Automatic marker-free estimation methods for the axis of rotation in sub-micron X-ray computed tomography

Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 Con en s lis s a ailable a ScienceDi ec Tomog aphy o Ma e ials and S uc u es jou nal homepage: www.jou nals.else ie .com/ omog aphy-o -ma e ials-and-s uc u es Au oma ic ma ke - ee es ima ion me hods o he axis o o a ion in sub-mic on X- ay compu ed omog aphy Ma ek Zemek a,1 , Jakub Šalplach a a , Tomáš Zikmund a,⁎ , Kazuhiko Omo e b , Yoshihi o Takeda b , Pe e Obe a c,d , Joze Kaise a a Labo a o y o X- ay mic o and nano compu ed omog aphy, Cen al Eu opean Ins i u e o Technology, B no Uni e si y o Technology, Pu kyňo a 656/123, B no 612 00, Czechia b Rigaku Co po a ion, 3-9-12, Ma suba a-cho, Akishima-shi, Tokyo 196-8666, Japan c Ins i u e o Physics o he Czech Academy o Sciences, Na Slo ance 1999/2, P ague 182 21, Czechia d Rigaku Inno a i e Technologies Eu ope s. .o., Za Radnicí 868, Dolní B ežany 252 41, Czechia ARTICLE INFO Keywo ds: Compu ed omog aphy Ro a ion Axis Tuning- o k a i ac Au oma ic ABSTRACT Misalignmen o he o a ion axis causes se e e a i ac s in X- ay compu ed omog aphy. Calib a ion o his pa ame e is o en insu icien o sub-mic on esolu ion measu emen s and needs o be co ec ed du ing he pos -p ocessing. This co ec ion can be accele a ed by a ious au oma ic me hods. These a y in mechanisms and pe o mance, making hem sui able o di e en use-cases. This wo k summa izes exis ing au oma ic me hods o es ima ing he o a ion axis in X- ay compu ed omog aphy, wi h a ocus on sub-mic on applica- ions. Some o he me hods a e implemen ed and compa ed in he con ex o a labo a o y sub-mic on scanne o demons a e p ac ical conside a ions o his ask. 1. In oduc ion X- ay compu ed omog aphy (CT) is a ool o he non-des uc i e imaging o in e nal s uc u es o samples [1–4,5]. A CT measu emen consis s o scanning a se ies o p ojec ions o e a ange o angles and p ocessing he acqui ed da a by omog aphic econs uc ion o yield c oss-sec ional images ( omog ams) o scanned objec s [6]. The aw da a p oduced by a mode n labo a o y CT scan can be iewed ei he as indi idual, usually wo-dimensional p ojec ions o as sinog ams, which display a single ow o p ojec ion da a a all acqui ed angles (Fig. 1) [6]. Con empo a y scanne s a e capable o eaching sub-mic ome e (sub-mic on CT) [7] o e en highe (nanoCT) [8] esolu ions. The geome ic alignmen o scanne componen s (Fig. 1) is a sig- ni ican ac o a ec ing he quali y and unce ain y o CT measu e- men s [9]. The mos c i ical alignmen pa ame e is he posi ion o he axis o o a ion (AoR) [10,11] and i s p ojec ion on o he de ec o . Tomog ams a e se e ely deg aded by cha ac e is ic uning o k s eak a i ac s in hal -scan da a [6,10] o double edges in ull-scan da a [6] when he physical posi ion o he AoR does no coincide wi h he AoR assumed du ing he econs uc ion. These a i ac s dec ease in magni- ude as he assumed AoR posi ion app oaches i s ue loca ion (Fig. 2) bu e en ela i ely small e o s can cause a su icien dis u bance o ende he esul ing omog ams unusable o any u he analysis. Such a i ac s should hus be educed as much as possible o minimize hei nega i e impac on he diagnos ic po en ial o images. The AoR can gene ally be calib a ed using specialized phan oms and p ocedu es be o e pe o ming a scan [11]. The phan oms a e usually made o me al wi es [11] o o he high-con as objec s. The AoR is lo- calized by e alua ing he symme y o he phan om as i o a es on he sample s age [11]. Co ec ion o he AoR is hen done by ha dwa e o so wa e ools like adjus ing he o a ional s age o shi ing he acqui ed da a, espec i ely. Howe e , his calib a ion is o en insu icien a he high esolu ions o sub-mic on CT because o he limi ed p ecision o he sample s age [14,15] and a ein oduc ion o he AoR misalignmen while moun ing he sample o due o he mal expansion [16,17]. The posi ion o he AoR mus he e o e be localized and co ec ed a e scanning by analyzing and p ocessing he acqui ed p ojec ion da a. The AoR can also be es ima ed pos -scan in a manual o au oma ic ashion. The es ima ion p ocess can be enhanced using iducial ma ke s placed on he sample [18]. Typical iducial ma ke s a e small pa icles (compa able in size o a single pixel [18]) wi h a high con as [19,20], bu o he objec s like hin wi es [18] may also be used. Fea u es o he sample s age assembly wi hin he ield o iew (FoV) can also se e as e e ences o he AoR posi ion in some cases [21]. The disad an age o h ps://doi.o g/10.1016/j. ma e .2022.100002 Recei ed 6 Augus 2022; Recei ed in e ised o m 9 Decembe 2022; Accep ed 18 Decembe 2022 A ailable online 26 Decembe 2022 2949-673X/© 2022 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). ]]]] ]]]]]] ⁎ Co esponding au ho . E-mail add esses: [email p o ec ed] (M. Zemek), [email p o ec ed] (T. Zikmund). 1 0000-0002-3236-4111 iducial ma ke s is ha hey can obscu e sample s uc u es and com- plica e bo h he sample p epa a ion and he da a p ocessing [18,22,14,20] This means ha i is p e e able o es ima e he AoR using unal e ed p ojec ion da a whene e possible. A manual AoR es ima ion consis s o an ope a o in e ac i ely changing he assumed posi ion o he AoR in an e o o minimize a i ac s in he econs uc ed omog ams (Fig. 2) [16,17]. This is a labo ious p ocess ha is p one o e o s, bu a human ope a o can also le e age hei expe ience and in ui ion o deal wi h complex cases [17]. The au oma ic es ima ion mimics his p ocess by sco ing quan i iable e ec s o AoR shi s in images using objec i e me ics. This can inc ease he da a h oughpu , lowe he ope a o ’s wo kload, and p o ide objec i e and epea able esul s. An au oma ic AoR es ima ion is o en implemen ed as a s and-alone p ocessing s ep o as a pa o a mo e gene al scheme o co ec ing geome ic misalignmen s. An es ima e o he AoR posi ion can also be ob ained as a by-p oduc o some algo i hms o a pe -p ojec ion mo- ion co ec ion [18,22]. This is because he AoR misalignmen is in essence a special case o he gene al mo ion p oblem in which he sample and he s age a e misaligned by a cons an ho izon al shi in all p ojec ions. This a icle