Advances in Crystallographic Image Processing for Scanning Probe Microscopy

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1 Advances in Crystaographic Iage Processing for Scanning Probe Microscopy P. Moeck Nano-Crystaography Group, Departent of Physics, Portand State University, 1719 SW 10 th Avenue (SRTC), Portand, OR 97201, USA, This book chapter reviews progress in crystaographic iage processing (CIP) for scanning probe icroscopy (SPM) that has occurred since our description of the technique was first put into open access in this book series in the year The signa to noise ratio in a kinds of experienta iages of ore or ess reguar 2D periodic arrays is significanty enhanced by CIP and the technique is independent of the type of recording device. In the SPM iaging context, CIP can be understood as an a posteriori sharpening of the effective experienta scanning probe tip by coputationa eans. It is now possibe to reove utipe scanning probe ini-tip effects in iages fro 2D periodic arrays of physica objects that either sef-assebed or were created artificiay. Accepted within the scientific counity is by now aso the fact that SPM tips can change their shape and fine structure during the operation of a icroscope and, thereby, obfuscate the recorded iages in systeatic ways. CIP restores uch of the seared out inforation in such iages. The adaptation of a geoetric Akaike Inforation Criterion fro the robotics and coputer vision counity to the unabiguous detection of 2D transation syetries enabed uch of our recent progress. In the ain body of this book chapter, we discuss this adaptation and briefy iustrate its utiity on an exape. Keywords: Scanning Probe Microscopy; geoetric Akaike Inforation Criterion; 2D Bravais attices; crystaography 1. Introduction Crystaographic Iage Processing (CIP) is an averaging technique originay deveoped for the anaysis of bioogica acrooecues iaged with transission eectron icroscopes (TEMs) [1-3]. Key to this kind of averaging is the presence of utipe copies of identica objects with we defined two-diensiona (2D) point syetries that are arrayed in a ore or ess reguar 2D periodic anner within an iage that was obtained by a noisy iaging process. Individua defects in the array, such as oecuar or atoic vacancies, for exape, do not present obstaces to the successfu appication of CIP when they are not too nuerous. The CIP averaging technique can, however, be appied to experienta iages fro genuiney paracrystaine [4] arrays ony as a zero order approxiation. It is typicay the structure (incuding the shape, size and syetry) of the icroscopic object itsef that is of interest to science. A 2D periodic array of this object then ony serves the purpose of aking the averaging within CIP feasibe. As a resut of the averaging, the signa to noise ratio of the iage intensity is significanty enhanced so that concusions on the object structure (shape and size) can be drawn with ore confidence. These concusions can in turn be reated to the object's physica properties or bioogica functions. The syetry of the objects in a CIP processed/syetrized iage is deterined by the pane syetry group that has been appied. As far as increasing signa to noise ratios in experienta iages fro 2D periodic arrays is concerned, CIP is far superior to traditiona Fourier fitering [5] because a of the pane syetries that the array ideay possesses are taken into account. Crystaographic iage processing for scanning probe icroscopy (SPM) is conceptuay equivaent to an a posteriori sharpening of the effective experienta scanning probe tip procedure a processing step that is desirabe for any kind of noisy SPM iage fro a 2D periodic array of objects. The point spread function of a SPM or TEM can be derived on the basis of CIP in a straightforward anner fro iages that were recorded fro 2D periodic caibration sapes [6-8]. The inverse of this point spread function can then be epoyed to correct subsequenty recorded iages [8]. In the case of scanning transission eectron icroscopy, CIP aows for the reova of so caed eectron probe tai artifacts fro atoic resoution iages. Portand State University s Nano-Crystaography Group pioneered the adaptation of CIP to the anaysis of noisy 2D periodic iages that were recorded with scanning probe icroscopes [6-14]. The underying object arrays were coposed of either eta-organic or organic oecues on strongy or weaky interacting substrates [8-11] or the square esas of a coercia SPM caibration sape [8]. Whie the forer objects were iaged with scanning tunneing icroscopes (STMs), the coercia caibration sape was iaged with an atoic force icroscope (AFM) in order to deonstrate the infuence of different recording paraeters on this instruent s point spread function [8]. For practica purposes, we found that either square or circuar seections of arrays in experienta iages encopassing severa tens to about one hundred repeating objects sufficed. The atheatica foundations of CIP are coprehensivey discussed in refs. [2,3]. For the convenience of the readers, we discussed soe of the crystaographic foundations of CIP in refs. [6,7] and in the appendices to refs. [12,14]. Note that there are aso two Master of Science theses on this subject [8,11] openy accessibe (directy fro within this book chapter), which feature good descriptions of the atheatica and crystaographic foundations as we. 503

2 In recent years, researchers in the SPM counity began to appreciate that the recording tip of a SPM can change during the operation of the icroscope [15,16] and, thereby, obfuscate detais in the iages in systeatic ways. Significant iage obfuscations were observed for exape in a STM study and it was suspected that the tip ay have picked up oecues fro the 2D periodic array [15]. Whie an AFM tip was scanning across an inorganic crysta under water over severa hours, atoic resoution detais in the recorded iages were suddeny ost. Subte atoic rearrangeents at the tip of the AFM were suspected to be the cause of that [16]. Our own recent deveopents [11-14] are particuary usefu for correcting STM iages for the iaging artifacts of doube and utipe scanning probe ini-tips (or a bunt SPM tip in other words). These deveopents were enabed by our adaptation of a geoetric Akaike Inforation Criterion (AIC) [17-23] to the unabiguous detection of the 2D Bravais attice that an iage of a 2D periodic array woud ost ikey possess in the absence of experienta recording noise. As deonstrated with atheatica rigor in refs. [12,14], the recording of a 2D periodic iage with utipe scanning probe ini-tips cannot change the transation syetry (and 