Uncertainty Determination for Dimensional Measurements with Computed Tomography

Transcription

1 Uncertainty Determination for Dimensional Measrements with Compted Tomography Kim Kiekens 1,, Tan Ye 1,, Frank Welkenhyzen, Jean-Pierre Krth, Wim Dewlf 1, 1 Grop T even University College, KU even Association A. Vesalisstraat 13, 3000 even, Belgim, wim.dewlf@grop-t.be Katholieke Universiteit even, Mechanical Engineering Department, Division PMA Celestijnenlaan 300B, 3001 Heverlee, Belgim, jean-pierre.krth@mech.kleven.be Abstract De to the availability of increased comptational power combined with better X-ray sorce and detector technologies, the obtainable resoltions now allow to se CT-machines not only for material and medical science, bt also for dimensional metrology. However, the transition from visalization to measring reqires calibration steps to ensre that measrements are traceable to the nit of length. There is still a lack of proper standards to calibrate CT machines and to check their performance for dimensional metrology. In addition, there is still room for a better systematic approach for measrement ncertainty determination. Therefore, this paper presents a method to determine the ncertainty of dimensional measrements with indstrial CT machines based on the ISO-GUM. Sbseqently, some of the ncertainty factors are qantified based on a series of systematic measrements. Keywords: Uncertainty evalation, compted tomography, dimensional metrology, traceability Introdction Uncertainty evalation Determining the ncertainty of dimensional CT measrements is challenging, de to the nmeros factors inflencing the CT accracy and ncertainty; e.g. workpiece, measrement procedre, scanning parameters, ser, No correct statement of the measrement ncertainty of those measrements exists, since CT systems are not traceable to the nit of length. Some stdies have estimated the ncertainty calclation of CT measrements, based on simlations [1], experiments [, 3, 4], or a combination of both [5]. Schmitt et al. conclde that the procedre sing calibrated workpieces is the most promising to investigate measrement ncertainty for CT. Analytical stdies of the measrement ncertainty of CT systems are mostly based on the procedre described in the gideline ISO/TS [6]. The expanded ncertainty is divided in three main ncertainty contribtors: the standard ncertainty de to the calibration, the standard ncertainty de to the measrement procedre and the standard ncertainty reslting from the workpiece. Additionally, a bias is added to take the systematic errors into accont. Sbcategories of those main ncertainty contribtors are rarely made [7]. After a description of the basic measrement procedre, a first main part of this paper presents a theoretical stdy of the CT measrement ncertainty, based on the described measrement procedre and an analytical eqation based on the ISO-GUM [8]. In a second part the main ncertainty contribtors are identified and illstrated with some examples, based on experimental measrements with different test objects. Measrement procedre inclding voxel size and edge correction calibration (Figre 1) 183

