Development of an Optical Displacement Transducer for

Development of an Optical Displacement Transducer for Routine Testing of Asphalt ConcreteDevelopment of an Optical Displacement Transducer for Routine Testing of Asphalt Concrete
Tomer Hamam1; Eyal Levenberg2; and Lihi Zelnik-Manor3
Abstract: Routine mechanical characterization of asphalt concrete is performed under small-strain levels with on-specimen linear variable displacement transducers (LVDTs) as deformation measuring devices. An optical LVDT was conceptually proposed and evaluated in this study to serve as a viable noncontact alternative to physical LVDTs. The envisioned device consists of a pair of low-end low-resolution grayscale cameras, each monitoring a virtual gauge point, i.e., a small untreated surface area of the tested specimen. The gauge length is the distance between the two virtual gauge points, and the sought-after information is their differential in-plane translation. Digital image correlation techniques were employed for the measurement, operated on the natural material texture without requiring speckle coating. As a first step toward evaluating the concept, the study explored both the precision and the accuracy that may be achieved with one low-resolution image sensor. A calibration scheme was also offered for introducing object-scale dimensions into the analysis. From this predevelopment study it is concluded that the envisioned optical LVDT is viable, rendering the idea worthy of consideration. DOI: 10.1061/(ASCE)MT .1943-5533.0001570. © 2016 American Society of Civil Engineers.
Author keywords: Asphalt concrete; Material characterization; Linear variable displacement transducer (LVDT); Digital image correlation; Precision and accuracy.
Introduction
The most advanced pavement design and analysis procedures are based on continuum mechanics principles (ARA 2004; Zhou et al. 2010; Ullidtz et al. 2010). Intrinsically, these procedures place con- siderable emphasis on the accurate mechanical characterization of asphalt concrete (AC), especially under small-strain levels relevant to service conditions of the order of 10−4. This study is motivated by the desire to simplify the deformation instrumentation technique for AC specimens in routine characterization. A noncontact optical linear variable displacement transducer (LVDT) is conceptually proposed for this purpose.
Fig. 1 schematically presents a single unit of the envisioned device, along with the measurement setup. A typical cored AC specimen is displayed on the right-hand side of the figure; the specimen is cylindrical, 150 mm tall, and 75 mm in diameter, supported at the bottom. Awide variety of aggregate types, shapes, and sizes can be clearly observed on the mantle producing a rich optical texture. Two gauge points are shown attached to the material surface; in actuality, these are small metallic parts, up to 10 mm in diameter, to which a single LVDT is fastened. The envisioned optical LVDT is shown on the left-hand side; it has two noncontact sensors, each pointing directly at the specimen and aiming for a virtual gauge
point (VGP). The VGPs are depicted by flat circular markers that merely indicate small untreated surface areas similar in size to physical gauge points. The vertical spacing between the VGPs is the gauge length, the desired measurement is their differential in-plane translation, and the dashed lines indicate that a line of sight is required by the setup.
In general terms, the envisioned optical LVDT should be devel- oped to offer similar measurement capabilities, performance characteristics, and pricing level compared to a physical LVDT. At the same time, being noncontact in nature, the device should greatly simplify routine instrumentation, shunning many of the shortcom- ings in the current practice.
Background
Current Practice
Measuring deformations in routine AC characterization involves three key experimental elements [Witczak et al. 2002; AASHTO T342 (AASHTO 2011); AASHTO T322 (AASHTO 2007)]: (1) utilization of on-specimen instrumentation as a means to annul machine compliance issues and ensure correct readings (Scholey et al. 1995; Kaloush et al. 2001); (2) performing replicate measurements covering different locations of the tested specimen in an effort to better capture the bulk properties of an essentially particulate material (Weissman et al. 1999; Levenberg and Uzan 2004; Kim et al. 2009; Velasques 2009; Coleri and Harvey 2011); and (3) utilization of LVDTs, each bridging two so-called gauge points with spacing of the order of 100 mm (Witczak et al. 2002; Kim 2009). The relatively large gauge length is needed for homogenizing the true underlying heterogeneous response.
LVDTs are relatively inexpensive motion sensors that deliver real-time readings; they are reliable and robust and suited for both quasi-static and oscillatory testing. A typical LVDT consists of a fixed frame of coils surrounding a movable core; the electromagnetic field in the coils is altered predominantly by axial core
1Ph.D. Student, Georgia Tech, 827 Turpin Ave. SE, Atlanta, GA 30312. E-mail: [email protected]
2Assistant Professor, Faculty of Civil and Environmental Engineering, Technion–Israel Institute of Technology, Room 729, Rabin Building, Technion City, Haifa 3200003, Israel (corresponding author). E-mail: [email protected]
3Associate Professor, Faculty of Electrical Engineering, Technion– Israel Institute of Technology, Room 959, Meyer Building, Technion City, Haifa 3200003, Israel. E-mail: [email protected]
Note. This manuscript was submitted on June 21, 2015; approved on December 15, 2015; published online on March 21, 2016. Discussion per- iod open until August 21, 2016; separate discussions must be submitted for individual papers. This paper is part of the Journal of Materials in Civil Engineering, © ASCE, ISSN 0899-1561.
