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A new approach to the three-dimensional quantification of angularity using image analysis of the size and form of coarse aggregates
Engineering Geology 91 (2007) 254 – 264 www.elsevier.com/locate/enggeo
A new approach to the three-dimensional quantification of angularity using image analysis of the size and form of coarse aggregates J.R.J. Lee, M.L. Smith ⁎, L.N. Smith Machine Vision Laboratory, Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Frenchay Campus, Bristol BS16 1QY, UK Received 6 November 2006; received in revised form 16 February 2007; accepted 28 February 2007 Available online 7 March 2007
Abstract This paper describes a system for the acquisition and analysis of 3D data from the surfaces of coarse aggregate particles. The technique uses laser triangulation to acquire data from the upper hemispheres of particles passing along a conveyor belt. Methods of determining particle size and form in three dimensions are described and a new approach to the quantification of angularity is presented. The algorithm uses mathematical morphology to provide a geometrically meaningful interpretation of particle shape. The relative advantages of 2D and 3D analysis of aggregate particles are discussed, and results are presented which demonstrate the validity of this approach. © 2007 Elsevier B.V. All rights reserved. Keywords: Aggregates; Particle size, shape and angularity; Laser triangulation; Mathematical morphology
1. Introduction The application of image analysis techniques to the study of aggregate shape properties has become increasingly prevalent in recent years. It has long been acknowledged that the size distributions, form, angularity and surface texture of aggregates significantly impacts on the performance of concrete and asphalt mixtures. It is also generally accepted that the techniques currently used in industry to determine these properties are at best somewhat laborious. Furthermore, they are often subjective and provide only indirect measures of the properties of interest. Several authors have identified the ambiguities associated with particle sieve sizes and the corresponding dependent measures ⁎ Corresponding author. E-mail address: [email protected] (M.L. Smith). 0013-7952/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2007.02.003
of elongation and flakiness (Persson, 1998; Kwan et al., 1999; Mora and Kwan, 2000; Lanaro and Tolppanen, 2002). The methods used to determine angularity are similarly limited (Wilson et al., 1997; Masad and Button, 2000). For example, the traditional measure described in BS812 assumes a causal relationship with packing density, defining an angularity number as the amount by which the proportion of solid volume in a packing density test falls below 67%. This somewhat arbitrary figure is thought to represent an optimal packing density given a sample of equally sized, perfectly spherical particles. However, it is hard to see how variations in size distribution, elongation and flakiness would not have equally marked effects on packing density as variations in angularity. It is not then surprising that researchers and industry continue to look to image processing technologies for alternatives. One of the major difficulties encountered
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here is that of reconciling the results of an image-based analysis with those of the traditional industry techniques. For example, having developed a direct measure of angularity or surface texture using image analysis, one can only validate it using the indirect measures that one believed to be flawed or subjective in the first place! The problem is compounded when the properties of interest are arbitrarily defined. This applies particularly to the measurement of particle angularity, or conversely roundness. The European standard EN 933-5 describes how angularity should be determined by visually assessing what percentage of the particle surface is considered to be crushed or broken. This will yield different results from the Wadell approach of examining the relative radii of corners (Wadell, 1935) or from comparison with the similarly derived Powers scale (Powers, 1953), and different again from the results of a packing density test as described in the previous paragraph. Similarly, research into image-based analysis of aggregate properties has generally attempted to approximate angularity with the measurement of other morphological properties. Several researchers have attempted to assess angularity using measures of compactness, by calculating the perimeter-to-area ratio (in two dimensions) or the surface area-to-volume ratio (in three dimensions) of the particle (Wilson et al., 1997; Mora and Kwan, 2000; Masad and Button, 2000; Masad et al., 2001; Manohar and Sridhar, 2001; Pons et al., 2002). The rationale here is that the circle (or in three dimensions, the sphere) is the most compact shape, in that it has the shortest perimeter for a given area. As the shape becomes more angular the perimeter-to-area ratio increases. Masad and Button (2000) use this method at low and high resolutions to gain a measure of both angularity and surface texture, on the premise that the perimeter is more sensitive to variations in angularity at low resolution and to variations in texture at high resolution. Wilson et al. (1997) attempt to compensate for the effects of aspect ratio on compactness by comparing perimeter-to-area ratios with an equivalent ellipse rather than a circle. Masad et al. (2001) also use the equivalent ellipse to provide a normalised measure of angularity. They propose an angularity index that compares the radii of the particle to those of an equivalent ellipse at a fixed number of angular increments, a technique that proved useful in predicting the rutting resistance performance of a hot-mix asphalt. The fundamental limitation of these approaches is that compactness and angularity are two different properties. Whilst there is undoubtedly some correlation, a shape does not necessarily become more angular as it becomes less compact. The validity of the measure further degrades when one considers that the
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particle boundary is generally represented using square pixels — a common problem with image-based measures of compactness is that in many cases they will actually return a square, or in three dimensions a cube, as the most compact shape (Bribiesca, 1997; Bribiesca, 2000). Another approach is to compare the overall shape of the particle to that of a reference shape. Lanaro and Tolppanen (2002) compare the frequency spectra of the cross-sectional profiles of coarse aggregate particles to a number of geometrically simple reference shapes, whilst Podczeck (1997) used measures of deviation from squares, triangles and circles in order to characterise powder particle morphology. Whilst these techniques are useful in determining and representing the underlying shape of a particle, it is not clear how useful they are in providing explicit measures of properties such as angularity. The most intuitive method would appear to be the mathematical morphology approach taken by Masad and Button (2000), in that the resistance of a particle boundary to a series of erosion and dilation operations is directly dependent on the sharpness of the edges and corners. The primary weakness of this approach is that it lacks rotational invariance, in that the use of a simple square structuring element means that an image of a square particle with sides aligned with the axes of the grid would lose no area whatsoever. Kalliomäki et al. (2005) recently developed a more significant mathematical morphology algorithm that uses progressive erosions using circular structuring elements of increasing diameter to the same effect. In this paper we extend the mathematical morphology approach to three dimensions and show how scale, rotation and orientation independence can be attained by replacing the square structuring element with an ellipsoid that is adaptively determined by the size and aspect ratios of the particle. We discuss the relative merits of two-dimensional and three-dimensional imaging in the case of coarse aggregate and present an effective means of 3D data acquisition. In addition to the angularity algorithm we show how size and form can be recovered in a relatively straightforward manner. The limitations of this approach are discussed and opportunities for further work are identified. 1.1. Context The overall aim of this research is to establish technologies and methodologies that will enable development of an automated petrographic analysis system for aggregate. Such a system would be capable of performing many of the geological tests that are currently performed in laboratories. The analysis of the morphologies and
