Using Multiple 2D Projections To Characterize 3D Irregular Particles

by Michael Taylor
Manager, Materials Research

The primary purpose of this study was to determine how multiple 2D projections of a particle could be used to estimate features of its 3D geometry. Individual, irregular rock particles were imaged from multiple (usually 13, sometimes 64) selected directions using a digital camera. The images were analyzed quantitatively using commercially available image-analysis software. Although many measures can be taken from each2D view this study was limited to the area of the projection; how the area varies with direction; and how the results can be used in an inverted procedure to estimate the geometrical properties of the 3D particle from its projected area. Since the volume of each particle was of great interest, but could not be measured directly, it was elected to obtain the volume by an independent means. Each rock was weighed and the weight multiplied by a density (obtained from the regular and frequent values obtained by the process control department of the quarry). Fully crushed rocks from 4 different size groups were examined. River-rounded rocks were also imaged and the differences were easily detectable by the imaging method.

126 rocks and 4 regularly shaped objects were imaged and about 2500 images were analyzed. The statistics from the individual particles were then used to estimate the properties of large assemblies of rocks. The three properties that could be estimated were (a) size (both true volume and “sieve size”) (b) shape and (c) surface area.

The results show that potential uses for the multiple view technique include (a) estimation of the size distribution (psd) of products when using image analysis systems to view rock products online, (b) identifying and characterizing rocks from different sources (c) estimating the surface area of assemblages of particles (d) providing a numerical and unambiguous characterization of the size and shape of particles. Such information should be of great help to designers of baserocks, asphalts and concretes.

Construction materials consist of assemblages of very large numbers of irregular particles. The behavior of these assemblages exerts a central influence on the mechanical behavior of the end products that are made from them.

Many studies have shown that the mechanical behavior of real materials is strongly influenced by (a) the geometry of individual particles and (b) their interactions. Thus geometry has become a necessary tool for understanding materials and a prerequisite for scientific attempts at improving their applications. While the present paper was motivated by studies of construction materials such as concrete, asphalt, and baserocks, geometry plays a similar role in related fields such as soils, sands, filter materials, pharmaceuticals, catalysts, foodstuffs and superconductors. Geometry determines the space filling characteristics of particles and the extent, the shape and the tortuosity of the voids between them. Geometry also controls the number and the angles of the contacts between particles: these determine the mechanical response of the assemblage. All these parameters are of interest in many fields.

Materials that contain extremely high numbers of particles may be studied on either the gross level (samples of the size used in typical applications) or, in a more analytical approach, using groups of very few (or even a single) particles. Treatment of very large numbers of particles requires the use of statistics. This paper discusses methods that use both approaches.

In construction projects the selection of aggregates for baserocks, concretes and asphalts involves the determination of the volumes of the various component size groups that are to be used.

For concretes, economics, and some functional concerns, suggest that as much of the space as possible be filled with aggregate. However the transported mass must permit handling and consolidation: consequently a balance must be struck between cost and “placeability.”

In asphalt, the objective is to leave sufficient voids in the aggregate matrix (typically 15 to 18% of total volume) to allow for the chosen amount of binder (10%–15%) while leaving enough for a chosen volume of air (about 5%).

For (unbound) bases the desire is for an aggregate matrix strong enough to support the expected loadings, to be stable for long periods, and (sometimes) to permit water to flow through the layer at desired flow rates.

The selection of the percentages of each size component has been a central concern of researchers and users for over a century, and continues today. However there is little consensus on numerical approaches and most design methods rely upon empirical data and/or trial and error. The author believes one of the reasons for this may be the application of one set of rules to materials from many locations. This approach ignores the variation in the geometry of aggregates from different sources. This was one of the stimuli for the present study.

Once the volumes of constituents are selected it is necessary for these to be converted to weights. This is because of the interesting fact that weight is very much easier to measure and control than volume. Thus the density of the material is an additional and essential property required by designers and suppliers.

Consideration of these requirements indicates that the following geometrical properties of particles are of interest.

  • Volume
  • Shape
  • Surface area
  • density
  • surface texture
It is an interesting fact that no generally acceptable test method for measuring (estimating) any of these properties currently exist. The most dependable test is that for density and even this remains insufficiently simple, fast or accurate for most engineering purposes. As a result, researchers continue to re-address definitions and to seek new equipment to improve the estimation of these properties.

The current paper addresses, almost exclusively, volume, shape and surface area. The density was obtained independently. It is hoped to address the other parameters in a later paper.

