Abstract
X-ray micro-tomography (XMT) is increasingly used for the quantitative analysis of the volumes of features within the 3D images. As with any measurement, there will be error and uncertainty associated with these measurements. In this paper a method for quantifying both the systematic and random components of this error in the measured volume is presented. The systematic error is the offset between the actual and measured volume which is consistent between different measurements and can therefore be eliminated by appropriate calibration. In XMT measurements this is often caused by an inappropriate threshold value. The random error is not associated with any systematic offset in the measured volume and could be caused, for instance, by variations in the location of the specific object relative to the voxel grid. It can be eliminated by repeated measurements. It was found that both the systematic and random components of the error are a strong function of the size of the object measured relative to the voxel size. The relative error in the volume was found to follow approximately a power law relationship with the volume of the object, but with an exponent that implied, unexpectedly, that the relative error was proportional to the radius of the object for small objects, though the exponent did imply that the relative error was approximately proportional to the surface area of the object for larger objects. In an example application involving the size of mineral grains in an ore sample, the uncertainty associated with the random error in the volume is larger than the object itself for objects smaller than about 8 voxels and is greater than 10% for any object smaller than about 260 voxels. A methodology is presented for reducing the random error by combining the results from either multiple scans of the same object or scans of multiple similar objects, with an uncertainty of less than 5% requiring 12 objects of 100 voxels or 600 objects of 4 voxels. As the systematic error in a measurement cannot be eliminated by combining the results from multiple measurements, this paper introduces a procedure for using volume standards to reduce the systematic error, especially for smaller objects where the relative error is larger.
Highlights
X-ray micro-tomography (XMT) is a popular technique for the non-destructive qualitative and quantitative investigation of the internal structure of objects
The boundaries of features rarely coincide with the boundaries of the regular voxel grid, leading to the “partial volume effect” at the interface, where voxels have an intermediate bulk composition, and there is some uncertainty in the exact boundary location (Ketcham and Carlson, 2001; Stock, 1999)
The systematic error in the grain volume will come about from effects such as an error in the threshold used, while the random error will come about due to effects such as the change in the partial volume effect due to the specific location of the mineral grain relative to the voxel grid, which will change from scan to scan and from grain to grain
Summary
X-ray micro-tomography (XMT) is a popular technique for the non-destructive qualitative and quantitative investigation of the internal structure of objects It has been widely applied across material science (Puncreobutr et al, 2012; Stock, 1999), engineering (Aydoğan et al, 2006; Ghorbani et al, 2011; Ketcham and Carlson, 2001) and biological sciences (Yue et al, 2011) to provide quantitative data about the structure and morphology of 3D objects and features within them (crystals, pores, fractures etc.). The detector size was 2000 Â 2000 pixels, giving a linear resolution of approximately 17 μm for the magnification selected We chose this example as there are a large number of mineral grains within the image volume and these grains are known to have a wide volume distribution. The first section of this paper gives a short description of this algorithm as the data generated from it is the source of the statistical analysis
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