Correlation of 2D and 3D Particle Properties with Simulated Particle Imaging Dataset Thomas Buchwald, Ralf Ditscherlein, Urs A. Peuker Highlights Correlation of 2D and 3D Particle Properties with Simulated Particle Imaging Dataset Thomas Buchwald, Ralf Ditscherlein, Urs A. Peuker •Workflow for simulation of static and dynamic image analysis is presented. •Particle characteristics for several solids types are determined. •Influence of image analysis methods on shape factors are shown. •A correlation for 3D sphericity from 2D shape factors is found. •A correlation of particle width from 2D measures is found. Correlation of 2D and 3D Particle Properties with Simulated Particle Imaging Dataset Thomas Buchwald, Ralf Ditscherlein, Urs A. Peuker Institute of Mechanical Process Engineering and Mineral Processing, TU Bergakademie Freiberg, Agricolastr. 1, 09599, Freiberg, Germany Abstract Particle size and shape characteristics are commonly measured with two- dimensional (2D) imaging techniques, two of which are static or dynamic imagingtechniques. These2Dparticlecharacteristicsneedtobeappliedto particulateprocesseswheretheymodelthree-dimensional(3D)processes. The correlation between 2D and 3D particle characteristics is therefore necessary,buttheknowledgeisstilllimitedtoeithermathematicallysimple shapes or specific sets of investigated bulk solids. A particle dataset consisting of six bulk solids measured with X-ray microscopy was used to simulate the results of 2D imaging techniques to createadatabasetotestthecorrelationbetweensetsofparticlecharacteris- tics. Thedatasetthuscreatedoffersthepossibilitytostudythecorrelation between characteristic values and robustly predict the 3D properties of bulk solids measured with 2D measurement techniques. Several correla- tions are determined, including 2D shape factors vs. Wadell’s sphericity (3D), and Feret diameters (2D) vs. particle width (3D). Keywords: imaging techniques, static image analysis, dynamic image analysis, circularity, sphericity, shape factors, equivalent particle size, particle characteristics, correlation Preprint submitted to Powder Technology August 15, 2024 1. Introduction 1 The characterization of particles regarding size and shape is essential 2 for most particulate processes. Advances in measuring techniques have 3 made the CT measurement of bulk solids and resulting particle-discrete 4 datasets possible, enabling new methods of analyzing, e.g., separation 5 processes [1, 2]. 6 However, tomographic measurement is a time-consuming and costly 7 process, so the characterization of bulk solids in everyday industrial and 8 laboratory applications is mostly done with other well-established tech- 9 niques. For the measurement of particle size and shape in orders of 10 magnitude from 1 µmto10 mm, static and dynamic image analysis are 11 widely used, and have often replaced traditional sieve analysis [3, 4, 5]. 12 Furthermore, inline particle measurements are becoming more abundant 13 in research and industry [6, 7]. 14 Wadellintroducedtheconceptofsphericitytoaccountforaparticlesed- 15 imentation velocity deviating from the sedimentation velocity of a sphere 16 [8, 9]. Sphericity has since been used by many researchers and practition- 17 ers to represent particle shape in a single value. But Wadell recognized 18 that the true sphericity for single particles might be hard to come by – it 19 was even deemed unmeasurable by peers [10] – so he proposed the mea- 20 surementoftheprojectionofaparticleatrestandanalternativedefinition 21 for sphericity from it (Eq. 14). 22 The classical approach by Zingg to classify particles into shape cate- 23 goriesbytheratiosoftheirprincipaldimensions(elongationandflatness) 24 isstillwidelyinuseandhasbeenrecentlyimplementedinaparticleshape 25 analysistool[11,12]. 2Daspectratios,alongwithcircularityandconvexity, 26 are recognized in the literature as meaningful shape descriptors [13]. 27 Since Wadell, many people have investigated how 2D imaging tech- 28 niquesmayaccuratelydescribethe“true”,3Dparticleshape[14,15,16,17, 29 18, 19]. In many ways, this study tries to retrace the steps of Bagheri et al. 30 [20],whocomparedcomputedtomographymeasurementswithprojection 31 images to find correlations to accurately describe 3D shape. Whereas be- 32 fore a particle’s three principal dimensions (length, width, and thickness) 33 weredefinedasperpendiculartoeachother,withlengthbeingthedimen- 34 sion between the two points on the particle furthest from each other, the 35 authors propose the determination from the two projections with mini- 36 mum(forthickness)andmaximumareas(forwidthandthickness). Their 37 2 results are interesting, while lacking statistical robustness because of the 38 small sample size. 39 Totrytoovercomethetime-consumingtaskofmeasuringparticleswith 40 computed tomography, several researchers have shown how to simulate 41 realistic3Dparticledata. Theirworkutilizesrandomfields[21]andspher- 42 ical harmonics [22]. Additional work has been done on reconstructing 3D 43 particlesfrom2Dprojectionsusingconvolutionalneuralnetworks[23,24]. 44 This happens in recognition of the approach of capturing single particles 45 from multiple angles to describe the 3D particle shape [25, 26]. The other 46 approachistoquantifyparticleshapeaccuratelyonlyinthestatisticalsense 47 bymeasuringenoughparticlestohaveagoodestimateofthemeanparticle 48 shape of a given bulk solid [13]. 49 In this study, we take the second approach by asking how well 2D 50 descriptorscandescribe3Dparticleshape. Thetextcomprisestwodistinct 51 parts. Thefirstpartisconcernedwithanexpansivedatasetof3Dparticles 52 provided by the PARROT particle database1[27] and the simulation of 53 both static and dynamic image analysis. The resulting dataset is publicly 54 available (see Supplementary Data) and it is our hope that it can serve 55 as a foundation for investigation of many effects that accompany image 56 analysis and that have yet to be properly quantified. The second part tries 57 to correlate some 3D properties with 2D properties determined from the 58 simulated particle projections. This part of the study, sections 4.1 through 59 4.2, is meant to prove how meaningful the developed dataset is. 60 2. Materials and Methods 61 2.1. Particle Characteristics 62 Theterm particlecharacteristic asusedinthistextincludesallparameters 63 that can describe the size and shape of a particle. It comprises three sub- 64 groups:geometricproperties ,equivalentdiameters andshapefactors . Geometric 65 properties can be directly measured from the 2D or 3D representation of 66 agiven particle. Equivalentdiameters aretypically diametersofthe circle 67 (2D)orsphere(3D)thatshareoneofthegeometricalpropertiesofthepar- 68 ticle. Finally, shape factors are mostly ratios of two different geometrical 69 properties,oneofwhichmaybecalculatedfromtheparticle’sconvexhull. 70 1parrot.tu-freiberg.de 3 2.1.1. 2D Measures 71 These geometric properties can all be derived directly from the projec- 72 tion or section of a particle in any direction (Fig. 1b); therefore, they are 73 applicable to all 2D imaging techniques, like static and dynamic image 74 analysis. 75 (a)Orthogonalviewof3Dsurface (b) Vectorized contour (c) Convex hull (d) Pixelized projection (e) Euclidian distance map with contours of same distance from projection border (f) Minimum enclosing circle (black) and maximum inscribed circle (white) Figure 1: Illustration of methods for generation of 2D descriptors for particle shape In the current study, only the vector representation of the silhouette 76 image is used. The accuracy of the calculated parameters therefore only 77 dependsontheresolutionoftheoriginal3Dsurfacemeshandthemarching 78 cubesprocedure[28]withwhichitwasproducedfromthevoxelrepresen- 79 tationsthatthemselvesoriginatedinthereconstructedtomographyimage 80 stack. Thecontourisvoxelizedsolelytosimplifythecalculationofbound- 81 ingcircles,enablingtheuseofstandardPythonlibraries. Bothpixelization 82 and orthogonal projection images, as shown in Fig. 1a, offer possibilities 83 for testing the effects of image resolution and roughness measurement, 84 respectively. 