Particles may be divided into two classes depending upon whether or not they have regular geometry. A regular particle is here defined as one whose geometry can be completely described with a few terms and measurements (e.g. sphere, cube, ellipsoid). Such shapes are convenient as conceptual models and to indicate approaches that might prove helpful in more general cases. However most particles of interest in engineering are irregular. Their properties such as size, shape, surface area, and volume must be defined, must be measurable and (to be of any use) must be quantifiable. There are two methods by which this may be done.

Voxel representation. It is convenient to imagine a single particle as represented by an assemblage of voxels (3D elements). It is also convenient to imagine (without loss of generalization) that these voxels are cubical. Hence a single particle can be represented by erecting a “box” around it and subdividing the box into (equally sized for convenience) voxels. For illustrative purposes let the box have 100 by 100 by 100 voxels and have sides oriented parallel to x, y and z in a chosen orthogonal system. The box is thus divided into a million voxels. If each voxel is allocated a binary index, m, which is assigned as 0 for space and 1 for solid (meaning the voxel is within the solid body) then the entire geometry of any body is represented by a matrix with one million 4-d entries.

With this representation the entire geometry of the particle is captured. Such representations are possible with modern tomographic equipment.

This method emphasizes that (if the voxels are chosen small enough) each particle has a unique size, shape and surface area.

The voxel approach allows quantitative evaluation of size, surface area and volume (and other properties), as follows.

To compute the size of one particle, or to compare the size of two, one merely counts voxels with m=1.

To compare surface area one counts those voxels (with m=1) AND which have 1, 2, 3, 4 or 5 faces adjacent to a voxel with m = 0. (it is simpler, computationally, to eliminate (a) all voxels with m = 0 plus (b) all voxels with m = 1 AND with all 6 faces adjacent to a voxel with m=1). It may be noted that, in general, the accuracy of the measured surface area with a voxel representation is less than that for size, because the use of cubical voxels at surfaces usually involves a larger error than for volume.

To compare shape is more complex since 3D shape involves the spatial arrangement of the solid voxels. Shape thus depends upon the values of x, y and z for each solid voxel in the body. Therefore each body also has a unique shape.

The most complete description of shape is to keep the entire matrix (one million 4-d elements in the example cited). Current developments in computer science may make this a practical method soon, provided the number of particles required by the problem at hand is not too large.

A less data intensive, but therefore less accurate, method for characterizing shape, and for comparing the shape of two objects, has been suggested by Taylor (2000). In outline it involves the following steps.

  1. obtain the voxel representation of one particle in a chosen coordinate system say x, y, z.
  2. compute the coordinates of the center of volume.
  3. compute the orientation of the three principal axes of volume.
  4. compute the magnitude of the 3 principal moments of volume.
  5. “normalize” the 3 moments in step 4 by dividing each by the size of the particle. This removes the influence of size on the results found in step 4.
  6. The resulting 3 normalized moments serve as an estimator of shape. Clearly they contain less information than the matrix with hundreds of thousand of entries, but they may serve some useful role in characterizing shape.

The chances of exact congruence of two particles for any geometric quantity is initially small and becomes more so as the number of voxels increases and/or the size of the particle increases. Hence, provided the voxels are small enough, the axes in step 3 (and the quantities in steps 4 and 5) are unique to each particle and thus serve not only to quantify its geometry but also to serve as an identifier of the particle.

Thus to compare the shape of two particles (or to confirm that one has selected a given particle from a group—not an easy task otherwise) one can compare the results of step 5 for the two particles.

This approach is explained in more detail in Taylor (1)

Fourier representation: an alternate approach is made possible by the use of the Fourier approach but in 3D rather than the familiar 2D. This approach has been used to great effect by Garboczi (2) at NIST. In this approach it is first necessary to obtain the3D spatial coordinates of several points on the surface of the irregular particle. Equations for the numerical values of the coordinates of any point on the surface are then estimated by representing the coordinates by (infinite) trigonometric series. In use, the infinite series are necessarily truncated. The accuracy is increased as more terms are included. However the number of terms that can be included is limited by the number of points originally surveyed since it is from these that the coefficients in the series representation are computed.

It can be seen that the voxel representation is an “internal” representation while the Fourier approach uses an “external” discrete surface survey. The accuracy of the voxel method is related to the number of voxels chosen and the accuracy of the tomographic equipment. The accuracy of the Fourier approach is controlled by the number of surface points surveyed, the accuracy of the measuring equipment, and the number of terms used in the equations.