2D Bravais attice) in the iage. Note that both ref. [14] and the 2010 book chapter [6] entioned in the abstract can be accessed openy fro within this book chapter. Aso note in passing that geoetric AICs are known to err a itte bit on the side of caution in coparison to both geoetric Bayesian Inforation and geoetric Miniu Description Length criteria, which coud be epoyed as aternatives [21]. This eans that geoetric AICs have in rea word appications a sight tendency to favor odes with saer deviation easures for a given set of sophistication/generaity easures. Such odes are in this book chapter the ower syetric 2D Bravais attices and their corresponding pane syetry groups. Cobined with the traditiona pane syetry deviation quantifiers [3,6-8,11] of CIP, our geoetric AIC is the key enaber for the correction of utipe scanning probe ini-tip effects in experienta SPM iages. We wi, therefore, in this book chapter concentrate on this criterion and iustrate its utiity on an exape. For the reader s convenience, we aso provide a brief description of the traditiona pane syetry deviation quantifiers of CIP. A good coege eve textbook introduction to crystaography in 2D is ref. [24]. The reference text on this subject is ref. [25]. 2. The three traditiona pane syetry deviation quantifiers of CIP In order to deterine the pane syetry to which a 2D periodic iage and with it the underying ore or ess reguar array of objects ost ikey beongs, one traditionay utiizes Fourier coefficient (FC) apitude (A res ) and phase ange (φ res ) residuas [3,6-8,11]. These residuas quantify how uch an unprocessed iage deviates fro a syetrized (CIP processed) one, and thus serve as figures of erit for deterining which pane syetry group best odes both the experienta iage and the underying sape that has been iaged. There is, however, currenty no fuy objective way to use these two residuas (deviation quantifiers) to assign the correct pane syetry group to a noisy iage. This is because higher syetric pane groups (such as p4) possess a higher utipicity of the genera position (per priitive unit ce) than their subgroups (such as p4, c2, and p2). These subgroups are fored fro their supergroup by the reova of certain syetry operations. Reations such as this are known as incusion reations. Whenever there are incusion reations, one cannot sipy base a decision on which ode best describes the data in the absence of noise by a deviation quantifier aone [17]. This is because the weakest ode, in our case the pane group with the owest syetry, wi aways have the saest distance easure (regardess on how it is defined). Appication of a geoetric AIC [18-23] aows one to ake decisions on the basis of inforation theory because the sophistication/generaity of a of the odes to be ranked enter the cacuations in an appropriate way. In the absence of such a criterion, one woud generay concude that a particuar pane syetry group is the ore ikey group whenever the FC residuas of an iage are not significanty arger for that group than for its respective subgroups. There is however no quantification of what significanty arger ay ean in genera or in any particuar case. In addition to the FC apitude and phase ange residuas, it is aso custoary to utiize the so caed A o /A e ratio [3,6-8,11] for those six pane syetry groups that possess systeatic absences [25]. This ratio is defined as the apitude su of the FCs that are forbidden by the pane syetry but were nevertheess observed, A o (subscript o for odd), divided by the apitude su of a other observed FCs, A e (subscript e for even). For pane syetry groups pg, c, p2g, p2gg, c2, or p4g, this ratio is hepfu for infored pane syetry ode seections because it is zero for strict syetry adherences. For noisy 2D periodic iages, a arger vaue of this ratio akes it ore unikey that the corresponding group is the correct pane syetry. Because of incusion reations [17], the appication of this pane syetry deviation quantifier is aso not fuy objective. Note that the three traditiona pane syetry deviation quantifiers of CIP are a based on Fourier transfors so that decisions are ade in reciproca space. 3. Constraints for our geoetric AIC Within the robotics and coputer vision counities, ode seections invoving non-disjoint casses of odes often epoy geoetric AICs because these criteria dea with incusion reations propery and their appication supports unabiguous decisions in the presence of Gaussian noise of ean zero [18-23]. Whie the utiate goa of appying a geoetric AIC in conjunction with CIP woud be to unabiguousy identify pane syetries in noisy 2D periodic 504

3 iages, we present here the basics of a geoetric AIC procedure which unabiguousy deterines the underying 2D transation syetry in such iages. These transation syetries are represented by the five types of attices that are coony used by crystaographers and are known as the 2D Bravais attices [25], see Fig. 1. We adapted for our purposes a geoetric AIC that was originay deveoped by Kenichi Kanatani and coworkers [22,23] for the cassification of a coputer ouse drawn quadriatera as one of the quadriateras with at east the constraints of a trapezoid (trapeziu outside of North Aerica). In the Eucidean pane, the positions of the vertices for any such quadriatera (i.e. trapezoid, paraeogra, rectange, rhobus, and square) is subject to one or ore of the constraints isted in Fig. 2. Within a set of constraints, the reations are agebraicay independent. obique, p rectanguar, op square, tp b b three hexagona rhobuses cobine to one hexagon hexagona, hp 60 or 90 a priitive subunit of the rectanguar centered Bravais attice, oc a Fig. 1 (a-d) Unit ce shapes for the four priitive 2D Bravais attices. (e) Shape of the priitive subunit (red) of the rectanguar centered 2D Bravais attice. The shapes of these unit and subunit ces correspond to the shapes of quadriateras with two or ore geoetric constraints on their vertices (Fig. 2). The points in (a-d) are attice points. The crystaographic standard conventions [25] for the four priitive unit ces are that the x-axis points downwards and the y-axis point to the right. This resuts in a unit ce ange of γ > 90 for the paraeogra in (a) and γ = 120 for the hexagona rhobus in (d). The ange γ is by crystaographic convention neither 60 nor 90. The coon nae of the quadriatera is given in each pane. The crystaographic nae of the corresponding Bravais attice and its standard abbreviation are given at the botto of each pane. As outined in section 4, a few changes suffice to turn this set of panes into representations of 2D Bravais attices in reciproca (Fourier) space. Modified after a sketch in ref. [11]. Fig. 2 Geoetric constraints on the vertices of quadriateras in the Eucidean pane. One pair of opposite sides is parae to each other in a trapezoid. This constitutes a geoetric constraint. The quadriateras that we are interested in (see Fig. 1) possess ore than one constraint and incusion reations with respect to their shapes (see Fig. 3a). Modified after sketches in ref. [11]. A square, for instance, is subject to constraints {1, 2, 3, 4} in Fig. 2. A rectange is ony subject to a subset of the square s constraints, i.e. {1, 2, 3}. This constitutes an incusion reation [17]. We took four types of constraints for the quadriateras isted above, a invoving cross products or dot products, fro refs. [22,23] and used the to define the shapes of the unit ces of three of the four priitive 2D Bravais attices. For the case of a specia rhobus with 505