85 4 Area and perimeter. Both the projection area 𝐴pand the perimeter 𝑃pare 86 calculatedbymethodsprovidedbythe Shapely package,directlyfromthe 87 projection contour, as shown in Fig. 1b. Because of the inherent fractal 88 behavior of many real solids’ surfaces, the perimeter is much less robust 89 than the projection area for smaller particles. Still, the effect of measure- 90 mentresolutionwillbemorepronouncedinthedeterminationofthe(3D) 91 surface area, where surface roughness comes more into effect than in the 92 2D case [29]. 93 ConvexHull. Theconvexhullisdeterminedusingamethodofthe Shapely 94 polygonobjectthatcontainsthecontour. Fortheconvexhull,botharea 𝐴c95 and perimeter 𝑃care determined. 96 Feret Diameters. Minimum and maximum Feret diameters are determined 97 by brute force: the projection contour is rotated in 500 steps between 98 0◦and 180◦, and the boundaries in both axis directions are determined. 99 Thesmallestmeasureddistancebetweenboundarieswillbetheminimum 100 Feret diameter 𝑥Fe,min, while the largest distance will be the maximum 101 Feret diameter 𝑥Fe,max. The two measures, 𝑥Fe,minand𝑥Fe,max, are shown 102 in Fig. 2a. As can be seen, the two Feret diameters are not necessarily at 103 arightangle,whichiswhytwo additional Feretdiametersaredetermined: 104 𝑥Fe,min90and𝑥Fe,max90, which are perpendicular to the 𝑥Fe,maxand𝑥Fe,min, 105 respectively. 106 (a) Maximum 𝑥Fe,maxand minimum 𝑥Fe,minFeret diameters (b)Maximum 𝑥Fe,maxandperpendicular 𝑥Fe,min90 Feret diameters Figure 2: Illustration of different definitions of Feret diameters The use of perpendicular Feret diameters serves two purposes. Firstly, 107 forstaticimageanalysis,themaximumandminimumFeretdiameterswill 108 5 be very close to the length and width of a particle, respectively (Fig. 2b). 109 Secondly,the(true)minimumFeretdiameter 𝑥Fe,minanditsperpendicular 110 Feret diameter 𝑥Fe,max90will, in most cases, be very close to the actual 111 dimensions of the oriented bounding box, i.e., the bounding box of least 112 area. 113 Minimum Enclosing Circle. The diameter of the minimum enclosing circle 114 𝑑ecbelongs to the circle that has the least area while still containing the 115 entire projection contour (Fig. 1f). While dedicated Python packages for 116 the task of determining this measure exist, such as miniball , here, the 117 computer vision library OpenCV was used [30]. 118 For the calculation of 𝑑ec, the contour needs to be transformed into an 119 array first, equivalent to a pixel representation (Fig. 1d). The pixelization 120 is achieved with the Python library scikit-image , which contains the 121 polygon method that generates pixel coordinates inside a given polygon. 122 To increase the accuracy of 𝑑ec(and𝑑ic), the contour coordinates are 123 scaled up by a factor of 2 before pixelization, significantly affecting the 124 results of both the center coordinate of the circle as well as its radius. 125 Further scale-up is not considered necessary, or even useful, because the 126 original 3D mesh does not offer more resolution anyway. 127 Maximum Inscribed Circle. The determination of the maximum inscribed 128 circle𝑑icfor a 2D contour, as well as the maximum inscribed circle for a 129 3D surface, is not straightforward. For the authors, none of the methods 130 found in literature were computationally efficient and more robust than a 131 simple brute-force bisection algorithm. 132 A solution to efficiently determine the maximum inscribed circle was 133 found by discretization of the contour, i.e., pixelization. The method uses 134 the Euclidean distance transform as implemented in the Python library 135 scipy[31]. Thetransformcalculatesthedistanceofeachobjectpixelfrom 136 the background (Fig. 1e). The pixel that contains the highest value after 137 the transform will be the center of the maximum inscribed circle, while 138 the corresponding pixel value will be 𝑑ic/2, i.e., the radius of the circle. 139 The Euclidean distance transform is computationally inexpensive and is a 140 relatively simple method for determining the maximum inscribed circle, 141 asittransformstheproblemfromvectorspacetopixelspace. Thisreduces 142 thecomplexityoftheproblemsignificantly,albeitatthecostofbeingonly 143 as accurate as the pixel dimensions allow. 144 6 It has since been found that the method described here has been used 145 in other particle-related research [32, 33]. 146 2.1.2. 3D Measures 147 VolumeandSurfaceArea. Bothvolumeandsurfaceareaarepropertiesofthe 148 trimesh objectthatcontainstheparticlemesh,soitisdefinedbyfunctions 149 already implemented by the package. 150 However,usingthesurfaceareaofthecomputedmeshleadstoanerror 151 in the determination of Wadell’s sphericity (cf. Eq. 8), where a maximum 152 value of 𝜓Wa≈0.92is reached even for highly spherical particles like 153 soda-lime glass. This maximum sphericity value of 𝜓Wa≈0.92stems 154 from the conversion of the particle volumes from voxel representation to 155 a triangular surface. The marching cubes algorithm interpolates between 156 the edges of the voxels to smooth the surface, depending on the number 157 andconfigurationofadjacentsolidvoxels[34,35]. Theresultingerrorwill 158 be8 %to9 %[36]. In comparison, the error in sphericity determination 159 fromthevoxelrepresentationforaspherewouldbe >30 %,becauseofthe 160 greatly exaggerated surface area. 161 Analternativedeterminationofsurfaceareaisachievedwiththecollec- 162 tionofplugins MorphoLibJ forImageJ[37]. Fromit,the ParticleAnalysis3D 163 plugin is used which computes the surface area of a 3D object with an N- 164 dimensionalextensionoftheCroftonformula[38,39]. Notethattheorigi- 165 nalJavalibrarieswereused,accesseddirectlyinPythonthrough Pyjnius. 166 The accuracy of the method was tested by producing incrementally larger 167 spheres in discrete voxel representations and meshing them, using both 168 voxel representation and mesh for the calculation of surface area and par- 169 ticle volume. The result is shown in Fig. 3, where Wadell’s sphericity 170 has been calculated using all three permutations of equivalent diameters 171 calculatedfromtwovolume( 𝑥V,Eq.2)andtwosurfacearea( 𝑥S,Eq.3)def- 172 initions:𝑥V,voxelusesthevolumeequaltothenumberofvoxels, 𝑥V,meshthe 173 volumecontainedinthemeshproducedbythemarchingcubesalgorithm, 174 𝑥S,croftonthe approximation of surface area with the 3D Crofton formula, 175 and𝑥S,meshthe surface area of the mesh directly. 176 It can be seen that using volume and surface area of the mesh leads 177 to a final sphericity value 𝜓Wa<1, even for spheres of diameters larger 178 than a hundred voxels. Using the information of the voxel representation 179 directly,asusedin MorphoLibJ willresultinsphericityvalues 𝜓Wa>1for 180 smaller spheres, which is also counterintuitive. The underlying problem 181 7 (a) Calculated sphericity values over number of voxels of sphere𝑁vxand volume-equivalent diameter 𝑥V (b) Voxelized sphere with a diameter of six voxels (c) Marching cubes– meshed surface of the voxel object in Fig. 3b Figure 3: Sphericity values for different definitions of particle surface area and volume, illustration of marching cubes meshing result is that not every voxel must be fully filled by the particle and the volume 182 approximated by counting the voxels is therefore too high. If the volume 183 of the mesh is used instead of the volume of the voxelized particle, i.e., 184 𝑥V,meshinsteadof𝑥V,voxel,theresultingsphericityvalueswillapproachthe 185 limitof𝜓Wa→1,withsmallspheresobservingsphericityvalues 𝜓Wa<1. 186 As this definition of sphericity, as shown by the green points in Fig. 3a, is 187 the most intuitive and realistic one, the following strategy for 3D particle 188 property determination is recommended and used in this study: 189 •particle volume 𝑉pis determined directly from the particle mesh, 190 •particlesurfacearea 𝑆pisdeterminedwiththe3DCroftonformulaas 191 implemented in MorphoLibJ . For this, the particle mesh is voxelized 192 again. 193 Specific Surface Area. A combination of volume and area, specific surface 194 area is an important measure for all sorts of processes involving heat, 195 moment, or mass transfer. It is defined as: 196 𝑆V=𝑆p 𝑉p(1) In contrast to most other particle properties, specific surface area will 197 decreasewithincreasingparticlevolume. Asexplainedabove,thesurface 198 8 area𝑆piscalculatedfromthevoxelizedsurface,whiletheparticlevolume 199 𝑉pis computed from the meshed surface. 