Each method requires considerable “surveying” work. The voxel method requires tomography. The Fourier approach requires accurate ranging.

Which method is preferred, or is more efficient, for a given purpose must await further research work. It should not be difficult to design a “head to head” test given a well defined objective function. It is hoped that this can be done in the near future.

Tomography and surface scanning both require equipment that is sophisticated and expensive and thus currently limited to facilities such as Universities and large research laboratories. Until such machines are more widely available it is natural to seek alternate methods to study 3D geometry. This was one motivation for the current work: the other was the application of imaging methods to estimate size in an on line system (see hereinafter).

Using multiple 2D views of a particle to deduce some of its 3D geometry is not new. The views are termed “projections” or “shadows” by various authors.

A very brief outline of previous research in this area includes the following.

  • Cauchy (3): in one of the earliest studies encountered the French mathematician proved that for convex bodies the average projected area equals 0.25 times the surface area. A convex body is defined as one for which all straight lines, joining (arbitrarily chosen) pairs of surface points, remain within the body.
  • Underwood (4): later proved that for non convex bodies Cauchy’s result becomes a minimum. This suggests that if both the true surface area and the average projected area could be measured then the ratio of the two might serve as a measure of “convexity.”
  • Bodziony and Kraj (5): laid the mathematical foundations for projections by developing equations for the statistical properties of the projections of a (elemental) plane. They showed the following. To measure the area of the projections it is sufficient to view the elemental plane over the half sphere. The distribution of area is uniform (i.e. there is no preferred direction). The expected value of the area is the same for all directions. For independent directions of view the variance of area is inversely proportional to the number of directions (i.e. the more directions the better). For k directions the variance reaches a minimum if the k directions are chosen as the vectors normal to the faces of the corresponding regular polyhedron (i.e. the polyhedron with k faces). When k is 2 or 3 the variance is a minimum if the viewing directions are orthogonal to each other.
  • Umhauer and Gutsch (6): constructed a device for obtaining 3 orthogonal views simultaneously and showed that frequency distributions for “size” based upon 3 views were much narrower than those using one direction only. However their samples were principally regular bodies.
  • Laurentini (7): in a series of studies related to machine vision, he created definitions and methods for identifying shapes within computerized images. Some improvements were made by considering views from 3 directions taken simultaneously.
  • Vickers (8), and Vickers and Brown (9): wrote a series of papers in which they developed mathematical expressions for computing the distribution of the area of projections. Their work was also limited to regular shapes.

Based upon previous work it was elected to construct a device that could permit a photograph to be made of a particle from any direction (see Figure 1). The device consisted of a rotating table mounted on a base marked so that the angle of rotation (defined as longitude—see below) could be measured. On the perimeter of this table was secured a support with a hollow shaft. The support rotates with the table. The shaft was provided with a central hole designed to receive a spindle. The spindle could be fixed with a set screw and once the spindle was secured the shaft could be turned through a measured rotation (defined as latitude—see below). The rock to be photographed was glued to the outer end of the spindle, and the length of the spindle adjusted so that the rock was centered over the rotating table.

The base plate was also fitted with a mounting screw to hold a digital camera in a fixed position. The center of the camera lens was the same height as the center of the rocks when held it the spindle. In this way each rock could be mounted in the device and then viewed from any angle of latitude and longitude.

In Figure 1 the object will be centered over the boss on the left side of the table. The camera is mounted on one of the three holes shown on the right. The current position of the holding device will block the view of the camera and hence is not used during actual viewing.

Each rock was identified by imprinting a number upon it and by storing it in a labeled plastic bag. Thus individual rocks could be re-selected for future additional work if necessary. Rocks were photographed individually using a digital camera (initially Kodak LS200 later LS 430). A commercial image analysis program (Photoshop®) was used to “extract” the outline of each rock and to “binarize” the image (make all pixels within the boundary black, and all outside it white). The program is also able to count the number of pixels contained therein (this varied from about 10,000 to about 50,000 for the rocks chosen). In this initial study only the area of the views was considered. The definition of the boundary and the extraction of data were performed by hand. Future work will be conducted using automatic features for these functions.

The rock is imagined to be located at the center of a transparent sphere. The camera may be placed at any point of the surface but always points to the center. Thus if the sphere is imagined as the Earth with Greenwich as the zero of longitude and the equator as the zero of latitude then any direction of view can be defined by the latitude and longitude of the camera position.

In the experiments it was simpler to fix the position of the camera; to rotate the rock; and to use the rotation of the table as longitude and that of the spindle as latitude.