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sizes of rocks, which is the subject of this paper, would form an important function of the device; however it would also perform a number of further machine vision tests that relate to the engineering properties of the aggregate. This would include a spectrographic analysis of the particles, since the mineralogical composition of aggregate materials is critical for engineering applications such as concrete production where resistance to chemical alteration must be considered. The intention is for the final system to be mobile, so that petrographic analysis could be performed on location at a quarry. The availability of an online petrographic analysis system is expected to provide aggregate producers with a number of benefits, including reduced costs and very significant reductions in the time required for the rock analysis. 2. 2D or 3D analysis? Much of the research to date has concentrated on the analysis of projected particle boundaries, i.e. one attempts to make deductions on the shape of a particle from its silhouette. The first problem encountered here concerns the measurement of the overall form of the particle. In order to provide robust data on size, elongation and flakiness one must recover the three principal dimensions of the particle. Clearly this is not possible from a single two-dimensional projection. As such, a number of researchers have devised innovative ways around this. Kwan et al. (1999), Mora and Kwan (2000) supplement the visual information obtained with the weight of an aggregate sample in order to derive a ‘mean thickness’ for the group of particles, which can be used to give a measure of the average flakiness of the sample. Stereological methods are also widely used to statistically deduce three-dimensional particle size distributions from two-dimensional analyses (Sahagian and Proussevitch, 1998; Coster and Chermant, 2001). However, these are best applied to a random crosssection (thin section) through the material sample rather than to the projected perimeters under discussion here. Brzezicki and Kasperkiewicz, 1999 developed a system whereby particles are analysed on a curved corrugated grid, or more specifically a ‘fragment of a cylindrical surface with a series of parallel indentations’. Two light sources are arranged in such a way that the camera captures two perpendicularly projected shadows of each rock, allowing flakiness as well as elongation to be calculated directly. Maerz (1998), Maerz and Lusher (2001) use two orthogonal cameras to capture both side and overhead views of rocks moving along a conveyor, a development that has been implemented in commercial applications. This also allows direct calculation of
elongation and flakiness on the assumption that a particle will rest in its most stable position — i.e. the height observed in the side elevation should represent the smallest dimension of the particle. Whilst these approaches provide effective means of determining three-dimensional form to varying extents, boundary analysis approaches suffer from far more fundamental limitations when one attempts to measure more complex shape properties such as angularity and surface texture. The most dangerous assumption in these cases is that the observed planar projection represents a genuine cross-section of the particle. In the case of a simple three-dimensional object such as a sphere, this assumption will hold. Assuming a parallel projection, the projected boundary of a sphere will always be its cross-sectional great circle. However in the case of more complex objects this is no longer the case. If one considers a drill bit for example, a silhouette projected along the axis of the bit will show a circle. A silhouette projected orthogonally to the axis of the drill bit will show a shape close to a rectangle. There is no possible projection that will reveal the complex, helical nature of the bit. This is because rather than viewing a genuine cross-section of the object, we are viewing multiple overlapping, partially occluded crosssections. The same applies to aggregate particles, particularly the more angular specimens, where the effect is typically to oversimplify the particle shape. As such, there is a growing recognition of the increased level of robustness afforded by three-dimensional data acquisition. Garboczi (2002), Garboczi and Bullard (2004) use X-ray tomography on concrete and cement samples to recover complete three-dimensional data on the constituent aggregates, allowing a more direct comparison between the morphological properties of the aggregates and the performance of the composite. Lanaro and Tolppanen (2002) use laser triangulation, scanning aggregate particles from multiple viewpoints in order to recover the entire surface. The authors acknowledge that their technique cannot be extended to industrial quality control in its current form, suggesting instead that scanning could be limited to the upper hemispheres of aggregate particles moving along a conveyor belt given that the upper and lower hemispheres ‘usually exhibit the same geometrical properties’. This is similar to the approach developed herein. We consider that acquiring 3D information from the visible surfaces of passing aggregate particles offers a useful compromise between the two-dimensional approach which offers simplicity but lacks rigour, and the full three-dimensional approach which offers robustness but lacks efficiency. In the following section we present an overview of the laser triangulation system before discussing the processing of the acquired data.
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3. Acquisition Laser triangulation is used to recover three-dimensional data from the surfaces of the aggregate particles. A laser line is projected vertically down onto a conveyor belt such that the line is perpendicular to the direction of motion, as illustrated in Fig. 1. The camera will then capture an image of the laser line as it intersects the belt or a passing particle. The image of the laser line is distorted by the profile of the rock particle, as illustrated in Fig. 2. Trigonometry is used to relate the observed position of the laser line in the acquired image to the corresponding impact position of the laser line in 3D space. The mathematics differ slightly from the traditional laser triangulation derivation (see for example Besl, 1998; Amann et al., 2001), in that we wish to define a coordinate system with its origin at the interception point of the camera axis, the conveyor belt and the laser line. The plane of laser light seen in Fig. 1 thus becomes the yz plane in the coordinate system, with the z-axis rising vertically upwards and the y-axis lying on the conveyor belt and perpendicular to the direction of motion. This simplifies the analysis, as we know that every point we capture lies in this plane. The remaining x-coordinate is then simply the relative position of the conveyor belt when the image was acquired. We describe the positions of pixels in the image using a second coordinate system, where i and j are the horizontal and vertical coordinates respectively and the origin lies at the optical centre of the image. We first consider the calculation of z-values, by looking at the imaging system in the xz plane. This is shown in Fig. 3. The known parameters are f (the focal length of the camera lens), θ (the angle formed between the camera
Fig. 2. Captured image of laser impact position on particle surface.
axis and the laser plane), d (the distance between the lens and the conveyor belt along the camera axis) and j (The vertical position of any given point on the laser line in the image). The parameter u describes the distance between the laser impact position and the camera lens and v describes the distance between the corresponding point on the image plane and the camera lens. From the sine rule it can be shown that: z¼
dsinh sinð180−h−/Þ
ð1Þ
Where: tan/ ¼
j f
ð2Þ
In order to calculate the corresponding y-coordinate of a given point, the analysis is as shown in Fig. 4. From Fig. 3 and the sine rule, we know that: u¼
dsinh sinð180−h−/Þ
And from Pythagoras' theorem: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ f 2 þ j2
ð3Þ
ð4Þ
Then from the similar triangles shown in Fig. 4 we can deduce that: u y¼ i v
Fig. 1. Laser diode and camera position.
ð5Þ
Eqs. (1)–(5) thus allow us to describe the acquired cross-section of a particle in terms of a set of 3D coordinates, where the x-coordinate is determined by the incremental distance moved by the conveyor belt between images being captured. By concatenating these cross-sections we are able to generate a dense set of coordinates describing the entire visible surface of the particle. In practice, in order to avoid regions of
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Fig. 3. z-value analysis.
occlusion where points on the surface are not visible to both the camera and the laser, a second camera is added to the system. This is placed symmetrically opposite to the first camera about the laser plane, as shown in Fig. 5. The surface data captured by each camera is then merged to provide a complete description of the upper hemisphere of the particle (clearly the underside of the
particle will remain invisible to both cameras). This is then most conveniently represented as a range intensity image, as illustrated in Fig. 6. The image is constructed in the xy plane, as if the rock was being viewed from above. The intensity of the image at any given point is a function of the height of the particle at that point. In this case each pixel, and each unit change in intensity value, represents 0.1 mm. This is an efficient way to represent our acquired data as the range intensity image uses no more memory than a standard 2D image, whilst containing genuine 3D data.