© ASCE 04016066-1 J. Mater. Civ. Eng.
J. Mater. Civ. Eng., 04016066
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movements, which results in a corresponding voltage signal (Nyce 2004). Thus, LVDTs are analog devices with no theoretical resolution limitation. However, limited resolution does transpire from imperfections in the detection circuitry and from analog-to-digital signal conversion. Ultimately, with adequate signal conditioning, proper calibration, and tight temperature control, top-notch LVDT systems exhibit precision and accuracy levels of the order of 0.1 μm.
Despite their advantages and popularity, certain difficulties and risk of incorrect readings arise when instrumenting AC specimens with LVDTs (e.g., Daniel et al. 2004): (1) drilling or gluing activ- ities, sometimes both, are required for fixing gauge points to the specimen surface; these are labor intensive, prolong the testing du- ration, and may result in erroneous gauge length; (2) the number of LVDTs that can be attached onto a single specimen is practically limited owing to space restrictions, constraining the possible number of replicate measurements; (3) at elevated temperatures, when the tested material is very soft, the combined self-weight of gauge points, mounts, and LVDTs can generate drift in the test data; (4) LVDTs themselves are sensitive to temperature changes and magnetic interference; (5) core movement may be mechanically ob- structed by friction when improperly installed or when gauge points either move out of plane or rotate during an experiment; (6) LVDT wires require careful handling and isolation from mechanical vibration sources, or recorded data will be contaminated by noise; and (7) in fast-rate experiments, the combined mass inertia of all involved components (i.e., frame of coils, core, gauge points, and mounts) can become influential to the point where actual specimen response is incorrectly captured by the instrumentation owing to, for example, amplification, attenuation, or phase lagging. Because of the focus on small deformations, the limited measurement range of LVDTs was not listed here as a drawback.
Optical LVDT
Much like a physical LVDT, the envisioned noncontact device has the task of measuring the differential in-plane movement between a pair of VGPs with submicron precision and accuracy. Additional experimental capabilities, ordered according to their relative importance, include the following: (1) allowing for free choice in selecting monitored points and, therefore, flexibility in adjusting the gauge length; this is desirable because it would maintain a similar- ity with physical LVDTs since gauge points can be glued to a specimen at any location; (2) allowance for fast sampling rates of the
order of 100 readings per second—and preferably much faster; this is needed because small-strain characterization tests are often oscillatory with frequencies as high as 50 Hz and because oversam- pling offers means for improving measurement precision; (3) deliv- ery of measurements in real time for immediate feedback during setup and testing; this is desirable because it would enable monitoring and intercepting problems before and during an experiment; and (4) flexibility in varying the distance between the optical LVDT and the tested specimen; this ability is needed for convenience and also for dealing with space restrictions in environmental chambers.
The requirement for noncontact measurement in combination with the rich optical texture of cored or cut faces of AC innately suggests a digital image correlation (DIC) application. DIC is rou- tinely practiced in the arena of experimental mechanics, primarily as a full-field measurement method (Peters and Ranson 1982; Chu et al. 1985; Grédiac and Hild 2012). The technique is based on imaging speckle patterns painted over a tested specimen from an external station and comparing pictures taken in different experimental stages. The shape and progression of the deformation field is found by dividing the imaged region into subsets (or patches) that are essentially small pixel matrices, and performing pairwise correlation across images.
A measurement quality of the order of 0.01 pixels is potentially achievable with DIC algorithms for in-plane translated speckle images (Tian and Huhns 1986; Sutton et al. 1988; Zhou and Goodson 2001; Zhang et al. 2003). Accordingly, achieving a 0.1 μm reading similar to LVDTs means that pixel sizes should correspond to approximately 10 μm in object space. In this case, capturing a pair of VGPs spaced 100 mm apart with a single camera requires an imaging sensor of the order of 100 megapixels. Such a huge pixel matrix is not currently offered by DIC cameras; even if made available, it would be associated with a larger physical size (Goldstein 2009), slow frame rate, and demand long analysis times. At present, DIC techniques are applied to AC materials for measuring surface strain fields or visualizing cracking (Seo et al. 2002; Chehab et al. 2007; Birgisson et al. 2009; Yi-qiu et al. 2012). In these studies, the images are of the order of tens of millimeters in size, and the real- ized displacement accuracy is inferior compared with LVDTs by at least one order of magnitude.
In light of the aforementioned discussion, it is suggested to utilize low-resolution grayscale cameras as noncontact sensors for the optical LVDT. Each camera would point and image a single VGP with a rudimentary sensing matrix of the size order of 0.1 megapixels. The low pixel count enables fast acquisition (i.e., higher frame rate), holds potential for real-time calculations, and ensures competitive pricing. It is important to note that only the average in-plane translation of the imaged areas is of interest, i.e., the distortionwithin the VGPs is of no practical importance. Such averaging mimics what occurs when a physical metallic gauge point is glued to a specimen surface and mechanically prevents it from deforming. Also, imaging a small region is advantageous for cylindrical specimens because small regions on curved surfaces appear nearly planar.