4 interior ange of 60 (and attice defining ange of 120 per crystaographic convention [25]), which is of iportance to our particuar appication as it represents the hexagona Bravais attice, Fig. 1d, we cae up with a fifth constraint as isted in Fig. 2. Our fifth constraint aows one to distinguish between rhobuses with and without an interior ange of 60 or 90. The shape of the priitive "subunit" of the rectanguar centered Bravais attice [25] is represented by a genera rhobus (with interior ange (γ ) other than 60 or 90, Fig. 1e), for which the geoetric shape constraints were aready given in refs. [22,23]. Figure 3a represents the geoetric hierarchy of quadriateras arranged by the tota nuber of geoetric constraints. The shapes of five types of quadriateras (in the upper part of Fig. 3a) correspond exacty to the five possibe shapes of units and subunits of 2D periodic arrays, Fig. 1. It is straightforward to construct an incusion reation diagra that is equivaent to Fig. 3a for the syetries of the four priitive unit ces and the one subunit ce of the 2D Bravais attices in purey crystaographic ters, Fig. 3b. A pane syetry group containing a hoohedra 2D point syetry is coony referred to as the hoohedra pane syetry group and is aso the pane syetry group of the corresponding 2D Bravais attice [25]. For the incusion reation diagra of these hoohedra pane syetry groups, the non-transationa syetry operations which are generators for the group, i.e. which cobine with the group s transation vectors to generate a syetry operations of the group, can be taken as syetry constraints on the crystaographic unit and subunit ces. a b b a a = b + (½ ½) >90 because trapezoids are not centrosyetric, they cannot represent unit ces in genuiney crystaine arrays paracrystaine arrays [4], where buiding bocks are siiar but not identica, pane syetries are broken, neighboring buiding bocks are reated to each other by pair distribution functions, and FC peaks in the apitude part of the FT are broadened proportionay to the square of the refection order Fig. 3 (a) Incusion reation diagra of the geoetric hierarchy of quadriateras. The nuber and ist of constraints (fro Fig. 2) on the shape of the quadriateras are given for each eve of the hierarchy. Connectors between quadriateras indicate that the constraints of the botto quadriatera are a subset of the constraints of the top quadriatera. (b) Hoohedra pane syetry group incusion reation diagra of the crystaographic hierarchy of the 2D Bravais attices. Pane syetry groups which contain a hoohedra point group are isted aong with the 2D Bravais attice type. Sets of non-transationa group generator operators (and their ocation within the standard crystaographic unit ce [25,26]) that highight the incusion reations are aso given. Note that a unit or subunit ces in (b) are centrosyetric, i.e. contain two-fod rotation points at position (0,0). Whie (a) refers to the shape of one unit or subunit ce individuay, (b) refers to the shapes of a such ces coectivey as two transations are invoved. Entities beow the dashed (bue) ines at the botto of both panes are outside the scope of traditiona CIP. Modified after sketches in ref. [11]. For the rectanguar centered Bravais attice with pane syetry c2, the generating point syetry operations at the provided ocations in Fig. 3b cobine with three transation, i.e. (1,0), (0,1), and (½,½), rather than the usua two for a priitive Bravais attices. The centering has the effect of doubing the unit ce area in direct space, but does not constitute a crystaographic constraint by itsef. (Endnote [26] continues the discussions of the rectanguar centered unit ce.) The priitive subunit of a attice with pane syetry group c2 possesses the shape/syetry of a genera rhobus with an interna ange (γ ) other than 60 or 90, see Figs. 1e and 3a. A square and a hexagona rhobus can be considered to be specia kinds of rhobuses with the added constraint of an interna ange of 90 or 60 (attice defining ange of 120 [25]), respectivey. Restricting the interna ange to one of these two vaues aows one to ove up in the hierarchy in Fig. 3a. Conversey, reaxing the interna ange constraint eads to a descent in this hierarchy. 506

5 The shapes of four of the quadriateras that are subject to up to four of the five geoetric constraints on their vertices in Fig. 3a (as isted in Fig. 2) correspond directy to the shapes/syetries of the unit ces of four of the five 2D Bravais attices, Figs. 1a-d and 3b. For references to the fifth, the rectanguar centered 2D Bravais attice, Fig. 1e, we utiize the above entioned priitive subunit [26]. The unit and subunit ce shapes/syetries in which we are interested in can, thus, be defined in ters of either geoetric constraints on the vertices of the corresponding quadriateras (Figs. 2 and 3a) or constraints due to the non-transationa generators of the pane syetry hoohedries (Fig. 3b). What is iportant is that the nuber of constraints is identica in both types of hierarchy trees. Note that it is not iportant for our appication of a geoetric AIC how, exacty, the constraints are foruated. Of iportance is ony that the odes for the experienta data, which are ranked by the geoetric AIC with respect of their ikeihood of representing the transation syetry of noisy iage data, are not disjoint. In other words, these odes need to possess a hierarchy of constraints with the ore sophisticated (syetric) ode possessing one ore constraint than the ess syetric ode, whie aso possessing a of the constraints of the ore genera ode. 4. Geoetric AIC for the unabiguous detection of 2D Bravais attices To quantify deviations fro transation syetry odes, we utiize a five eber set of east-squares