Fig. 4. y-value analysis.
Fig. 5. Prototype triangulation system.
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Fig. 8. Determination of minimum bounding box.
Fig. 6. Range intensity image of coarse aggregate particle.
This representation will provide the foundations for all our subsequent analysis. 4. Processing 4.1. Determination of size and form Having acquired the surface data from a particle, the first processing requirement is to determine the length, width and height of the aggregate particle. As we know nothing of the underside of the rock we have to use the assumption that the rock will rest in its most stable position and so, in line with the approach of Maerz
(1998), Maerz and Zhou (1999), Maerz and Lusher (2001), we measure height orthogonally upwards from the belt. The system is calibrated such that the conveyor belt is the reference height level of 0, so we simply record the maximum z-value obtained from the particle. As the height is as such orthogonal to the image plane, the recovery of length and width can be reduced to a two-dimensional problem. A threshold operation is first applied to the range intensity image in order to segment the particle from the background, thus reducing the data to a binary representation as shown in Fig. 7. Length and width can then be determined by rotating the image to find the minimum bounding box, as illustrated in Fig. 8. Given that the length, width and height of the particle have now been recovered, measures of elongation and
Fig. 7. Original range image (left) and result of threshold at conveyor belt height level (right).
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flakiness can be expressed as individual aspect ratios as per previously published literature. 4.2. Determination of angularity The two basic operations of mathematical morphology are erosion and dilation. The erosion of an image f (i, j) with a structuring element g(x,y) involves translating the structuring element to every point on the image and taking a ‘minimum of differences’, thus: ð f HgÞði; jÞ ¼ minf f ði−x; j−yÞ −gðx; yÞ : ði−x; j−yÞ; ðx; yÞaZ 2 g
ð6Þ
Similarly, for a dilation we translate the structuring element to every point on the image and take a ‘maximum of sums’, thus: ð f PgÞði; jÞ ¼ maxf f ði−x; j−yÞ þgðx; yÞ : ði−x; j−yÞ; ðx; yÞaZ 2 g
ð7Þ
A morphological opening consists of an erosion followed by a dilation, and can be expressed as ( f Θ g) ⊕ g This is the operation used by Masad and Button (2000) in the previous mathematical morphology approach to particle analysis. It should however be noted that these expressions, which describe the actions of a multi-level structuring element on a multi-level image, differ from those typically used to describe the binary morphological operations used in the research mentioned above. The concepts of multi-level or greyscale mathematical morphology, as described in the early authoritative literature (Serra, 1982), are based around the visualisation of both the image and the structuring element as three-dimensional surfaces. As such, a standard intensity image, denoted as a function f (x,y) in twodimensional space, is considered in greyscale morphology to be a set of points [x,y, f (x,y)]. In this case, as the grey levels in our range-intensity image represent genuine height data, this can of course be interpreted more literally. Thus, performing a morphological opening using a structuring element with grey values representing those of a hemisphere can be considered as analogous to rolling a ball all over the underside of the surface represented by the pixel values of the image. If the surface is smooth with relatively low curvature, the ball will make contact with the surface at every point. However, the ball will be unable to make contact with surface points contained within protrusions of high curvature, such as sharp edges and corners. The result of the morphological opening is then the union of every path that the ball may follow, or more formally the union
of the translations of the hemispherical structuring element to every point on the surface. This is illustrated by Fig. 9. This particular morphological opening was originally described by Sternberg (1986) as a means of smoothing discontinuities and removing noise from standard greyscale images. In this case it is suggested that the morphological opening of a range-intensity image could provide an indication of particle angularity, given the selection of an appropriate structuring element, by analysis of the proportion of particle volume lost to the procedure. This effectively simulates the wear processes by which aggregate particles become rounded, by reducing each to a well-rounded state. It should be noted that only convex features are modified by the morphological opening, any concavities on the surface of the particle remain unaltered. The selection of a suitable structuring element is now considered. Firstly, recall that we wish to quantify angularity by analysing the proportion of the entire particle volume lost to the operation, and that angularity is traditionally measured as the sharpness of edges and corners relative to the overall size of the particle (Wadell, 1935; Powers, 1953). Intuitively then, the size of the structuring element needs to be adaptively determined according to the size of the particle under analysis. Secondly, the Powers/Wadell approach describes the
Fig. 9. a — Cross section of image as surface. b — Application of morphological opening. Regions shown in black are inaccessible. c — Result of operation.
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measurement of the angularity of a corner with respect to the maximum inscribed circle in a given projection. A spherical structuring element will therefore be unsuitable unless the particle has neutral elongation and flakiness ratios, as a circle of the correct size for one projection may be inappropriate for another. As such, an ellipsoidal structuring element with aspect ratios equal to those of the particle is used. The principal axes of the structuring element remain aligned with the principal axes of the
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particle throughout the operation, thereby ensuring proportionality from any two-dimensional projection. Experimental work has shown that a structuring element with principal axes 1/3rd of the length of those of the particle gives optimal results. Fig. 10a to f shows line profiles taken from rangeintensity images of gravel particles exhibiting various levels of angularity. The y-axes of the graphs represent the height of the particle and the x-axes represent an
Fig. 10.
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Fig. 11. Average percentage volume loss vs. angularity number.