Since an infinity of views are possible it was necessary to select a number of views that is sufficiently large to ensure a satisfactory degree of confidence in the statistics and yet not much larger so as to avoid wasting resources. Bodziony and Kraj (5) showed that the variance in the results is improved rapidly as the number of views is increased from 2 to 6 but provided no information for numbers greater than this.

Whatever the number of views, it is necessary to select the values of latitude and longitude so that the results are not biased relative to direction. This can be accomplished by selecting equal steps in longitude coupled with equal steps in (sine of latitude) as explained e.g. in Russ (1999).

Using the half sphere, it was elected to perform two series of tests. The first was designed to minimize the number of directions: 4 angles of latitude and 4 of longitude were selected (resulting in 13 independent views). This permitted a relatively large number of rocks to be measured. The second series used 8 longitudes and 8 latitudes (giving 65 independent views) which was anticipated would provide better statistics but which permitted only a few rocks to be measured.

The viewing directions used in each series are shown in Tables 1 and 2.

For calibration purposes two cubes and two spheres were also imaged. The cubes chosen were dice (so that individual faces could be identified if desired) and the spheres were glass marbles. The diameter of the marbles and the side of the dice were both 15mm (0.6 inches).

In addition to these regular bodies a small rectangular “target” was constructed and mounted on a stand so that it could be placed next to the rock sample and viewed in the same image. This was about 25mm (1 inch) square and inscribed with a grid of orthogonal lines spaced 2.5mm (0.1 inches) apart. Since the area of objects in the photographs were measured by counting pixels this served as the primary calibration device.

Rocks were selected from two commercial quarries: the first group included crushed particles of sizes ranging from approx. 25mm (1 inch) to approx. 3 mm (0.1 inch). The rocks are sold in four size groups. Samples were selected at random from each of these size fractions. All these products were 100% crushed. The second group of rocks were from a quarry that produced river run material that was almost 100% naturally formed. The different types were chosen to investigate whether there were any differences between crushed and naturally sorted rocks.

The numbers of samples of each type are listed in Table 3.

It is desirable to know the volume of each rock. Since rocks are not truly convex it was not possible to rely upon an estimate using Cauchy’s formula. Hence an independent measure of volume was made by weighing each sample and using a density value to convert to volume. The company owning the quarries conducts regular laboratory testing for density so the history of density was available (with both mean and standard deviation). The density values used were from the testing date closest to the taking of samples.

It may be noted that the samples are divided into size classes since that is how these products are specified in the market place. However standard sieve procedures do not separate rocks by weight or volume or size (as defined above). The classes are reported in this way in this paper merely for convenience: as will be seen hereinafter the arbitrary size ranges may be ignored when individual rocks are studied.

For each rock there are 13 (or 65) views. For each view the projected area was measured and then a histogram plotted for the 13 or 65 results. Note that the qualifier “projected” will be omitted hereinafter, for convenience.

Of the many quantities that other authors have suggested as useful measures of shape two were selected to obtain comparisons. The two selected (without any strong reasons) were (a) the diameter of the equal volume sphere, EVS (the diameter of a sphere with the same volume as the particle) and (b) the side length of the equal volume cube, EVC (the side length of a cube that has the same volume as the particle). The independently estimated volume of each rock was used in these calculations.

These two parameters are presented for each rock.

From the 13 views the following statistics were calculated (using Excel®).

  • Maximum area
  • Minimum area
  • Average area
  • Standard deviation of area
  • ratio max/min area
Since there were 81 crushed particles and 20 river-run particles there are 101 histograms and sets of statistics. Only a single sample, selected at random, is given here (Table 4). Full data may be found in the M.S. thesis of Lau (10).

The information from tables such as these were collated for each rock size and for each rock type (crushed or rounded) and the results summarized for each group.

Notation: For convenience in what follows (and for plots), volume is denoted V, surface area SA and projected area (individual) as pa; average pa for one particle apa. For size groups the average projected area is denoted gapa and the leading “g” used for other group properties where obvious.

The results for 2 spheres and 2 cubes are shown in Tables 5 and 6. Note that cube number 1 was attached to the spindle in the center of a face while number 2 was attached at a corner. With a limited number of views it was expected that number 2 would show both a larger maximum area and a large a ratio of max/min area. These expectations are confirmed.