Conceptually, the most closely related study to the current research is the work of Huang et al. (2010) wherein a dual-camera arrangement was suggested to replace traditional strain gauges for Portland cement concrete characterization. The study employed two high-resolution cameras spaced 100 mm apart, each imaging a small rectangular surface area (3.4 × 4.3 mm) of the tested specimen. The device was validated experimentally, by quasi-statically compressing cylindrical and cubical concrete specimens, measuring their Young’s modulus with standard gear, and comparing the outcome to that obtained from the dual-camera arrangement. In the study, the concept was deemed viable based on the resulting moduli values and spread. However, it should be noted that the work did
Fig. 1. Envisioned optical LVDT for routine testing of asphalt concrete
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not mention actual camera resolutions, optical attributes of lenses, or calibration method employed for converting image-space units into object-space (physical) units. Also, authors reported that speckle spraying was required because inferior results were obtained when reading the natural material texture.
Speckle Coating
A key experimental feature common to the vast majority of DIC cases is that the specimens are precoated (prior to testing) with an artificial speckle pattern. Speckle patterns serve as a carrier of deformation information for DIC algorithms and as such are considered a necessity (Sutton et al. 2009). Typically, the procedure entails covering the specimen with white color followed by spray painting black dots on it. This technique is a clear drawback when considering routine testing of AC specimens. Moreover, it has also been shown that speckle characteristics affect DIC performance (Lecompte et al. 2006; Haddadi and Belhabib 2008; Pan et al. 2010; Stoilov et al. 2012; Crammond et al. 2013).
Not many studies are reported in the technical literature in which natural material textures, with no surface preparation, were read by a typical DIC algorithm. Marcellier et al. (2001) measured plane displacement and strain fields in stretched human skin. The images were captured with an 8-bit grayscale 0.3 megapixel camera (752 × 582 sensor); the imaged area was approximately 40 × 50 mm in size. A bilinear interpolation scheme was applied, and the achieved subpixel precision was only 0.34. More recently, Sjögren et al. (2011) analyzed the deformation behavior of graphite cast irons. They used a commercial DIC code for tracking the strain field under tensile conditions. The images were captured with a 1.3 megapixel camera (1,392 × 1,040 sensor); the imaged area was only 1.5 × 1.0 mm in size. Study results were presented and discussed, but without any mention of the precision attained.
Tracking natural material texture has also been reported in connection with the mechanical characterization of cellular materials, such as polymeric foam (Wang and Cuitino 2002; Roux et al. 2008) and glass wool (Bergonnier et al. 2005; Hild et al. 2002). In Wang and Cuitino (2002) a rigid body translation test was performed in which a block of foam placed on a motorized linear table was imaged in different positions. A monochrome 0.3 megapixel digital camera (768 × 484 sensor), equipped with a telecentric lens, was used; the object was lit with a fiber optic illuminator. An iterative search algorithm was employed to minimize the sum of squared differences between subsets; several different subset sizes were in- vestigated with two subpixel interpolation schemes. A standard deviation of 0.04 pixels in translation detection (i.e., precision) was reported with bicubic interpolation and subset sizes of 15 × 15 pixels or larger; imaged area dimensions were not mentioned. In Bergonnier et al. (2005) crimped glass wool was imaged with a 1.3 megapixel grayscale camera; each pixel in the image was approximately 0.15 mm in object-space size. A second image was prepared from the original, numerically shifted by 0.5 pixels. A DIC algorithm was applied to calculate the imposed displacement with different subsets and for different subset sizes. It was reported that the lowest standard deviation, of 0.015 pixels, was produced with 64 × 64 pixel subsets. Further information regarding bias or interpolation scheme was not included.
Study Objective
There are three basic challenges associated with the envisioned device: (1) avoiding the necessity of speckle coating, which is an obvious disadvantage for routine testing; (2) utilizing low-end grayscale cameras with low-resolution imaging sensors to enable
fast acquisition and interpretation, as well as competitive pricing; and (3) obtaining sufficiently precise and accurate deformation measurements in AC specimens comparable to LVDTs.
Compliance with these fundamental challenges is considered a first and essential step toward realizing the optical LVDT concept. Accordingly, the objective herein is to quantify the error spread (precision) and error bias (accuracy) by which a natural AC texture can communicate small translation information to a DIC algorithm. The entire investigation is based on an analysis of low-resolution images of cut AC faces.
Approach and Methodology
The approach taken here to pursuing the stated aim was to analyze image pairs having exact and a priori known subpixel translation magnitudes. At this early stage of concept evaluation it was decided to generate the translations digitally and not experimentally (Reu 2011). This approach focuses the investigation on displacement detectability while bypassing experimental issues that can potentially affect the outcome, such as image noise, lighting conditions, lens distortions, mechanical vibrations of the setup, and out-of- plane specimen movements (Haddadi and Belhabib 2008; Pan et al. 2009; Bornert et al. 2009).