ocuses on me hods aimed speci ically o p i- ma ily a he au oma ic AoR es ima ion by p ocessing he scanned da- ase s wi hou he aid o dedica ed phan oms o iducial ma ke s. The ci cula scan ajec o y ( he sou ce and he de ec o midpoin bo h o a e in a single plane) is he mos widely used ajec o y in mos la- bo a o y CT applica ions, and i will be e e ed o in his wo k unless s a ed o he wise. Simila me hods a e also documen ed o ela ed modali ies such as laminog aphy [23] o op ical p ojec ion omog aphy (OPT) [24], bu hese a e beyond he scope o his e iew. AoR es ima ion me hods gene ally all in o ou dis inc ca ego ies as summa ized in Table 1. The i s a e he me hods based on he Fig. 1. Illus a ion o a ypical CT scan geome y, showing key scanne componen s (bold), basic geome ic pa ame e s (i alicized), and di e en iews o aw and p ocessed da a, which a e ele an o AoR es ima ion. Fig. 2. Simula ed image econs uc ed om p ojec ions om a ullscan (360 ∘ ) and hal scan (180 ∘ ) ange, showing he e ec s o double edges and uning o k a i ac s, espec i ely. The 257 × 257-pixel es image was c ea ed in Ma lab by gene a ing, smoo hing, and h esholding uni o mly dis ibu ed andom alues. Bo h p ojec ion da a (scanned in 0. 5 ∘ inc emen s) and omog ams we e c ea ed using he ASTRA oolbox [12,13]. Table 1 O e iew o au oma ic AoR es ima ion me hods g ouped by ca ego y. Cen e o mass Opposi e p ojec ion Sinog am symme y Tomog am e alua ion egis a ion e alua ion Aze edo e al.[16] (1990) Olande [25] (1994) Liu, Malcolm[26] (2006) B une i, De Ca lo[27] (2004) Hogan e al.[18] (1993) Pan e al.[17] (2012) Pa el e al.[28] (2008) Walls e al.[24] (2005) Jun, Yoon[29] (2017) Yang e al.[30] (2015) Liu[31] (2009) Dona h e al.[32,33] (2006) Li e al.[34] (2010) Dong e al.[35,36] (2013) Yang e al.[37] (2012) Yang e al.[38] (2017) Yang e al.[23] (2013) Cheng e al.[39] (2018) Vo e al.[40] (2014) Zhou e al.[21] (2021) Meng, Wu[41] (2017) Lin e al.[42] (2019) Ma e al.[43] (2020) Vo e al.[44] (2021) Vacek, Jacobsen[45] (2022) M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 2 cen e -o -mass o ela ed concep s in he p ojec ions, whe e a well- de ined poin is iden i ied in he p ojec ion da a, acked h oughou he scan, and aligned wi h he AoR posi ion. The second ca eego y consis s o opposi e p ojec ion egis a ion app oaches in which wo p ojec ions aken a opposi e iews a e aligned and hei displacemen is used o es ima e he AoR. Sinog am symme y e alua ion me hods also use edundancies in he acqui ed da a bu include addi ional p o- cessing s eps in o de o es ima e he AoR in a wide ange o geome- ies. The las ca ego y includes omog am e alua ion me hods which e alua e econs uc ed CT slices wi h a ious assumed AoR posi ions. Each o hese ca ego ies is discussed in sec ion 2 and ca ego y-spe- ci ic limi s and condi ions a e also compiled and ou lined. A p ac ical example o implemen ing au oma ic AoR es ima ion me hods is also included in sec ion 3. This example is p esen ed in he con ex o a labo a o y sub-mic on CT scanne wi h a quasi-pa allel geome y. 2. Theo y 2.1. Cen e -o -mass me hods Cen e -o -mass (CoM) me hods include some o he oldes AoR es i- ma ion p ocedu es in he li e a u e [16], wi h se e al au ho s ci ing hei ela i e obus ness o noise [16,29] and as compu a ion imes [16] as hei main s ong poin s. Aze edo e al. [16] a e among some o he i s au ho s o desc ibe a ma ke - ee AoR alignmen me hod. The cen e o mass in hei app oach is calcula ed as a weigh ed mean o pixel posi- ions o a p ojec ion, wi h he weigh s being he abso p ion alues e- co ded by he co esponding pixels [16]. I is impo an o pe o m his calcula ion a e applying a loga i hmic ans o m [18,29] o linea ize he aw X- ay in ensi y da a acqui ed by he CT scanne acco ding o he Lambe -Bee law. The esul ing poin is hen ea ed as a iducial ma ke which can be acked h oughou he scan. A sine cu e is i ed on o he de ec ed poin s and i s o se can be used o es ima e he AoR [16]. Hogan e al. [18] p esen a simila app oach o co ec ing he pe -p o- jec ion mo emen , which is hen adap ed and adjus ed by Olande [25] o speci ically es ima e he AoR misalignmen . Jun and Yoon [29] pub- lished a mo e ecen me hod whe e hey used he cen e o a enua ion (CoA) o de ec a ixed poin in pa allel-beam p ojec ion da a and align his poin om each p ojec ion o a common line [29]. The CoA is a ela ed concep o he cen e o mass, bu i is de i ed speci ically o objec s exp essed as unc ions o he mass a enua ion coe icien a he han elying solely on he measu ed X- ay abso p ion alues [29]. CoM me hods a e only sui able o pa allel-beam (o quasi-pa allel) scans wi h a ci cula ajec o y [31]. They canno be used in he p e- sence o la e al da a unca ion whe e he sample is no en i ely con- ained wi hin he scan FoV and ex ends pas he le o igh edge o he p ojec ion da a [16,29,30], which is a common occu ence in applica- ions such as egion-o -in e es omog aphy [6]. A he e ogeneous de- ec o esponse [25,27,32], X- ay beam ins abili y [27], and nonlinea e ec s such as beam ha dening [16,40] will also nega i ely impac he esul s o me hods in his g oup. The pe o mance o CoM is also im- pai ed i di ac ion and e ac ion a e no negligible [21,40]. Sys- ema ic single-pixel e o s such as de ec i e pixels can also skew he esul s [16]. Mos o hese me hods a e addi ionally hinde ed by low con as [16,27] and some sou ces [25,40,27,32] ha e called he o- bus ness o noise o his ca ego y in o ques ion. 2.2. Opposi e p ojec ion egis a ion Opposi e p ojec ion egis a ion (OPR) app oaches da e almos as a back [25] as he CoM ca ego y in he li e a u e. These me hods assume ha wo p ojec ions aken 180 ∘ apa a e mi o images which can be aligned when one o hem is lipped along he di ec ion pe - pendicula o he AoR. The speed o hese me hods is among hei bigges s eng hs [25]. The OPR me hods a e also ela i ely obus o noise acco ding o some sou ces [25,30]. Olande [25] compa es wo pa allel-beam p ojec ions using he mean squa e di e ence (MSD) me ic, which can be implemen ed in bo h he spa ial domain and he phase componen o he Fou ie spec um. An op imiza ion scheme is applied o ind