arbitrary line across the image plane. The values on the axes are arbitrary as the images were resized prior to the operation. The black region shows volume lost due to the morphological opening. It should be noted that the volume loss does not necessarily correspond entirely to the profile shown. Corners and edges that are not visible in these projections will still have an impact as this is a three-dimensional operation. However, it can clearly be seen that there is a broad correlation between the angularity of the particles and the proportions of volume lost. 5. Results and discussion The measurement of particle size was validated manually using callipers. It was demonstrated that the system is capable of measuring the three principal dimensions of a particle to within ±0.1 mm. This was the intended accuracy and is entirely a function of the design, i.e. the resolution of the camera, the range of particle sizes and hence the magnification used, the quality of the laser and the manufacturing tolerances of the conveyor belt and drive equipment. Elongation and flakiness ratios can implicitly be calculated with similar precision. In some cases a lower resolution may suffice, for example when measuring angularity the range-intensity image is rescaled to a resolution of 0.5 mm/pixel. This provides sufficient resolution for the analysis of angularity whilst removing the effects of textural features and improving the efficiency of the processing. The measurement of angularity, as discussed earlier, is more difficult to validate due to the subjectivity of the manual classification. A test sample consisting of two hundred morphologically diverse gravel particles in the 8–32 mm size interval was graded against the Powers scale by geologists at the Slovenian National Building and Civil Engineering Institute (ZAG), Ljubljana and Petromodel ehf, Reykjavik. The particles were then
passed through the laser triangulation system and the ellipsoidal morphological opening was applied. Fig. 11 shows the average percentage volume loss per particle plotted against the angularity value manually determined using the Powers technique. Note that 1 represents the ‘very angular’ category and 6 represents the ‘well rounded’ category. Fig. 11 shows that there is a reasonable linear relationship between the volume lost due to the ellipsoidal morphological opening and the geologists’ classification of particle angularity. One point of concern was that the variance in the results appears to be very high, as shown in Fig. 12. It appears that the standard deviation increases almost linearly with increasing angularity. This suggests that the deviations can be explained by the difficulties in objectively classifying particle angularity, as it is far easier to distinguish between a well rounded particle and a rounded particle than it is to distinguish between an angular particle and a very angular particle (Powers, 1953). It was initially hoped that the system would be able to recover surface texture along with the size, form and angularity of the aggregate particles. This however has proved to be difficult due to laser speckle noise at the acquisition stage. Speckle noise is a phenomenon whereby coherent light hitting a surface is reflected back from varying distances. The interaction of the reflected light can lead to both constructive and destructive interference, the result being that the projected pattern consists of bright and dark regions. This causes problems in the recovery of 3D data, as the brightest point in the cross section of the laser line will no longer necessarily correspond to the centre of the line. Thus this translates into noise in the recovered surface data. In the case of aggregate particles, this problem is exacerbated for two reasons. The influence of laser speckle would usually be reduced by using a narrower
Fig. 12. SD of average percentage volume loss vs. angularity number.
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laser line width. However, when using focusable laser diodes there is a trade-off between the required depth of field and the minimum attainable laser line width. As such, a line width fine enough to recover the surface texture of an aggregate particle would not remain in focus over the depth of fields necessary to recover variations in form. In the case of a fixed laser line scanning material passing along a conveyor belt, the required depth of field extends from the impact position of the line on the belt to the height of the largest particle one expects to encounter. This limitation is compounded by the range of colours and reflectances encountered in construction aggregates, as the apparent width of the laser line in the acquired images is also dependent on the aperture settings of the cameras. Thus an aperture setting which results in images of fine lines on black particles will also result in images of very coarse lines on white particles. The resultant effect of all these issues is that the magnitude of the surface noise caused by laser speckle exceeds the magnitude of the surface variations due to surface texture, making quantification of surface texture fundamentally difficult. Lanaro and Tolppanen (2002) avoid this problem when using a laser to scan aggregate material as the scanning equipment is mobile, thus reducing the depth of field constraint, however as discussed previously this makes application in an industrial environment difficult. Possible solutions included the use of laser speckle reduction hardware or the use of a broadband white light source. However, the surface texture of coarse aggregate is considered to be less critical than particle size distributions, form and angularity to the performance of concrete and asphalt. 6. Conclusions The research described in this paper has shown how laser triangulation can be applied effectively to recover three-dimensional information from the upper surfaces of aggregate particles moving along a conveyor belt. Particle size, form and angularity can be quantified accurately from the acquired data. This approach offers a useful compromise between the accuracy limitations of two-dimensional image analysis and the application limitations of a full three-dimensional acquisition. Simple techniques for the recovery of particle size and form have been described and a new algorithm for the quantification of particle angularity has been presented. The ellipsoidal morphological opening is geometrically intuitive and allows angularity to be quantified directly rather than by inference from other
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properties. Results demonstrate the validity of this method. This approach offers potential for useful application to real time quality monitoring and control in industrial environments. The research activities were undertaken as part of a broader collaborative project which ultimately aims to combine morphological data with mineralogical information obtained using spectroscopy, thus providing a fast, accurate and complete analysis of the fundamental properties of aggregate materials. 7. Future work The primary aim should be to develop a means of simultaneously quantifying surface texture in addition to size, form and angularity from the acquired 3D data. As discussed in the previous paragraphs, this will involve investigating alternative means of structured light projection that may alleviate the limitations imposed by laser speckle noise. The effects of colour variations encountered in aggregate materials, and the refractive nature of certain minerals, on the recovery of surface texture should also be investigated. The other challenge is to increase the throughput of the device. At present particles are analysed as they pass underneath the laser and cameras in single file. The development of a system capable of extracting rigorous morphological information from material passing in bulk would bring the technology a step closer to widespread industry acceptance. Acknowledgements The authors wish to express their gratitude to Petromodel (Iceland) ehf., RMC Aggregates (UK) Ltd. and the Slovenian National Building and Civil Engineering Institute (ZAG) for their collaboration and support in this research. References Amann, M.-C., Bosch, T., Lescure, M., Myllylä, R., Rioux, M., 2001. Laser ranging: a critical review of usual techniques for distance measurement. Optical Engineering 40 (1). Besl, P.J., 1998. Active, optical range imaging sensors. Machine Vision and Applications 1, 127–152. Bribiesca, E., 1997. Measuring 2-D shape compactness using the contact perimeter. Computers and Mathematics with Applications 33 (11), 1–9. Bribiesca, E., 2000. A measure of compactness for 3D shapes. Computers and Mathematics with Applications 40, 1275–1284. Brzezicki, J.M., Kasperkiewicz, J., 1999. Automatic image analysis in evaluation of aggregate shape. Journal of Computing in Civil Engineering 123–128 (April).
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Coster, M., Chermant, J.L., 2001. Image analysis and mathematical morphology for civil engineering materials. Cement and Concrete Composites 23, 133–151. Garboczi, E.J., 2002. Three-dimensional mathematical analysis of particle shape using X-ray tomography and spherical harmonics: application to aggregates used in concrete. Cement and Concrete Research 32, 1621–1638. Garboczi, E.J., Bullard, J.W., 2004. Shape analysis of a reference cement. Cement and Concrete Research 34, 1933–1937. Kalliomäki, I., Vehtari, A., Lampinen, J., 2005. Shape analysis of concrete aggregates for statistical quality modelling. Machine Vision and Applications February. Kwan, A.K.H., Mora, C.F., Chan, H.C., 1999. Particle shape analysis of coarse aggregate using digital image processing. Cement and Concrete Research 29, 1403–1410. Lanaro, F., Tolppanen, P., 2002. 3D characterization of coarse aggregates. Engineering Geology 65, 17–30. Maerz, N.H., 1998. Aggregate sizing and shape determination using digital image processing. Centre For Aggregates Research (ICAR) Sixth Annual Symposium Proceedings, St Louis, Missouri April 19–20, pp. 195–203. Maerz, N.H., Lusher, M., 2001. Measurement of flat and elongation of coarse aggregate using digital image processing. Transportation Research Board 80th Annual Meeting, January 7–11, Washington, D.C. Maerz, N.H., Zhou, W., 1999. Flat and elongated: advances using digital image analysis. Centre for Aggregates Research (ICAR) Seventh Annual Symposium Proceedings, Austin, Texas, April 19–21, pp. B1-4-1–B1-4-12. Manohar, B., Sridhar, B.S., 2001. Size and shape characterization of conventionally and cryogenically ground turmeric (Curcuma domestica) particles. Powder Technology 120, 292–297.