Subsequently, a numerical procedure was followed in which a typical AC specimen was initially imaged with a 10 megapixel grayscale camera. This camera and associated lens optics were of high quality that did not conform to the optical LVDT vision. Next, different plane translations in two perpendicular directions were digitally applied to the images. These translations were imposed in integer pixel steps, which obviated the need for interpolation or any other subjective handling. Finally, the translated high-resolution images were downscaled by a factor of 100 (i.e., 10-fold in each direction). The outcome was a set of 0.1 megapixel photos containing known subpixel translation magnitudes. These low-resolution images conform to the optical LVDT vision because they approximate the acquisition performance of a low-end camera. A similar image-generation approach for subsequent precision assessment was reported in Kelly et al. (2007) and in Debella-Gilo and Kääb (2011).
The described process was repeated for three imaged areas that differed in size. The aim here was to cover a range of practical VGP dimensions: small (≈ 0.5 mm), medium (≈ 3 mm), and large (≈ 10 mm). These targeted sizes were simply achieved by imaging the same AC specimen from three different distances. The resulting images were then processed by a DIC algorithm, from which the precision and accuracy of displacement detection could be quantified. Later on, the physical (object-space) dimensions of the pixels in the three magnification levels were measured. Calibration was achieved with the DIC algorithm by analyzing a sequence of images in which the specimen was physically displaced at known intervals by an accurate piston.
Image Acquisition and Manipulation
The entire investigation presented here made use of a cylindrical AC specimen fabricated in the laboratory from a typical mixture. Aggregates for all sieve sizes were composed of quarried and crushed limestone; binder was unmodified (neat) and classified as PG-70. The aggregate blend was dense graded with a 19 mm maximum size; binder content was 4.5%. The material was first mixed under 170°C, then oven-conditioned for 2 h at 155°C, and finally densified in a gyratory compactor. After an overnight cool-down to room temperature, the gyratory specimen was cored
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and trimmed to yield a cylindrical specimen, 100 mm in diameter and 180 mm tall, with an air void content of approximately 4.5%. Diamond coring or saw cutting of compacted mixtures is standard practice for obtaining AC test specimens; these actions always yield surfaces with rich optical textures.
The top base of the AC cylinder was glued to the piston of a single-axis electromagnetic load frame (Instron ElectroPuls E10000, High Wycombe, U.K.). This piston can be instructed to perform minute up or down submicron movements, with displacements measured by a built-in optical encoder that has an accuracy level of the order of 0.01 μm. A grayscale camera [IDS Imaging Development Systems (Obersulm, Germany) uEye model UI5490RE] was positioned close to the E10000, fixed to a tripod. The image sensor in this camera is based on CMOS technology; it has a 3,840 × 2,748 pixel matrix, each 1.67 × 1.67 μm in size, re- porting an 8-bit value for each pixel (12-bit internally). Additionally, the camera was equipped with an Infiniprobe (0–8 ×∞− 18 mm) zoom lens along with a blue LED ring light for illuminating the imaged area; uEye Cockpit software was used for picture acquisition. Refer to Fig. 2 for a schematic diagram of this setup.
The E10000 piston was moved up or down, vertically displacing the glued AC specimen in front of the camera. Images were acquired at several displacement levels when the specimen was mo- tionless. This entire procedure was repeated for three different distances—20, 40, and 80 mm—measured from the front of the zoom lens to the specimen surface. The adjustments in distance and tuning of the lens focus were done manually, and for this reason the imaged locations differed for different magnification levels.
Table 1 lists the applied piston displacements during which images were acquired in each of the three imaging distances. As can be seen, eight photos were taken at the largest magnification (Images S0 to S7), with the specimen translated by up to 6 μm relative to the initial position. Eleven photos were taken at the intermediate zoom level (images M0 to M10) with piston movements up to 10 μm. Seven pictures were acquired for the largest imaging distance (Images L0 to L6), with a maximal specimen translation of 20 μm.
Images S3, M4, and L4 are displayed in Figs. 3–5, respectively. These three images were randomly selected from the available set in each magnification level to serve as a basis for synthetic translations and subsequent numerical investigation. Fig. 3 shows Image S3, obtained from the closest camera distance of approximately 20 mm (i.e., the largest magnification level). The dimensions of the imaged zone in the figure are approximately 0.7 × 0.5 mm, corresponding to a small VGP. The width of the picture is aligned with the circumference of the AC cylinder, and the height is aligned with the cylinder’s length. Bright spots seen in this figure can be as- cribed to limestone aggregates that were cut during coring or naturally exposed as a result of imperfect binder coating; the darker areas indicate binder-coated aggregates. Closely spaced parallel grooves are noticeable in the image; these are scratch marks resulting from the core drilling process.