distance easures J i (i = ), one for each 2D Bravais attice. This type of distance easure quantifies how uch experienta iage data differs fro its ode counterparts, i.e. fro the five crystaographic transation syetry odes, Figs. 1 and 3b. In order to dea with incusion reations within this set of odes propery, we utiize a geoetric AIC. Usefu reations are derived fro the equation for this geoetrica AIC: (i) the ratios of J residuas that aow for the seection of a ode with respect to the other odes within the sae hierarchy branch (Fig. 3), (ii) the inforation content of an iage with respect of the process of seecting a geoetric ode, and (iii) the confidence eve for the preference of a geoetric ode fro a set of odes with incusion reations. Note that our unabiguous identification of transation syetry in an experienta iage fro a 2D periodic array of objects resuts fro both the appication of a set of east-squared distance easures and their usage within the appicabe geoetric AIC. It is irreevant whether the geoetric AIC is epoyed in direct or reciproca (Fourier) space. For coputationa efficiency, we prefer to define our transation syetry deviation quantifiers in reciproca space. The shapes and syetries (or types) of unit or subunit ces of Bravais attices are, of course, unaffected by Fourier transfors between these two spaces. The ajor change required in Fig. 1 to represent 2D Bravais attice in reciproca (rather than direct) space woud be to repace a references to 120 with 60 and vice versa. Stateents such as γ > 90 in Fig. 1 woud need to be changed into γ* < 90. The ower-eft or ost-eft vertex of each of the quadriateras in Fig. 1 coud then represent the position of the (0,0) FC peak in the Fourier transfor (FT) apitude ap. 4.1 The five eber set of transation syetry deviation quantifiers It is not ony practica to deterine both the positions for the vertices of the unit or subunit ces and the uncertainties in these positions in reciproca space, but aso highy advantageous. This is because both the positions of the (1,0), (0,1), and (1,1) FC peaks and their (peak) shapes in the FT apitude ap resut fro transation averaging over a of the seected unit or subunit ces in an experienta iage. We have straightforward coputationa access in the apitude ap of a FT to the average vaues of the reciproca unit (a*, b* and γ*) or subunit (a *, b * and γ *) ce paraeters incuding their error bars. (An endnote entions a crystaographic pecuiarity of the two attice points containing rectanguar centered Bravais attice in reciproca space that is of no further consequence to our procedure [27].) Five syetrized unit or subunit ce shapes (and the corresponding attice paraeters) are obtained in reciproca space (fro the sae experienta data) by CIP enforcing the hoohedra pane syetry group that beongs to each of the 2D Bravais attices [28]. A pane syetry group is enforced by averaging the syetry reated FCs accordingy. As a resut, FC peak heights change in the FT apitude ap. New high order FC peaks ay aso appear. Another resut of the syetrization process is that peaks in the FT apitude ap change their positions [28]. These changes naturay affect the positions of the (1,0), (0,1), and (1,1) FC peaks in the FT apitude ap, resuting in the needed five eber set of unit or subunit ce paraeters in reciproca space and, hence, a syetrized unit or subunit ce shape set. The appied CIP procedures ensure that a unit or subunit ces have the sae area. On the basis of the derived reciproca space unit or subunit ce paraeters, we cacuate a five eber set of transation syetry deviation quantifiers that needs to be considered as part of a geoetric AIC for the decision on the underying transation syetry of a 2D periodic array for which we have an experienta iage. As wi becoe cear beow, the absoute agnitude and physica unit of the two (reciproca attice) transation vectors per unit or subunit ce in the set is not iportant because ony ratios of J vaues need to be considered within our geoetric AIC procedure. Trapezoids cannot represent unit or subunit ces of 2D Bravais attices because they are not centrosyetric. The trapezoid s shape (and its pace in the geoetric hierarchy of quadriateras, Fig. 3a), is, however, usefu to us because it is obtained by extending or shortening one side of a paraeogra (or any of the other quadriateras that we are interested in) by shifting one vertex forward or backward aong the ine of its side, see Fig. 4a. We can use the trapezoid s shape to sape the uncertainty in the deterination of the positions of the above entioned (1,0), (0,1), 507

6 and (1,1) FC peaks in the apitude aps of the five syetrized FTs that we obtained fro the appied CIP procedures as entioned above [28]. The necessary extra shifts of the vertices (which ake trapezoids) out of the quadriateras (that represent both the average unit or subunit ces in reciproca space and the priitive transation units of our five transation syetry odes) can be taken to atch fu widths at haf axiu of the (1,0), (0,1), and (1,1) FC peaks in the set of syetrized FT apitude aps. This idea is iustrated in Fig. 4a on a paraeogra. Variations in the shapes of these FC peaks are due to (direct space) irreguarities in the 2D periodic array and iage, see endnote [29], and infuence the saping of the positions of these peaks. The quadriateras that are defined by the (0,0), (1,0), (0,1), and (1,1) FC peaks in the FT apitude ap of a 2D periodic iage are on theoretica grounds aways centrosyetric. A tota of 16 trapezoids coud be produced fro a centrosyetric quadriatera by shifting vertices as described above (and shown in Fig. 4a) in order to faciitate the saping of FC peak positions. The reciproca space equivaents of 12 of the are currenty utiized by us for the cacuation of our five eber set of transation syetry deviation quantifiers J, see endnote [29] as we as appendix A of ref. [11] for the appicabe equations. Figure 4b iustrates the distances (d i ) fro the vertices of experienta data, x i, of an average unit or subunit ce in an FT apitude ap of an experienta 2D periodic iage to the corresponding ode vertices, x i, of one of the quadriateras that possess one of the shapes of the unit or subunit ces of the four higher syetric 2D Bravais attice odes (Figs. 1b-e). The su of the squares of these distances is our transation syetry deviation quantifier: x2' + x3 x3' + x4 x4' J = x (1) We use prie notation for our ode unit or subunit ce shapes to differentiate the fro the experienta unit ce shape, which is given without pries, Fig. 4b, and for generaity taken to be a paraeogra. What is