2 In the first step of a CT measrement, some hndreds or even thosands of D X-ray images of the object are taken from different anglar positions in the CT machine. In the second step, those images are reconstrcted into a 3D voxel model, e.g. sing the software CTpro. Afterwards, the data analysis can be performed, and dimensional measrements can be extracted. Data captation Reconstrction Data analysis e.g. Nikon CT-machine e.g. software CT-pro Figre 1: Different steps of a CT measrement e.g. software VolmeGraphics Before the reslting length of a CT measrement is known, the voxel model has to be rescaled for the correct voxel size. The resoltion (voxel size) is mainly determined by the position of the workpiece between the X-ray sorce and the detector (i.e. magnification). The closer the object is to the sorce, the better is the resoltion and the smaller are the voxels. Voxel size calibration can be carried ot in different ways. A first, commonly sed option is to se a calibration object. This calibration object is either scanned simltaneosly with the measrement object, or scanned separately at the same position of the magnification axis. The rescale factor is calclated as the ratio of the CT and the reference measrement of the calibration length (e.g. the distance between the center points of two spheres). Another option is to se a known distance on the workpiece itself to rescale the other featres of the CT model. After rescaling for the correct voxel size, an edge detection step is needed making a segmentation between material voxels and backgrond voxels based on a locally or globally optimized grey vale selection. In this paper, an advanced (locally adaptive) edge detection method has been sed ( Stdio Max.1). The reslt of this step is a 3D model that allows to measre the different featres of the object. The advanced edge detection algorithm sed for the measrements in this paper varies the edge grey vale locally to correct for artifacts that inflence the measrement. Nevertheless, even for monomaterial objects, this advanced thresholding method is often inadeqate to find the correct edge, ths introdcing an ncertainty on the measrement reslt. As proposed by Kiekens et al. [9] an additional calibration step can be sed to correct locally for this wrong edge determination, reslting in an edge correction step. Notice, however, that sch edge correction step is not reqired for all distances. A distinction has to be made between edge independent and edge dependent distances. For edge independent distances, the measrement reslt is independent of the chosen grey vale for the segmentation between material and the srronding air (e.g. Figre left), hence no edge correction is reqired. For other distances, however, the measrement reslt is heavily dependent on the edge determination step (e.g. Figre right), hence a potential edge offset needs to be corrected. Figre : Edge INdependent (left) and edge dependent (right) distances Analytically, this rescaling and edge correction can be formlated as follows: = α CT + (1) where is the measrement reslt and α is a rescaling factor to correct the size of the voxels applied on CT, the measred length on the ncorrected CT model. The edge correction term is defined as Δ. 184

3 In Eqation (1), the rescale factor α is the ratio of the calibration length, measred by a reference measrement instrment, e.g. a CMM), reslting in cal,ref, and the measrement of this length by compted tomography, where this distance is measred for the same position of the magnification axis as CT. cal,ref = CT + () Eqation () represents the analytical eqation for the main calibration steps performed on a CT measrement reslt. In order to minimize errors on the rescale factor, it is recommended to rescale on a distance that is as long as possible. PART 1: Theoretical evalation of CT ncertainty This section presents a theoretical evalation of the ncertainty of dimensional measrements with compted tomography, based on a GUM-based analytical ncertainty eqation and on the measrement procedre described above, and more specifically, by Kiekens et al. [9]. In general, the combined standard ncertainty on the measred length can be calclated according to the ISO-GUM [8], based on Eqation (3), where the inpt qantities x i with associated ncertainty xi are spposed to be ncorrelated: N = i 1 x = i xi Applying Eqation (3) on Eqation () yields: + ( 1) CT cal,ref CT cal,ref = + cal,ref + CT (4) cal, CT Basically, the ncertainty on a CT measrement consists of for main ncertainty contribtors, the ncertainty on the reference measrement of the calibration length ( cal,ref ), the ncertainty on this length in the CT measrement ( ), the ncertainty on the CT measrement of the workpiece ( CT ) and the ncertainty on the edge offset term ( Δ ). Each term consists of several sbterms. Dependent on the characteristics of the measred distance, on the calibration method, and on the measrement procedre, some of the (sb)terms of Eqation (4) will be very small or zero, as will be exemplified in Part. PART : Identification of the ncertainty contribtors This section elaborates on the for ncertainty contribtors of Eqation (4) and illstrates some of these with nmerical vales based on experimental measrements with varios test objects. 1 st term: Uncertainty on calibration length cal,ref as measred by calibration device ( cal,ref ) The ncertainty on the reference measrement cal,ref is an ncertainty on a measrement with a conventional (reference) measrement instrment, sch as a CMM or an Abbe comparator. It can therefore be determined based on the available standards for the reference measrement instrment sed. Accrate manfactring and reference measrement of the calibration object can allow redcing this term to very small vales. Eqation 4 allows appreciating that this is especially tre when the calibration length is relatively large in comparison with the actal measrand. nd term: Uncertainty on calibration length as measred by CT device ( ) (3) 185