he minimum MSD which p esumably co esponds o he op imal AoR posi ion [25]. Pan e al. [17] use c oss-co ela ion (in he spa ial o Fou ie domain) o es ima e he shi be ween wo opposi e p ojec ions. They hen com- pensa e he da ase by hal he ound shi o adjus he AoR posi ion [17]. Yang e al. [30] sugges se e al o he AoR es ima ion me hods in he OPR ca ego y, including he phase co ela ion o egis a ion using image keypoin de ec ion and desc ip ion algo i hms [30]. The p inciples o OPR can also be applied o o he han pa allel- beam geome ies wi h some addi ional condi ions and p ocessing. Fan- beam p ojec ion da ase s can be ebinned ( ea anged) o pa allel-beam p ojec ions [25], bu his is only possible in he cen al slice o he cone- beam da a scanned along a ci cula ajec o y [46]. Mo e ad anced concep s can also be applied o accommoda e he cone-beam da a and specialized ajec o ies such as helical CT [47]. These concep s include di e ences along PI-lines [47] (pa ame ic in e al lines [48]) o epi- pola consis ency condi ions [49] in he p ojec ion da a. All o hese concep s gene ally equi e mo e han a single pai o p ojec ions o ind ma ching opposi e da apoin s due o he cone-beam geome y. This means ha al hough hey can be used o c ea e a syn he ic pai o p ojec ions o use in an OPR alignmen , hey a e mo e o en applied in he SSE app oaches discussed in sec ion 2.3 o u he le e age in- o ma ion in he da a a all scan angles. A la ge amoun o me hods ela ed o hese concep s has been used o sol e he mo e gene al p oblem o mo ion co ec ion [47,49]. The wo ks o Pa el e al. [28] and Liu [31] apply p inciples e y close o he OPR in he an-beam and cone-beam geome ies. Bo h a e based on a di ec compa ison o indi idual opposi e da apoin s [31] o en i e p ojec ions [28]. Howe e , hese me hods ope a e p ima ily on sinog ams and use da a a all a ailable angles ins ead o only one o se e al p ojec ion pai s. The o e all cha ac e is ics o hese app oaches co espond he bes wi h he SSE ca ego y and bo h me hods a e hus u he desc ibed in sec ion 2.3. Mos OPR me hods a e limi ed o pa allel-beam CT geome ies [34] and geome ies whe e he sample size is negligible compa ed o i s dis- ance om he adia ion sou ce [25]. Typical cone-beam geome ies may cause a no iceable di e ence in he geome ic magni ica ion be ween sample ea u es close o he sou ce and ea u es u he away om i . This can cause wo opposed p ojec ions o be di e en enough so ha hei egis a ion ails which leads o an inco ec ly es ima ed AoR. Using only a subse o he p ojec ion da a (usually only wo p o- jec ions) makes he OPR me hods ulne able o sample mo emen [17,30,40] and o ixed-pa e n op ical de ec s such as un esponsi e pixels [40]. Insu icien con as [40] o a lack o dis inc ea u es [32] can also nega i ely in luence he esul s o he OPR. Some sou ces also dispu e he obus ness o OPR o noise [38]. The accu acy o he esul s also su e s i he wo egis e ed images a e no exac ly 180 ∘ apa , which is a eal possibili y depending on he scan se up [11]. Ne e - heless, mos AoR es ima es end o be loca ed a mos wi hin se e al pixels o he ue AoR posi ion unless scan condi ions a e ex emely ad e se [17,30]. This makes he OPR me hods sui able as a quick and ough i s es ima e be o e applying a mo e sophis ica ed algo i hm [25,17,30]. 2.3. Sinog am symme y e alua ion Sinog am symme y e alua ion (SSE) me hods exploi he pe iodic na u e o he p ojec ion da a h ough compa isons be ween symme- ical/ edundan da a poin s. This ca ego y expands on he concep s in oduced by he OPR me hods by employing addi ional p ocessing in he sinog am domain a he han ope a ing on indi idual p ojec ions. This makes he SSE much mo e sui able o an-beam and cone-beam geome ies han he OPR. M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 3 This is he mos di e se o he ou g oups as i ea u es me hods which ope a e in bo h he spa ial and equency domains. The spa ial- domain SSE can u he be spli based on whe he an algo i hm employs any iden i ica ion o poin s o a eas o in e es in he sinog am. This leads o a o al o h ee sub-ca ego ies: segmen a ion-based SSE, com- pa ison-based SSE, and equency domain SSE. Me hods in he SSE ca ego y a e gene ally as [31,34,40] and many o hem a e inhe en ly well-equipped o deal wi h noise and o he luc ua ions [31,37] by using da a om all a ailable iews du ing he AoR es ima ion. Segmen a ion-based SSE me hods simpli y he AoR es ima ion p o- blem by ans o ming he inpu sinog am in o a educed bina y e- p esen a ion, which leads o simple and easy- o-implemen AoR es i- ma ion algo i hms. The pe o mance o hese me hods su e s in low- con as da a [26,31] and in he p esence o subs an ial unsha pness [26] o la e al da a unca ion [31]. These me hods also equi e he inpu da a o be ull-scan and hey will gene ally ail when applied o sho e scan p o ocols [26,34]. Liu and Malcolm [26] loca e he le - mos and igh mos edges o a sinog am using edge de ec ion and es- ima e he AoR loca ion as he midpoin be ween hese wo ex ema. Li e al. [34] ake an app oach in which he sinog am is bina ized using an algo i hm such as O su’s me hod [50]. The mean o ay posi ions abo e he h eshold is hen aken as he AoR es ima e [34]. Ma e al. [43] combine he segmen a ion-based and he compa ison-based SSE in o a hyb id me hod. A ough AoR es ima ion is accomplished by segmen ing he inpu sinog am using he mean sinog am alue as he h eshold and localizing he cen e o he segmen ed da a co esponding o he scanned sample. The inpu sinog am is hen spli in o wo 180 ∘ sec ions, which a e co ela ed o ind a e ined es ima e o he AoR [43]. App oaches ha can be ca ego ized as he compa ison-based SSE a e summa ized below. These me hods p ocess he inpu sinog am da a i sel a he han con e ing hem o a simpli ied ep esen a ion. They gene ally e alua e di e ences o symme ic da apoin s o a gi en AoR using app op ia e simila i y measu es. The co ec AoR posi ion is es- ima ed by op imizing hese measu es. This subca ego y also assumes ull-scan inpu da a [37,23,40,42] and i can be sensi i e o misalign- men o he componen s o he scanne [23]. I ends o o e be e pe o mance in low-con as [31] o noisy [37] da a. La e al unca ion is also no an issue he e. In ac , compa ison-based SSE me hods can ope a e only on a na ow s ip o he sinog am as long as he AoR posi ion is con ained wi hin his s ip [31,42]. Howe e , hese me hods can s ill ail when he e a e no enough p ominen ea u es in he si- nog am da a [37]. The SSE me hods a e also sensi i e o sys ema ic e o s such as ing a i ac s in p ac ice. This is because e oneous di - e ences be ween co esponding da a poin s educe he simila i y o hese poin s, which compa ison-based SSE elies on. Liu [31] no es he edundancy o measu emen s in an-beam ull scans and de i es a me hod ela ed o he PI-line p inciple [47,48] men ioned in sec ion 2.2. An X- ay akes he same pa h h ough he sample wice in a an-beam ull scan, only in opposi e di ec ions. The posi ions o hese co esponding measu emen s in he p ojec ions can be calcula ed o a ce ain AoR posi ion. A sum o squa ed di e ences o all hese edundan measu emen s is hen compu ed o e alua e he sui abili y o each AoR es ima e. Minimizing his sco e wi h espec o he AoR posi ion leads o an es ima e o he ue AoR [31]. Pa el e al. [28] compa e opposi e p ojec ions in a cone-beam geo- me y. They no e ha he opposi e p ojec ion pai s will show di e - ences due o he cone-beam magni ica ion, bu posi ha his should no a ec he es ima ion [28]. The me hod is based on selec ing a single ow (sinog am) o a cone-beam da ase and compa ing all opposi e pai s o p ojec ion ec o s wi hin i . The compa ison is pe o med by lipping one o he ec o s, shi ing i by di e en amoun s, and inding he shi ha minimizes he oo -mean-squa e di e ence be ween he pai o ec o s. An a e age o shi s calcula ed o all opposi e pai s is assumed as he AoR es ima e [28]. This p ocess can be epea ed o a ange o ows in he da a, which he au ho s use o addi ionally es i- ma e he AoR il [28]. Yang e al. [37] op o ebin he ull-scan an-beam sinog am da a o a pa allel-beam geome y. They hen spli he ebinned sinog am in o wo 180 ∘ hal es, mi o he second hal -sinog am o e e se he p o- jec ions wi hin, and apply he c oss-co ela ion o ind shi s be ween co esponding p ojec ions o he wo hal es. The mean alue o hese shi s is hen di ided by wo, leading o an es ima e o he dis ance be ween he cu en midpoin o he da ase and he es ima ed AoR [37]. Yang e al. [23] u he s eamline his app oach by omi ing he ebinning s ep and summing all he p ojec ion da a in a an-beam si- nog am o p oduce a single ec o o alues. This ec o is ideally symme ic a ound he AoR posi ion. The au oco ela ion o his ec o e eals he amoun o shi needed o align he da ase ’s midpoin wi h he AoR [23]. Meng and Wu [41] in es iga e he an-beam geome y and come o a simila conclusion as Liu [31] in e ms o he edundancy o he ull- scan da a. A pai o edundan ays a a ce ain dis ance has a well- de ined angula in e al be ween hem. This in e al is exac ly 180 ∘ only o he ays in e sec ing he AoR. The sinog am is hus spli in o wo 180 ∘ hal es and he c oss-co ela ion coe icien o each co e- sponding p ojec ion loca ion om bo h hal es is calcula ed [41]. Ap- plying he c oss-co ela ion on columns o he sinog am ins ead o i s ows allows he au ho s o a oid he ebinning s ep necessa y in he me hod o Yang e al. [37]. Maximizing he alue o he co ela ion coe icien yields he es ima ed AoR posi ion [41]. Compa ison-based SSE me hods a e also sui able o specialized scanning p o ocols such as o se scans [42], whe e he de ec o o he AoR is delibe a ely displaced o one side o he p ojec ion da a and addi ional p ocessing is applied o he da ase o ex end he FoV [42]. Lin e al. [42] p esen a me hod o such o se -scan da a wi h a dis- placed AoR. The cen al ay ( he ay in e sec ing he AoR) is no longe pe pendicula o he de ec o in such scans. This causes mos o he exis ing AoR es ima ion me hods o ail [42]. The me hod bea s e- semblance o he au ho s’ ea lie wo k [23], bu i adds a ans o m o he da a on o a i ual de ec o ha is pe pendicula o he cu en AoR es ima e. The ans o med da a on bo h sides o he AoR a e summed o p oduce wo ec o s o alues ha a e expec ed o be mi o ed a ound he AoR. The a iance o he di e ence be ween hese ec o s se es as a cos unc ion which is minimized o es ima e he co ec AoR posi ion. Vo e al. [44] employ an app oach ha is sligh ly di e en o he p e- ious ones, as i is ailo ed o he pa allel-beam o se -scan da a. The ull-scan sinog am is di ided in o 180 ∘ hal es and he second hal is lipped. The o e lap be ween bo h hal es is hen de e mined by op i- mizing an image co ela ion me ic o na ow egions in bo h hal -si- nog ams. The hal -sinog ams a e hen s i ched in o one wide hal -scan sinog am wi h he AoR posi ioned in he middle o he esul ing da ase [44]. The equency domain SSE is a small subca ego y which ne e - heless o e s some unique p ope ies compa ed o he p e ious ca e- go ies. These p ope ies include possibili y o use wi h hal -scan da a [40] and an e ec i e pe o mance in noisy o o he wise ad e se con- di ions [40,45]. The me hod o Vo e al. [40] is designed speci ically o he pa allel-beam hal -scan da a. A hal -scan sinog am is con e ed o an a i icial ull-scan one by copying and lipping he sinog am and conca ena ing he o iginal and he lipped copy. The AoR misalign- men s will cause discon inui ies in he s acked sinog am and co e- sponding equency componen s in i s wo-dimensional equency spec um. These componen s can be easily dis inguished om he e- quency coe icien s belonging o he genuine s uc u es o he sinog am based on hei loca ion wi hin he spec um. Minimizing he o al magni ude o equency componen s in hese loca ions by shi ing he wo sinog am hal es wi h espec o each o he will lead o an es ima e o he AoR posi ion [40]. Vacek and Jacobsen [45] sum oge he op- posi e p ojec ions in a ull-scan sinog am o p oduce e en ec o s symme ic a ound he AoR. The misalignmen o he AoR o ms a amp in he phase componen , which can be con e ed back o a shi alue o M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 4 he AoR in he spa ial domain. The in luence o noise on he algo i hm is minimized by examining only he low- equency componen s, bu he me hod can ail in he p esence o low- equency luc ua ions i i he p ojec ions a e unca ed [45]. 