical response to an imposed stress causes a departure from this target pattern in the deformation image and therefore indicates the presence of a manufacturing defect. The location of the anomaly in the deformation image indicates the defect's approximate location, and the overall size of the anomaly is an indication of its severity. Due to the sensitivity of the method, and because the technique directly images mechanical response in the presence of manufacturing flaws rather than directly imaging the flaws themselves, manufacturing defects that do not result in an anomalous deformation image represent flaws with minimal impact on mechanical performance. These flaws may, in many cases, be safely disregarded for the purpose of qualifying the component for effective operation over its lifetime. An additional advantage of the approach, therefore, is a reduction in the fraction of manufactured product that is unnecessarily rejected.

Laser speckle formation
An optically rough object is one whose surface microrelief is greater than the wavelength of illuminating light. When such an object is illuminated by a diffused laser, or any coherent light source, a grainy optical field referred to as a "laser speckle pattern" is observed [17]. This pattern is the result of the coherent superposition of countless wavelets scattered randomly due to the microrelief of the scattering surface. This random scattering replaces the original phase relationship of the impinging wave with randomly phased wavelets. The spatial distribution of the resulting optical amplitude field is a stochastic, non-deterministic array of smoothly varying speckles ranging from low to high amplitude. Due to this formation mechanism, clusters of speckles can be used to uniquely characterize a local region of the underlying surface. Any deformation or movement of the object surface (in-plane and/or out-of-plane) results in an associated movement of the speckle pattern. This behavior then allows for a non-contact method for characterization of strain and is the basis for a family of techniques known as speckle interferometry [17][18][19][20] that have been widely employed in experimental mechanics research. An example of a typical speckle pattern is shown in Fig. 1.
The amplitude of an individual speckle is a function of the resulting overall interference of its contributing interfering wavefronts. The speckle appears dark if the cumulative total interference is destructive, and the speckle appears bright if the cumulative interference is constructive. Because this speckle effect results wherever there exists a superposition of optical waves, and since the optical wavefront itself propagates through three-dimensional space, a speckle pattern is a three-dimensional phenomenon that fills the space between the scattering surface and the imaging system. Individual speckles are generally ellipsoid in shape with the largest dimension of the ellipse in the direction of the primary scattering [17]. Two types of speckle patterns can be observed: objective and subjective. An objective speckle pattern is generated by placing an imaging surface directly in the path of the scattered wavefront, allowing the speckle pattern to form directly on its surface. In this case, the average transverse size of the speckle (i.e., the size in a direction 1 3 orthogonal to the primary direction of propagation) is given [18] by where λ is the wavelength of the illumination, l 1 is the distance from the scattering surface to the imaging plane, and D 1 is the maximum dimension of the illuminated scattering surface. The average size of the speckle parallel to the primary direction of the scattering is given [18] by In both cases, the average speckle size is fixed according to the imaging geometry itself. In contrast, a subjective speckle pattern is one which is generated via an optical imaging system with a lens. In this case, the size of the speckle in the transverse direction is given [18] by where q is the distance from the lens to the imaging plane and D 2 is the diameter of the input aperture of the lens. From Eq. 3, it can be seen that the size of the speckle is inversely proportional to the aperture diameter, thereby allowing for an easy way to control its size (hence the term "subjective").
The three-dimensional nature of a laser speckle pattern allows for an optical system to focus on an arbitrary plane in the space between the illuminated surface and the optical system itself. The speckle pattern will be altered randomly as the focal plane is displaced, but a high-contrast speckle pattern can always be obtained. Furthermore, as the illuminated surface is rotated out-of-plane it behaves as a mirror, converting the out-of-plane motion of the surface into in-plane translation of the speckle pattern at some offset focusing plane. Therefore, speckle patterns can be used to determine both in-plane displacement and out-of-plane tilt and deflection, depending on the optical configuration of the imaging system.