Masad, E., Button, J., 2000. Unified imaging approach for measuring aggregate angularity and texture. Computer-Aided Civil and Infrastructure Engineering 15, 273–280. Masad, E., Olcott, D., White, T., Tashman, L., 2001. Correlation of fine aggregate imaging shape indices with asphalt mixture performance. Transportation Research Record 1757, 148–156. Mora, C.F., Kwan, A.K.H., 2000. Sphericity, shape factor, and convexity measurement of coarse aggregate for concrete using digital image processing. Cement and Concrete Research 30, 351–358. Persson, A.-L., 1998. Image analysis of shape and size of fine aggregates. Engineering Geology 50, 177–186. Podczeck, F., 1997. A shape factor to assess the shape of particles using image analysis. Powder Technology 93, 47–53. Pons, M.-N., Vivier, H., Delcour, V., Authelin, J.-R., Paillères-Hubert, L., 2002. Morphological analysis of pharmaceutical powders. Powder Technology 128, 276–286. Powers, M.C., 1953. A new roundness scale for sedimentary particles. Journal of Sedimentary Petrology 23, 117–119. Sahagian, D.L., Proussevitch, A.A., 1998. 3D particle size distributions from 2D observations: stereology for natural applications. Journal of Volcanology and Geothermal Research 84, 173–196. Serra, J., 1982. Image Analysis and Mathematical Morphology. Academic Press, London. Sternberg, S.R., 1986. Grayscale morphology. CVGIP 35, 333–355. Wadell, H., 1935. Volume, shape and roundness of quartz particles. Journal of Geology 43, 250–280. Wilson, J.D., Klotz, L.D., Nagaraj, C., 1997. Automated measurement of aggregate indices of shape. Particulate Science and Technology 15, 13–35.
A new approach to the three-dimensional quantification of angularity using image analysis of the size and form of coarse aggregates J.R.J. Lee, M.L. Smith ⁎, L.N. Smith Machine Vision Laboratory, Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Frenchay Campus, Bristol BS16 1QY, UK Received 6 November 2006; received in revised form 16 February 2007; accepted 28 February 2007 Available online 7 March 2007
Abstract This paper describes a system for the acquisition and analysis of 3D data from the surfaces of coarse aggregate particles. The technique uses laser triangulation to acquire data from the upper hemispheres of particles passing along a conveyor belt. Methods of determining particle size and form in three dimensions are described and a new approach to the quantification of angularity is presented. The algorithm uses mathematical morphology to provide a geometrically meaningful interpretation of particle shape. The relative advantages of 2D and 3D analysis of aggregate particles are discussed, and results are presented which demonstrate the validity of this approach. © 2007 Elsevier B.V. All rights reserved. Keywords: Aggregates; Particle size, shape and angularity; Laser triangulation; Mathematical morphology
1. Introduction The application of image analysis techniques to the study of aggregate shape properties has become increasingly prevalent in recent years. It has long been acknowledged that the size distributions, form, angularity and surface texture of aggregates significantly impacts on the performance of concrete and asphalt mixtures. It is also generally accepted that the techniques currently used in industry to determine these properties are at best somewhat laborious. Furthermore, they are often subjective and provide only indirect measures of the properties of interest. Several authors have identified the ambiguities associated with particle sieve sizes and the corresponding dependent measures ⁎ Corresponding author. E-mail address: [email protected] (M.L. Smith). 0013-7952/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2007.02.003
of elongation and flakiness (Persson, 1998; Kwan et al., 1999; Mora and Kwan, 2000; Lanaro and Tolppanen, 2002). The methods used to determine angularity are similarly limited (Wilson et al., 1997; Masad and Button, 2000). For example, the traditional measure described in BS812 assumes a causal relationship with packing density, defining an angularity number as the amount by which the proportion of solid volume in a packing density test falls below 67%. This somewhat arbitrary figure is thought to represent an optimal packing density given a sample of equally sized, perfectly spherical particles. However, it is hard to see how variations in size distribution, elongation and flakiness would not have equally marked effects on packing density as variations in angularity. It is not then surprising that researchers and industry continue to look to image processing technologies for alternatives. One of the major difficulties encountered
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here is that of reconciling the results of an image-based analysis with those of the traditional industry techniques. For example, having developed a direct measure of angularity or surface texture using image analysis, one can only validate it using the indirect measures that one believed to be flawed or subjective in the first place! The problem is compounded when the properties of interest are arbitrarily defined. This applies particularly to the measurement of particle angularity, or conversely roundness. The European standard EN 933-5 describes how angularity should be determined by visually assessing what percentage of the particle surface is considered to be crushed or broken. This will yield different results from the Wadell approach of examining the relative radii of corners (Wadell, 1935) or from comparison with the similarly derived Powers scale (Powers, 1953), and different again from the results of a packing density test as described in the previous paragraph. Similarly, research into image-based analysis of aggregate properties has generally attempted to approximate angularity with the measurement of other morphological properties. Several researchers have attempted to assess angularity using measures of compactness, by calculating the perimeter-to-area ratio (in two dimensions) or the surface area-to-volume ratio (in three dimensions) of the particle (Wilson et al., 1997; Mora and Kwan, 2000; Masad and Button, 2000; Masad et al., 2001; Manohar and Sridhar, 2001; Pons et al., 2002). The rationale here is that the circle (or in three dimensions, the sphere) is the most compact shape, in that it has the shortest perimeter for a given area. As the shape becomes more angular the perimeter-to-area ratio increases. Masad and Button (2000) use this method at low and high resolutions to gain a measure of both angularity and surface texture, on the premise that the perimeter is more sensitive to variations in angularity at low resolution and to variations in texture at high resolution. Wilson et al. (1997) attempt to compensate for the effects of aspect ratio on compactness by comparing perimeter-to-area ratios with an equivalent ellipse rather than a circle. Masad et al. (2001) also use the equivalent ellipse to provide a normalised measure of angularity. They propose an angularity index that compares the radii of the particle to those of an equivalent ellipse at a fixed number of angular increments, a technique that proved useful in predicting the rutting resistance performance of a hot-mix asphalt. The fundamental limitation of these approaches is that compactness and angularity are two different properties. Whilst there is undoubtedly some correlation, a shape does not necessarily become more angular as it becomes less compact. The validity of the measure further degrades when one considers that the
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particle boundary is generally represented using square pixels — a common problem with image-based measures of compactness is that in many cases they will actually return a square, or in three dimensions a cube, as the most compact shape (Bribiesca, 1997; Bribiesca, 2000). Another approach is to compare the overall shape of the particle to that of a reference shape. Lanaro and Tolppanen (2002) compare the frequency spectra of the cross-sectional profiles of coarse aggregate particles to a number of geometrically simple reference shapes, whilst Podczeck (1997) used measures of deviation from squares, triangles and circles in order to characterise powder particle morphology. Whilst these techniques are useful in determining and representing the underlying shape of a particle, it is not clear how useful they are in providing explicit measures of properties such as angularity. The most intuitive method would appear to be the mathematical morphology approach taken by Masad and Button (2000), in that the resistance of a particle boundary to a series of erosion and dilation operations is directly dependent on the sharpness of the edges and corners. The primary weakness of this approach is that it lacks rotational invariance, in that the use of a simple square structuring element means that an image of a square particle with sides aligned with the axes of the grid would lose no area whatsoever. Kalliomäki et al. (2005) recently developed a more significant mathematical morphology algorithm that uses progressive erosions using circular structuring elements of increasing diameter to the same effect. In this paper we extend the mathematical morphology approach to three dimensions and show how scale, rotation and orientation independence can be attained by replacing the square structuring element with an ellipsoid that is adaptively determined by the size and aspect ratios of the particle. We discuss the relative merits of two-dimensional and three-dimensional imaging in the case of coarse aggregate and present an effective means of 3D data acquisition. In addition to the angularity algorithm we show how size and form can be recovered in a relatively straightforward manner. The limitations of this approach are discussed and opportunities for further work are identified. 1.1. Context The overall aim of this research is to establish technologies and methodologies that will enable development of an automated petrographic analysis system for aggregate. Such a system would be capable of performing many of the geological tests that are currently performed in laboratories. The analysis of the morphologies and