2.4. Tomog am e alua ion Tomog am e alua ion (TE) me hods model he beha io o a human ope a o du ing manual AoR es ima ion. They sco e econs uc ed c oss-sec ional images using me ics which e alua e changes in he da a quali y caused by he AoR shi s. Op imiza ion o such me ics ideally yields he ue AoR posi ion. Ope a ing on he CT slices makes he TE me hods independen on he used acquisi ion geome y [32]. The TE me hods also end o be obus o luc ua ions in he p ojec ion in ensi y [32]. B une i and De Ca lo [27] obse e ha he AoR misalignmen smea s in ensi ies o da apoin s o e neighbo ing pixels. They use he numbe o non-ze o (non-backg ound) poin s in he omog am as a me ic o minimize in o de o es ima e he AoR. Walls e al. [24] and Dong e al. [35,36] iew he AoR misalignmen as a cause o blu which dec eases he a iance o image alues. Thei me hod is hus based on maximizing he a iance o alues in he econs uc ed omog am as a unc ion o he AoR posi ion [24]. This me ic is sensi i e o unca ion a i ac s on he edges o he FoV [24], s eaks caused by limi ed iews [21], and o he image a i ac s which a i icially inc ease he image a iance. Zhou e al. [21] expand upon he use o his me ic by i s segmen ing he image in o backg ound and o eg ound and only com- pu ing he me ic in he la e . The au ho s epo ha his imp o es he pe o mance o he me ic in measu emen s wi h a poo signal- o-noise a io and a limi ed numbe o p ojec ions [21]. Dona h e al. [32,33] sugges h ee dis inc me ics which all yield an es ima e o he ue AoR posi ion when minimized. The disc e e e sions hese me ics a e: he sum o absolu e alues o pixels in he omog am, he sum o nega i e alues, and he en opy o he g ayscale his og am o he omog am. The i s wo me ics a e suppo ed by ma hema ical p oo s in he o iginal wo k [32] and hei use is limi ed o cases whe e only posi i e alues a e expec ed in he omog am da a. The hi d en opy-based me ic is a heu is ic, bu i can also be applied o omog ams wi h nega i e alues [32]. The obus ness o hese me- ics o noise can be inc eased by a e aging se e al neighbo ing sino- g ams be o e econs uc ion, bu he me hod may s ill exhibi lowe p ecision i he signal- o-noise a io in omog ams is oo low [33]. Yang e al. [38] conside he AoR es ima ion as an image classi i- ca ion p oblem. The ask is o label omog ams as aligned o misaligned based on whe he hey we e econs uc ed wi h he ue AoR posi ion o no . The au ho s ain a con olu ional neu al ne wo k on a da abase o pa ches o manually classi ied omog ams [38]. Tomog ams e- cons uc ed a a ange o AoR posi ions a e hen p esen ed o he ained ne wo k, which p edic s he images wi h ea u es mos in- dica i e o an accu a e alignmen [38]. One limi o his me hod is ha i may ha e subop imal esul s o da ase s which ha e e y di e en cha ac e is ics and ea u es om he aining da a [38]. Cheng e al. [39] e alua e he quali y o omog ams using hei o al a ia ion (TV). A i ac s caused by a misaligned AoR inc ease he TV, which in u n means ha minimiza ion o his alue yields an es ima e o he co ec AoR. The in luence o noise on he TV me ic is educed by applying a smoo hing il e o he omog am [39]. The au ho s also no e ha he image should con ain enough in o ma ion (quan i ied using en opy) o his me ic o p o ide eliable esul s [39]. TE me hods can be enhanced by any a i ac educ ion o denoising algo i hms o he use o ad anced econs uc ion algo i hms. Howe e , his u he inc eases he complexi y and compu a ion ime ela i e o o he AoR es ima ion ca ego ies. The high ime-cos o hese me hods can be especially conce ning when dealing wi h la ge da ase s [34,30,40,42,45], bu esea che s ha e used a ious echniques o e- duce he o al calcula ion ime. These echniques include he use o op imiza ion algo i hms [39] and coa se- o- ine app oaches [35,39]. A p ealignmen using a coa se bu as me hod can also be used o acqui e an ini ial AoR es ima e and a mo e p ecise TE me hod can hen be applied only in a na ow ange a ound his es ima e [35,43,21]. 2.5. Me hods wi h AoR es ima ion as a seconda y ai Some o he wo ks e e enced in sec ions 2.1 o 2.4 combine he AoR es ima ion wi h he es ima ion and/o co ec ion o o he pa ame e s such as he AoR il [28,30], angula e o s [39], mo ion o he sample [18,28], o o he geome ical pa ame e s [32]. Some au ho s use p ope ies o he acqui ed da ase s o e en mo e complex and holis ic alignmen app oaches. These algo i hms o en all in o one o he ca- ego ies ou lined in sec ions 2.1 o 2.4, bu hey pe o m a mul i a ia e op imiza ion o he scan geome y ins ead o localizing only he AoR posi ion. I is common o such me hods o es ima e h ee and mo e pa ame e s such as he sou ce-objec dis ance, slan o he de ec o , and e ical misalignmen s. Examples o such algo i hms include wo ks by Viskoe [51] (mos ela ed o he CoM and TE app oaches), Ky iakou e al. [52] (TE), Pane a e al. [53] (SSE), and Kings on e al. [54] (TE). Rep ojec ion-alignmen algo i hms we e i s used in elec on o- mog aphy [55] and la e applied o CT [56]. This g oup o me hods is closely ela ed o he AoR es ima ion, bu dis inc in se e al aspec s. These app oaches i e a i ely co ec he pe -p ojec ion d i and he sample mo emen [14] o misalignmen s in he scan geome y [57]. The e is a la ge a ie y o di e en app oaches in his g oup [22], and hei p ima y applica ion is omog aphy wi h esolu ions in he o de o ens o nanome e s and less [15,22], whe e he sample can exhibi non- negligible mo ion h oughou a scan. The la es o hese app oaches join ly econs uc , ep ojec , and compa e da a o he o iginal p o- jec ions o es ima e and co ec he d i o samples [22,58] and in some cases e en hei de o ma ion [8] in a single calcula ion amewo k. These ep ojec ion me hods can also co ec sys ema ic shi s o he AoR, essen ially a special case o sample d i , as a by-p oduc o he i e a i e p ocess [22]. Ano he no able alignmen me hod p esen ed by [59] uses com- pa ison o image ea u es in opposi e p ojec ions o es ima ing he misalignmen o he AoR pi ch, oll, and posi ion. The scale-in a ian ea u e ans o m [60] is used in his wo k, bu o he ea u e desc ip o s and ex ac o s [61] can also be used [59]. Fea u es ex ac ed om pai s o he opposi e p ojec ions a e ma ched using a b u e o ce compa ison. The AoR posi ion and o he misalignmen s can hen be deduced om spa ial ela ionships o ma ching ea u es [59]. This me hod di e s om he pos -scan me hods in sec ion 2.2 because i is used p io o a scan o i e a i ely adjus he sample s age and ensu e he alignmen o he ha dwa e i sel . 