The use of speckle pattern offset focusing for tilt detection is illustrated in Fig. 2a, which shows the case where the surface area is illuminated by a collimated source, while the imaging sensor is positioned at the Fourier transform plane [21] of the lens. It has been shown [20] that in this configuration, any in-plane motion of the speckle registered by the sensor is due entirely to out-of-plane rotation of the illuminated surface. In-plane motion of the scattering surface will primarily cause the speckle pattern to decorrelate-only tilting will manifest as coordinated speckle motion in this configuration. Additionally, from Fig. 2a, it can be seen that the sensitivity of the method is proportional to the distance between the illuminated surface and the imaging system. In practice, exploiting this characteristic to gain or reduce sensitivity is cumbersome as it involves altering the physical layout of the imaging system. Furthermore, the requirement that the illumination be collimated may require the use of very large and expensive optics. For these reasons, this method of inspection is not widely used. Figure 2b illustrates an alternative optical configuration for detecting out-of-plane rotation initially proposed by Gregory [22]. If a diverging wavefront is used for object illumination, Gregory showed that speckle motion resulting from out-of-plane rotation can be isolated from inplane motion by configuring the optical system such that it focuses on a virtual source behind the surface at a distance corresponding to the optical path length between the real point source and the object surface. In practice, this can be arranged by placing a mirror on the object surface and adjusting the optical system so that the light source is in focus. As with the previous method shown in Fig. 2a, in-plane motion of the scattering surface does not result in organized speckle motion but, rather, speckle pattern decorrelation. Since this method relaxes the requirement for collimated source illumination, it is much more practical to implement. Unfortunately, the requirement of a fixed focal plane once again requires that the optical layout-in this case, the positioning of the light source-be modified in order to modulate the system's sensitivity. Figure 2c shows a third possible arrangement for out-ofplane rotation detection, as proposed by Chiang and Juang [23], that eliminates the need for cumbersome and expensive collimating optics while simultaneously allowing for variable sensitivity without modifying the physical optical layout. As previously described, a laser speckle pattern is a three-dimensional phenomenon that fills the space between an illuminated surface and an imaging system. By focusing on an offset plane within this space, out-of-plane rotation is converted to in-plane displacement of the speckle field. From Fig. 2c, it can be seen that the greater the offset focal distance from the scattering surface, the greater the in-plane speckle motion (as registered by the imaging sensor) due to out-of-plane rotation. Therefore, by simply altering the location of the front focal plane of the imaging system, the sensitivity of the technique to out-of-plane rotation can be easily modulated, while the rest of the system remains fixed. If the angle of the out-of-plane rotation is small enough such that it's tangent can be approximated by its magnitude, the relationship between the degree of defocus and the sensitivity (as given by the in-plane translation of the speckles) is essentially linear [23]. The drawback to this method, however, is the inability to fully decouple in-plane motion with out-of-plane rotation, as both will result in motion of the speckle pattern. In practice, the ambiguity that arises from this mixed-mode behavior can be dealt with in one of two ways: (1) incorporate a second camera to image the speckle pattern very close to the surface of the object as in Fig. 1 (thereby minimizing its sensitivity to out-of-plane rotation) and use it to simultaneously monitor the in-plane displacement behavior of the speckle pattern for the purposes of gauging its contribution to the total motion registered on the defocused sensor or (2) apply the technique only in situations where the geometry of the object and the nature of the applied stress guarantee that out-of-plane motion will be the dominant deformation mode. This latter option is the approach adopted in this work.

Holographic interferometry
Since the advent of the laser, a large family of techniques has been cultivated to leverage its convenience, power, and stability as a coherent light source for the purposes of optical interferometry. These techniques are particularly useful in the field of experimental mechanics (the experimental determination of stress and strain) and can be separated into two broad categories: holographic interferometry and speckle interferometry. Holographic interferometry involves superimposing an object's reconstructed wavefront-captured via a hologram of the object in its reference state-with its instantaneous wavefront after the application of stress [20,[24][25][26]. The interference of these two wavefronts results in a fringe pattern that encodes the three-dimensional deformation of the object with fractional wavelength sensitivity. The method is full-field and non-contact and can be applied to any object provided that its surface is not purely specular. An example optical layout for the generation of holograms via the classic Leith-Upatnieks side-band Fresnel method [27][28][29] is shown in Fig. 3a. The laser source is split into two beams: an object beam that illuminates the object and an off-axis reference beam that directly illuminates the photographic plate in which the hologram is stored. The interference between the object wave scattered from the object surface and the reference wave is recorded by the hologram plate which, when developed, acts as a diffraction grating. By illuminating the hologram plate with the reference wave, the original object wavefront is reconstructed [30][31][32]. Alternatively, by illuminating the hologram plate with the reference wave while simultaneously maintaining illumination of the object as it is subjected to stress, the interference between the reconstructed object wavefront stored in the hologram and its instantaneous deformed wavefront can be visualized as a fringe pattern. This is shown in Fig. 3b, where a pressure cylinder has been pressurized after recording a hologram of its unpressurized (reference) state. The deformation of the resulting fringe pattern due to the mechanical influence of the clamps on either side of the cylinder at its base is clearly visible. Drawbacks to this form of interferometry include an extreme sensitivity to environmental vibrations and thermal drift which makes the method difficult in practice, as well as the general difficulty in interpreting the fringe pattern to determine true quantitative displacements, which limits the potential for automation.