Fig. 4 shows Image M4, obtained from a distance of 40 mm (intermediate magnification). The dimensions of the imaged zone are approximately 2.5 × 1.8 mm, corresponding to a medium VGP size. In this size scale the parallel coring marks are still noticeable. The edge of an exposed aggregate is seen as the bright zone in the top central part of the image. Other portions of the image (e.g., lower right corner) are predominantly dark with very limited brightness contrast. These near-uniform textured areas, which can randomly appear on the surface of any AC specimen, pose a chal- lenge for the DIC algorithm. Fig. 5 shows Image L4, obtained from a distance of approximately 80 mm (smallest magnification level). The size of the imaged zone is approximately 9.1 × 6.5 mm, corresponding to a large VGP size. In this size scale the coring scratches are barely noticeable. Also, at this scale the overall shape and natural geologic texture of some of the aggregates become visible. The image is slightly darker compared with the previous two because of the greater distance from the ring light.
The high-resolution (3,840 × 2,748 pixels) images shown in Figs. 3–5 were individually translated by integer pixel steps. This was done in two perpendicular directions in a range of 0 to 10 pixels and resulted in a grid of 121 different in-plane translation com- binations. A perimeter 10 pixels wide was subsequently trimmed
Fig. 2. Schematic diagram of experimental setup
Table 1. Applied E10000 Piston Movements for Three Different Imaged Distances
Imaging distance ≈20 mm small VGP (largest magnification)
Imaging distance ≈ 40 mm medium VGP (intermediate magnification)
Imaging distance ≈ 80 mm large VGP (smallest magnification)
Image identifier Movement (μm) Image identifier Movement (μm) Image identifier Movement (μm)
S0 0.0 M0 0.0 L0 0.0 S1 0.8 M1 1.0 L1 1.0 S2 1.5 M2 2.0 L2 2.0 S3 2.1 M3 3.0 L3 4.0 S4 3.2 M4 4.0 L4 8.0 S5 4.0 M5 5.0 L5 15.0 S6 5.0 M6 6.0 L6 20.0 S7 6.0 M7 7.0 — — — — M8 8.0 — — — — M9 9.0 — — — — M10 10.0 — —
Note: VGP = virtual gauge point.
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from all photos, and the resulting 3,820 × 2,728 pixel images were downscaled by a factor of 100 to a 382 × 273 pixel matrix (i.e., 10-fold in each direction). This size reduction was achieved with a bicubic B-spline interpolation method in three successive steps (1=2, 1=2, and 1=2.5). The outcome was three sets of reduced-resolution images, each set linked to a different VGP size and containing an array of subpixel translations in the range of 0 to 1 pixels at 0.1 pixel intervals. The full set of images acquired when the AC cylinder was displaced by the E10000 piston (Table 1) was also downscaled according to the aforementioned procedure. Because no synthetic translation was applied, there was no need to trim the perimeters, and the final low-resolution image size was 384 × 275 pixels. This image set contains calibration information, linking pixel dimensions in image space to physical dimensions in object space.
Downscaled versions of the images in Figs. 3–5 are shown in Fig. 6. Note that the main texture features observed at high resolution are visually retained despite the reduced image quality. Also included in Fig. 6 are the corresponding pixel intensity histograms. The abscissa in each histogram represents grayscale values between 0 (black) and 255 (white); the ordinate represents frequency of oc- currence, i.e., pixel counts normalized by the total number of pixels in the image. These histograms clearly indicate that pixel values do not equally span the full intensity range, i.e., there is room for intensity stretching and equalization algorithms (though not carried out in this study). All three distributions, while associated with an entire image, closely resemble speckle pattern histograms. The two close-range photos are characterized by a unimodal, bell-shaped distribution, whereas at the largest imaging distance some cluster- ing emerges around two separate intensity values. In Berfield et al. (2007) it is argued that patterns characterized by bimodal distributions are less effective for DIC application.
Accuracy and Precision Investigation
Digital Image Correlation
The theory governing DIC is discussed in many textbooks and articles and has many derivations and flavors. The most basic for- mulation was applied in this study—the Lucas–Kanade tracker algorithm for simple in-plane translation (Lucas and Kanade 1981). Consider two digital images, F and G, composed of a grid of pixels with discrete grayscale intensity levels. Pixel locations within an image are identified by positive integer indices i and j; the associated gray levels are given by the scalar functions Fði; jÞ and Gði; jÞ. Assuming Image G is a pure translation of Image F (or vice versa), the following expression holds: GðiþΔi; jþΔjÞ ¼ Fði; jÞ, whereΔi andΔj denote the translation magnitudes in the i and j directions; these translations are not confined to integer values.