4 The ncertainty on the CT voxel size calibration length which shold be an edge independent distance consists of both a random error and a systematic error component (see Eqation 5). The former covers npredictable variations in repeated observations of the measrand. If the latter is cased by an effect that is well nderstood and its size is significant relative to the reqired accracy of the measrement, a correction Δ can be applied to to compensate for sch systematic error, rather than acconting for sch systematic error by adding an ncertainty component,systematic. = + (5),random, systematic Standard ncertainty de to random error cal,ct,random The random error component,random represents the repeatability of the relevant part of the CT measrement process for edge independent distances. When the X-ray CT data of the calibration length and of the measrement object originates from one single data capitation, hence also one single reconstrction step, the ncertainty on originates solely from the repeatability of the data analysis step. This is the case when the calibration object and the measrement object can be positioned together on the rotary table of the CT machine, or when an edge independent distance of the workpiece itself can be sed as calibration distance. After reconstrcting the voxel model, the measrement of the calibration length in the analysis software (here Stdio Max.1) is repeated n times. Data captation e.g. Nikon CT-machine Reconstrction e.g. software CT-pro Figre 3: Repeatability within the data analysis Data analysis n measrements e.g. software VolmeGraphics Based on these n independent measrements of the calibration length on the reconstrcted CT model, the experimental variance of the mean of this distance can be calclated as the variance of those n measrements q : s n 1 1 ( ) ( ) q q q where q n n 1 = (6) = 1 is the arithmetic mean or average of the n independent observations q in the software s can be on one single reconstrcted voxel model. The experimental variance of the mean ( ) q sed as a measre of the ncertainty on the calibration length in the CT measrement de to random error, when only one data captation is needed for both the measred and the reference length. This ncertainty is ( q ) s( q ) = cal, random (7) In contrast of doing a complete measrement, inclding data captation, performing different measrements in the software on one reconstrction only takes some mintes, and so is feasible to do for every measrement. However, this ncertainty also can be estimated based on previos measrements (according to the ISO-GUM [8] type B evalation of standard ncertainty). When the standard deviation s(q ) is calclated based on a series of calibration measrements, this vale can be sed as an estimate of the standard ncertainty. In case the calibration object and the measrement object cannot be positioned together on the rotary table of the CT device, two consective data captations with respective reconstrction steps are 186

5 reqired. This implies that additional ncertainties are introdced, de to the repeatability of the data capitation step (CT device) and of the reconstrction algorithms: the voxel size of the CT model for the calibration object might not completely eqal the voxel size of the CT model for the measrement object. Data captation Reconstrction Data analysis e.g. Nikon CT-machine p measrements e.g. software CT-pro Figre 4: repeatability of the complete measrement proces e.g. software VolmeGraphics The best estimate of the variance of the arithmetic mean q of p different data capitations (with the same measrement settings) is the experimental variance of the mean given by s p 1 1 ( ) ( ) q q q p p 1 = (8) meas= 1 where q represents the different measrements (inclding data captation). The associated standard s q, here referred to as ncertainty is the experimental standard deviation of the mean ( ) ( q ) cal, random meas, representing the ncertainty de to the random error on the calibration length in the CT measrement where different measrements are needed. cal, random ( q ) s( q ) meas = (9) In practice, one will not perform mltiple measrements to estimate the standard ncertainty de to different data captations for each and every measrement. We propose to base the ncertainty on an a priori estimation of s(q ) instead of on s( q meas ) (i.e. se a so-called type B evalation, according to the ISO-GUM [8]). In sitations where a type A evalation (evalation based on statistical methods) is based on a comparatively small nmber of statistically independent observations, a type B evalation of the standard ncertainty can be as reliable as a type A evalation. The standard ncertainty on the calibration length is simply this standard deviation, calclated based on a series of calibration measrements ( q ) s( q ) cal, random meas =. The ncertainty on the calibration length de to random errors always is smaller than 1 µm. Standard ncertainty de to systematic error cal,ct,systematic The second term in Eqation (5) covers all ncompensated systematic errors, i.e. all ncompensated repeatable errors that case the voxel size to be non-constant throghot the voxel model. For example, a misalignment between the detector and the rotation axis in a CT machine can lead to voxel sizes that are dependent on the position of the measrand. Figre 5 illstrates this effect on the measrement of a test object that was designed with three rows of 18 steel spheres with a diameter of 4 mm ± 1 µm. The different spheres were in contact, allowing to measre an edge independent distance, i.e. the distance between the centre points of two adjacent spheres. Plastic is sed to hold the spheres, withot introdcing additional artefacts, except for the first and the last sphere, which were exclded from the analysis. Figre 5 shows that the distance between the sphere center points are 7 µm meas 187