Single-beam speckle interferometry and CASI
In parallel to the development of holographic interferometry was the development of an alternative family of techniques that exploited the unusual qualities of a laser speckle pattern to perform full-field non-contact analysis with high sensitivity. Most of the methods developed, such as electronic speckle pattern interferometry (ESPI) [33,34], are similar to holographic interferometry in their use of two beams. However, driven by the desire to simplify the implementation of these methods, singlebeam approaches soon emerged. These were an attempt to address the complexity associated with the need for two incident beams by replacing them instead with two speckle images produced by a single beam. The fundamental method that was initially developed was a single-beam The resulting fringe pattern encodes the local in-plane displacement of the speckles speckle diffraction technique that arose from the work of Burch and Tokarski [35] and later refined by Archbold and Ennos [36]. This technique is illustrated in Fig. 4. It involves illuminating an optically rough object with a diverging laser beam to generate a speckle pattern (as in Fig. 1), which is then recorded on a photographic negative. The object is then subjected to some form of stress, and a second speckle pattern is recorded on the same negative, generating what is referred to as a double-exposure specklegram. If the strain experienced by the object is not overly large, the result is a specklegram consisting of locally translated clusters of speckles. These speckles act as tiny, paired apertures on the photographic negative that, when illuminated by a plane wave (such as a collimated laser), cause the beam to diffract. If an imaging screen is placed at a sufficient distance from the opposite side of the negative to that which is being illuminated-or an imaging lens is used-an interference pattern consisting of parallel fringes is observed. These are essentially the same as Young's fringes resulting from a dual-slit setup. The pitch of the fringes is inversely proportional to the local displacement, and the fringe rotation indicates the direction of the displacement to within a ± 180-degree ambiguity (the direction of displacement is perpendicular to the major axis of the fringe). The displacement at the point of illumination is given by where λ is the illumination wavelength, F l is the focal length of the optical processing step, M is the system's optical magnification, and P f is the fringe pitch. By rastering the analysis beam across the specklegram, the entire displacement field can be obtained. The pointwise filtering process of a double-exposure specklegram is illustrated in Fig. 4. With the advent of affordable CCD cameras, Chen and Chiang, Chen, and Chen et al. [37][38][39][40] demonstrated that the entire process of single-beam speckle diffraction analysis could be automated while simultaneously retaining all of its advantages over holographic interferometry. The algorithm they proposed, computer-aided speckle interferometry (CASI), replaces the double-exposure specklegram with two discrete digitized speckle patterns taken in succession. Likewise, the local analysis beam of Fig. 4 is replaced by local subimage pairs extracted from their parent images, while a mathematical model is used to simulate the fringe-forming interference of the single-beam diffraction method to extract the translation of the local speckle cluster. Consider two subimages of equal dimensions extracted from two speckle patterns, before and after the application of stress: where u and v are the displacements in the x-and y-direction, respectively, and n(x, y) is an additive uncorrelated noise component that is assumed to be small and is often therefore disregarded. The frequency domain representation (via Fourier transform) is written as for the first subimage and for the second subimage. A new spectrum, in the form of a normalized cross-power spectrum of the two subimage spectra, is obtained by where ε * ε indicates the complex conjugate. A second Fourier transform gives rise [40] to the secondary frequency domain (ξ,η) representation of this new spectrum: By computing the magnitude of G( , ) , an expanded impulse function centered at the displacement point (u, v) in the second spectral domain is obtained. By locating the crest of this single impulse function, the direction and magnitude of the deformation between speckle patterns are uniquely determined. A detailed exposition of the derivation can be found in [40].
In practice [41], implementation of the CASI algorithm involves partitioning the input speckle patterns into a grid of subimage pairs (subimage size is typically 32 × 32 or 64 × 64 pixels) and applying the above mathematical model to determine the mutual displacement between any given subimage pair. In doing so, the two-dimensional displacement field is resolved.

Methodology
With the optical and mathematical origins of laser speckle patterns and the CASI algorithm having been described, we now illustrate how the two can be used in combination for post-build validation of cellular truss core components. The method proposed here is similar to shearography [15,16] in that both approaches generate contour fringes representing surface deflection slope isoclines that result in response to stress. The method described in this work, however, benefits from some key advantages when compared to shearography. While both methods are true interferometers, shearography realizes the interference optically via a shearing camera whose sensitivity is a function of the illumination wavelength and is, therefore, essentially fixed. In comparison, the combination laser speckle and CASI method mathematically model the interference within a computer, thereby freeing the optical system and allowing adjustment of the offset focusing distance to be used as a means to easily modulate system sensitivity for a given application. Furthermore, this method is easily assembled, utilizes simple optics requiring little expertise to arrange, is adaptable to objects of almost any size, can be implemented with readily available commercial software, and has low capital equipment costs.
Consider a cellular truss core component such as the one illustrated in Fig. 5. If subjected to differential hydrostatic pressure-either by introducing pressure into the interior of the component or by placing the (sealed) component in an evacuated environment-the unsupported surfaces will deflect in response to the hydrostatic stress. If the surface of the component is illuminated by an optical system like that shown in Fig. 2c, the out-of-plane deflection resulting from the hydrostatic stress is translated to in-plane displacement of the speckle field at some offset focal plane between the surface and the camera lens. For any given offset plane, a larger surface slope will always result in greater observed in-plane translation of the speckle pattern, and by simple geometry, it can be seen that this effect is magnified as the focal offset distance from the surface grows. Internal cellular truss core layouts are typically regular and periodic. Therefore, the deflection pattern can be expected to be regular and periodic as well. If we consider the component surface to be a distributed linear continuum that gives rise to a deflection field described by some function f (x, y) that is differentiable everywhere, then, the in-plane speckle displacements will result in correlation fringes that are spatially related to the gradient vector of this deflection amplitude function, specifically where i and j are unit vectors in the x -and y-direction, respectively. The gradient vector is a function of both directional vectors ⃗ i and ⃗ j , and therefore, the overall magnitude of the surface gradient is computed as Consider an in-plane two-dimensional displacement field D derived from CASI analysis of an offset-focused speckle pattern (and therefore a function of the in-plane displacement response of the speckle pattern): where u and v are the translations of a local cluster of speckles in the x-and y-directions, respectively, and are linearly proportional to the partial out-of-plane deflection derivatives in the offset-focused configuration used in this study: In-plane translation of offset-focused speckle pattern in response to surface deflection of internal truss core component (due to the application of hydrostatic stress) It follows that the deflection gradient magnitude is directly proportional to the Euclidean distance of the in-plane displacements: From the previous discussion, it can be seen that the slope field can be indirectly visualized by computing the Euclidean distance of the two-dimensional displacement field of an offset-focused speckle pattern and visualizing these values as a grayscaled image (we say "indirectly" because it is the Euclidean distance that is ultimately being visualized here and not the true quantitative value of the slope per se). Herein, we refer to such an image as the proportional slope image. It is generated via the application of differential stress, followed by CASI processing of an offset-focused speckle pattern.