A typical distribution of pa with direction can be seen in Figure 2 below: this is for a rock with 65 views. The distribution for a sphere is a straight line equal in magnitude to (0.25 × π×d2). All other shapes will show some variation with viewing direction. Thus the variability in pa might be used as an indicator of the deviation from sphericity. Presumably a wider variation is indicative of greater irregularity in shape while the rate of change of area with direction may be considered as a possible measure of “angularity.” The author is aware that these terms are not well defined. Some future researcher may be able to develop more quantitative measures from this type of data.

The data for all rocks from one size group may then be assembled (typically 25 rocks). Table 7 below is a typical result.

The last two columns of Table 7 were computed to assist in a study of how to estimate shape with a factor that is independent of size (volume) as explained hereinafter.

The data from all size groups are assembled in Table 8

By comparing crushed and rounded materials (of the same size group) it can be seen that the image information is clearly different. The apa for rounded particles is considerably less than that for crushed: this was anticipated. However it was not anticipated that the standard deviation for the rounded particles would be greater than for crushed. No explanation can be offered at this time (a partial reason might be the smaller number of particles used: 10 rounded versus 25 crushed, but this does not seem fully adequate). Since crushed particles are required in some asphalt applications the imaging method might form a useful test to pre-qualify aggregate sources.

The information in Table 8 can also be used to assist in an image-based size classification system. If the “range” of the projected area for a given size group is defined by the limits (mean minus one standard deviation) and (mean plus one s.d.) the data from Table 8 can be used to construct Table 9 below.

It can be seen that these limits provide a clear discriminator for size since the ranges do not overlap (within the arbitrary, but reasonable) limits imposed. Thus, for example, if a single rock is shown to have an apa of 1.2 sq. cm then it is almost certainly from the size group 12 by 6 mm. However in an imaged based systems it is usually not possible to obtain the apa of a rock. Two limitations usually apply. First, it is more common, and easy, to obtain the projected area as seen from only one direction. Second, except for the rocks on the top layer the rocks are often partially obscured by others. Thus even the single pa is not known directly; a partially obscured (popa) area is the best that one can obtain. It is possible to proceed further by making two assumptions. First, that that the true pa of an “obscured” rock can be estimated from its shape. Second, that the statistics of the pa vs. direction plot can be used to compute a probability for the apa from the single observed pa.

The first of these is complex and is presently under study. The second is relatively straightforward.

It may be noted that the range of apa for each of the four size groups studied here are so widely separated that it may be possible to use simply the obscured pa as a discriminator. For rock sizes that are closer together the problem will become more complex.

The results using 13 views and 65 views of two (randomly selected) individual rocks are compared to show the confidence limits for the two cases.

From these limited results it can be seen that the 13 results do not seem to introduce any serious bias in the measures. The 64 views obviously give better accuracy and standard deviation but the 13 views are reasonable. It is, of course, necessary to determine what precision is needed in any given application.

This ratio may be used as a measure of the shape of the particles. However it must be noted that a single number is necessarily a relatively crude measure, its only advantage being simplicity. To determine whether this ratio may be of practical use for a given purpose further research is necessary. It is planned to compare these results with those reported by Jahn () using what is essentially a 3D caliper.

It may be noted that this ratio was remarkably close to 2 for all rocks tested in this study. No explanation for this could be surmised. It was also surprising that the largest value for this ratio was observed for a rounded rather than a crushed: this was unexpected.

It was recognized that the size groups defined by e.g. ASTM C 33 are both arbitrary and relatively “coarse.” Hence one major objective in this research was to determine if the volume (size) of a particle could be estimated from 2D views. If so, the classification of a large number of particles could be made much more precise.

Some relationships that can be speculated upon are:

  1. How is the apa for a single particle related to its volume?
  2. Is (apa)/volume a useful statistic?
  3. Is (apa/vol.66) a useful statistic?

The first speculation arises from the desire to predict volume from a projected area. However to plot apa against volume is of limited use because both the projected area and the volume of a rock are size dependent. This leads to speculation 2 in which the influence of size is removed by dividing apa by volume. A possibly more persuasive argument is that the area varies (approx.) as the square of the “size” while the volume will vary as its cube. This leads to speculation 3 above. However when plotted on log scales all the relationships listed above result in (approx.) straight lines. Hence the differences between suggestions 2 and 3 will merely alter the slope of the line and they may be considered essentially equivalent.