6 larger at the top than at the bottom, for each of the three colmns. This is significantly more than the variations de to random errors, discssed above. Distance between centre points of adjacent spheres [mm] 4,00 4, ,999 3,998 3,997 3,996 3,995 3,994 Colmn 1 Colmn Colmn 3 3, Nmber of measrement (from top to bottom) Figre 5 : Illstration of systematic error on calibration length inflence of position (top-bottom) De to the reprodcibility of the effect, it is clear that this concerns a systematic error. Nevertheless, straightforward compensation is difficlt since the magnitde of the observed deviations depends also on the position along the magnification axis. Therefore, it is advisable to perform a machine calibration step at different magnifications on beforehand, in order to estimate the related,systematic. However, withot knowing the exact relationship, often one can estimate the bonds (pper and lower limits a - and a + ) of the interval inclding all possible measrements in a particlar case. If there is no specific knowledge abot the possible vales within the interval, one can assme that it is eqally probable for a measred vale to lie anywhere within it (a niform or rectanglar distribtion of possible vales). The expected vale x i is the midpoint of this interval and a first estimate of the associated standard ncertainty is (according to the ISO-GUM 4.3.7): ( x ) a 3 i =. When a component of ncertainty determined in this manner contribtes significantly to the ncertainty of a measrement reslt, it is prdent to obtain additional data for its frther evalation. Another example of a systematic error can be a temperatre change, casing volmetric expansion of the measrement object, hence a measrement error. However, if the coefficient of thermal expansion and the temperatre difference are known, compensation is possible. Moreover, if the calibration object and measrement object are made from the same material, this ncertainty contribtor atomatically redces to zero. It is important to notice that the effect obtained from a Type B evalation (the systematic error), is inclded as an independent component of the total ncertainty to prevent doble-conting of ncertainty components. The portion of the ncertainty that contribtes to the observed variability (random error) is already inclded in the component of the ncertainty obtained from the statistical analysis of the observations (ISO-GUM ). 3 rd term: Uncertainty on the CT measrement of the workpiece CT Basically, the ncertainty on the CT measrement of the workpiece consists of the same sbterms as the CT measrement of the calibration object. + CT = (10) CT,random CT, systematic Nevertheless, this term shold be inclded separately, since the qantitative vales can be very different. For example, if the measred object contains a distance between two flat srfaces, the repeatability of this distance can be an order of magnitde bigger than the repeatability of measring 188