200 Convex Hull. The convex hull is another property of the trimesh object, 201 from which both volume 𝑉cand surface area 𝑆ccan be calculated. 202 Aligned Bounding Box. In this study, a bounding box defines the main di- 203 mensions of the particle. The aligned bounding box defines the length 𝑙, 204 width𝑤,andthickness 𝑡tobethelongest,intermediate,andshortestedge 205 lengths. Thisapproachiscongruentwiththedefinitionofparticledimen- 206 sions by Krumbein, who measured orthogonal lengths starting with the 207 longestonefoundontheparticle[40],whichisequivalenttothemaximum 208 Feret diameter. 209 Thealignedboundingboxiscreatedbytranformationoftheparticleso 210 thatitsprincipalaxesofinertiaalignwiththecartesiandimensionvectors 211 (Fig.4b). Thenecessarytransformisagainapropertyofthe trimesh object 212 containingtheparticlemesh. Afterthetransformation,theboundingbox, 213 again, is a property of the trimesh object (Fig. 4d). 214 The definition of 3D particle dimensions in this way also makes it 215 possibletodirectlycomparemeasurementswithstaticimageanalysissim- 216 ulation results. When the maximum Feret 𝑥Fe,maxand the perpendicular 217 Feretdiameter 𝑥Fe,min90(Fig.2b)areused,theywillbeidenticalwithlength 218 𝑙and width 𝑤for the aligned particle (section 3.1). For stable positions, 219 section3.2, 𝑥Fe,maxshouldstillreflectactualparticlelength 𝑙,while𝑥Fe,min90220 should differ somewhat because the true particle width is oriented at an 221 angle to the projection direction, i.e., surface normal. 222 Bagherietal.favouredtheuseofuncorrelatedFeretextremaforthede- 223 terminationofparticledimensionstoreduceoperatorerror[20]. However, 224 with most modern measurement setups particle dimensions are seldom 225 determined manually, and determination of a minimum Feret diameter 226 for compact projections may still be difficult if done manually anyway. 227 Oriented Bounding Box. The oriented bounding box is again calculated by 228 trimesh foragivenparticlemeshandrepresentstheboundingboxofleast 229 volumethatstillcontainsthewholemeshsurface(Fig.4c). Thedimensions 230 of the oriented bounding box are determined from the Cartesian coordi- 231 nates after applying the inverse transform on the bounding box, since the 232 orientedboundingboxislikelytobeatrandomanglestowardtheCartesian 233 axes, even if the particle was first aligned to its principal axes of intertia. 234 9 (a) 3D particle in its original position (b)3Dparticleof4aafterapplyingtheprincipal axis alignment transform (c) Oriented bounding box for particle in 4a, 𝑉=121 486 (d) Bounding box along Cartesian axes for aligned particle in 4b, 𝑉=146 740 Figure 4: Illustration of the two different definitions for bounding boxes, volumes given in axis units 10 Fig.5showscomparisonsofthedimensionsofalignedboundingboxes 235 and oriented bounding boxes for all investigated particles. The oriented 236 boundingboxhasonaveragesmallerdimensionsthanthealignedbound- 237 ingbox. Theeffectincreasesforthelongerdimensions: lengthwillmostly 238 besmallerfortheorientedboundingbox,whereasthereisamorerandom 239 scatter for thickness. 240 (a) Thickness (b) Width (c) Length Figure5: Comparisonofdimensionsdeterminedbyalignedandorientedboundingboxes On average, the oriented bounding box will be 14 %smaller than the 241 aligned bounding box for compact particles. In contrast, the oriented 242 bounding box will only be 12 %smaller for mica particles which, because 243 of their flat nature, should, in their aligned position, already be closer to 244 thesmallestboxpossible. Finally,soda-limeglasssphereshaveonaverage 245 oriented bounding boxes that are only 5.5 %smaller. 246 Thealignedboundingboxispreferredhereovertheorientedbounding 247 box because of its congruence with Krumbein’s definition who chose it 248 because it is easier to understand and determine by the practitioner. 249 Bounding Spheres. The minimum bounding sphere again is a property of 250 the mesh object defined by the trimesh library, so the diameter of the 251 minimum enclosing sphere 𝑑esis determined in a single line of code. A 252 visualization of both bounding spheres is found in Fig. 6. 253 The maximum inscribed sphere is approximated as the maximum in- 254 scribedcircleinthe2Dcase. Inbothcases,thefunction distance_transform_edt 255 from the scripylibrary [31] is used to calculate the Euclidean distance 256 transform to find the pixel/voxel that is furthest from the particle sur- 257 face. Thismaximumvaluewillbethediameterofthemaximuminscribed 258 sphere𝑑is. 259 11 (a) Minimum enclosing sphere (b) Minimum inscribed sphere (c) Voxelized particle from original surface in Fig. 4a Figure6: Illustrationofbothminimumenclosingsphereandmaximuminscribedsphere InordertoperformtheEuclideandistancetransform,thesurfacemesh 260 needs to be discretized into a voxel representation (Fig. 6c). The voxeliza- 261 tion is also done with methods provided by trimesh, and, as with the 262 2D case, at a scale factor of 2, which increases the accuracy of the diame- 263 ter estimation significantly. Care must be taken to produce a filled voxel 264 representation: most voxelization algorithms will only return solid voxels 265 wherethesurfaceofthemeshtouches. Anextrastepisinvolvedtofillthe 266 hollow discretized surface with scipy’s method binary_fill_holes . 267 2.1.3. Equivalent Diameters 268 Severalpropertiesin2Dand3Dcanbecomparedtothatoftheidealized 269 shapes,acircleintwoandasphereinthreedimensions. In3D,thediameter 270 of a sphere can be calculated that has the same volume 𝑉pas that of the 271 particle. This diameter will be called the volume-equivalent diameter: 272 𝑥V=3r 6𝑉p 𝜋(2) Inthesamesense,thediameterofthespherethathasthesamesurface 273 area𝑆pas that of the particle (surface-equivalent diameter) is: 274 𝑥S=r 𝑆p 𝜋(3) In two dimensions, the particle properties volume 𝑉pand surface area 275 𝑆preduce to projection properties, projection area 𝐴pand perimeter 𝑃p. 276 12 Thediameterofthecirclethathasthesameareaastheprojectionarea,the 277 area-equivalent diameter, is: 278 𝑥A=r 4𝐴p 𝜋(4) Lastly, the perimeter-equivalent diameter is the diameter of the circle 279 that has the same perimeter as that of the particle projection, defined as: 280 𝑥P=𝑃p 𝜋(5) 2.1.4. Shape Factors 281 Shapefactorsarederivedfromtwoorthreeoftheparticlepropertiesor 282 equivalentdiametersintroducedabove. Allshapefactorsdescribedbelow 283 are dimensionless, which means they can be used to good effect to find 284 correlations between 2D projections and 3D particle properties. 285 Length Ratios. Flatness𝑡/𝑤and elongation 𝑤/𝑙have been used before in 286 Fig. 8d to classify particles into shape categories. 287 In 2D, two more length ratios are used in this study. First the aspect 288 ratio is defined as the ratio of minimum and maximum Feret diameter: 289 AR=𝑥Fe,min 𝑥Fe,max(6) As discussed before, the two Feret diameters often at an angle ≠90◦. 290 Because the 3D particle dimensions are defined by their bounding boxes, 291 theyarenecessarilyatarightangletoeachother. Itthereforemakessense 292 to define an additional, orthogonal aspect ratio of perpendicular Feret 293 diameters: 294 AR90=𝑥Fe,min90 𝑥Fe,max(7) Sphericity. Severalshericitydefinitionsexist,someofthemfundamentally 295 different from each other, but for all of them, the sphericity 𝜓 < 1for 296 particles deviating from a sphere. 297 TheoriginaldefinitionofsphericitycomesfromWadellforapplication 298 on sedimentary particles [8]. Wadell defined sphericity as the ratio of the 299 13 surfaceareaofasphereofequalvolumeasthatoftheparticletotheactual 300 surface area of the particle: 301 𝜓Wa=𝑆sp 𝑆p= 𝑥V 𝑥S2 (8) 𝑆spis the surface area of the sphere having the same volume as the 302 particle. 303 Krumbeindefinedasphericitybycomparingagivenparticletoatriaxial 304 ellipsoid[40]. Afterdeterminingthelongestdimensionoftheparticle,the 305 secondlongestdimension perpendicular tothefirstisdetermined,withthe 306 third dimension being perpendicular to the other two. In this sense, the 307 threedimensionsareequivalenttolength 𝑙,width𝑤,andthickness 𝑡ofthe 308 bounding box of the principally aligned particle, as described in section 309 2.1.2. 