The first relationship may be examined in Figure 4 where (average projected area) is plotted against (volume) for each particle. In this figure the various (sieve) size groups are distinguished by identifying symbols. It will be noted that there is a certain amount of overlap between adjacent size groups. This points out that sieve analysis does not separate particles by volume. Just what sieving does accomplish is a complicated question, but it is clear that it does not separate simply by size, if size is defined as volume. However the imaging method is not constrained by “sieve groups.” With the volume of each particle estimated by independent means, all points can be plotted without regard to sieve size, as shown in Figure 4. The usefulness of figures such as this is that once a statistically valid sample size has been plotted in this fashion it may be relied upon for any future samples (real or imaginary) for that particular product. This suggests that an imaging system can offer an improvement over sieving.

Information such as figure 4 may also be used to estimate the true volume of each particle, and hence, by extension, to estimate the solid volume occupied in a mass of base rock, asphalt, or pcc. Present methods for estimating volume do not account for the different shapes of disparate aggregate sources. The proposed method would account for shape automatically. The result should be an improvement, and hence economy, in volumetric analysis of all construction materials.

To investigate the second speculation (that it is reasonable to use the measure (apa/volume) in order to remove the influence of size upon the shape), Figure 5 was plotted.

Since the range of volumes is large it is necessary to use Log scales in Figures 4 and 5.

Note that, while statistical bounds should be shown for each point, they are not shown in Figures 4 and 5 only because they would obscure the plot. They may be obtained from the data sheets.

The relationships between geometrical quantities for regular objects such as cubes and spheres are well defined and are plotted on the two Figures. The spherical shape has a minimum average projected area for a given volume (3÷(4×radius) where r is the radius). Hence the plotted lines for spheres in Figures 4 and 5 should represent a minimum. However it will be noted that some points for the irregular objects fall below these lines. This must be attributed to some combination of (a) measurement error (believed small) (b) the number of views selected.

It will be noted in Figure 5 that the plot includes about 120 points for particles whose sizes range from 0.25 inches (6 mm) to 1.5 inches (38 mm). Some of these rock fractions were crushed and some rounded. The points all fall close to a single line (with the possible exception of the rounded 0.75 inch (19mm) material. It is tempting to deduce that the shape of the particles does not depend upon size. While the quantity (average projected area/volume versus volume) may not be a “precise” measure of shape it is nonetheless probably true that this ratio would change if the shape changed. This is of interest since some studies on comminution have suggested that the shape will change with size. It is often suggested that shape becomes more “rounded” as the particle is made smaller; the argument being that a small particle has undergone more crushing blows and as a result “has had more corners knocked off.” The influence of crushing parameters (equipment, speed etc) is an interesting field of study and the current technique may offer some advantages in such studies.

  1. For scientific analysis the tomograph offers the best real 3D information but until tomographs are more widely available 2D views may offer an acceptable alternative in many situations.
  2. Multiple 2D views can provide useful results for estimating 3D properties of bodies.
  3. Although limited, this preliminary study indicates that the number of views needed to characterize a rock product for sizing is not unduly large. Comparison of 13 versus 64 views provides some data to guide future researches.
  4. The ratio (max/min area) of views is a crude but simple measure of shape. Further tests, using a wider variety of shapes, will assist in further evaluation of this potential measure.
  5. The plot of (apa/v) versus volume may be used to characterize any particular source of rock or rock/crusher combination. One advantage of this plot is that it records measurements of each individual particle. Current methods “lump” all particles from one sieve size. Each quarry/crusher combination can be so characterized (at least until a better characterization tool is developed) and the results can then be applied to all future samples of this product with confidence.
  1. Test a wider range of “shapes.” A possible practical application would be to contact researchers who have tested “flaky” aggregates in the many studies of “flat and elongated” particles (these have been stimulated by Superpave mix design procedures). Such studies would also serve to compare the shape parameters used in the two fields.
  2. In this study measurements were limited to the area of the projections only. Further useful information might be obtained if other parameters were also measured. Candidates include perimeter, moments of the area, etc.
  3. Extend the database to obtain more information on the statistical variation in properties of the projections with the number of views. In this study 13 views were compared with 65. The differences in the properties (mean and standard deviation) were not substantial. The appropriate number for any parameter is related to the use to which the data will be put. More statistics will assist in this decision.
  4. Use the data in a commercial system that uses imaging to predict psd for real materials. Imaging programs contains algorithms that attempt to predict the 3D geometry using (usually only one) 2D view. The data from this series of tests (plus additional tests as needed) should help improve the accuracy of such algorithms.
  5. Use a tomograph to obtain additional detailed geometrical information. A tomograph will provide at least 4 advantages. The volume of the body will be estimated more accurately (provided sufficient voxels are used). The surface area can be estimated and calculated. The moment measures mentioned above can be calculated. The density of each voxel can be estimated so that a true (average) density can be obtained for each rock. These advantages will provide a rich source of data.
  6. Use tomographic data to define shape quantitatively and demonstrate the usefulness of the measure by tests on real materials.