The task of the DIC algorithm is to inferΔi andΔj given a pair of images contaminated by random noise. To achieve this, the sum of squared differences (SSD) in gray value intensities between subsets taken from both images is first defined:
SSDðx; yÞ ¼ X i;j
½Gðiþ y; jþ xÞ − Fði; jÞ2 ð1Þ
in which x and y = trial translation magnitudes; i = summed over i0 − N to i0 þ N; and j = summed over j0 − N to j0 þ N, where i0 and j0 are the central coordinates of the considered subset with dimensions ð2N þ 1Þ × ð2N þ 1Þ. The SSD is a scalar statistic that is sensitive to the agreement level between the subsets being compared. A set of ðx; yÞ values that minimizes the SSD function is
Fig. 3. High-resolution AC surface image: small VGP (0.7 × 0.5 mm)
Fig. 4. High-resolution AC surface image: medium VGP (2.5 × 1.8 mm)
Fig. 5. High-resolution image of AC surface: large VGP (9.1 × 6.5 mm)
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taken as the best estimate for the actual translations Δi and Δj associated with the ði0; j0Þ subset. The minimization is done iteratively, assuming xð0Þ ¼ yð0Þ ¼ 0 and then performing repeated updates until the SSD converges to a minimum. The update procedure between iterations ðnÞ and ðnþ 1Þ is xðnþ1Þ ¼ xðnÞþ ΔxðnÞ and yðnþ1Þ ¼ yðnÞ þΔyðnÞ; the increments ΔxðnÞ and ΔyðnÞ are computed by a closed-form expression attained from applying a Gauss–Newton minimization algorithm (e.g., Baker and Matthews 2004):
" ΔxðnÞ ΔyðnÞ
∂GðnÞ ∂x ðF − GðnÞÞ
ð2Þ
in whichGðnÞ ¼ Gðiþ yðnÞ; jþ xðnÞÞ, F ¼ Fði; jÞ, and the summa- tion range for i and j is as defined earlier covering the associated subset.
Computations
The DIC investigation initially focused on calculating the synthetic grid of 121 translations in each of the 3 magnification levels. The aim here was to perform error analysis by comparing known and DIC-inferred translations. The results of this investigation are obtained and reported in pixel units. Later on, the DIC investigation focused on analyzing the sequence of AC images obtained in between piston movements (Table 1). Here, advantage was taken of the accurate axial movement of the E10000 piston to introduce a physical scale into the images and subsequently calculate their magnification levels. In both investigations, rectangular subsets containing 4,225 pixels (¼ 65 × 65) were employed. Subset size affects DIC performance because it controls the texture information included in the computations. The size choice herein was guided by previous studies (e.g., Yaofeng and Pang 2007; Pan et al. 2008),
which demonstrated that a 65 × 65 pixel matrix typically generates consistent results while balancing computational effort. This choice was further reinforced from observing that subset entropy level did not increase with further size enlargement. A bicubic interpolation scheme was used to evaluate Image G at noninteger coordinates, as required in Eqs. (1) and (2).
Given that images were translated purely relative to each other, either digitally or experimentally, inferred translations from different subsets should yield identical results. However, because the natural material texture is analyzed, the brightness range and pattern randomness varies across subsets in a given image, unlike speckle patterns. These characteristics influence the smoothness of the search path for optimal translation and therefore generate nonuniform performance. To better illustrate this point, Fig. 7 com- pares the SSD function [Eq. (1)] of two different 65 × 65 subsets taken from Fig. 4 following digital imposition of a five pixel translation in two perpendicular directions. Subsequent to downscaling, the imposed vertical and horizontal translations become 0.5 pixels. The two different subsets are shown on the right-hand side, and their associated SSD surfaces are shown on the left-hand side. The theoretical agreement peak (i.e., minimal SSD) is expected at the central point, indicated by a cross marker, with coordinates (0.5,0.5). To produce these surfaces, 250,000 values were computed in each case (i.e., a grid of 500 × 500 calculation points) em- ploying a bicubic interpolation scheme; both contour lines and color scaling were included to better convey the shape of the surfaces. As can be clearly observed, the top subset is associated with a smooth SSD surface, whereas the bottom subset generates a rough and jagged surface with several competing optimal points. Operat- ing the DIC algorithm with such subsets can potentially lead to inaccurate results.
One option for circumventing this issue is to perform an image search to identify and reject potentially poor performing subsets. This could result in a user alert system, requesting that the optical LVDT be moved, i.e., instructing the user to select a different VGP. This type of prescreening was not pursued here, first, because of the desire to allow maximum flexibility in choosing the monitored
Fig. 6. Low-resolution versions of images in Figs. 3–5 (upper row); corresponding pixel intensity histograms (lower row)
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points, and second, because a simple, reliable, and computationally cheap subset evaluation criterion is not readily available. In turn, the approach followed was statistical in nature, based on repeated DIC calculations for every pairwise comparison of images with different subsets. If each subset is viewed as an independent translation sensor, then multisensor fusion techniques can be used to generate an improved estimate of the actual translation (Hackett and Shah 1990). In this connection, a choice was made to employ 250 subsets for every magnification level. Such a large number generated sufficient data for interpretation and for identifying and rejecting outliers, if needed. The 250 coordinate sets of ði0; j0Þ were randomly selected to ensure objectivity in image coverage because no guidelines exist for performing a more intelligent subset selection. Moreover, and in line with recent studies (e.g., Mazzoleni 2013; Pan 2013), Gaussian prefiltering was employed before com- puting the derivatives required by Eq. (2). For this purpose a rota- tionally symmetric filter was employed over a 7 × 7 pixel matrix with a standard deviation of 2.5 pixels; gradients were subsequently computed with Sobel convolution kernels.