310 𝜓Kr=3r 𝑤𝑡 𝑙2(9) Another definition for sphericity has been defined by Sneed and Folk 311 as𝜓SF=3p 𝑡2/(𝑤𝑙)[41], but will not be used in this study. 312 Another sphericity definition is the ratio of the diameters of the two 313 boundingspheres,i.e.,maximuminscribedspheretominimumenclosing 314 sphere [42]: 315 𝜓bs=𝑑is 𝑑es(10) Lastly,Hofmannappliedtheconceptofstatisticalentropytotheparticle 316 shape description [43]: 317 𝜓Ho=1 ln(1/3)3Õ 𝑖=1𝑝𝑖ln𝑝𝑖, (11) where𝑝𝑖=𝑑𝑖 𝑑1+𝑑2+𝑑3,𝑑1=𝑙,𝑑2=𝑤, and𝑑3=𝑡. 318 Hofmann’s sphericity is supposed to be the most representative mea- 319 sure for the prediction of particle settling velocity [44]. 320 14 Circularity. “Circularity” is the name chosen according to the definitions 321 of Wadell [9] for the 2D equivalent of sphericity, basically a “projection 322 sphericity”,sometimesalsocalled“roundness”[45]. Likesphericity,circu- 323 larityapproachesavalueofoneforparticlesthatcloselyresemblecircular 324 shapes and will decrease in value for particles becoming less compact. 325 The original circularity definition as ratio of perimeter of the area- 326 equivalent circle to the actual projection perimeter is due to Wadell [9]. 327 Wadell stressed that circularity and sphericity are fundamentally differ- 328 ent from roundness in the sense that roundness is a mesoscopic measure 329 and circularity is a macroscopic measure. In other words, circularity and 330 sphericityshow shapedeviations,whereasroundnessshows surfacedevia- 331 tions. 332 𝜓c=𝑃c 𝑃p=𝑥A 𝑥P=s 4𝜋𝐴p 𝑃2p(12) The square of circularity 𝜓cis called the form factor and is equivalent 333 to the “roundness” factor defined by Cox [46, 47, 48]. 334 FF=4𝜋𝐴p 𝑃2p(13) Because one early criticism of 𝜓Wawas the difficulty of measurement, 335 Wadell proposed more easily attainable circularity measure: 336 𝜓c,Wa=𝑥A,stable 𝑑ec(14) Intheaboveequation, 𝑥A,stableisthediameterofacircleofequalprojec- 337 tionareaasthatofagivenparticle atrest,i.e.,lyingonasurfaceinastable 338 position.𝑑ecis, as per previous definition, the diameter of the minimum 339 enclosing circle. 340 Another method of defining circularity is through both bounding cir- 341 cles, i.e., the radius of the maximum inscribed circle 𝑑icand the radius of 342 the minimum enclosing circle 𝑑ec: 343 𝜓c,bc=𝑑ic 𝑑ec(15) Eq. 15 is the square of the circularity definition by Riley [45]. 344 15 Solidity. Asameasureofconcavity,asolidityfactor 𝑆xcanbecalculatedin 345 both 2D and 3D. It compares the actual particle volume or projection area 346 toitsconvexhulls. Iftherearenoconcavities,thesoliditywillbe1andthe 347 particle or projection will be its own convex hull. 348 𝑆x,3D=𝑉p 𝑉c(16) 𝑆x,2D=𝐴p 𝐴c(17) Convexity. Anothermeasurefordeviationfromaconvexobjectisthecon- 349 vexity,forwhichthesymbol 𝐶xisused. Itcomparesthesurfaceofparticle 350 or projection directly to the convex hull. 351 𝐶x,3D=𝑆c 𝑆p(18) 𝐶x,2D=𝑃c 𝑃p(19) 2.2. Particle Datasets 352 2.2.1. Acquisition 353 Thesolidsparticledatausedinthisstudywaspreparedpreviouslyfor 354 thestatedpurposeofprovidingreference3Ddatasets. Amethodologywas 355 developedtoproduceisolated,i.e.,non-touching,particlesinawaxmatrix 356 [49, 50]. Tomographic reconstruction of X-ray microscopy measurements 357 ofthesewaxmatricesoffersthepossibilitytoeasilysegmentandextractthe 358 single3Dparticles. Theparticledataisavailableintheformoftheoriginal 359 reconstructedtomographystacksaswellassingleparticlesurfacesinSTL 360 format in the dedicated particle database PARROT [27]. 361 VTKfilesthatrepresentcroppedvoxel-basedregionsofinterestforev- 362 eryparticlefromthetomographicreconstructionswereusedtorecalculate 363 STLmeshesfortheparticles,assomeSTLsurfacesinthePARROTdataset 364 were not watertight, which would have led to problems in later analysis. 365 The STL data used in this study is available in the supplementary data. 366 Table 1 gives an overview of the six solids of which particle surface 367 data has been used. The particle size distributions are shown in the form 368 of cumulative sums in Fig. 7. Thee solids are typically in a particle size 369 16 rangebetween 50 µmto300 µm. TheX-raymicroscopymeasurementswere 370 performed for a final voxel size, i.e., edge length, of 2 µm. 371 Table 1: Used particle systems, provided in the PARROT particle database [27] type production process particle size particles aluminium oxide crushing 55 µmto200 µm1571 dolomite calcination and crushing90 µmto200 µm642 soda-lime glass spray drying 150 µmto300 µm602 limestone dry milling 55 µmto200 µm1271 mica comminution and magnetic separation90 µmto300 µm415 quartz crushing <200 µm 1656 Figure 7: Number-base ( 𝑄0) and volume-based ( 𝑄3) particle size distributions for the six solids provided by the PARROT database 2.2.2. Description 372 The properties of the six solids (cf. Table 1) are shown in Fig. 8. From 373 the plot of sphericity 𝜓Waover volume-equivalent diameter (Eqs.8 and 2) 374 17 in Fig. 8a, it can be seen that four solids—quartz, limestone, dolomite, 375 and aluminum oxide—are clustered in the same area with relatively high 376 sphericityvaluesof 𝜓Wa>0.5. Thesoda-limeglassparticlesarethelargest 377 andalsohavethehighestsphericityvalues. Thehighsphericityvaluescan 378 be traced to the production process by spray drying, resulting in mostly 379 sphericalshapes. Incontrast,themicaparticlesshowverylowsphericities. 380 (a) Sphericity vs. particle size (b) Width vs. thickness (c)Explanationofparticleshapeclassificationchartin Fig. 8d; isolines show values for Krumbein’s spheric- ity𝜓Kr (d) Zingg’s particle shape classification chart for as- pectratiosofthethreeparticledimensions,elongation 𝑤/𝑙and flatness 𝑡/𝑤 Figure 8: Properties of the six particle datasets InFig.8d,theparticlesareplottedalongtwoaspectratios,flatness 𝑡/𝑤 381 andelongation 𝑤/𝑙,whichmakesclassificationaccordingtoparticleshape 382 possible.𝑙,𝑤, and𝑡are the three main dimensions of a particle: length, 383 width, and thickness, respectively, defined by the aligned bounding box 384 (cf. section 2.1.2). The plot was first introduced by Zingg and later devel- 385 18 opedbyKrumbeinandJanoo[11,40,51]. Fig.8cservesasanexplanation, 386 also showing isolines for sphericity, though Krumbein’s sphericity defini- 387 tionisused,cf.Eq.9. Alternativedescriptorsfortheparticleshapegroups 388 "disc,""cubic,"and"rod"are"oblate,""compact,"and"prolate,"respectively 389 [52]. 390 Soda-lime glass spheres are expectedly clustered at values close to one 391 for both aspect ratios, while the majority of particles of the other solids 392 are mostly compact and could be classified as cubic and slightly rod- or 393 disc-shaped, depending on their particular flatness or elongation values. 394 In contrast, the mica particles are very flat and may be classified as disc- 395 and blade-like. 396 Fig. 9 provides an example for each of the four categories according to 397 the Zingg classification chart. The examples also serve to give an impres- 398 sion of what the different solids look like. While most of the limestone 399 and quartz particles can be classified as compact/cubic, the two particles 400 shown in Figs. 9a and 9d can be clearly identified as belonging to their re- 401 spective categories of disc- and rod-like. The soda-lime glass particles are 402 mostlynear-perfectspheres,resultingintheaforementionedhighspheric- 403 ityvalues. Themicaismostlyflakyinnature,resultinginverylowflatness 404 values,aneffectthatcanbepredictedfromtheplotofwidthvs. thickness 405 in Fig. 8b. 406 Becausetheirpropertiesareverysimilar,thegroupofquartz,limestone, 407 dolomite, and aluminum oxide will be grouped as "compact particles" in 408 section 3. 409 2.3. 2D Imaging Simulation 410 2.3.1. Static Image Analysis 411 Static image analysis, as defined by ISO 13322-1, involves image acqui- 412 sitiontodetermineparticlesizewheretheparticlesarenotmovingagainst 413 the axis of the optical equipment [53]. If a particle is large enough that 414 adhesion forces with respect to the surface it is resting on are negligible, 415 the particle will orient itself in a position in which at least its longest di- 416 mensionismeasurable. Twopossibilitiesforthesimulationofstaticimage 417 analysis were calculated: 418 •alignment of the principal inertia vectors on the Cartesian axes and 419 •alignment of the particle in one of its stable resting positions. 