2.6. Me hods aimed p ima ily a o he modali ies Yang e al. [62] applied an AoR es ima ion me hod in compu ed laminog aphy using an app oach simila o he au ho s’ p e ious wo k aimed a CT [23]. This app oach is no s ic ly limi ed o compu ed laminog aphy, bu he oblique scan geome y equi es a sligh ly di - e en heo e ical app oach when compa ed o CT [62]. All p ojec ion da a a e summed o c ea e a single image whose ows o m e en ec o s symme ical a ound he AoR. The au oco ela ion o he ows yields an es ima e o he AoR posi ion [62]. The AoR es ima ion is also essen ial in ansmission op ical p ojec- ion omog aphy (OPT), an imaging modali y closely ela ed o CT which uses isible ligh ins ead o X- ays [24]. Key di e ences be ween CT and OPT include he much mo e signi ican e ec o e ac ion and sca e ing and a shallowe dep h o ield in he la e [24]. Se e al au oma ic AoR es ima ion me hods aimed p ima ily a OPT ha e been published [24,63,36,35,43]. Many o hese a e also applicable in CT, and so hey a e included in he o e iew in sec ions 2.3 and 2.4. M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 5 3. Implemen a ion o axis-o - o a ion es ima ion Di e en au oma ic AoR es ima ion me hods a y in obus ness, p ocessing ime, and equi emen s placed on he inpu da a. The choice o an app op ia e me hod o be implemen ed in a CT da a p ocessing wo k low mus be conside ed ca e ully and in con ex . This example shows how AoR es ima ion me hods may be implemen ed in he con ex o a labo a o y sub-mic on CT scanne wi h a quasi-pa allel geome y. This in e media e geome y combines ai s o bo h pa allel-beam and cone-beam CT. I is a ela i ely unde - ep esen ed geome y in he exis ing li e a u e on he au oma ic AoR es ima ion, so his example p o ides a use ul amewo k o p o iding addi ional and com- plemen a y insigh o p e ious publica ions. The Rigaku nano3DX scanne [64] used in his example u ilizes a quasi-pa allel geome y wi h a powe ul sou ce a a la ge dis ance om he sample and a ela i ely sho sample-de ec o dis ance. The e- sul ing X- ay beam has a e y shallow cone angle and a low geome ic magni ica ion o a ound 1.02 × o mos egula measu emen s. X- ays a e de ec ed by he XSigh Mic on LC X- ay CCD came a. Rays a e con e ed o isible ligh by a scin illa o and hen magni ied on o he de ec o a ay using mic oscope op ics. The scanne is equipped wi h a Mic oMax-007 HF X- ay sou ce wi h in e changeable coppe (Cu) and molybdenum (Mo) a ge s ope a ing a 40 kV/30 mA and 50 kV/24 mA, espec i ely. Measu emen s wi h he Mo a ge use a 0.1 mm Al il e and no il e ing is applied in Cu measu emen s. 3.1. Model da ase selec ion Fi e model cases (Table 2) we e chosen o co e a ange o ypical applica ions o he used scanne . The collec ion ea u es da ase s wi h low con as and high noise (case 1), homogeneous objec s (case 2), beam ha dening (case 3), signi ican ly unca ed p ojec ions (case 4), as well as a da ase wi hou any signi ican complica ing cha ac e is ics (case 5) (Fig. 3). The e is also he possibili y o sample mo emen ha is no iceable o e he du a ion o a scan bu negligible in-be ween p ojec ions. All es da ase s con ain 800 p ojec ions wi h 1648 × 1250 pixels each, which a e scanned o e a hal -scan ange (180 ∘ ). 3.2. Posi ional accu acy o he AoR es ima e AoR es ima ion me hods need o be accu a e, which means ha he es ima e o an e ec i e me hod mus be close o he ue alue. Howe e , he e is no uni e sally accep ed le el o he AoR accu acy. Some sou ces ega d he accu acy o 0.5 pixels as su icien [17], while o he s ci e a shi o as li le as 0.4 pixels unaccep able [23,37] and aim o a maximum de ia ion o se e al hund ed hs o a pixel [32]. This migh be because he magni ude o uning o k a i ac s is undamen- ally in luenced by aspec s such as noise, con as , sample shape and mo emen , econs uc ion algo i hm se ings, and mo e. An app o- p ia e accu acy le el mus he e o e be de e mined on a case-by-case basis. Fo his example – a sub-mic on CT wi h a quasi-pa allel geo- me y and ela i ely low-con as da a - a maximum e o o 0.5 pixels was deemed app op ia e based on p ac ical es s (Fig. 4) as well as some o he li e a u e [17,27,24]. 3.3. Tes ed es ima ion me hods The selec ion o sui able AoR es ima ion me hods was i s na owed down based on he cha ac e is ics o he used CT scanne . The high p obabili y o unca ion due o he small FoV o sub-mic on CT made he CoM ca ego y imp ac ical. Mos SSE me hods we e also excluded due o hei incompa ibili y wi h hal -scan da a. Fou app op ia e me hods we e inally selec ed o u he e alua ion: . •M1, an OPR me hod which uses he g adien c oss-co ela ion [40,65]. •M2, he SSE me hod o Vo e al. [40] in which a sinog am is du- plica ed, lipped, and s acked on o he o iginal wi h a a ying AoR. Spec al analysis o his s ack e eals discon inui ies be ween he wo sinog ams which a e ideally minimal a he ue AoR. Table 2 O e iew o model cases o es ing selec ed au oma ic AoR es ima ion me hods in he case o he nano3DX CT scanne . The used X- ay sou ce has ixed se ings o each a ge ma e ial, which a e men ioned in sec ion 3. Case Sample Ta ge ma e ial Exposu e Voxel size FoV wid h Cone angle 1 Fibe - ein o ced polyme Mo 8 s 0.53 μm 873 μm 0.09 ∘ 2 Ruby ball Mo 10 s 1.03 μm 1697 μm 0.18 ∘ 3 Li-ion ba e y ca hode Mo 40 s 0.53 μm 873 μm 0.09 ∘ 4 Too hpick Cu 23 s 0.53 μm 873 μm 0.09 ∘ 5 Fo amini e a mic o- ossil Mo 20 s 0.53 μm 873 μm 0.09 ∘ Fig. 3. Sample CT slices o model da ase s shown wi h enhanced con as . Case 1 and i s de ail is shown in (A) and (F), case 2 in (B) and (G), case 3 in (C) and (H), case 4 in (D) and (I), and case 5 in (E) and (J). M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 6 •M3, a TE me hod