Clearly, the magnitude of the computed Euclidean distance is dependent on multiple factors, including the object material and geometry, the degree of differential pressure being applied, the degree of offset focusing, and the presence of any local manufacturing defects that might compromise the local strength of the part. Of these, the only parameters that can be readily adjusted are the differential pressure and the offset focusing distance, which directly controls the system sensitivity and needs to be sufficiently large for visualization. Likewise, the differential pressure needs to be sufficient to induce a detectable slope deformation field for a given optical configuration, but not so large as to induce new flaws. There is no universal method for pre-optimizing these two parameters since they depend purely on the component under inspection. They must be chosen empirically and guided by experience.
Based on the previous discussion, the approach for internal truss core inspection advocated here consists of the following steps: (1) Mount the component to be inspected in a manner similar to that depicted in Fig. 2c, with its surface normal coinciding with the camera's optical axis. (2) Adjust the camera focus to an offset plane between the object surface and the camera (object specific). (3) Capture a speckle pattern of the object in its reference (unstressed) state and again after the application of differential pressure (object specific). (4) Process image pair via CASI algorithm and compute the proportional slope field via Eq. 13. (5) Generate a proportional slope image by assigning a grayscale value to the gradient data. (6) Post-process proportional slope image for geometric anomalies and irregularities.

Test object
To illustrate the defocused laser speckle inspection method, an internal truss test object was prepared. The test object was designed to imitate an AM aerospace component with internal lattice supports. A honeycomb core optical mirror, similar to that shown in Hilpert et al. [3], was used as the foundation for the object model, with a honeycomb web thickness of 0.61 mm. The honeycomb truss was designed with perforations in the webbing so that unprocessed build material could be removed. This also allowed the use of internal pressurization for the speckle inspection. (Alternately, the object could be placed in an evacuated chamber for inspection.) The test object was designed with an integrated hose barb so that hydrostatic pressure could be introduced into the interior. It was fabricated out from titanium (TI 6Al-4 V) by Protolabs Corp. using the direct metal laser sintering process. Figure 6 shows an exploded view of the test object, while Fig. 7 lists the various dimensions and wall thicknesses in inches.
Internal features were incorporated into the design to simulate potential defects that may occur during the manufacturing process. Figure 8 shows the computer-aided design (CAD) model with front face removed to highlight the three simulated internal defects that were built into the part: feature "a" mimics an internal crack, feature "b" represents a partially bonded interface, and feature "c" simulates a missing layer of material or a completely disbonded wall-face interface.

Immersion ultrasound analysis
The honeycomb test object was inspected using a commercially available scanning (6-axis) immersion ultrasound system (MISTRAS Group, Inc.). We used a spherically focused immersion sensor with a center frequency of 20 MHz, 9.5 mm aperture, and 35 mm focal length. The part was scanned in pulse-echo mode through its thickness with the sensor directed at the top of the part proximate to the simulated defects ("top plate"). The height of the sensor (z-axis) was adjusted so that the backwall echo of the "top plate" was maximized, which sets the focal point of the beam on the back of the "top plate." The sample was scanned using a unidirectional grid in the x-y plane, where the grid spacing was 0.254 mm 2 . At each inspection location on the x-y scanning grid, 10 waveforms were averaged, digitally filtered with a bandpass filter 5-35 MHz, digitized at 250 MHz, and recorded as a single amplitude time trace (A-scan).
When the sensor is located over a portion of the top plate that does not have supporting structure connected to it, the sound bounces between the front and backwalls of the top plate and multiple acoustic echoes are observed, as shown in the A-scan in Fig. 9a labeled "No Support." Alternatively, when the probe is located over a portion of the "top plate" that has lattice bonded to its underside, the ultrasound leaks into the strut material and the backwall echo is greatly diminished, as seen in the A-scan in Fig. 9a labeled "Supported." The UT data is processed by normalizing each A-scan such that the peak absolute magnitude is unity and occurs at time = 0 μs. Then, the root square sum (RSS) is calculated between t = 0.6 and 2.6 μs since this is when the echoes from the "top plate" backwall occur. The RSS is shown on the color scale in the C-scan image, Fig. 9b. Here, the location of the honeycomb lattice network is clearly observed by the regions with a reduced RSS.
The mid-lattice crack defect A, shown in Fig. 8(a), is not detected in the UT C-scan. This is expected since the defect lies midway through the lattice, and no ultrasound echoes were observed from sound that propagated through the lattice and reflected off the mid-lattice hole or the "back plate." Because of the focused beam and the size and geometry of the struts, any ultrasound that leaks into the struts does not get coherently reflected back to the sensor. The simulated layer bonding defect C, shown in Fig. 8(c), was detected in the UT C-scan as shown in Interestingly, the simulated partial fusion bond flaw B, shown in Fig. 8(b), was not detected in the ultrasound C-scan. It is likely that the transducer spot size was too big compared to the percent of unbonded strut area. With further data processing it may be possible to detect this feature with statistical significance; however, this was not the focus of the present work.