iportant for cacuations of J vaues is that the ode unit or subunit ce area is aways kept constant and atches that of the experienta unit ce area. (This is ensured by the CIP syetrization procedure described above, which deivers a set of reciproca attice or priitive subattice paraeters with error bars [29].) The uncertainties of deterining the position of three of the vertices of each ode quadriatera (corresponding to (1,0), (0,1) and (1,1) FC peak positions) are saped as discussed above, but the saping coud be iproved by the utiization of other schees. We currenty cacuate 12 preiinary j vaues in an anaogous anner to reation (1) for each fina vaue of our transation syetry deviation quantifier. A fina J vaue is obtained for each of the five odes by averaging over the corresponding preiinary j vaues. Error propagation cacuations ead fro the position uncertainties of three vertices of non-centrosyetric quadriateras (i.e. trapezoids) in reciproca space, via error bars on the reciproca attice or subattice paraeters, to error bars on each of the five transation syetry deviation quantifiers J. a π-γ* b (0,0) exp * (1,0) x 1 = x 1 ' x 2 ' x exp 2 Fig. 4 (a) Paraeogra fro which two trapezoids are created by shifting a vertex forwards or backwards parae to one of its sides. These kinds of trapezoids are used for the cacuation of preiinary j vaues that contribute to a fina J vaue. The Gaussian distribution function inset at the shifted vertex iustrates that a trapezoid s shape sapes the position of that vertex. (b) Iustration of our transation syetry deviation quantifier J (as obtained fro the corresponding preiinary j vaues). The sus of the squared differences between the positions of the (0,0), (1,0), (0,1), and (1,1) FCs in the apitude ap of the FT of an experienta iage and the positions of their ode counterparts in FT apitude aps (of versions of that iage) that were syetrized to hoohedra pane syetry groups by CIP [28] are east-squares easures of the deviation of the experienta data fro the five possibe transation syetries. For specificity, the vertex distances between an (experienta) paraeogra and a (ode) square of the sae area are shown, but the iustration of J for the other four cases is anaogous. Modified after siiar sketches in ref. [11]. Since there are five 2D Bravais attices (in both direct, Fig. 1, and reciproca space) there is in our procedure a five eber set of J residuas that quantify deviations fro the crystaographic transation syetries. In other words, the set of J residuas are east squares distance easures for how uch the shape of an average unit or subunit ce of a 2D periodic iage differs fro the shape of the corresponding quadriateras that represent the priitive transation entity of the corresponding 2D Bravais attices. Ony ratios of J vaues wi be of interest for the appication of the geoetric AIC beow so that we noraize the set of transation syetry deviation quantifiers by dividing a ebers of the set by the J vaue for the paraeogra, which wi typicay be the saest. When we assue that the reevant FC peak positions are (at east in a first order approxiation) subject to independent Gaussian noise of ean zero and variance σ 2, it foows that the ratio J/σ 2 ust approxiatey obey a χ 2 distribution with rn n degrees of freedo [18,20]. x 3 ' * x 3 (0,1) exp x 4 ' x 4 (1,1) exp 508

7 4.2 The geoetric AIC and paraeters for our geoetric ode seection procedure For coparing two non-disjoint odes S and S, with residuas J and J reative to the sae experienta iage data, we deterine which ode is the better fit by coparing their geoetric AIC vaues (rather than their J residuas aone). We do this (as aready entioned in section 2) because any set of residuas wi favor the ess constrained (ore genera/ess sophisticated) ode whenever there are incusion reations [17]. (S is a ore sophisticated/syetric ode (superscript for ore) than S (superscript for ess) and possesses the higher nuber of constraints L. L is appicabe to ode S so that L > L ). The geoetric AIC for a ode S is defined as foows [18,20]: 2 AIC ( S) = J + 2( DN + n) σ (2) The data has N degrees of freedo and is subject to independent Gaussian noise with standard deviation σ and ean zero. The ode S is represented by a anifod of diension D with n degrees of freedo. The geoetric AIC easures the predictive capacity of a geoetric ode in inforation theory ters [18-21]. If the iage data that needs to be judged with the hep of the geoetric AIC is the positions of the vertices of an arbitrary quadriatera (under no constraint) in the Eucidian pane, then each vertex has 2 degrees of freedo, giving the entire data a degree of freedo N = 2 ties 4 = 8, which is aso the diension of the data space. An arbitrary quadriatera can, thus, be defined as a singe point in an 8-diensiona data space. If a ode S for the data has additiona constraints, it wi define a sub-anifod of the data space with diension D and n degrees of freedo (i.e. the degrees of freedo of a vector which paraeterizes the L constraints). For exape, a square ode can be defined as a point, constrained to ie on the surface of a anifod, S, which is a inear subspace of the 8-diensiona data space. The co-diension of a inear subspace, r, is defined as foows: codiension (subspace) = diension (data space) diension (subspace). In particuar, the sub-anifod representing the ode space for squares possesses diension D = 8 4 = 4 (fro D = N L where L is the nuber of constraints which can be geaned fro Fig. 3), degree of freedo n = 4 (n = L), and co-diension r = 8 4 = 4 (r = N D). Siiary, a rectanguar ode possesses D = 5, n = 3, and r = 3. Tabe 1 ists the paraeters of the appicabe AIC for the five quadriatera shapes that we are interested in as odes for transation syetric unit or subunit ce shapes. Tabe 1 Geoetric paraeters that enter into the cacuation of the appicabe geoetric AIC for our unabiguous transation syetry identification procedure. The degree of freedo (or diension) of the data space (N) is eight. ode diension (D) degree of freedo (n) co-diension (r) # of constraints (L) paraeogra / p2 / p rectange / p2 / op genera rhobus / c2 / subunit of oc square rhobus or rectange / p4 / tp hexagona rhobus / p6 / hp Ratios of J residuas that aow for the seection of one non-disjoint ode over another When S is a ore sophisticated ode than S (as obtained by iposing an additiona syetry/geoetric constraint on ode S ), it is shown in refs. [18,20] and backed up by experienta/siuated evidence [19,21] that, in the first order, the foowing is an unbiased estiator of the squared noise eve for both odes: 2 J σ = r N n (3) where r is the co-diension of the ode and the reation is vaid as ong as rn n is arger than zero. As the squared noise eve is the sae for both (the and the ) odes, we do not need to estiate it, but it needs to be Gaussian distributed noise of ean zero. For other types of noise, reations (2) and (3) ay either be fufied approxiatey or the whoe geoetric AIC approach ust be generaized [18,20]. A ore constrained ode S wi be favored over a ess sophisticated ode S if it has a saer geoetric AIC, AIC ( S ) < AIC( S ) (4a) For AIC ( S ) > AIC( S ) (4b) the ess constrained ode wi be favored over the ore sophisticated ode. No decision on which ode is favored is possibe for AIC ( S ) = AIC( S ) (4c) Cobining equations (2), (3) and (4a), we are eft with the foowing reation: J 2( D D ) N + 2( n n ) < 1+ J r N n (5) which ust be true for the ore sophisticated ode S to be favored over the ess syetric ode S [18,20]. 509