Results and Error Analysis
Displayed in Figs. 8–10 are computed translations generated by the DIC procedure superposed over the 121 synthetic grid translations.
Fig. 8 corresponds to a small VGP (Fig. 3), Fig. 9 to a medium VGP (Fig. 4), and Fig. 10 to a large VGP (Fig. 5). The results from the 250 individual subsets in each translation case are seen as clusters of closely spaced circular markers. Accordingly, each figure contains 30,250 DIC computations (¼ 121 × 250). In general terms, poor performing subsets are associated with less accurate translation results; for a cluster this shows up as a larger spread and as bias relative to the associated target grid point. Within each of the three figures there is noticeably better DIC performance for integer pixel translations (i.e., four corners of the grid) compared with all other synthetic targets. Moreover, across figures (i.e., comparing different magnification levels), there is no noticeable difference in performance between the small and medium VGP cases (Figs. 8 and 9); in the large VGP case (Fig. 10) the clusters exhibit a slightly wider spread.
As a way to quantify the DIC performance, the accuracy and precision of subsets and clusters were statistically analyzed. Accu- racy was quantified based on translation errors relative to the grid points; precision was quantified based on the standard deviation in the errors. For each individual cluster, horizontal and vertical errors in pixel units were defined for every subset as the DIC translation (in either direction) minus the associated grid translation. A graphi- cal depiction of such a computation for one cluster (i.e., 250 subsets) is illustrated by the scattergram in Fig. 11. The abscissa
Fig. 7. Visual comparison of SSD surfaces associated with two different subsets (medium VGP)
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denotes horizontal errors and the ordinate denotes vertical errors; a large cross marker placed over the coordinate origin indicates the point of zero error or perfect accuracy, and a solid rectangular marker designates the centroid of the error cluster. In this example the standard deviations in the vertical and horizontal directions are 1.17 × 10−3 and 1.14 × 10−3 pixels, respectively. The coordinates of the centroid (−5.01 × 10−4, 1.12 × 10−3) are the bias error of the cluster.
A similar analysis was carried out to quantify the DIC performance in Figs. 8–10. Separately at each magnification level the horizontal and vertical errors were computed for all subsets across all grid points. The top two charts in Fig. 12 show the percentile ranking of the resulting errors. Each chart refers to a different direction (either vertical or horizontal) and includes three curves, indicating data associated with a small VGP (solid line), medium VGP (dotted line), and large VGP (dashed line). As can be seen, the abscissa values span a rather narrow range, between −0.01 pixels and 0.02 pixels. The steepness of the curves is indicative of the associated precision. In this respect the small and medium VGP cases seem slightly superior the the large VGP case. The standard deviations (precision) and the bias error (accuracy) are reported in Table 2 for the two directions. With reference to the upper part of the table, it can be seen that the performance is directionally independent and exhibits very good precision and accuracy levels of the order of 0.003 pixels. The two bottom charts in Fig. 12 depict the same information as discussed earlier, but based on cluster averages and not on individual subsets. While accuracy naturally re- mains unaffected, as can be graphically observed, simple averaging improves the precision level. The outcome of this averaging is also reported in Table 2. The improvements are on the order of 10% for the small and medium VGP magnification and 40% for the large VGP.
The physically translated image sets (Table 1) were analyzed using the previously described DIC procedure. The inferred translations between image pairs were calculated by simple averaging of the individual subset results. The outcome for the three magnification levels is presented in Fig. 13. The abscissa denotes E10000 piston movements in a range of 0–20 μm; the ordinate denotes the associated DIC translation. Each circular marker indicates a computed translation between an image pair. To generate this figure, the first four images in each magnification level separately served as references for subsequent images, allowing more translation information to be extracted from the limited set originally acquired. Three oblique trend lines were fitted to the data using
Fig. 8. DIC results for small VGP; computed translations of individual subsets (circular markers) superposed over synthetic grid translations (cross markers)
Fig. 9. DIC results for medium VGP; computed translations of individual subsets (circular markers) superposed over synthetic grid translations (cross markers)
Fig. 10. DIC results for large VGP; computed translations of individual subsets (circular markers) superposed over synthetic grid translations (cross markers)
Fig. 11. Error scattergram example for one cluster composed of 250 subsets (image-space units)
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a least-squares approach, each forced to pass through the origin of the coordinate axes. The agreement level of the DIC-computed translations with straight lines was statistically significant, with R-squared better than 0.99. Such a method can serve as a calibration procedure for the optical LVDT. The visible scatter, especially in the medium VGP case, is attributed to vibrations present during image acquisition. The solid line is associated with the largest magnification level (imaging distance of 20 mm); it indicates that each pixel corresponds to 1.70 × 1.70 μm in object space. The dotted
line is associated with the intermediate magnification level (imaging distance of 40 mm); it indicates that each pixel corresponds to 6.52 × 6.52 μm in object space. Finally, the dashed line is associated with the smallest magnification level (imaging distance of 80 mm); it indicates object-space pixel dimensions of
Fig. 12. Statistical investigation of horizontal and vertical translation accuracy: (a) all subsets; (b) cluster averages
Table 2. Results of Digital Image Correlation Error Analysis in Image- Space Units (Refer to Fig. 12)
Direction and error measure Small VGP Medium VGP Large VGP
Based on all subsets (pixels) Horizontal precision 3.06 × 10−3 2.94 × 10−3 3.91 × 10−3 Horizontal accuracy 2.66 × 10−3 3.18 × 10−3 3.66 × 10−3 Vertical precision 2.53 × 10−3 3.22 × 10−3 3.86 × 10−3 Vertical accuracy 3.51 × 10−3 3.11 × 10−3 3.76 × 10−3
Based on cluster averages (pixels) Horizontal precision 2.75 × 10−3 2.53 × 10−3 2.11 × 10−3 Vertical precision 2.27 × 10−3 2.98 × 10−3 2.36 × 10−3
Note: VGP = virtual gauge point.