X-ray computed tomography analysis
X-ray CT was performed using an in-house-built X-ray system. This system uses a Yxlon MGC41 X-ray generator and a PerkinElmer 1621 AN14 ES detector. The X-ray source was operated at 300 kV peak energy and 2.3 mA, using a 6-mm Cu filter. Using a magnification of approximately 1.27, a total of 1800 projections were acquired at equally spaced angles as the object was rotated 360°. For each projection, 12 frames were averaged with exposure times of 1.3 s to generate the digital radiograph at that orientation. The 3D volume was reconstructed using Livermore tomography tools (LTT) [42] with an effective voxel size of 157 µm. The three flaw locations were then inspected at various angles and cross-sections to gauge their visibility. Figure 10 shows representative results, with the flaws circled. Figure 10a is a cross-section through the centerline of the sidewall holes at the site of flaw A, Fig. 10b is the image of a cross-section plane containing flaw B, and Fig. 10c is the image of a cross-section through the plane containing flaw C. Based on the CT imagery, all three flaws are verified to be presented and as designed, though not necessarily easily identifiable; flaw C is revealed only by a subtle grayscale shift that, in practice, may escape post-build inspection (particularly if the component is large).

Real-time holographic interferometry analysis
Prior to implementing the experimental methodology outlined in Section 3.1, real-time holographic interferometry was used to generate a ground truth deformation image of the test object under differential pressure conditions. The test object was rigidly mounted in a vertical position in a standard side-band Fresnel optical arrangement for hologram (re) construction. Holographic plates from LitiHolo™ Corp. were used for hologram recording. These were 2-mm glass plates with a pre-applied 16-µm photopolymer film (C-RT20 instant holographic film) on a 60-µm plastic substrate, and plate dimensions of 6 in. × 8 in. (width × height). The holographic film is photopolymerizing, such that no wet chemical processing was needed. This allows the plate to remain in place during photo-exposure and subsequent reconstruction, thus creating ideal conditions for real-time holographic interferometry. The camera used in this study was a FLIR Oryx ORX-10G-310S9M CMOS camera with resolution of 6464 × 4852 pixels (W × H). The camera's optical axis was positioned normal and centered relative to the surface of the honeycomb test object, with the holographic plate positioned approximately 5 in. in front of the camera lens such that it filled the camera field of view. A Coherent Sabre™ argon laser emitting at the 514 nm line was used as the illumination source. Figure 11 illustrates the optical arrangement.
A hologram of the test object in its reference state (i.e., zero differential pressure) was recorded by exposing the hologram plate for approximately 2 min. Pressure was then slowly introduced into the interior of the test object using a manual air pump, while the fringe formation, as imaged by the CMOS camera, was observed on the computer monitor in real time. This process continued until clear fringes were visible, at which point the pressure increase was arrested. The final differential pressure was 40 psi. The fringe pattern that formed in response to this pressure encodes the surface deflection and is shown in Fig. 12a. The deflection fringe pattern consists of a sequence of low-frequency fringes corresponding to the overall shape distortion, superimposed with a high-frequency fringe pattern corresponding to the honeycomb internal truss deflection pattern. The high-frequency pattern has the primary diagnostic value.
To estimate the out-of-plane deflection resulting from the hydrostatic pressure, a line profile of grayscale values was extracted and plotted. To minimize any contribution from the global (low-frequency) fringe pattern, the region chosen for profile extraction was selected such that it was fully contained within a low-frequency fringe, with the fringe's direction parallel to the line profile path. The underlying region is shown highlighted in Fig. 12a, along with the interpretation of the high-frequency honeycomb fringe pattern in Fig. 12b. From the geometry of the honeycomb component, it is known a priori that the region corresponding to a vertical interior support will correspond to a local deflection minimum, while the central unsupported portion overlying each honeycomb cell will correspond to a local deflection maximum. Referring to the grayscale overlay plot in Fig. 12b, it can be seen that the central portion of a bright fringe corresponds to the local deflection maximum and then gradually transitions to a dark fringe as one traverses from left to right, ultimately reaching a minimum grayscale value before reversing direction and briefly climbing again. This  reversal of direction is the result of the optical path length between the reference and deformed wavefronts exceeding 180° phase shift and is mirrored about the long axis of a hexagonal cell wall, a feature which is known a priori to be a local deflection minimum. The total cumulative phase shift from the center of the hexagonal cell to its supporting wall consists of the initial 180° phase shift from maximum to minimum, plus the reversal portion which is approximately 25% the height of the main lobe (25% of a 180° phase shift or 45°). Therefore, the total pressure-induced angular phase shift from the center of the honeycomb cell to the vertical support is approximately 225° or ~ 63% of the source wavelength. Considering the 514-nm illumination wavelength, this corresponds to an out-of-plane deflection of approximately 320 nm.
The diagnostic utility of holographic interferometry for this application can be evaluated by examining Fig. 13, where the three areas overlying the internal flaws are shown in detail. Flaw C (see Fig. 8) is clearly visible as an irregularity in the fringe pattern. Flaws A and B, however, are not readily apparent despite the sensitivity of the method, leading to the conclusion that these two flaws do not sufficiently alter the mechanical behavior of the component as to become visible as fringe pattern irregularities at these stress levels.