8 For the five quadriateras in the upper part of Fig. 3a, equation (5) reduces with variabe substitutions according to section 4.2 (as suarized in Tabe 1) to the foowing inequaity: J L L < 2 J L (6a) where L is the nuber of constraints on the ore syetric (sophisticated/constrained) ode (superscript ) and L is the nuber of constraints on the esser (i.e. ore genera) ode (superscript ). For coparing the shapes of two quadriatera odes with an incusion reation with respect to the sae iage data, the ore constrained ode wi be favored if reation (6a) is satisfied [22,23]. The ess constrained ode wi be favored otherwise, i.e. when J L L > 2 J L (6b) 4.4 Inforation content of an iage and confidence eve of a geoetric AIC based preference On the basis of the geoetric AIC for two quadriatera odes with an incusion reation, the inforation content, K, of an experienta iage with respect to the process of ode seection was introduced in refs. [19,20] as: AIC( S ) r N n J 2( D N + n ) K = = ( + ) AIC( S ) (2D + r ) N + n J r N n (7) Whenever the inequaity (4a) is fufied, K is saer than unity and the ore sophisticated ode represents the iage data better than the ess constrained ode. In the specia case of K = 1, no decision can be ade if the ore sophisticated ode or the ess syetric ode is the better representation of the iage data. This case corresponds to J L L = 2 J L (6c) and the confidence eve, C, of a decision in favor of either ode is zero percent. The axia (100 %) confidence eve C ax for a decision that the ore syetric ode represents the iage data better than the ess constrained ode is given by the condition J = J, which resuts in the reation: AIC( S ) N(2D + r ) + 2n n cri K = = AIC( S ) N(2D + r ) + n (8) for the critica inforation content fro which onward such a decision is possibe. This entity can be utiized for the noraization of the axia confidence eve to certainty (100 %), yieding: (1 K ) C = 100% (1 Kcri ) (9) Equation (9) is just an ad-hoc definition, whereby confidence eves of ess than 50 % do not ipy that reation (6a) and concusions that foow for it are no onger vaid. Our confidence eve reation is intended to support the enta assessent of the J /J ratio region within which reation (6a) is indeed vaid, i.e. between unity and the nuerica vaue on its right hand side. Low confidence eves ean according to reation (9) that one is cose to (but sti beow) the axia J /J ratio where the ore syetric ode is sti favored over the esser ode. At ow confidence eves, one needs to be extra cautious if the assuptions behind the appication of the geoetric AIC are reay justified. Possibe appications of the geoetric AIC-ratio based inforation content concept are outined beow. 4.5 Towards autoated procedures that obtain the best possibe signa to noise ratio in 2D periodic iages As pointed out in ref. [20], reation (7) can serve as a continuous easure on the preferabiity of one ode over another within soe dynaica process. For the recording of iages fro 2D periodic arrays with known transation syetry in the ow eectron dose ode in a dynaica transission eectron icroscope (DTEM), for exape, one coud increase the eectron dose graduay, in an increenta anner, whie testing the recorded preiinary data using reations (7) to (9). In this way, one coud deterine whether or not increased dosing shifts these easures toward or away fro the correct (a priori known) concusion. The signa to noise ratio in these data woud be expected initiay to iprove continuousy with increasing eectron dose so that the confidence eves of reations (9) wi tend towards a axia obtainabe vaue. Increasing the eectron dose further in increents after this axia vaue is obtained does not ake sense and coud be an indication that the sape is now starting to be daaged structuray by dissipation of the energy that the eectron bea deposited. This energy dissipation woud resut in continuousy worsening signa to noise ratios as the eectron dose is further increased. Siiar effects woud be obtained if the sape were drifting whie the eectron dose is increased increentay. Again one shoud stop increasing the eectron dose when the confidence eve for the decision in favor of the (a priori) known transation syetry begins to decine. 510

9 In ters of an appication in robotics and coputer vision [20], a robot coud activey contro a caera so that the inforation content of continuousy recorded iages tends to its axia vaue in dependence of externa infuences such as changing scenery or ighting conditions. On a DTEM, a coputer with "eectron vision" coud decide if further increasing the eectron dose is ikey to ake the signa to noise ratio better or worse. When the robot on the DTEM tries to axiize the inforation content of the so far accuuated iage data with respect to a geoetric AIC-ratio based decision in a transation syetry ode seection process by using reation (9), it aso tries to find the axia signa to noise ratio for the (fina) recorded iage on the basis of the (a priori) known 2D Bravais attice of the sape. Whie this vision of future deveopents is in ine with eerging trends in aterias inforatics [30], future DTEM appications wi probaby require generaized geoetric AICs that can dea with Poisson noise. Such appications coud then aso be supported by a future geoetric AIC for fu pane syetry ode seection. 