Fig. 13. Scale calibration based on images of a physically translated AC specimen (Table 1)
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23.71 × 23.71 μm. Combining the previously calibrated pixel sizes with the DIC performance from Table 2 suggests that submicron precision and accuracy are realistically achievable for the envisioned device.
Conclusions and Remarks
Motivated by the need to simplify AC instrumentation for routine small-strain characterization, this paper put forward an idea for an optical LVDT. Three fundamental predevelopment challenges were addressed: (1) performing DICwithout speckle coating, (2) utilizing a low-resolution grayscale image sensor, and (3) obtaining sufficiently precise and accurate measurements. It was found that DIC analysis applied to artificially translated images exhibited variabil- ity in error performance across different subsets. Simple averaging of inferred translations led to dramatically improved error characteristics. Doing this is equivalent to assuming that all subsets have identical variance and therefore should be weighted equally. In this connection, two additional schemes for outlier identification were attempted: (1) the RANSAC iterative algorithm (Fischler and Bolles 1981) and (2) the Mahalanobis generalized distance (De Maesschalck et al. 2000). Both methods produced marginal improvements compared with simple averaging, which did not warrant the added computational effort. Ultimately, DIC translations were detected with a precision and accuracy of approximately 0.003 pixels (Table 2). This level of error performance was similar for three different magnifications and for two in-plane perpendicular translation directions.
Under more realistic experimental conditions, some degradation in measurement performance is expected for many reasons, e.g., image noises, lens distortions, or mechanical vibrations. However, even if precision and accuracy degrade relative to the synthetic conditions by one order of magnitude, i.e., from 0.003 to 0.03, submicron translations comparable to LVDTs could still be realistically measured by the envisioned device. This conclusion is valid for a 0.1 megapixel resolution sensor imaging VGP in a size range of 0.5 to 10 mm. As inexpensive low-end cameras advance toward 1.0 megapixel resolution, additional margin of error will be made available to offset imperfect testing environments.
Overall, the three basic challenges associated with the optical LVDT were met: (1) speckle coating appeared redundant because the natural texture of cut AC faces carried excellent DIC information; (2) low-resolution imaging provided sufficient translation information of VGPs for subsequent analysis; and (3) the fusion of computed translations across subsets produced error characteristics comparable to those of physical LVDTs. Consequently, the find- ings in this study are deemed favorable, rendering the envisioned device worthy of consideration.
Future Work
The path toward realizing the optical LVDT concept calls for car- rying out additional numerical and experimental investigations, as well as building a working prototype. Future numerical work should focus on optimizing computational cost and increasing the reliability of the system, with an eye kept constantly on real- time application. To achieve this, there is a need to revisit several heuristic decisions taken in this pilot work: (1) choice of DIC algorithm type, (2) choice of number of analyzed subsets in an image, (3) strategy for selecting subset locations in an image, (4) choice of subset size, (5) method of subpixel interpolation, (6) technique for outlier detection and rejection (e.g., Hodge and Austin 2004), and (7) fusing method for combining translations delivered by different
subsets. With respect to experimental work, once a working prototype is assembled, there is a need to commence accumulating practical experience: (1) establish a calibration scheme, (2) evaluate object illumination options, (3) carry out comparative measurements with physical LVDTs, and (4) test different AC mix types and specimen configurations to ensure robustness.
Acknowledgments
The authors wish to thank Mr. Alexey Kharitonov from OpteamX Industrial Cameras Ltd. for allowing access to the imaging equip- ment and for providing setup guidance as well as experimental assistance. The financial support of Netivei Israel—National Transport Infrastructure Company Ltd. is also acknowledged.
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