Offset-focused CASI analysis
With the presence of the build flaws confirmed via UT (Section 4.1) and CT (Section 4.2) and having established the ground truth deflection behavior via holographic interferometry (Section 4.3), we now introduce the CASI analysis of the offset-focused speckle pattern. As in the holographic interferometry analysis, the honeycomb test object was mounted vertically and illuminated off-axis with a Coherent Sabre™ argon laser at the 514-nm line. The illumination angle is almost arbitrary given a sufficiently defuse surface: one needs only to avoid specular highlighting to achieve a high-quality speckle pattern. The camera was positioned approximately 48 in. from the test object such that it filled the vertical field of view. The offset focus distance was chosen empirically, similar to the differential pressure level for the holographic interferometry. That is, the interior of the object was pressurized to 40 psi, and the computer screen was monitored for visible speckle motion (analogous to monitoring for fringe formation in the case of holographic analysis). If no appreciable motion was observed, the differential pressure was returned to zero, the offset focusing distance increased by 1 in., and the test object was repressurized. This procedure was repeated until reasonable speckle motion (on the order of 5-10 pixels) was observed in response to the differential pressure. In practice, where the mechanical response of the object is not known a priori, this method is effective for calibrating the optical offset parameter. In this case, the final offset distance chosen for the honeycomb test object under 40 psi differential pressure was approximately 15 in. from the surface of the object (see Fig. 14).
Two images were collected: a reference image of the test object at zero differential pressure and a "dynamic" image at 40 psi internal hydrostatic pressure. This image pair was processed via the CASI algorithm as described in Section 3.1. Subimage pairs of 32 × 32 pixels/subimage were extracted and processed on a 4-pixel grid (every 4 pixels in the vertical and horizontal directions) to generate the u and v displacement fields on the offset focal plane resulting from the hydrostatic pressure. The Euclidean distance resulting from the u and v displacements at each analysis point was then computed according to Eq. 14. Finally, a 16-bit linear grayscale (0 for minimum Euclidean distance, 65,535 for maximum distance) was assigned to the data and used to generate the proportional slope image. The result is shown in Fig. 15a. When compared to the results achieved with holographic interferometry (Fig. 12a), the interior honeycomb cell layout is much more apparent. The presence of the lowfrequency fringe pattern in Fig. 12 tends to obscure honeycomb cell boundaries (which also appear more circular in the holographic interferogram). The low-frequency fringes in the holographic interferogram have the added effect of periodically inverting the local high-frequency fringe pattern of the honeycomb cell structure, making automated (or even manual) comparisons between honeycomb cells difficult. In contrast, the lack of low-frequency global fringes in the CASI result of Fig. 15a facilitates honeycomb cell boundary identification. In addition to superior internal layout visualization, the offset-focused speckle method is very tolerant of off-axis imaging geometries. Figure 15b illustrates the results achieved when the imaging axis is rotated 45° relative to the surface normal. The internal honeycomb pattern maintains high visibility, while sidewall deformation can now be simultaneously visualized. The clarity of the result serves to emphasize the ease of optical setup: non-planar components are as amenable to inspection by this method as are planar ones. In other words, the arrangement of the optical components is by no means critical since the generated imagery is evaluated qualitatively in any case for the presence of flaws.
To illustrate the results that can be achieved under even lower differential pressures, the previous analysis was repeated for both holographic interferometry and offset-focused CASI but at 10 psi hydrostatic pressure. Results for both conditions (10 psi and 40 psi loadings) are shown in Fig. 16a (10 psi) and b (40 psi) for holographic interferometry and Fig. 16c (10 psi) and d (40 psi) for offset-focused CASI. The holographic interferogram of Fig. 16a shows that the honeycomb pattern and flaw C are barely visible at 10 psi differential pressure. By inspection, the same subregion as that depicted in Fig. 12 appears to undergo approximately 1/4-wavelength cycle from the honeycomb support to the center of the adjacent cells. We estimate the out-of-plane deflection in this case to be on the order of 130 nm. By contrasting the 10 psi holographic interferogram of Fig. 16a with the 10 psi result generated via the offset-focused CASI method in Fig. 16c, it can be seen that the interior honeycomb layout is easily visible with good cell boundary definition, and flaw C is as apparent in the 10 psi image as it is in the 40 psi image.
As in Fig. 13, the magnified view of the regions directly overlying the three build flaws is shown in Fig. 17. Flaw C is easily visible in the proportional slope image, whereas flaws A and B are again undetectable. Just as in the holographic interferometry analysis, any influence flaws A and B have on the mechanical behavior of the test object appears to be minimal under the current loading conditions when analyzed via the offset-focused CASI method.

Morphological post-processing
Having seen how a simple optical inspection method can be used for flaw detection, we now address the post-processing of the data imagery. The quality of the proportional slope image (Fig. 15) is high enough that post-build validation can likely be satisfied Fig. 16 Holographic interferograms for a 10 psi differential pressure and b 40 psi differential pressure. Offset-focused CASI images for c 10 psi differential pressure and d 40 psi differential pressure through simple manual image inspection. We acknowledge, however, that in some cases a more rigidly defined and repeatable methodology may be desired. For such cases, a straightforward approach based on particle/cell analysis is proposed. Such workflows are routine in biological and materials science research where the need to identify particles, cells, and grain structure in optical and scanning electron micrographs is common [43][44][45]. As a result, many commercially available software packages exist not only for processing images directly, but also for software development kits to facilitate custom line-of-business applications. One commonly used application is ImageJ™, an image processing package originally created for processing biomedical images [46,47]. We here use the built-in functionality of ImageJ to illustrate the general approach with the caveat that, in the context of this application, any heuristic workflow will be highly specific to the component under inspection and most likely require iterative refinement. Shape description is a rich field of image processing research (see, for example [48]), and comprehensive coverage of this topic is beyond the scope of this work. However, for the purposes of demonstration, we include a hypothetical workflow to illustrate the general approach. The process begins with an image binarization thresholding operation that splits the image into two categories: cells/particles and empty space. In this case, the particles are the regions overlying the honeycomb cells that respond to the imposed hydrostatic stress and are visualized in the proportional slope image. The morphology and geometric regularity of these features will reflect the underlying structure of the honeycomb core, while deviations from expected behavior will be a result of the influence of an internal build flaw. Therefore, the binarization thresholding process is the most crucial step in the workflow (other than the generation of the proportional slope image itself) and is usually determined empirically for a given component and optical setup to maximize diagnostic quality of morphological processing to follow. In this work, a locally adaptive thresholding operation proposed by Phansalker et al. [49] was used. The thresholding procedure is a local operation and takes the form.