5. Guideines for the appication of our geoetric AIC Taking up our exape fro section 3, we see fro Fig. 3 that a square s shape is subject to four geoetric (or crystaographic) constraints, whie there are ony three such constraints on a rectange s shape. Then according to reation (6a), the square attice is to be favored on inforation theoretica grounds over the rectanguar attice as transation syetry for an experienta iage as ong as the residua J for the square ode is saer than 5 / 3 J, the residua for the rectanguar ode. So a square woud be the correct choice for the unit ce shape despite its typicay arger residua as ong as reation (6a) is fufied and there are ony negigiby sa systeatic errors in the experienta data to which the geoetric AIC ethodoogy is appied. To dea with the excusion of the other three possibe unit ce shapes, reations (6a) and (6b) need to be epoyed repeatedy. Confidence eves are to be cacuated with reations (9). For exape, in order to concude that the shape of a unit ce is indeed a square, it ust be preferred per geoetric AIC over both a rectange s and a genera rhobus shape, which in turn have to be both preferred over a paraeogra s shape. The genera rhobus, on the other hand, ust aso be preferred over a hexagona rhobus in this case. It needs to be ephasized that there is no need for a rue of thub or any arbitrariy introduced (subjective) threshod to copare the transation syetry deviation quantifiers for two transation syetry odes in order to find out which one of the is ore ikey to present the data in the (extrapoated) absence of iaging induced noise. We have instead fuy objective additive ters within the geoetric AICs that account for the odes sophistication/generaity. Any kind of threshod that is externa to the probe is, therefore, not needed. We are justified to choose a square ode or a rectanguar ode for the transation syetry depending on whether the inequaity reations (6a,b) are satisfied or not, respectivey. In addition, we even have with reation (9) a robust inforation theory based easure for the confidence of our seection in favor of the ore syetric ode. Pugging the characteristic vaues for the square shaped unit ce and the rectanguar shaped unit ce fro Tabe 1 into reations (6a), (7), (8), and (9), and assuing that the ratio of their two J residuas is 4 / 3, i.e. sighty higher than haf way in the J to J ratio interva that aows for decisions in favor of the square unit ce, we obtain a confidence eve of approxiatey 49 %. Note that this vaue being saer than 50 % does by no eans ipy that the rectange shoud be chosen over the square as unit ce ode of the experienta data! Reation (6a) ceary andates the opposite! A ow confidence eve of about 15 % and saer eans that one shoud be cautious in preferring the square over the rectange as average unit ce shape and perhaps reanayze if reevant systeatic errors (that need to be corrected as far as this is possibe before the appication of the geoetric AIC) are reay negigibe in coparison to rando Gaussian errors of ean zero that are deat with effectivey by the geoetric AIC. With the hope of boosting the confidence eve, one coud try to devise a better FC peak position saping schee that does not invove trapezoids and ake the transation syetry deviation quantifiers ore accurate and precise by that route [29]. For high confidence eves, on the other hand, one does not need to be too concerned about the accuracy and precision of the J /J ratios. We wi present two exapes of the unabiguous identification of square Bravais attices in section 6 beow. The exape iage that we wi show expicity invoves siuated data in the context of the reova of a bunt SPM tip artifact. Due to that data having been siuated, confidence eves of geoetric AIC based decisions on the underying transation syetry ode wi be essentiay 100 %. The other exape deas with experienta iage data that has aready been anayzed in ref. [11], so that there is no need to show the corresponding iage here. 6. Reova of utipe ini-tip iaging artifacts in STM iages Athough ony a siuated exape, Fig. 5 suffices for our purpose of showing how our unabiguous transation syetry detection procedure enhances traditiona CIP. The p4-syetry in Fig. 5a is syetricay perfect because we iposed this syetry on an experienta neary-p4 STM iage [31] using the CIP progra CRISP [32]. In Figure 5b we have artificiay constructed an iage akin to what one woud see with three STM ini-tips shifted ateray with respect to each other, constituting a bunt tip, and siutaneousy scanning the sae sape surface. Obfuscated iages such as Fig. 5b have been discarded (or isinterpreted ) in the past, but CIP presents an 511

10 aternative to recover the correct otif inforation fro the. Our geoetric AIC procedure can be appied to such iages and CIP can be used for the purpose of their subsequent crystaographic averaging which resuts in the sharpening of the effective experienta scanning probe tip and restoration of the key features of the 2D periodic otif, Fig. 5c. We find fro visua inspection that the un-obfuscated sape/iage in Fig. 5a, and the obfuscated iage in Fig. 5b (as we as the CIP processed iage, Fig. 5c, of course), possess the sae transation syetry, which is that of the square 2D Bravais attice. Whie the obfuscation due to utipe SPM ini-tips cannot affect the transation syetry, it ay odify the point syetry of the 2D periodic otif in the iage significanty. This eans that whie the subiages of the individua 2D periodic array objects ay becoe seared out beyond recognition, they wi sti be arranged in the very sae 2D periodic anner throughout the whoe iage. These facts are iustrated by Fig. 5b. Tabes 2a and 2b ist the traditiona pane syetry deviation quantifiers of CIP fro the appication of the CRISP progra [32] to this figure. a Fig. 5 (a) A 512 by 512 pixe siuated sape/iage whose p4-syetry is known by design (on the basis of an experienta STM iage in ref. [31]). The area of this iage is approxiatey 80 n 2 and contains a sufficient nuber of 2D periodic objects for CIP appications to be eaningfu. (b) Siuation to ode what three non-coinear ini-tips (or a bunt STM tip in other words) woud have recorded fro this sape, constructed in Photoshop. (c) CIP reconstruction of (b) with pane syetry p4 enforced. b c (a) p2 p11 p11 p1g1 p11g p2 p2g p2g p2gg c11 c11 c2 p4 p4 p4g A res n.h ϕ res A o /A e n.h. n.h. n.h n.h n.h. n.h. 1.2 (b) p3 p31 p31 p6 p6 A res ϕ res A o /A e n.h. n.h. n.h. n.h. n.h. Tabe 2. Traditiona pane syetry deviation quantifiers fro the appication of CRISP to Fig. 5b. The traditiona pane syetry deviation quantifiers for group p4 are given in bod face. The etters n.h. stand for not hepfu (because of being 0 aways) (a) Non-hexagona pane syetry groups and (b) hexagona pane syetry groups. Due to our sape/iage (Fig. 5a) having been siuated assuing both idea iaging conditions and a perfecty reguar square array of oecues, the noraized J vaues of Fig. 5b are essentiay unity for a of the