420 19 (a) disc (limestone) (b) cubic (soda-lime glass) (c) blade (mica) (d) rod (quartz) Figure 9: Examplesfrom the datasets for particlesbelonging to the fourshape categories of Figs. 8c and 8d with solid type in brackets 3D manipulation of the provided STL files was done with the Python 421 library trimesh,which,asthenameimplies,focusesontriangularmeshes 422 [54]. The trimesh packageoffersoptionsforboththeproceduresnamed,in 423 particularamethodthatreturnsalistofthemostlikelystablepositionsof 424 a given mesh, containing both the necessary transform and the respective 425 probability of the particle settling in this position. Any resting positions 426 with a probability 𝑝>0.1were used for further 2D analysis. Because 427 highly spherical particles can easily have no positions of especially high 428 restingprobability,foreachparticle atleastthetwomostprobablepositions 429 werecalculated. Fig.10givesanexampleofthestablepositionsofaparticle 430 and the resulting projections, in this case in 𝑧-direction, i.e., onto the 𝑥𝑦- 431 plane. 432 The imaging simulation involves getting the projection perpendicular 433 to the plane that acts as the resting surface when calculating the stable 434 position transforms ( 𝑥𝑦). For the mesh aligned along its principal inertia 435 20 vectors,theprojectioniscalculatedperpendiculartotheplanethatcontains 436 the two major inertia vectors: when considering the aligned particle in 437 Fig. 4b, the projection would be in direction of the 𝑥vector, onto the 𝑦𝑧 438 plane. 439 The subsequent procedure involves a custom function that calculates 440 the orthogonal projection of the triangular mesh onto a plane defined by 441 a given normal. With a given plane normal, the particle is first rotated to 442 the correct position, and a projection transform is performed onto the 𝑦𝑧 443 (𝑥-axis) plane (Fig. 1a). The projected triangles are then transformed into 444 asingle2DpolygonusingthePythonpackageShapely[55]. Thus,asingle 445 contour is returned which can be used for further analysis. The relevant 446 code can be found in the supplementary materials, see section 6. 447 In principle, the effects of image resolution may be investigated by 448 scaling the projection and calculating a masked array that represents the 449 pixelimage. However,pixelizationinthissensehasonlybeenusedforthe 450 calculation of the enclosing and inscribed circles, cf. section 2.1.1. 451 2.3.2. Dynamic Image Analysis 452 Incontrasttostaticanalysis,dynamicimageanalysisisconcernedwith 453 the image acquisition and analysis of moving particles [56]. Particles are 454 therefore imaged in random orientations, unless the flow is highly turbu- 455 lent. Depending on the setup, particles may be imaged more than once if 456 theyarenotfastenoughtoleavethefieldofview. Inmanydynamicimage 457 analysis devices, these images will be taken as separate particle entities, 458 while devices exist that track the particle while moving through the field 459 of view to measure as many rotations as possible, e.g., the Camsizer 3D 460 (Microtrac). 461 The procedure to produce a projection image is mostly the same as 462 before, except that the particle is first rotated randomly. For every parti- 463 cle, three random orientations were used to produce projections, thereby 464 increasing the number of simulated data points. Of course, the number 465 of projections can be increased at will; however, to stay in line with the 466 number of projections achieved through the stable positions as described 467 intheprevioussection,threepositionswereconsideredsufficient. Alarger 468 datasetoftenprojectionsperparticlehasbeenproducedaswell;however, 469 the results achieved with it are the same as with the smaller one. 470 21 (a)𝑝=0.403 (b)𝑝=0.315 (c)𝑝=0.167 (d) (e) (f) Figure 10: Stable position of the particle shown in Fig. 4 with the respective occurrence probabilities (Figs. 10a, 10b, and 10c) and resulting projection silhouettes along 𝑧-axis (Figs. 10d, 10e, and 10f) 3. Simulation Results 471 With the methods given above, a dataset was produced that contains 472 thepropertiesofthe3Dparticlesandpropertiesoftheirrespectiveprojec- 473 tions,producedinaligned,resting,andrandomorientations. Thealigned 474 orientationstablenaturallycontainsasingleprojectionperparticle,so 6157 475 intotal. Thestableorientationstablecontains,onaverage,threeprojections 476 per particle for a total of 19 720projections, though the absolute number 477 per particle varies, cf. section 3.2. For random orientations, every particle 478 produced three projections, for a total of 18 471. 479 3.1. Aligned Orientation 480 The aligned projection dataset is in many ways the simplest one and is 481 used for verification of the analysis methods which are then used for the 482 datasets of stable orientations and random projections. Because there 483 22 is exactly one aligned projection for every particle, there are as many 484 projections as particles in the complete dataset of all solids, 6157in total. 485 Because of the large number of particles, any effects found are considered 486 at leastinteresting , though maybe not statistically robust. 487 Overall, 49particle characteristics are calculated and used to build a 488 correlationmatrix, 25comprising3Dand 24comprising2Dmeasuresand 489 descriptors. Table B.1 lists all particle characteristics. The characteristics 490 have been grouped into categories for easier comprehension of the corre- 491 lation matrix. 492 AresultingmatrixofPearsoncorrelationcoefficientsisshowninFig.11a. 493 Theresulting 49×49gridcontainsmanyduplicatesaswellasareasnotnec- 494 essarily interesting, like the correlation of 2D and 3D parameters against 495 themselves. The more interesting part of the matrix is the upper right or 496 lower left quadrant, where the correlations between 2D and 3D character- 497 isticsareshown. ThisiswhyFig.11bonlyshowstheupperrightquadrant. 498 From Arabic numbers and Roman numerals a specific characteristic may 499 be determined with Table B.1. 500 Furthermore, only the compact particles (quartz, limestone, dolomite, 501 and aluminium oxide) are used to calculate the correlation matrices. This 502 is to avoid errors from the highly spherical soda-lime glass particles, as 503 discussed in section 2.2.2, and the much higher scatter introduced by the 504 plate-like mica. 505 Becausesomecorrelationsarenotlinear,e.g.,betweenequivalentdiam- 506 etersandspecificsurfacearea,theSpearmanrankcoefficientischosenover 507 the Pearson correlation coefficient. When comparing the two correlation 508 matrices of Fig. 11, the choice of the Spearman rank coefficient (Fig. 11b) 509 indeed results in much higher values. Values greater than zero will sig- 510 nify a positive correlation, whereas, if rarely, negative values will signify 511 negative correlations. 512 The comparison of different geometric measures and/or equivalent 513 diameters will result in very high correlations, as observed by the much 514 more pronounced coloring of the upper left area of Fig. 11b. The brighter 515 regionsoflesscorrelationareallinareaswhereshapefactorsarecompared 516 with geometric measures, equivalent diameters, or other shape factors. 517 This behavior should be expected, because geometric properties all scale 518 with absolute particle size, while shape factors are limited to the unit 519 interval[0,1]. 520 The most fruitful task is to search for high coefficient values where 521 23 (a)Correlationmatrixshowingstandardcorrelationcoefficient betweenallcomputedparticlecharacteristics forallparticles ;3D characteristics before, 2D after the dashed red line (b) Spearman rank correlation coefficient ma- trixfor compact particles only Figure 11: Correlation matrices for particle characteristics determined from aligned pro- jections; Fig. 11b only shows the first quadrant (upper right) of the complete correlation matrix, with dashed red lines separating geometric properties and equivalent diameters from shape factors (cf. Table B.1 for a list of parameters) 2D and 3D shape factors are correlated, which is the lower right area of 522 the correlation matrix in Fig. 11b. Because of the definition of particle 523 dimensions via the bounding boxes, elongation 𝑤/𝑙(18 in Fig. 11b) will 524 correlate very well with aspect ratio AR(17), though the correlation with 525 AR90(18) naturally is perfect. 