where T(x,y), m(x,y), and s(x,y) are the threshold, mean, and standard deviation, respectively, for a local window centered about pixel (x,y), R is the dynamic range of standard deviation (0.5 for a normalized image), k is a constant in the range [0.2,0.5], and p and q are constants assigned with the values of 3.0 and 10.0, respectively, as recommended by Phansalkar et al. [49]. A radius of 15 pixels was found to generate the best result in isolating honeycomb cells.
Following binarization, a morphological erosion + dilation operation-where the object's outer contour is sequentially eroded and dilated again-was applied to remove any residual interconnections between honeycomb cells that may have resulted from the local thresholding operation. Erosion and dilation are common morphological operations that function heuristically and are described using set notation. An erosion operation can be expressed as where B is the binarized (thresholded) image, S is a binary structuring element (both are treated as sets defined on a cartesian grid), and S xy is the structuring element translated so that its origin is located at position (x,y). Equation 17 can be read as "the erosion operation that results from B being eroded by structuring element S consists of a set of points (x,y) (i.e., binary image E) where S xy is fully contained within B." The inverse of the erosion operation is dilation, which is expressed as Equation 18 can be read as "the dilation operation that results from B being dilated by structuring element S consists of a set of points (x,y) (i.e., binary image D) where the intersection of S xy and B is not empty." In this work, erosion and dilation are used sequentially on the thresholded proportional slope image. This sequence of operations is commonly referred to as "opening" since it results in the removal of thin connecting sections between cells, isolating them within the image. This operation is expressed as In this work, a circular structuring element with diameter of 5 pixels was used. Best results were achieved by applying the opening operation twice in succession. The final binarized image is shown in Fig. 18a. This result was then analyzed using ImageJ, and a table of statistics was generated (global cell count and location, area, circularity, min/ max Feret's diameter, perimeter, etc.). The result of this analysis is depicted in Fig. 18b, where random colors have been assigned by ImageJ to differentiate between individual honeycomb cells identified by the morphological procedure.
At this stage, the morphological processing workflow becomes highly application-specific. The general approach is one of utilizing statistical descriptors to probe the proportional slope image for build flaws by inspecting morphological properties of individual cells. In this work, it is the regularity of the cell shape as reflected in the proportional slope image-and any observed departure from that regularity-that is used as a flaw identifier. This criterion is an obvious choice due to a priori knowledge of the internal truss geometry. When applied to the inspection of components where the geometry of the internal supporting structure is unknown, however, this cannot be assumed to be a reliable metric. In most cases, it is unlikely that a single descriptor would be sufficient for a robust flaw detection algorithm (for example, a feature may satisfy the required circularity, but its size may be much larger than expected). Therefore, multiple properties should be inspected for anomalies. In some cases, however, single properties can be very useful. One such property is the location of each individual cell, as this is an indication of the regularity of the underlying truss structure. This is typically determined by computing a cell's centroid, which is simply the arithmetic mean of a cell's X and Y coordinates within the global coordinate system. For example, the x and y centroid coordinates for a given cell can be easily calculated as where u and v are the x and y coordinates of a given pixel within the cell and A is the set of all pixels comprising the cell. An example of this is shown in Fig. 19a where, based on the result of Fig. 18b, the location of each honeycomb cell's centroid is indicated on a cartesian coordinate plane. A missing cell, or a cell in an unexpected location, can be automatically flagged as a candidate flaw. An additional property that can be diagnostically useful for regular patterns is Feret's diameter (maximum caliper dimension), which gives the maximum span distance of a cell without regard to its orientation and is computed algorithmically [50]. A histogram of all Feret's dimensions for the detected honeycomb cells is shown in Fig. 19b. The vast majority of cells have Feret's dimensions falling within the range of 9-10 mm, which is a reasonable result based on the dimensions shown in Fig. 7. In addition, there are numerous cells with smaller Feret's dimensions. These correspond to the partial honeycomb cells that are located around the perimeter of the component. Lastly, there exists one outlier at the upper end of the scale (Feret's dimension 19.04). This corresponds to the double honeycomb cell flaw (flaw C) that can be identified in the proportional slope image.
There are many descriptors that can be derived from morphological filtering. Here, we have only discussed two. The properties that are most pertinent to a particular inspection application are governed by the internal support structure configuration of the manufactured component. The single most important requirement of this methodology is to obtain a high-quality proportional slope image that reflects the local underlying mechanical response, followed by proper thresholding to generate a representative binarized image. Processing overhead for these operations is usually less than a second.