All of the photographic work and analysis was conducted by Mr. Tony Lau as part of the requirements for the Masters Degree in Civil and Environmental Engineering at the University of California, Davis. This extensive and repetitive work was a conducted with diligence, great good humor and small pay. Professors John Harvey and John Bolander offered support and invaluable guidance. The help of the staff is gratefully recognized by both authors.

The project would not have been possible without the financial support of the Granite Rock Company of Watsonville California (through the good offices of the owner Mr. Bruce W. Woolpert).

Bond, F.C. (1952) Chem.Eng. 59
Bond, F.C. (1954) Chem.Eng. 61
Cauchy A. (1850) “Memoire sur la rectification des courbes et de la quadrature des surfaces courbes” Mem. Acad. Sci. Paris 22, 3. Also in Oevres Completes Vol 1 (1908).
Lau, T (2002) “Using 2D projections to characterize 3D Particles” Thesis presented to the Faculty of the University of California, Davis in partial fulfillment of the requirements for the degree Master of Science in Civil Engineering June 2002.
Laurentini A. (1995) “How far 3D shapes can be understood from 2D silhouettes” IEEE transactions on Pattern Analysis and Machine Intelligence v17 No 2 Feb 1995 p188–193.
Mather, B. (1966) “Shape Surface Texture and Coatings” in ASTM STP 169A Significance Of Tests And Properties Of Concrete And Concrete Making Materials, Pub. American Society for Testing and Materials, Philadelphia 1966 pp571.
Price, W. (1966) “Grading and Surface area” in Significance Of Tests And Properties Of Concrete And Concrete Making Materials. Pub. American Society for Testing and Materials, Philadelphia 1966 pp571.
Russ J.C. and DeHoff R.T (2000) Practical Stereology Kluwer Academic.
Russ J.C. (1999) The Image Processing Handbook 3rd Ed. CRC Press pp771
Taylor M. (2002) “Quantitative Measures of Shape and Size of Particles” Powder Technology 124 (202) 94–100
Underwood E. E. (1970). Quantitative Stereology, Addison Wesley.
Vickers G.T. (1996) “The projected areas of ellipsoids and cylinders.” Powder Technology 86 195–200.
Vickers G.T. (1998) Powder Technology v 98 (1998) 250–257.
Vickers G.T. (2001) “The projected areas of ellipsoids and cylinders.” Proc. of the Royal Society of London. Series A. V 457 p283–306

With 4 locations of each angle there are 16 combinations: each of the 4 latitudes from column 2 is taken with each of the 4 longitudes in column 1. However 4 of these views are duplicates (all longitudes meet at the North Pole) so there are only 13 independent views.

Table 4-Sample Data Sheet
  • Rock ID: 6 of 25
  • type: crushed
  • size class: 12 by 6 mm (½ by ¼″)
  • weight: 1.66 gm
  • Density 2.809
  • computed volume: 0.591 cc = 0.03 cu ins
  • Diameter of equivalent volume sphere: 1.041 cm
  • Side of equivalent volume cube: 0.839 cm