526 Interestingly, elongation (18) also correlates well with Wadell’s alter- 527 nativecircularitydefinition 𝜓c,Wa(21)andtheboundingcirclescircularity 528 𝜓bc(22). Inasense, 𝜓bcformsakindofaspectratio,whichiswhyitscales 529 well with elongation: the inscribed 𝑑icand enclosed circle diameters 𝑑ec530 showgoodcorrelationwithminimum 𝑥Fe,minandmaximumFeretdiame- 531 ters𝑥Fe,max,respectively. Krumbeinsensiblytookthesquareofelongation 532 in his sphericity definition, Eq. 9, because elongation is a much better de- 533 scriptor for the overall change from the cubic shape than flatness. In this 534 sense,Krumbein’ssphericity(21)showspronounced,butnotashigh,cor- 535 relations with the aspect ratios (17, 18) and Wadell’s (21) and bounding 536 circles circularity (22). The relationship between 𝜓c,Waand𝜓bcis briefly 537 explored in Appendix A. 538 Afinalnotablecorrelationisfoundbetweenthe3D(24)and2Dsolidities 539 (23), indicating that 2D solidity is a good indicator of its 3D counterpart. 540 However, the correlation is not linear and found to be 𝑆x,3D≈𝑆𝑛 x,2D, with 541 24 𝑛=3to4. 542 3.2. Stable Orientation 543 As described in section 2.3.1, at least the two most probable resting 544 positions were used to produce projections. However, it is instructive to 545 plotthedistributionofstablepositionswithaprobability 𝑝>0.1persolids 546 type, as shown in Fig. 12a. 547 (a) Distribution of stable positions of all solids; dashedverticallinesindicatemeannumberofpo- sitions (b) Stable positions of a single limestone particle Figure 12: Stable positions of investigated solid particles for a position probability of 𝑝>0.1 Again, the soda-lime glass and mica particles clearly deviate from the 548 compactparticles(quartz,limestone,dolomite,andaluminiumoxide). The 549 compactparticlesonaveragehavethreetofourstablepositions. Thereare 550 someoutliersatsixandevensevenstablepositions. Onelimestoneparticle 551 isshowninitssevenstablepositionsinFig.12b. Incontrast,thesoda-lime 552 glass spheres have no stable positions 𝑝>0.1for80 %of particles. The 553 flaky mica particles expectedly find stable resting positions only on either 554 of their flat sides, and so obtain on average two stable positions. 555 For the simulation of static image analysis via stable positioning, the 556 correlation matrix in Fig. 13a exhibits a slight drop in very high correla- 557 tions. Thecorrelationsfoundbetweenshapefactorsforalignedprojections 558 (Fig. 11b) are still present though. 559 25 (a) Correlation matrix stable positions (b) Correlation matrix for random orientations Figure13: CorrelationmatricesofSpearmanrankcorrelationcoefficientsfor compactparti- clesandrandomorientations ;dashedredlinesseparategeometricpropertiesandequivalent diameters from shape factors (cf. Table B.1 for a list of parameters) Atypicalexampleofdecreasingcorrelationcoefficientsarethe3Dpar- 560 ticle widths (IV, 12/13) compared with minimum Ferets (12, 13) and 2D 561 bounding box width (14). The reason is that a particle at an angle will 562 notshowits truewidth 𝑤anymore. Comparedtothealigned projections, 563 the correlation between elongation (18) and aspect ratios (VI, 17/18) is 564 thereforeslightlydecreasing. Fig.14showsthecorrelationofseveralFeret 565 diameterswiththeirrespective3Dparticledimensions. Theperpendicular 566 definition of minimum Feret 𝑥Fe,min90scatters around the “true” particle 567 width, whereas the true minimum Feret 𝑥Fe,minsystematically underesti- 568 mates it.𝑥Fe,min90is therefore considered the more suitable estimate of 569 particle width. Because of the definition of elongation 𝑤/𝑙via the aligned 570 bounding box, it will be better estimated by the orthogonal aspect ratio 571 AR90then the unaligned aspect ratio AR. 572 Ofcourse,mostcorrelationsbetween2Dand3Dparticlecharacteristics 573 forstaticimageanalysis,aswasdiscussedinthisandtheprevioussection, 574 could have been found from careful thought experiments. Wadell based 575 his alternative sphericity definition (Eq. 14) on a projection of a particle at 576 restexactlybecauselengthandwidthshouldalwaysbemeasurableinthis 577 situation,andmostshapefactorsshouldscalewillwiththederivedaspect 578 ratio/elongation, as long as the particles are not deviating too much from 579 the cubic shape. 580 26 (a) Minimum Feret vs. width (b) Orthogonal minimum Feret vs. width (c) Maximum Feret vs. length Figure 14: Comparison of Feret diameters to 3D measures for all solids in stable positions (static image analysis simulation) 3.3. Random Orientation 581 Whencomparingthecorrelationmatricesofthestablepositionanalysis 582 (Fig. 13a) and that for dynamic simulation, i.e., projections of particles 583 at random orientations (Fig. 13b), the amount of correlation is notably 584 decreasing. 585 Mostly, the properties of the 3D convex hull, 𝑉c(2),𝑥V,c(4),𝑆c(6), 586 𝑥S,c(8), and𝑆V(9) scale well with projection area–related characteristics 587 𝐴c(1),𝑥A(2),𝐴c(3), and𝑥A,c(4). Additionally, the 3D convex hull’s 588 surface area (6, 8) correlates well with the 2D convex hull’s perimeter (6, 589 8). However, remember that the Spearman rank correlation coefficient is 590 used: correlations here need not be linear. 591 Incaseofthederivedshapefactors,theonlygoodcorrelationexistsbe- 592 tween3D(24)and2Dsolidity(23), 𝑆x,3Dand𝑆x,2D,respectively. Otherwise, 593 correlation between shape factors has decreased considerably. 594 In the following, relationships between 2D and 3D particle properties 595 are investigated more closely. These investigations serve as examples on 596 how the described dataset may be used for further insights by interested 597 researchers. 598 4. Correlations 599 4.1. Cauchy’s surface area formula 600 The relationship between projection area and particle surface area is 601 wellknownasCauchy’stheoremorCauchy’ssurfaceareaformula,[57,58]. 602 27 Theformulastatesthatthesurfaceareaofaconvexbody 𝑆p,cisfourtimes 603 the projection area averaged over several projections 𝐴p,c. 604 𝑆p,c=4𝐴p,c (20) Thistheoremcanbetesteddirectlyonthesimulateddata,notsomuch 605 toprovethetheorem,buttotestthevalidityofthedataset. Fig.15showsthe 606 relations of surface area and projection area, both for the actual particles 607 and their convex hulls. Note that single points are plotted, not actual 608 averagedvalues,soCauchy’stheoremmayonlyholdontheaverage,which 609 iswhylinearregressionlinesareincluded. Forthesoda-limeglassconvex 610 hulls (Fig. 15b), the value of 3.98is particularly close to the theoretical 611 value, while for the compact particles it is only slightly smaller at 3.90. 612 For mica particles, the values significantly decrease. The lower regression 613 valueformicaisexpected,asitisverylikelyforaflakyparticletoproduce 614 silhouettes of comparably lower projection area. 615 (a) Particle surface area vs. projec- tion area (b) Convex surface area vs. convex projection area Figure 15: Correlations of surface area and projection area for random orientations Fortherelationofactualparticlesurfaceandprojectionarea,i.e.,ofthe 616 non-convex shapes, surface area is underestimated for all particles: when 617 Cauchy’s theorem would be used, surface area would be predicted to be 618 smallerthantheactualvalue(s). Thistrendiscausedbytheruggedsurface 619 that cannot be properly estimated by projections, as most concavities will 620 be hidden in the projection image – an effect that can be much larger for 621 very rough particles [29]. 622 28 4.2. Particle Width–Feret Correlation 623 As mentioned in the introduction, dynamic image analysis is widely 624 used to replace sieve analysis. More often than not, data from a dynamic 625 image analysis system needs to be adjusted to account for divergence of 626 the measurements from a sieve analysis. When sieving is done on square 627 aperture sieve meshes, particles will be classified according to their in- 628 termediate dimension, width 𝑤. In static image analysis, particle width is 629 verywelldeterminedbytheminimumFerets, 𝑥Fe,minor𝑥Fe,min90(cf.Fig14). 