Estimated costs
Lastly, a cost comparison is shown in Table 1. A multitude of factors will affect the capital costs of each methodology. Therefore, these costs should be considered as approximate; they serve to illustrate the cost-effectiveness of the approach described in this work. As no commercially available system exists for the offset-focused CASI method, we have estimated the cost based on the authors' experience regarding development time and the equipment involved. The cost of an inspection system is heavily dependent on the size of the object that is to undergo inspection. For example, in the case of ultrasound, larger objects may require a greater number of independent axes and/or larger ultrasound immersion tanks, thereby increasing the cost of the system. In some cases, the tank may become too large to be practical. In the case of defocused CASI, larger objects may require higher resolution sensors and greater laser power for proper illumination, as well as potentially a larger vacuum chamber and pump (if one is being used). For the purposes of comparison, we assume a fully sealed object (the most likely type to be manufactured) of equal size to that shown in this paper, placed in a polycarbonate vacuum chamber for exposure to differential pressure (i.e., the interior of the object remains at manufactured/atmospheric pressure, but its environment is partial vacuum). In this case, the primary components of a defocused CASI system would be a polycarbonate vacuum chamber (~ $3 k), a roughing pump for vacuum generation (~ $4 k), a 100-mW laser for illumination (~ $1 k), and computer + machine vision camera (~ $3 k). Software costs cannot be reasonably estimated as they depend too heavily on how elaborate the software developed/purchased is-interested readers should add a value they consider reasonable given their own inspection needs.

Discussion and conclusions
We here propose a hybrid methodology for post-build verification and inspection of cellular truss core components. The method involves illumination by diffuse laser to generate a speckle pattern, which is then imaged using an offset-focused optical arrangement while the component is subjected to differential pressure conditions. A titanium honeycomb test component was fabricated with three simulated build flaws and then inspected using X-ray CT, immersion ultrasound, and the proposed offset-focused CASI. Of these methodologies, only X-ray CT could identify all three simulated flaws. Immersion ultrasound and offset-focused CASI were both successful at identifying the simulated layer bond flaw, but not the other two.
The most likely challenge one can expect to encounter when implementing this method is choosing an appropriate degree of defocus. It should not be assumed that the same offset focus distance can be applied across a broad range of manufactured components. The degree of motion of the speckle pattern is a function of both the defocus distance and the differential pressure. A manual (component-specific) calibration routine is suggested. This calibration can be performed easily by non-expert operators and proceeds as follows: it is recommended that a reasonable limit of differential pressure be established, as well as an initial defocus distance (this is application-specific, but a good initial value is on the order of 20 cm). With the lens focused on this offset plane, the differential pressure is then slowly applied while the speckle pattern is observed at increased magnification on the monitor. The point at which speckle motion becomes easily observable will establish the differential pressure for testing. If the pre-established pressure limit is reached but no motion has been observed, the offset focus distance should be increased by 50%, the differential pressure returned to zero, and the process is repeated until clear speckle motion is observed. The observed motion need not be large-scale but only on the order of 2x the average speckle size to achieve excellent results.
The offset-focused CASI method described in this work is not without its limitations and drawbacks. For example, the methodology conclusively identified the missing bond failure between the honeycomb wall and the top plate, but Table 1 Estimated system costs for the three methodologies compared in this paper. Due to the wide range of system capabilities and inspection requirements, these are rough estimates only * Software procurement/development not included

Methodology
Estimated capital equipment cost ($) Ultrasound immersion 200-500 k Computed tomography > 500 k Offset-focused CASI > 11 k* not the mid-level notch or the simulated intermittent bonding failure. We believe that, even with additional refinement, these last two simulated flaws would not be visible in the proportional slope image due to their minimal (or in this case, undetectable) impact on the mechanical performance of the part. This behavior highlights the fundamental difference in approach between offset-focused CASI and the other two imaging methodologies: X-ray CT and immersion ultrasound image the flaws directly, whereas offsetfocused CASI identifies flaws indirectly by directly imaging mechanical deformation. If a flaw's impact is so minor as to have no detectable impact on mechanical behavior, it cannot be detected by this approach. This is detrimental to post-build protocols that attempt a full accounting of all build flaws. However, if the primary objective of the verification process is to guarantee mechanical performance, the offset-focused CASI method is a potentially useful addition to the family of inspection methods, as it is the only one of the three in this study which is capable of providing such information directly. This can, in some cases, increase production efficiency by qualifying finished product that may otherwise be discarded due to the presence of a small, but inconsequential, build flaw. Furthermore, the method scales well both in terms of object size and data resolution, the latter being a function of the camera's sensor resolution. The analysis takes only seconds and, as we have demonstrated, can generate results of higher diagnostic quality than holographic interferometry. Sensitivity can be adjusted via the offset focusing distance, and while this study utilizes the CASI algorithm to process the speckle patterns in real time, any commercially available correlation software is likely to work as well if realtime performance is not necessary. Importantly, automated workflows can potentially be implemented using widely available image analysis and machine vision software, reducing the complexity of the computational implementation to (1) procuring the most suitable software and (2) determining (via experimentation) the appropriate experimental and processing parameters for a given component (most often, this would involve the method and magnitude of stress applied to the component). Lastly, while this work has focused on the inspection of a highly regular internal truss, it can also be applied to post-build characterization of irregular internal truss components. Taken together, these advantages indicate that CASI, in combination with defocused laser speckle imaging, can be a useful tool for post-manufacturing qualification of AM components.