7 well conditioned sphere centre distances, traditionally sed for voxel size calibration. Other type of distances (cylinders, planes,...) also can introdce systematic measrements, inclded in,ct,syst. 4 th term: Uncertainty on the edge offset term Δ For edge dependent distances an additional term shold be inclded, representing the ncertainty on the edge offset term. Even after an edge detection, based on an advanced algorithm, nmeros parameters are inflencing the ncertainty on this edge. The position of the correct edge is dependent on a nmber of parameters, which are not inclded in the available software algorithms. Different systematic measrements indicate that some of the parameters having an inflence here are srronding material, featre size, orientation, filtration, beam hardening correction and settings. Since some of these inflencing factor are clearly correlated, a smmation of the different ncertainties cased by those different inflences wold overestimate the total ncertainty. Bt, despite of this correlation, it is sefl to give some nmerical examples of the inflence of some of these terms, indicating the importance of this ncertainty contribtor. Dewlf et. al. [10] investigated the inflence of srronding material on the measrement of a steel cylinder. A systematic error arises de to the srronding material arond one part of the object. To illstrate this inflence on the measrement ncertainty, an accrate alminim cylinder with a nominal diameter of 8 mm ± 1 µm was srronded partly by a thick alminim hollow cylinder (d ot = 50 mm; d in = 40 mm). This big alminim cylinder introdces beam hardening problems, since the beam has other properties entering the inner cylinder at the top compared to the bottom, hindering a correct edge detection. The measrement was done withot any hardware or software filter. On Figre 6, representing the diameter of the inner cylinder over different slices, an obvios change in diameter can be observed were the inner cylinder leaves the oter one, introdcing an ncertainty on the edge offset term. 8,16 Diameter inner cylinder [mm] 8,14 8,1 8,1 8,08 8,06 8,04 8,0 8, Slice nmber from top to bottom Figre 6 : Accrate alminim pin srronded by a thick alminim ring, withot filter, BH 1 Conclsion This paper presents a method to calclate the measrement ncertainty of dimensional measrements with compted tomography, based on the analytical eqation that describes the calibration steps needed to calibrate a CT measrement (voxel size calibration and edge correction). For main ncertainty contribtors are identified. Besides the ncertainty on the reference measrement and the CT measrement of the calibration length, the ncertainty on the CT measrement of the workpiece and, for edge dependent distances, the ncertainty on the edge correction term contribte to the total ncertainty. Each of those terms can be sbdivided into different sbterms. Based on systematic measrements, some of them have already been identified, and have been inclded into the analytical eqation. Some (sb)terms need to be inclded for every CT measrement, bt depending on the measred distance and the measrement procedre sed, some of the sbterms cancel ot in some sitations. A series of systematic measrements was performed to qantify some of the sbterms, allowing to indicate the importance of these terms. In this paper, an illstration was given for the systematic errors 189

8 on edge independent distances as well as for the ncertainty on the edge offset term for edge dependent distances. At this moment, more systematic measrements are ongoing, based on different calibration objects with the objective to identify and qantify more of the different sbterms, for edge independent as well as for edge dependent distances. References [1] Wenig, P., Kasperl, S., Examination of the measrement ncertainty on dimensional measrements by X-ray compted tomography, ECNDT, We.3.3.1, 006. [] Bartscher, M., Nekamm, M., Hilpert, U., Neschaefer-Rbe, U., Härtig, F., Kniel, K., Ehrig, K., Stabe, A., Goebbels, J., Achieving traceability of indstrial compted tomography, Key Engineering Materials, Volme 437, p , 010. [3] Weckenmann, A., Krämer, Ph., Assessment of measrement ncertainty cased in the preparation of measrements sing compted tomography, XIX IMEKO World Congress Fndamental and Applied Metrology, isbon, Portgal, September 6-11, 009. [4] Müller, P., Hiller, J., Cantatore, A., De Chiffre,., Investigation of measring strategies in compted tomography, New technologies in manfactring 011, ISBN [5] Schmitt, R., Niggemann, C., Uncertainty in measrement for X-ray-compted tomography sing calibrated work pieces, Meas. Sci. Technol. 1, 010. [6] ISO/TS Geometrical Prodct Specifications (GPS) Coordinate measring machines (CMM): Techniqe for determining the ncertainty of measrement Part 3: Use of calibrated workpieces or standards. [7] Krth, J.-P., Bartscher, M., Carmignato, S., Schmitt, R., De Chiffre,., Weckenmann, A., Compted tomography for dimensional metrology, CIRP Annals - Manfactring Technology, Volme 60, Isse, p , 011. [8] ISO-GUM Evalation of measrement data Gide to the expression of ncertainty in measrement, 008. [9] Kiekens, K., Welkenhyzen, F., Tan, Y., Bleys, Ph., Voet, A., Krth, J.P., Dewlf, W., A test object with parallel grooves for calibration and accracy assessment of indstrial compted tomography (CT) metrology, Meas. Sci. Technol., (7pp), 011. [10] Dewlf, W., Tan Y., Kiekens, K., Vanherck, P., Sense and non-sense of beam hardening correction in CT-metrology, CRIP Annals, Volme 61/1, p ,