based on he sum o absolu e alues [32], which i e a i ely econs uc s CT slices in o de o minimize a cos unc- ion. The sum o absolu e alue was chosen as he cos unc ion o e o he s in [32] based on he au ho s’ ecommenda ion. •M4, a TE me hod based on he o al a ia ion [39] which ope a es in a simila manne o M3. O he TE me ics we e omi ed as hey a e based on simila concep s as M3 o M4. Me hods M1 and M2 a e sui ed o he pa allel-beam geome y, bu hei pe o mance in a quasi-pa allel geome y is unclea . The me hods a e s ill expec ed o ou pu es ima es close o ue AoR posi ions, bu hei p ecision may be impac ed. The esul s o M2 can also be in lu- enced by he amoun o unca ion in case 4 in pa icula . Me hods M3 and M4 a e no expec ed o be impac ed by he geome y, bu hey may be a ec ed by some o he a i ac s p esen in he es cases due in pa o he lack o any addi ional p ocessing applied o he da a. These me hods a e also expec ed o ake signi ican ly longe o compu e han M2 and especially M1. Me hod M1 equi es only a single-pass calcula ion, while me hods M2, M3, and M4 op imize hei esul i e a i ely. A simple g id sea ch in 0.5-pixel in e als in a su icien ly wide AoR ange was used o he i e a i e me hods. Da a we e la - ield co ec ed in he nano3DX ac- quisi ion so wa e and all u he p ocessing was pe o med using Ma lab R2020a. All es ima ions we e un on a Windows 10 machine wi h a six-co e AMD Ryzen 5 2600X CPU, an nVidia GeFo ce GTX 1050 Ti GPU, and 64 GB o RAM. Tomog ams we e econs uc ed ia he Feldkamp-Da is-K ess algo i hm [66] implemen ed in he ASTRA oolbox 1.9 [12,13] wi h a cosine il e cu o a 0.85- imes he max- imum equency. No o he p ocessing (such as a i ac educ ion) was applied on he omog ams. Mino adjus men s we e made in he im- plemen a ion o he selec ed me hods compa ed o hei o iginal pub- lica ions in o de o make he es ing seamless. These and o he de ails a e desc ibed in appendix A. Re e ence AoR alues o each case we e ob ained as median alues o a manual AoR es ima ion pe o med by en expe ienced ope a o s who we e checking wi h he 0.5-pixel accu acy le el de e mined abo e. Es ima es o he indi idual ope a o s a e lis ed in appendix B. Resul s o he es ed au oma ic me hods we e hen compa ed o he e e ence alues while conside ing he a ge accu acy le el. The manual es i- ma ion and all au oma ic me hods excep M1 we e pe o med on he sinog am closes o he cen e ow o he de ec o a ay [37]. This si- nog am con ained adequa e in o ma ion in all es cases, so a mo e sophis ica ed choice o slices such as he one desc ibed in [36] was no necessa y. 3.4. Es ima ion esul s and discussion Expe imen al esul s (Fig. 5) show ha none o he es ed me hods we e obus enough o handle all i e es cases wi h a su icien ac- cu acy. M1 and M4 exceeded he 0.5-pixel h eshold in case 5 and M2 and M3 did no each su icien accu acy in case 4. M2 also de ia ed om he e e ence AoR in case 1. Figu e 6shows esul s o he es ed me hods o case 4. Resul s o M1 and M4 a e wi hin he ± 0.5-pixel accu acy window and show no ob ious di e ences om he e e ence. The ou come o M2 is close o he e e ence and con ains no no iceable uning o k a i ac s, bu changes o some o he sample s uc u es can be obse ed. The esul o M3 is se e ely dis o ed. The esul s e eal some in e es ing and no immedia ely ob ious p ope ies o he es ed me hods. The unca ion in case 4 caused M2 o de ia e om he e e ence. This was expec ed as he pe o mance o he me hod is ied o he sample size [40]. Howe e , M2 also eached he second la ges o e all e o in case 1, p esumably due o he low con- as and high noise. The pe o mance o M2 and M1 in he es o he cases shows ha he me hods a e indeed compa ible wi h quasi-pa allel geome ies. Tomog aphic a i ac s we e a gene al conce n o M3 and M4. The unca ion in case 4 had he mos impac on M3 in pa icula . This was su p ising a i s , bu i does in ac seem o be consis en wi h he o iginal publica ion desc ibing M3 [32]. The au ho s men ion ha he Fig. 4. In luence o a misaligned AoR in case 2. E ec s o a posi i e and nega i e AoR misalignmen a e highligh ed by ed a ows. Only e y ain s eaks wi h minimal impac on subjec i e image quali y a e p esen wi h an AoR shi o 0.5 pixels, bu a a shi o 1 pixel s eaks a e al eady no iceable. Fig. 5. E o s o each es ed me hod in all es cases. A ba ending in an a ow indica es he alue is ou side he displayed ange. G een lines indica e he window o alues ha can be assumed as accu a e enough ( ± 0.5 pixels). M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 7 used cos unc ion is sui ed only o measu emen s wi h s ic ly posi i e a enua ion coe icien s [32]. This ende s i ine ec i e o measu e- men s o samples in a su ounding medium like wa e . The esul s shown he e and in appendix Csugges ha unca ion in he p ojec ion da a may ha e a simila e ec . The di e ences in he compu a ion ime be ween he ou me hods we e as expec ed. The wo TE me hods we e compu ed simul aneously, aking 26.8 s o p ocess when a e aged o e i e uns. Mos o his ime was aken up by he econs uc ion. The mo e ligh weigh M2 an o an a e age o 13.38 s. Compu a ion o he single-pass alignmen used in M1 was he as es a an a e age o 0.64 s. While hese un- imes a e indica i e o he ela i e ime demands o each me hod, hey can s ill be signi ican ly dec eased h ough ma hema ical and code op imiza ion. 4. Ou look The esul s sugges ha a gene al and ully au onomous applica ion o any es ed me hod is s ill limi ed, as he pe o mance o hese me hods depends s ongly on he cha ac e o he inpu da a. Di e en me hods a y in hei s eng hs and weaknesses. A comp omise be- ween he au oma ion, p ocessing speed, and eliabili y can hus be eached by a hyb id AoR es ima ion app oach using mul iple me hods. This has also been sugges ed in he p io li e a u e [17,30]. Ano he possible app oach may be o base he inal AoR es ima e on a consensus be ween he esul s o all applied me hods. Depending on he equi ed le el o au oma ion and p ocessing h oughpu , a manual check can be added o lea e he inal decision up o a human ope a o . Fo a la- bo a o y se ing wi h a lowe olume o p ocessed samples, his is an accep able comp omise