630 However, in dynamic image analysis it is only possible to know the three 631 main particle dimensions 𝑙,𝑤, and𝑡by imaging single particles from dif- 632 ferent angles or by tracking rotating particles while they fall through the 633 measurement system [26, 60]. 634 Usingthedatasetofrandomorientations,aspectratio ARwasfoundto 635 be the underlying variable explaining the deviation of both the minimum 636 𝑥Fe,minand maximum Feret diameters 𝑥Fe,maxfrom particle width 𝑤. The 637 following correlation was found to predict particle width very well: 638 𝑤≈𝑥Fe,max√ AR (21) Furthermore, because of the definition of aspect ratio, Eq. 6: 639 𝑤≈𝑥Fe,max√ AR=𝑥Fe,min√ AR=√𝑥Fe,max𝑥Fe,min (22) ThecorrelationdefinedbyEq.21isshowninFig.16a. Forthecompact 640 particles,theagreementbetween2Dand3Dparametersisquiteexcellent. 641 Evenforthenon-compactmicaparticles,thecorrelationholds,thoughthe 642 scatter is expectedly larger. Fig. 16b gives the relative deviation of width 643 estimates from actual particle width. For compact particles, the 95 %CI is 644 within a deviation of ±25 %, while the mean is within a ±5 %interval. 645 Therelevantparticledimensionforsieveanalysis,width 𝑤,istherefore 646 expected to be approximated well with any of the expressions of Eq. 22. 647 4.3. Sphericity Correlation 648 Circularity 𝜓c(Eq. 12) is often used in place of sphericity 𝜓Wa(Eq. 8) 649 becausetheformerismucheasiertomeasureinstaticordynamicimaging 650 setups[59]. Itwasthereforedeemedaworthwhileexercisetoseehowwell 651 circularity and sphericity correlate for the dynamic imaging simulation. 652 29 (a)95 %confidence ellipses (b)Relativeerroroverarea–equivalentdiameter for compact particles Figure 16: Correlation of Feret diameters with particle width 𝑤 4.3.1. Mica 653 Fig. 17 shows plots of Wadell’s sphericity 𝜓Waover circularity 𝜓c. The 654 firstinsightisinregardstoextremelysmallcorrelationvaluesoftheshape 655 factors in the correlation matrices (Fig. 13b): at first sight, there is only a 656 point cloud with no tendency whatsoever. 657 At second sight, because of the nature of the two shape factors, both 658 shouldbezeroforinfinitelystretchedobjectsandoneforspheres. Because 659 of this unique relationship, a linear regression needs no offset, i.e., should 660 start from zero. If a linear regression then returns a slope of one, the two 661 shape factors are perfectly correlated. Any spread in either direction is 662 then purely stochastic. The term “stochastic” here signifies the inherent 663 scatterofimagingparticlesatrandomorientations,notmeasurementerror. 664 From Fig. 17a it can be seen that the correlation between circularity 665 𝜓cand sphericity 𝜓Wais rather non-ideal, whereas the square root of 666 sphericityp 𝜓Waleads a much better linear regression slope of 1.06. 667 Another way to evaluate the two parameter pairs is to plot the 95 % 668 confidence interval (CI). Because the data is two-dimensional, confidence 669 ellipses are calculated, as shown in Fig. 17b, that include 95 %of particles 670 undertheassumptionthatthepointcloudisnormallydistributedforboth 671 variables. As shown, the center of the ellipse ofp 𝜓Wavs.𝜓c(in green) is 672 much closer to the equality line then the ellipse for 𝜓Wavs.𝜓c(in blue). 673 Squaringthenew-foundrelationshipgivessphericity 𝜓Waovertheform 674 30 (a) Linear regression results (b)95 %confidence ellipses Figure17: Correlationofsphericityanditssquarerootwithcircularity formicaparticles at random orientations factor FF=𝜓2 c: 675 𝜓Wa≈FF (23) Estimates of the shape factors are summarized for all solids in Table 2. 676 Interestingly,thecorrelationworksverywellfortheflakymica,whereasit 677 doesn’t fit nearly as well for the other solids. 678 4.3.2. Compact Particles 679 Starting from the relationship found for mica, Eq. 23, variables where 680 evaluated that could explain the deviation of the compact particles from 681 true equality. The best relationship, which is an extension of Eq. 23, is: 682 𝜓Wa≈FFp 𝜓c,bc/𝐶2 x,2D(24) The linear regression results and confidence ellipses for Eq. 24 are 683 shown in Figs. 18a and 18c. Except for the highly spherical soda-lime 684 glass, the relationship seems to slightly overestimate Wadell’s sphericity. 685 Additionally, Fig. 18e shows the relative error of Eq. 24’s estimates for 686 sphericityasarollingmeanoverprojectionarea–equivalentparticlediam- 687 eter𝑥Aand the corresponding 95 %CI. The relative error for the compact 688 particlesisagainshowingaslightoverestimationofsphericityforparticle 689 sizes,hintingatayettobefoundexplanatoryvariable. However,theerror 690 for the mean is below 10 %for all particle sizes. 691 AdownsideintheusabilityofEq.24forpredictionofWadell’sspheric- 692 ity is that both the form factor FF(Eq. 13) and convexity 𝐶x,2D(Eq. 19) 693 depend on accurate determination of the projection perimeter 𝑃p, which 694 31 (a) Linear regression results for Eq. 24 (b) Linear regression results for Eq. 25 (c)95 %CI ellipses for Eq. 24 (d)95 %CI ellipses for Eq. 25 (e) Relative error for compact particles over (projection) area–equivalent diameter for Eq. 24 (f) Relative error for compact particles over (projection) area–equivalent diameter for Eq. 25 Figure18: Correlationof2DshapefactorsandWadell’ssphericityforthedynamicimage analysis case ( random orientations ) 32 willleadtoresolutionproblemsforsmallparticles,asdemonstratedbythe 695 increased error for smaller particle sizes in Fig. 18e. 696 Therefore another correlation was developed which does not depend 697 on the accurate determination of perimeter. Instead, it only uses aspect 698 ration and bounding circles’ circularity: 699 𝜓Wa≈𝜓c,bc/3p AR90 (25) While this second correlation is simpler, it also leads to an improved 700 prediction of Wadell’s sphericity, as shown in Figs. 18b, 18d, and 18f. The 701 relative error for the mean stays below 2 %. The correlation may even be 702 useful for estimation of single particle sphericity values, because the 95 % 703 CI stays within a range of ±25 %. 704 Average sphericity predictions are summarized in Table 2. Eq. 25 is 705 clearly superior compared to Eq. 24. Only in the case of mica, Eq. 24 706 provides a better average estimate of sphericity, which may hint at the 707 correlationbeingmoreaccurateinthecaseofplate-likeparticles. However, 708 for mica even the form factor is a fairly accurate predictor of sphericity. 709 Table 2: Average sphericities 𝜓Wadetermined with the correlation equations 23, 24, and 25; intervals shown are the standard deviations from the mean equation 23 24 25 material 𝜓Wa(3D) FF FFp 𝜓c,bc/𝐶2 x,2D2𝜓c,bc/3√ AR90 quartz 0.73±0.05 0.63±0.16 0.69±0.09 0.71±0.08 limestone 0.73±0.05 0.58±0.16 0.65±0.11 0.69±0.08 mica 0.46±0.12 0.43±0.13 0.44±0.17 0.52±0.13 dolomite 0.69±0.04 0.59±0.12 0.68±0.09 0.70±0.07 soda-lime 0.91±0.05 0.78±0.18 0.93±0.11 0.93±0.09 Al2O3 0.63±0.05 0.54±0.13 0.61±0.10 0.65±0.08 Note that the predictive value is reasonably good because of the large 710 number of data points. If there had been only a handful of particles, the 711 final correlation would have been nearly impossible to find. Furthermore, 712 the predictive power may not hold for all types of solids. 713 Especially for Eq. 24, because of the use of convexity, effects of image 714 resolution could come into play which have not been part of this study, 715 33 but are planned to be investigated in the future. If surface roughness 716 significantlyincreases,surfaceareaeffectscouldbeunderestimatedby2D 717 convexity[29],becausethefractalbehaviorofsurfaceroughnesscannotbe 718 accurately determined by 2D imaging techniques. Because the resolution 719 of STL mesh, voxel image, and projection silhouette are directly linked 720 in this study, the correlation is expected to give sphericity values for the 721 resolution of the image acquisition. 722 Consider the following example. At a pixel dimension of 2 µm, Eq. 24 723 willgiveasphericityvaluethatisvalidatthesame(3D)voxelresolutionof 724 2 µm. Ifahighlyrugged3Dparticlesurfaceismeasuredatahigherresolu- 725 tion,i.e.,≪2 µmintheexample,particlesurfacewillincreaseandWadell’s 726 sphericity value decrease. Of course, this resolution effect is mostly mit- 727 igated by use of Eq. 25, which does not rely on projection perimeter at 728 all. 729 How could Eq. 25 be used in practice? It needs to be understood 730 that this correlation is not able to predict sphericities of single particles 731 accurately, which is apparent from the scatter in Fig. 18a. However, what 732 thecorrelationcanachieveisthepredictionofameansphericityforagiven 733 bulksolids,whichisinmostcasestheonlyneededinformation. Ofcourse, 734 as demonstrated by Fig. 18f, size-dependent sphericity determination is 735 possible in principle. 