non-hexagona 2D Bravais attices. Inequaity (6a) is, thus, fufied for a of the non-hexagona Bravais attices, where superscript stands for the square attice in the rounds of decisions that resuts in its preference over both the rectanguar (priitive) and the rectanguar centered attice (which foowed the rounds of decisions of the preference of both odes over the obique ode). This corresponds according to reation (9) to a confidence eve of essentiay 100 % for the preference of the square attice as ode for the transation syetry in Fig. 5b. The noraized J vaue for the hexagona ode with respect to the pseudo-experienta data in Fig. 5b is, on the other hand, as arge as 104.5, i.e. ore than two orders of agnitude arger than its counterpart for the rectanguar centered ode. Inequaity (6b) is obviousy fufied (as >> 5 / 3 ), so that there is no doubt at a that a genera rhobus is preferred over a hexagona rhobus. These kinds of (quaitative and quantitative) resuts are, of course, to be expected because the J residuas are, by definition, not affected by bunt scanning-probe tip effects. (We aso know, of course, fro the design history of Fig. 5b that it ust possess the transation syetry of the square Bravais attice.) The assessent of the three traditiona pane syetry deviation quantifiers of CIP is with our detection of the correct transation syetry sipified to a decision between the three pane syetry groups that are based on a square attice, i.e. p4, p4, and p4g. Evauating these quantifiers in Tabe 2a for these three pane syetry groups, it is cear that the underying pane syetry is p4, which we, of course, aso know to be true fro the design history of Fig. 5b. A typica resut for an experientay obtained (rather than siuated) iage fro an array of objects with an approxiate (rather than a ade perfect by syetrization) square 2D Bravais attice is that the noraized J vaues 512

11 for the non-hexagona transation syetries increase sowy with the nuber of constraints on the transation syetry odes, whie the corresponding vaue for the hexagona ode is about two orders of agnitude arger. The anaysis of the experienta STM iage fro ref. [31] that we utiized above to create Figs. 5a,b resuted, for exape, in the foowing noraized residuas: J p = 1.00, J op = 1.01, J oc = 1.10, J tp = 1.15, and J hp = [11,29]. We use here the crystaographic standard [25] two-etter abbreviations of the five 5 Bravais, see Fig. 1, as subscripts. The confidence eves for decisions in favor of the square ode over both the rectanguar (priitive) ode and the rectanguar centered ode are approxiatey 77 % and 93 %, respectivey. The rectanguar (priitive) ode is favored over the obique ode with a confidence eve of approxiatey 99 %. The confidence eve for the decision in favor of the rectanguar centered ode over the obique ode is approxiatey 90 %. It is, therefore, quite cear that the noisy experienta iage fro ref. [31] (that we utiized above to create Figs. 5a,b) and with it the underying oecuar array possesses a square Bravais attice, athough this ight be ony a pseudo-syetry [33]. Fortunatey, pseudo-syetries are soewhat rare in nature when arrays of physica objects sef-assebe and the syetries of the aggregates resut fro oca interactions between the atos and oecues. 7. Suary and Outook A detaied account of our adaptation of a geoetric AIC to the tasks of the identification of the underying transation syetry of experienta SPM iages of ore or ess reguar 2D periodic arrays of identica objects was given. Our set of J residuas, based on the positions of a few peaks in Fourier transfor apitude aps, is usefu as a east-squares distance easure of crystaographic transation syetries to be copared using a geoetric AIC. This is because unike the three traditiona CIP figures of erit, our J residuas are not affected by severe distortions of the periodic iage otif by recordings with utipe ini-tips (aso known as bunt tips). The appication of our geoetric AIC akes the identification of the transation syetry unabiguous, provided that reevant systeatic errors are negigibe and rando errors possess at east approxiatey a Gaussian distribution with ean zero. Identifying the underying transation syetry in 2D periodic iages iproves the syetry identification step in CIP anayses by reducing the possibe pane syetry groups to ony those copatibe with the 2D Bravais attice identified. Decisions as to which pane syetry ost ikey underies noisy experienta iage data fro ore or ess reguar 2D periodic arrays of sef assebed objects are, however, currenty not copetey objective because one has to rey there on the three traditiona pane syetry deviation quantifiers of CIP. Our future work wi invove deveoping a geoetric AIC procedure which can be used to identify the (fu) pane syetry group of a noisy experienta iage directy (rather than being iited to its transation syetry part). Acknowedgeents This work was supported by both Portand State University s (PSU s) Venture Deveopent Fund and the Facuty Enhanceent progra. A grant fro PSU s Internationaization Counci is aso acknowedged. Coaborative work with forer ebers of the author s/psu s Nano-Crystaography Group, i.e. his MSc students Bi Moon and Tayor Biyeu, as we as with his coeague (in PSU s Departent of Physics) Prof. Jack Straton is gratefuy acknowedged. Tayor Biyeu is aso thanked for creating the sketches that are shown here in odified for as Figs. 1 to 4. Professor Kerry Hipps of Washington State University at Puan is thanked for the raw STM iage that we used to create Fig. 5a. Professor Jack Straton is thanked for creating Fig. 5b. Professor eeritus Kenichi Kanatani of Okayaa University and two current ebers of PSU s Nano-Crystaography Group, i.e. PhD students Pau DeStefano and Andrew Depsey, are thanked for critica proof readings of the anuscript of this book chapter. References [1] Kug A. Iage anaysis and reconstruction in the eectron icroscopy of bioogica acrooecues. Cheica Scripta ; 14: (Proc. 47 th Nobe Syposiu, Kierkegaard P. editor, 1979, August 6-9, Lidingö, Sweden.) [2] Dorset DL. Structura Eectron Crystaography. New York and London: Penu Press; [3] Zou X, Hovöer S, Oeynikov P. Eectron Crystaography: Eectron Microscopy and Eectron Diffraction. Oxford: Oxford University Press; [4] Hoseann R, Hindeeh AM. Structure of Crystaine and Paracrystaine Condensed Matter, J. Macroo. Science - Physics B. 1995; 34: [5] Park S, Quate CF. Digita fitering of scanning tunneing icroscope iages. J. App. Phys. 1987; 62: [6] Moeck P. Crystaographic iage processing for scanning probe icroscopy. 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