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sizes of rocks, which is the subject of this paper, would form an important function of the device; however it would also perform a number of further machine vision tests that relate to the engineering properties of the aggregate. This would include a spectrographic analysis of the particles, since the mineralogical composition of aggregate materials is critical for engineering applications such as concrete production where resistance to chemical alteration must be considered. The intention is for the final system to be mobile, so that petrographic analysis could be performed on location at a quarry. The availability of an online petrographic analysis system is expected to provide aggregate producers with a number of benefits, including reduced costs and very significant reductions in the time required for the rock analysis. 2. 2D or 3D analysis? Much of the research to date has concentrated on the analysis of projected particle boundaries, i.e. one attempts to make deductions on the shape of a particle from its silhouette. The first problem encountered here concerns the measurement of the overall form of the particle. In order to provide robust data on size, elongation and flakiness one must recover the three principal dimensions of the particle. Clearly this is not possible from a single two-dimensional projection. As such, a number of researchers have devised innovative ways around this. Kwan et al. (1999), Mora and Kwan (2000) supplement the visual information obtained with the weight of an aggregate sample in order to derive a ‘mean thickness’ for the group of particles, which can be used to give a measure of the average flakiness of the sample. Stereological methods are also widely used to statistically deduce three-dimensional particle size distributions from two-dimensional analyses (Sahagian and Proussevitch, 1998; Coster and Chermant, 2001). However, these are best applied to a random crosssection (thin section) through the material sample rather than to the projected perimeters under discussion here. Brzezicki and Kasperkiewicz, 1999 developed a system whereby particles are analysed on a curved corrugated grid, or more specifically a ‘fragment of a cylindrical surface with a series of parallel indentations’. Two light sources are arranged in such a way that the camera captures two perpendicularly projected shadows of each rock, allowing flakiness as well as elongation to be calculated directly. Maerz (1998), Maerz and Lusher (2001) use two orthogonal cameras to capture both side and overhead views of rocks moving along a conveyor, a development that has been implemented in commercial applications. This also allows direct calculation of
elongation and flakiness on the assumption that a particle will rest in its most stable position — i.e. the height observed in the side elevation should represent the smallest dimension of the particle. Whilst these approaches provide effective means of determining three-dimensional form to varying extents, boundary analysis approaches suffer from far more fundamental limitations when one attempts to measure more complex shape properties such as angularity and surface texture. The most dangerous assumption in these cases is that the observed planar projection represents a genuine cross-section of the particle. In the case of a simple three-dimensional object such as a sphere, this assumption will hold. Assuming a parallel projection, the projected boundary of a sphere will always be its cross-sectional great circle. However in the case of more complex objects this is no longer the case. If one considers a drill bit for example, a silhouette projected along the axis of the bit will show a circle. A silhouette projected orthogonally to the axis of the drill bit will show a shape close to a rectangle. There is no possible projection that will reveal the complex, helical nature of the bit. This is because rather than viewing a genuine cross-section of the object, we are viewing multiple overlapping, partially occluded crosssections. The same applies to aggregate particles, particularly the more angular specimens, where the effect is typically to oversimplify the particle shape. As such, there is a growing recognition of the increased level of robustness afforded by three-dimensional data acquisition. Garboczi (2002), Garboczi and Bullard (2004) use X-ray tomography on concrete and cement samples to recover complete three-dimensional data on the constituent aggregates, allowing a more direct comparison between the morphological properties of the aggregates and the performance of the composite. Lanaro and Tolppanen (2002) use laser triangulation, scanning aggregate particles from multiple viewpoints in order to recover the entire surface. The authors acknowledge that their technique cannot be extended to industrial quality control in its current form, suggesting instead that scanning could be limited to the upper hemispheres of aggregate particles moving along a conveyor belt given that the upper and lower hemispheres ‘usually exhibit the same geometrical properties’. This is similar to the approach developed herein. We consider that acquiring 3D information from the visible surfaces of passing aggregate particles offers a useful compromise between the two-dimensional approach which offers simplicity but lacks rigour, and the full three-dimensional approach which offers robustness but lacks efficiency. In the following section we present an overview of the laser triangulation system before discussing the processing of the acquired data.
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3. Acquisition Laser triangulation is used to recover three-dimensional data from the surfaces of the aggregate particles. A laser line is projected vertically down onto a conveyor belt such that the line is perpendicular to the direction of motion, as illustrated in Fig. 1. The camera will then capture an image of the laser line as it intersects the belt or a passing particle. The image of the laser line is distorted by the profile of the rock particle, as illustrated in Fig. 2. Trigonometry is used to relate the observed position of the laser line in the acquired image to the corresponding impact position of the laser line in 3D space. The mathematics differ slightly from the traditional laser triangulation derivation (see for example Besl, 1998; Amann et al., 2001), in that we wish to define a coordinate system with its origin at the interception point of the camera axis, the conveyor belt and the laser line. The plane of laser light seen in Fig. 1 thus becomes the yz plane in the coordinate system, with the z-axis rising vertically upwards and the y-axis lying on the conveyor belt and perpendicular to the direction of motion. This simplifies the analysis, as we know that every point we capture lies in this plane. The remaining x-coordinate is then simply the relative position of the conveyor belt when the image was acquired. We describe the positions of pixels in the image using a second coordinate system, where i and j are the horizontal and vertical coordinates respectively and the origin lies at the optical centre of the image. We first consider the calculation of z-values, by looking at the imaging system in the xz plane. This is shown in Fig. 3. The known parameters are f (the focal length of the camera lens), θ (the angle formed between the camera
Fig. 2. Captured image of laser impact position on particle surface.
axis and the laser plane), d (the distance between the lens and the conveyor belt along the camera axis) and j (The vertical position of any given point on the laser line in the image). The parameter u describes the distance between the laser impact position and the camera lens and v describes the distance between the corresponding point on the image plane and the camera lens. From the sine rule it can be shown that: z¼
dsinh sinð180−h−/Þ
ð1Þ
Where: tan/ ¼
j f
ð2Þ
In order to calculate the corresponding y-coordinate of a given point, the analysis is as shown in Fig. 4. From Fig. 3 and the sine rule, we know that: u¼
dsinh sinð180−h−/Þ
And from Pythagoras' theorem: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ f 2 þ j2
ð3Þ
ð4Þ
Then from the similar triangles shown in Fig. 4 we can deduce that: u y¼ i v
Fig. 1. Laser diode and camera position.