be ween he con enience and objec i i y o au oma ic me hods and he con ol o a manual es ima ion. The numbe o AoR es ima ion me hods in he ecen li e a u e skews hea ily owa ds he SSE and TE ca ego ies. Ad ances in he compu e ha dwa e mean ha e en demanding ope a ions like omog aphic e- cons uc ion can be pe o med ela i ely quickly. This makes p ac ical use o he demanding TE me hods easible. A he same ime, he SSE and TE ca ego ies end o be he mos e sa ile in e ms o scan geome y and he cha ac e o he scanned da a, so i is likely ha hey will see mo e use in he u u e. The TE me hods in pa icula can employ ad anced image quali y me ics o be e eplica e he e alua ion o CT da a by a human obse e . The use o a i icial in elligence may also become mo e popula o AoR es ima ion, as i s is a majo end in mos o he a eas o image p ocessing. An example o his is he con olu ional neu al ne wo k used by Yang e al. [38] (men ioned in sec ion 2.4). The inc eased capabili ies o mode n compu e s also a o me hods o mo ion co ec ion and algo i hms o mul i-pa ame e geome ic alignmen 2.5. These me hods end o be mo e demanding han he specialized AoR es ima ion, bu hey also o e much mo e lexibili y. This is especially ue in non-s anda d scan geome ies and high- e- solu ion applica ions whe e he sample mo ion is a majo issue. 5. Conclusion The alignmen o he axis o o a ion in a CT scan is essen ial o a high-quali y omog aphic econs uc ion. The es ima ion o his posi- ion di ec ly om he scanned da a is o en he mos expedien ap- p oach o achie e his alignmen . The labo ious ask o es ima ing he AoR posi ion can be au oma ed using a ange o published me hods which all in o ou dis inc ca ego ies wi h speci ic s eng hs and limi s. These ca ego ies include cen e -o -mass me hods, opposi e p o- jec ion egis a ion, and sinog am symme y e alua ion, which all op- e a e on he p ojec ion da a o sinog ams. The inal ca ego y is omo- g am e alua ion. The choice o an app op ia e AoR es ima ion me hod o a pa icula applica ion should be ca e ully conside ed based on he cha ac e is ics o po en ial me hods. Mos published app oaches assume a speci ic scan geome y o angula ange. The omog am e alua ion ca ego y is un- ique in his ega d as i a oids any such assump ions by ope a ing on he econs uc ed da a. Howe e , i is also he mos compu a ionally demanding ca ego y. The egis a ion o opposi e p ojec ions appea s o be a e y quick and obus app oach o pa allel-beam and quasi-pa - allel geome ies. The sinog am symme y e alua ion and cen e -o -mass me hods a e mo e specialized, making hem mo e suscep ible o ail when assump ions abou he inpu da a a e no ul illed. A uly uni e sal and obus au oma ic AoR es ima ion me hod does no seem o exis a he momen , so he combined es ima ion o mul iple di e en me hods can be le e aged o an inc eased obus ness. A basic AoR es ima ion may also be inadequa e in non-s anda d CT applica- ions. Fo ins ance, high- esolu ion nanoCT scans show signi ican amoun s o mo emen be ween consecu i e p ojec ions. A mo e com- p ehensi e co ec ion o any mo ion o he sample and geome ic misalignmen s, such as one o he ep ojec ion-alignmen me hods, migh be mo e app op ia e in such cases, despi e he added compu a- ional complexi y. Decla a ion o Compe ing In e es The au ho s decla e ha hey ha e no known compe ing inancial in e es s o pe sonal ela ionships ha could ha e appea ed o in lu- ence he wo k epo ed in his pape . Fig. 6. Recons uc ion o he da ase in case 4. The same cu -ou a ea as in Fig. 3 (H) is shown he e in (A). De ails in (B) o (E) o he a ea ma ked by a ed squa e show di e en AoR posi ions es ima ed manually (B) and by he es ed me hods. A ows poin o a eas whe e di e ences be ween C-E a e he mos p ominen . M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 8 Acknowledgmen s Funding: We acknowledge CzechNanoLab Resea ch In as uc u e suppo ed by MEYS CR [g an numbe LM2018110]; and he B no Uni e si y o Technology [g an numbe s FSI-S-20-6353, CEITEC VUT- K-21-6956]. We hank membe s o he Labo a o y o X- ay mic o and nano compu ed omog aphy o p o iding he es da ase s o he pu poses o his wo k, and o lending hei ime o manually es ima e he AoR posi ions o he es da ase s. We also hank Ma ina Pořízko á and E a Zikmundo á o p oo eading a ious s ages o he manusc ip . Appendix A. Implemen a ion De ails o AoR Es ima ion Me hods This sec ion summa izes de ails o he implemen a ion o au oma ic AoR es ima ion me hods used in his s udy. The basic s uc u e o he implemen a ions is summa ized below in he o m o pseudocode. The e is a numbe o changes om he o iginal desc ip ions o hese me hods in o de o ensu e p ope unc ionali y in he con ex o he subsequen es s. These a e highligh ed in bold and b ie ly desc ibed. P ocessing and egis a ion a e lexible in me hod M1 (algo i hm A.1). In his implemen a ion, median il a ion is applied o inc ease obus ness o noise, and a 2D Tukey window is applied o make he subsequen c oss-co ela ion mo e eliable. G adien c oss-co ela ion [40,65] was chosen as a as and eliable egis a ion me hod. Algo i hm A.1. o me hod M1. Fo me hod M2 (algo i hm A.2), s eps we e aken o inc ease he obus ness o he algo i hm o nanoCT da a. A di e en σpa ame e was chosen o he Gaussian il e han in he o iginal publica ion. A ow no maliza ion s ep was also added; each ow o he sinog am is no malized o educe he e ec o a changing p ojec ion in ensi y h oughou he scan, which can p oduce alse discon inui ies in he s acked sinog am. Algo i hm A.2. o me hod M2. A simple g id sea ch op imiza ion was used o M3 and M4 in algo i hm A.3 o simpli y he code compa ed o he o iginal publica ions. G id sea ch wi h a s ep o 0.5 pixels was su icien ly p ecise o he pu poses o es ing, no o e ly ime-demanding, and po en ially mo e obus o local op ima han mo e sophis ica ed op imiza ion me hods. Compa ed o he o iginal desc ip ion in [32], he me ic used o he me hod M3 was simpli ied in a way ha does no a ec he esul s. A simple FBP- ype algo i hm wi h a Ram-Lak il e was used o he econs uc ion. M. Zemek, J. Šalplach a, T. Zikmund e al. Tomog aphy o Ma e ials and S uc u es 1 (2023) 100002 9