736 5. Validation Measurements 737 Dynamic image analysis measurements were performed to validate 738 the correlations found in sections 4.3 and 4.2. For the experiments, the 739 Fritsch Analysette 28 ImageSizer was used. An example image is shown 740 in Fig. 19a. The acquired images were evaluated externally with a Python 741 script,becausethemachine’ssoftwaredoesn’tcalculateallthenecessary2D 742 shape factors, in particular the bounding circle diameters. Only particles 743 withverysharpcontourswereextractedandparticlesthatwerenotclearly 744 isolated or touched the image boundaries were discarded. Fig. 19b shows 745 particlesisolatedfromtheexampleimage. DistributionsofminimumFeret 746 diameters and circularities as given by the ImageSizer analysis software 747 were found to be in excellent agreement with the distributions found by 748 the Python script. 749 All six solids used in this text were measured with the dynamic image 750 analysis system. However, only the compact particles were evaluated, 751 34 (a) Example image with extracted particles high- lighted in green (b) Extracted (sharp) particles Figure 19: Example image of dynamic image measurement of limestone because the mica particles reflected light extremely well, which led to 752 problemsinimageevaluation. Imagesarestillavailableinasupplementary 753 dataset, see below. 754 Fig. 20a shows the distribution of minimum Feret diameters 𝑥Fe,min755 extracted from the dynamic image measurements in comparison to the 756 3D particle widths for limestone particles. The discrepancy between the 757 two distributions is noticeably decreased by use of Eq. 22. Not only is the 758 averageparticlewidthestimatedwell,theoveralldistributionispredicted 759 well, with percentiles shown in Table 3; as expected, the predicted values 760 are a little further spread apart, with additional spread originating in 761 the random particle orientations that themselves produce an element of 762 dispersion. 763 Table 3: Quantiles for particle width estimators percentile 𝑤 𝑥 Fe,min𝑥Fe,min√ AR 10 65.0 50.9 62.6 20 71.7 57.7 70.4 50 95.3 75.9 94.9 80 131.4 114.7 138.1 90 153.2 138.0 168.8 Using log-spaced size classes for the actual and estimated particle 764 35 (a) Probability densities for limestone particles: 2D minimum Feret di- ameter𝑥Fe,minand estimated particle width 𝑥Fe,min/√ ARaccording to section 4.2 in comparison to actual 3D particle width 𝑤 (b) Estimated sphericities (Eq. 25) as function of particle width for limestone particles (c) Estimated sphericities for the investi- gated compact solids Figure 20: Validation of the correlations found in the text with dynamic image analysis measurements widths,particlesphericity 𝜓WaisestimatedwithEq.25,theresultofwhich 765 is shown in Fig. 20b. As shown, the actual sphericity per the 3D particle 766 data is slightly underestimated. However, the estimate is much more us- 767 ablethanusingcircularity 𝜓c. Infact,Eq.25isverymuchusableforallthe 768 investigated solids, with Fig. 20c showing the average sphericity values in 769 3D against the 2D estimates. 770 Intheory,thecorrelationsvalidatedherecouldhavebeenfoundthrough 771 separate, i.e., stand-alone, 3D and 2D measurements. However, the sim- 772 ulated dataset gives the direct relation between 2D and 3D particle char- 773 acteristics for every single particle, which makes the analysis much more 774 robust. Note that the use of the dataset is not limited to the findings of 775 thistext,butisexpectedtoholdmanymoreinsightsintotherelationships 776 between 2D and 3D particle measurements. 777 36 6. Conclusions 778 A collection of particle surface meshes, resulting from X-ray tomo- 779 graphicmeasurements,hasbeenusedtosimulatebothstaticanddynamic 780 image analysis. The results have been evaluated to find the highest cor- 781 relations between 2D and 3D geometric measures and shape factors. The 782 dataset and methods described prove to be physically accurate and are 783 tested against validation measurements. 784 Examples have been given of the potential insights this dataset may 785 generate. Predictive correlations for Wadell’s sphericity in 3D have been 786 foundthatareexpectedtopredictsphericityvalueswellforawiderangeof 787 particles using only 2D shape descriptors, provided that enough particles 788 are measured. The most suitable correlation was found to be Eq. 24: 789 𝜓Wa≈𝜓c,bc/3p AR90 In the same vein, a correlation for estimating particle width from 2D 790 Feret diameters has been determined, Eq. 21: 791 𝑤≈𝑥Fe,max√ AR=𝑥Fe,min/√ AR=√𝑥Fe,max𝑥Fe,min Confirmation experiments with a broader set of particles are planned 792 in the future. An inherent measurement artifact of image analysis, pixel 793 resolution, needs to be investigated, but is expected to be negligible for a 794 large enough number of pixels as proven by validation experiments ( ??). 795 The dataset, as provided in the supplementary data, offers the possi- 796 bility to discover numerous correlations and insights regarding geometric 797 measures and shape factors, as well as their relationships across two and 798 three dimensions. We encourage researchers to use the dataset for their 799 research questions and to shed light into questions that had long been 800 obscured by computational complexity. 801 Supplementary Data 802 Supplementary files are available in the Open Access Repository and 803 Archive for Research Data of Saxon Universities (OPARA): 804 https://doi.org/10.25532/OPARA-479 805 https://doi.org/10.25532/OPARA-587 806 https://doi.org/10.25532/OPARA-595 807 37 Thesupplementaryfilesenableuserstoreproduceimagingdatasetsas 808 used in this study and demonstrate the methods for acquisition of all par- 809 ticle characteristics for an example particle. Particle STL files are included 810 in the first entry above. The second entry extends the code by determi- 811 nation of surface area via MorphoLibJ and a dynamic imaging dataset of 812 ten random orientations per particle instead of the previous three. The 813 final entry includes results from dynamic image analysis measurements 814 for validation. Note that you need a working Python setup and that all 815 code is made available as Jupyter notebooks. 816 Acknowledgements 817 We would like to express our gratitude to the Institute of Foundry 818 TechnologyatTUBergakademieFreibergforlettingususetheirAnalysette 819 28 ImageSizer for image analysis measurements. 820 This research work was supported by the German Federal Ministry of 821 Education and Research through the research project InfraDatRec (grant 822 no. 03XP0426A). 823 Appendix A. Wadell’s Circularity 824 One correlation found, that is not necessarily expected, is between 825 Wadell’s alternative circularity definition 𝜓c,Wa(Eq. 14) and the bounding 826 circlescircularity 𝜓c,bc(Eq.15). Thoughitisacorrelationbetweentwo2D 827 shapefactors,itistoointerestingtoignore. Aspredictedbythecorrelation 828 matrixinFig.11a,wherethecorrelationisfoundinthelowerright(second) 829 quadrant between values 46 ( 𝜓c,Wa) and 47 ( 𝜓c,bc), there is a near-perfect 830 linearrelationship. However,tohavethetwocircularitiesdirectlycoincide, 831 Wadell’s circularity is squared, 𝜓2 c,Wa. The resulting correlation is shown 832 in Fig. A.21a. 833 Because of the definitions of the two circularities, it is found that the 834 area–equivalent diameter is directly related to the bounding circle diame- 835 ters. 836 𝑥2 A≈𝑑ic𝑑ec (A.1) The above relationship is shown in Fig. A.21b. 837 38 (a) Wadell’s circularity vs. bound- ing circles circularity (b) Area-equivalent diameter vs. bounding circles diameters Figure A.21: Correlation of Wadell’s alternative definition for circularity (Eq. 14) and bounding circles circularity (Eq. 15) Appendix B. Particle characteristics 838 Table B.1 gives a list of the particle characteristics as used in the corre- 839 lation matrices, Figs. 11 and 13. 840