ð5Þ
Eqs. (1)–(5) thus allow us to describe the acquired cross-section of a particle in terms of a set of 3D coordinates, where the x-coordinate is determined by the incremental distance moved by the conveyor belt between images being captured. By concatenating these cross-sections we are able to generate a dense set of coordinates describing the entire visible surface of the particle. In practice, in order to avoid regions of
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Fig. 3. z-value analysis.
occlusion where points on the surface are not visible to both the camera and the laser, a second camera is added to the system. This is placed symmetrically opposite to the first camera about the laser plane, as shown in Fig. 5. The surface data captured by each camera is then merged to provide a complete description of the upper hemisphere of the particle (clearly the underside of the
particle will remain invisible to both cameras). This is then most conveniently represented as a range intensity image, as illustrated in Fig. 6. The image is constructed in the xy plane, as if the rock was being viewed from above. The intensity of the image at any given point is a function of the height of the particle at that point. In this case each pixel, and each unit change in intensity value, represents 0.1 mm. This is an efficient way to represent our acquired data as the range intensity image uses no more memory than a standard 2D image, whilst containing genuine 3D data.
Fig. 4. y-value analysis.
Fig. 5. Prototype triangulation system.
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Fig. 8. Determination of minimum bounding box.
Fig. 6. Range intensity image of coarse aggregate particle.
This representation will provide the foundations for all our subsequent analysis. 4. Processing 4.1. Determination of size and form Having acquired the surface data from a particle, the first processing requirement is to determine the length, width and height of the aggregate particle. As we know nothing of the underside of the rock we have to use the assumption that the rock will rest in its most stable position and so, in line with the approach of Maerz
(1998), Maerz and Zhou (1999), Maerz and Lusher (2001), we measure height orthogonally upwards from the belt. The system is calibrated such that the conveyor belt is the reference height level of 0, so we simply record the maximum z-value obtained from the particle. As the height is as such orthogonal to the image plane, the recovery of length and width can be reduced to a two-dimensional problem. A threshold operation is first applied to the range intensity image in order to segment the particle from the background, thus reducing the data to a binary representation as shown in Fig. 7. Length and width can then be determined by rotating the image to find the minimum bounding box, as illustrated in Fig. 8. Given that the length, width and height of the particle have now been recovered, measures of elongation and
Fig. 7. Original range image (left) and result of threshold at conveyor belt height level (right).
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flakiness can be expressed as individual aspect ratios as per previously published literature. 4.2. Determination of angularity The two basic operations of mathematical morphology are erosion and dilation. The erosion of an image f (i, j) with a structuring element g(x,y) involves translating the structuring element to every point on the image and taking a ‘minimum of differences’, thus: ð f HgÞði; jÞ ¼ minf f ði−x; j−yÞ −gðx; yÞ : ði−x; j−yÞ; ðx; yÞaZ 2 g
ð6Þ
Similarly, for a dilation we translate the structuring element to every point on the image and take a ‘maximum of sums’, thus: ð f PgÞði; jÞ ¼ maxf f ði−x; j−yÞ þgðx; yÞ : ði−x; j−yÞ; ðx; yÞaZ 2 g
ð7Þ
A morphological opening consists of an erosion followed by a dilation, and can be expressed as ( f Θ g) ⊕ g This is the operation used by Masad and Button (2000) in the previous mathematical morphology approach to particle analysis. It should however be noted that these expressions, which describe the actions of a multi-level structuring element on a multi-level image, differ from those typically used to describe the binary morphological operations used in the research mentioned above. The concepts of multi-level or greyscale mathematical morphology, as described in the early authoritative literature (Serra, 1982), are based around the visualisation of both the image and the structuring element as three-dimensional surfaces. As such, a standard intensity image, denoted as a function f (x,y) in twodimensional space, is considered in greyscale morphology to be a set of points [x,y, f (x,y)]. In this case, as the grey levels in our range-intensity image represent genuine height data, this can of course be interpreted more literally. Thus, performing a morphological opening using a structuring element with grey values representing those of a hemisphere can be considered as analogous to rolling a ball all over the underside of the surface represented by the pixel values of the image. If the surface is smooth with relatively low curvature, the ball will make contact with the surface at every point. However, the ball will be unable to make contact with surface points contained within protrusions of high curvature, such as sharp edges and corners. The result of the morphological opening is then the union of every path that the ball may follow, or more formally the union
of the translations of the hemispherical structuring element to every point on the surface. This is illustrated by Fig. 9. This particular morphological opening was originally described by Sternberg (1986) as a means of smoothing discontinuities and removing noise from standard greyscale images. In this case it is suggested that the morphological opening of a range-intensity image could provide an indication of particle angularity, given the selection of an appropriate structuring element, by analysis of the proportion of particle volume lost to the procedure. This effectively simulates the wear processes by which aggregate particles become rounded, by reducing each to a well-rounded state. It should be noted that only convex features are modified by the morphological opening, any concavities on the surface of the particle remain unaltered. The selection of a suitable structuring element is now considered. Firstly, recall that we wish to quantify angularity by analysing the proportion of the entire particle volume lost to the operation, and that angularity is traditionally measured as the sharpness of edges and corners relative to the overall size of the particle (Wadell, 1935; Powers, 1953). Intuitively then, the size of the structuring element needs to be adaptively determined according to the size of the particle under analysis. Secondly, the Powers/Wadell approach describes the
Fig. 9. a — Cross section of image as surface. b — Application of morphological opening. Regions shown in black are inaccessible. c — Result of operation.
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measurement of the angularity of a corner with respect to the maximum inscribed circle in a given projection. A spherical structuring element will therefore be unsuitable unless the particle has neutral elongation and flakiness ratios, as a circle of the correct size for one projection may be inappropriate for another. As such, an ellipsoidal structuring element with aspect ratios equal to those of the particle is used. The principal axes of the structuring element remain aligned with the principal axes of the
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particle throughout the operation, thereby ensuring proportionality from any two-dimensional projection. Experimental work has shown that a structuring element with principal axes 1/3rd of the length of those of the particle gives optimal results. Fig. 10a to f shows line profiles taken from rangeintensity images of gravel particles exhibiting various levels of angularity. The y-axes of the graphs represent the height of the particle and the x-axes represent an
Fig. 10.