Table 2: Orientation of 72 Views (65 independent) Up arrow
Longitudes (°) Latitudes (°) Sine of Latitude
0 0 0
45 7.2 0.125
90 14.5 0.250
135 22.0 0.375
180 30.0 0.500
225 38.7 0.625
270 48.6 0.755
315 61.0 0.875
360 (same as zero) 90.0 1.00
Samples of Rocks tested with 13 independent views Up arrow
Samples Size (mm, inches) Number photographed
Crushed 37 by 19 (1.5 by 0.75) 25
Crushed 19 by 12 (0.75 by 0.50) 31
Crushed 12 by 6 (0.50 by 0.25) 25
Crushed 6 by 2 (0.25 by 0.08) 25
River run 19 by 12 (.75 by .5) 10
River run 12 by 6 (0.50 by 0.25) 10
Table 1: Orientation of 16 Views (13 independent) Up arrow
Longitudes (°) Latitudes (°) Sine of Latitude
0 0 0
30 19.5 0.33
60 40.8 0.67
90 90 1.00
Table 5: Spheres (Marbles): Summary of 13 views each of 2 marbles Up arrow
Ident (#) Weight (gm) Volume (cc) Average Area (Sq cm) Average Area/vol (1/cm) Max Area Min Area Ratio Max/min Area
1 6.54 .113 1.823 16.13 1.000
2 6.52 .113 1.823 16.13 1.000
Table 3: Calibration Samples Tested with 13 independent views Up arrow
Samples Size (mm, inches) Number photographed
Spheres (marbles) 15 (0.6) 2
Cubes (dice) 15 (0.6) 2
Table 8: Summary of average measurements for all rocks (13 views each rock) Up arrow
Source size (mm) n apa (sq cm) St dev Max/min St dev apa/vol (1/cm) St dev
crushed 6 25 0.270 0.08 2.07 0.33 4.213 1.71
crushed 12 31 1.168 0.18 1.966 0.46 1.679 0.23
crushed 19 25 4.670 0.92 1.955 0.39 0.872 0.13
crushed 37 25 7.84 2.06 1.956 0.49 0.50 0.19
rounded 12 10 0.408 0.18 2.321 0.59 2.185 0.55
rounded 19 10 3.629 1.81 1.720 0.28 0.823 0.48
Table 6 : Cubes (Dice): Summary of 13 views each of two dice Up arrow
Ident (#) Weight (gm) Volume (cc) Average Area (Sq cm) Average Area/vol (1/cm) Max Area (Sq cm) Min Area (Sq cm) Ratio Max/min Area
1 4.05 .216 3.385 16.13
2 4.00 .216 3.824 16.13
Table 7: Typical summary results for one rock size group Up arrow
12x6mm crushed rock (#) Weight (gm) Volume (cubic cm) Average Project area apa (square cm) Max/min apa/vol apa/v66
1 1.89 0.673 1.322 1.513 1.965 1.718
2 1.82 0.648 1.063 2.441 1.641 1.415
3 2.28 0.812 1.305 1.760 1.607 1.497
4 2.71 0.965 1.280 2.208 1.326 1.310
5 1.93 0.687 1.229 1.928 1.789 1.575
6 1.66 0.591 1.108 1.771 1.875 1.568
7 2.15 0.765 1.291 1.599 1.687 1.540
8 2.35 0.837 1.373 1.497 1.641 1.544
9 1.89 0.673 1.058 1.427 1.573 1.375
10 2.93 1.043 1.483 2.303 1.421 1.442
11 2.85 1.015 1.440 2.092 1.419 1.426
12 1.67 0.595 1.023 1.344 1.720 1.442
13 1.26 0.449 1.065 1.840 2.375 1.809
14 1.77 0.630 1.067 2.010 1.694 1.448
15 1.97 0.701 1.249 1.710 1.782 1.579
16 2.49 0.886 1.334 1.980 1.504 1.444
17 2.34 0.833 1.081 1.906 1.297 1.219
18 1.25 0.445 0.762 2.351 1.712 1300
19 1.81 0.644 1.035 2.798 1.606 1.381
20 1.85 0.659 0.966 2.370 1.467 1.273
21 1.82 0.648 1.037 1.545 1.601 1.381
22 1.45 0.516 0.981 3.332 1.900 1.517
23 1.68 0.598 1.114 1.863 1.863 1.564
24 1.71 0.609 1.116 1.596 1.832 1.548
25 2.37 0.844 1.426 1.966 1.691 1.596
Average 1.996 0.711 1.168 1.966 1.680 1.477
St Dev 0.449 0.160 1.176 0.457 0.228 1.135
c.var. 0.225 0.225 0.151 0.232 0.136 0.092
max 2.93 1.043 1.483 3.332 2.375 1.809
min 1.25 0.445 0.762 1.344 1.297 1.219
max/min 2.344 2.344 1.946 2.479 1.831 1.483
Table 4: Sample Data Sheet Up arrow
View Longitude Latitude Area (pixels) Area (square inches) Area (square cm)
1 0 0 18847 .139 .915
2 0 19.5 19797 .146 .961
3 0 41.8 21215 .157 1.030
4 0 90.0 15847 .117 .769
5 30 0 22072 .163 1.072
6 30 19.5 27003 .199 1.311
7 30 41.8 24439 .180 1.187
8 30 90.0 22739 .170 1.104
9 60 0 23929 .177 1.162
10 60 19.5 26938 .199 1.308
11 60 41.8 28068 .207 1.363
12 60 90.0 24942 .184 1.211
13 90 0 20824 .154 1.011
Average 22820 .169 1.108
St dev. 3548 .026 .172
Coef var 0.16
Max 28068 .207 1.363
Min 15847 .117 .769
Max/min 1.77