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Fig. 11. Average percentage volume loss vs. angularity number.
arbitrary line across the image plane. The values on the axes are arbitrary as the images were resized prior to the operation. The black region shows volume lost due to the morphological opening. It should be noted that the volume loss does not necessarily correspond entirely to the profile shown. Corners and edges that are not visible in these projections will still have an impact as this is a three-dimensional operation. However, it can clearly be seen that there is a broad correlation between the angularity of the particles and the proportions of volume lost. 5. Results and discussion The measurement of particle size was validated manually using callipers. It was demonstrated that the system is capable of measuring the three principal dimensions of a particle to within ±0.1 mm. This was the intended accuracy and is entirely a function of the design, i.e. the resolution of the camera, the range of particle sizes and hence the magnification used, the quality of the laser and the manufacturing tolerances of the conveyor belt and drive equipment. Elongation and flakiness ratios can implicitly be calculated with similar precision. In some cases a lower resolution may suffice, for example when measuring angularity the range-intensity image is rescaled to a resolution of 0.5 mm/pixel. This provides sufficient resolution for the analysis of angularity whilst removing the effects of textural features and improving the efficiency of the processing. The measurement of angularity, as discussed earlier, is more difficult to validate due to the subjectivity of the manual classification. A test sample consisting of two hundred morphologically diverse gravel particles in the 8–32 mm size interval was graded against the Powers scale by geologists at the Slovenian National Building and Civil Engineering Institute (ZAG), Ljubljana and Petromodel ehf, Reykjavik. The particles were then
passed through the laser triangulation system and the ellipsoidal morphological opening was applied. Fig. 11 shows the average percentage volume loss per particle plotted against the angularity value manually determined using the Powers technique. Note that 1 represents the ‘very angular’ category and 6 represents the ‘well rounded’ category. Fig. 11 shows that there is a reasonable linear relationship between the volume lost due to the ellipsoidal morphological opening and the geologists’ classification of particle angularity. One point of concern was that the variance in the results appears to be very high, as shown in Fig. 12. It appears that the standard deviation increases almost linearly with increasing angularity. This suggests that the deviations can be explained by the difficulties in objectively classifying particle angularity, as it is far easier to distinguish between a well rounded particle and a rounded particle than it is to distinguish between an angular particle and a very angular particle (Powers, 1953). It was initially hoped that the system would be able to recover surface texture along with the size, form and angularity of the aggregate particles. This however has proved to be difficult due to laser speckle noise at the acquisition stage. Speckle noise is a phenomenon whereby coherent light hitting a surface is reflected back from varying distances. The interaction of the reflected light can lead to both constructive and destructive interference, the result being that the projected pattern consists of bright and dark regions. This causes problems in the recovery of 3D data, as the brightest point in the cross section of the laser line will no longer necessarily correspond to the centre of the line. Thus this translates into noise in the recovered surface data. In the case of aggregate particles, this problem is exacerbated for two reasons. The influence of laser speckle would usually be reduced by using a narrower
Fig. 12. SD of average percentage volume loss vs. angularity number.
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laser line width. However, when using focusable laser diodes there is a trade-off between the required depth of field and the minimum attainable laser line width. As such, a line width fine enough to recover the surface texture of an aggregate particle would not remain in focus over the depth of fields necessary to recover variations in form. In the case of a fixed laser line scanning material passing along a conveyor belt, the required depth of field extends from the impact position of the line on the belt to the height of the largest particle one expects to encounter. This limitation is compounded by the range of colours and reflectances encountered in construction aggregates, as the apparent width of the laser line in the acquired images is also dependent on the aperture settings of the cameras. Thus an aperture setting which results in images of fine lines on black particles will also result in images of very coarse lines on white particles. The resultant effect of all these issues is that the magnitude of the surface noise caused by laser speckle exceeds the magnitude of the surface variations due to surface texture, making quantification of surface texture fundamentally difficult. Lanaro and Tolppanen (2002) avoid this problem when using a laser to scan aggregate material as the scanning equipment is mobile, thus reducing the depth of field constraint, however as discussed previously this makes application in an industrial environment difficult. Possible solutions included the use of laser speckle reduction hardware or the use of a broadband white light source. However, the surface texture of coarse aggregate is considered to be less critical than particle size distributions, form and angularity to the performance of concrete and asphalt. 6. Conclusions The research described in this paper has shown how laser triangulation can be applied effectively to recover three-dimensional information from the upper surfaces of aggregate particles moving along a conveyor belt. Particle size, form and angularity can be quantified accurately from the acquired data. This approach offers a useful compromise between the accuracy limitations of two-dimensional image analysis and the application limitations of a full three-dimensional acquisition. Simple techniques for the recovery of particle size and form have been described and a new algorithm for the quantification of particle angularity has been presented. The ellipsoidal morphological opening is geometrically intuitive and allows angularity to be quantified directly rather than by inference from other
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properties. Results demonstrate the validity of this method. This approach offers potential for useful application to real time quality monitoring and control in industrial environments. The research activities were undertaken as part of a broader collaborative project which ultimately aims to combine morphological data with mineralogical information obtained using spectroscopy, thus providing a fast, accurate and complete analysis of the fundamental properties of aggregate materials. 7. Future work The primary aim should be to develop a means of simultaneously quantifying surface texture in addition to size, form and angularity from the acquired 3D data. As discussed in the previous paragraphs, this will involve investigating alternative means of structured light projection that may alleviate the limitations imposed by laser speckle noise. The effects of colour variations encountered in aggregate materials, and the refractive nature of certain minerals, on the recovery of surface texture should also be investigated. The other challenge is to increase the throughput of the device. At present particles are analysed as they pass underneath the laser and cameras in single file. The development of a system capable of extracting rigorous morphological information from material passing in bulk would bring the technology a step closer to widespread industry acceptance. Acknowledgements The authors wish to express their gratitude to Petromodel (Iceland) ehf., RMC Aggregates (UK) Ltd. and the Slovenian National Building and Civil Engineering Institute (ZAG) for their collaboration and support in this research. References Amann, M.-C., Bosch, T., Lescure, M., Myllylä, R., Rioux, M., 2001. Laser ranging: a critical review of usual techniques for distance measurement. Optical Engineering 40 (1). Besl, P.J., 1998. Active, optical range imaging sensors. Machine Vision and Applications 1, 127–152. Bribiesca, E., 1997. Measuring 2-D shape compactness using the contact perimeter. Computers and Mathematics with Applications 33 (11), 1–9. Bribiesca, E., 2000. A measure of compactness for 3D shapes. Computers and Mathematics with Applications 40, 1275–1284. Brzezicki, J.M., Kasperkiewicz, J., 1999. Automatic image analysis in evaluation of aggregate shape. Journal of Computing in Civil Engineering 123–128 (April).
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