# Method for Volume of Irregular Shape Pellets Estimation Using 2D Imaging Measurement
## 2 Materials and Methods  
The amount of data received from analysis of the digital image processing is limited for determining the distribution of pellets. Depending on the material being analyzed, pellets can be of irregular shapes, and any approximation of such pellets may largely differ from their real surface. Two-dimensional (2D) image information always provides only two parameters of pellet size: length and width. Therefore, the precision of evaluation of its volume depends on the level of matching between the approximation model and real surface of a pellet. The contours of pellets are usually approximated using geometric forms that are most widely used in calculations, such as circles, ellipses or best fit rectangle (smallest area limited by rectangle that can define the shape of pellet) [33]. These models are used for defining the contour of a pellet, as well as drawing the largest possible shape (limited by the perimeter of a pellet) inside of a contour of a pellet (Figure 1). Such forms of approximation are characterized by limitedness, as in the case of an irregular pellet shape the area limited by its contour can be up to two times larger than the real area of the pellet (Figure 2a). This approximation form is the reason for discrepancy between measured volume and real measurement results. To account for such discrepancies, a model of the area (matching real area of a pellet) limited by the regular geometric shape is used (Figure 2b). Such methods of pellet shape evaluation are used with silhouette contour approximation using ellipses [34], because this geometric shape allows for better evaluation of roundness of the pellet, when compared to a circle shape. The ellipse contour provides information about longitudinal and transverse dimensions. This method of measurement is similar to the sieve method. In many laboratories, sets of sieves are used as a control equipment of examination. In this case, when distribution of pellets must be determined, the main parameter is the width of the ellipse that limits the area of the pellet. The width of the ellipse is the same as the diameter of the holes of the sieve. Matching between the holes of a sieve and the approximated form is presented in Figure 3.  
Figure 1: Models of pellet approximation: (**a,b**)—p pellet is approximated using a circle drawn around it and a circle drawn inside of it; (**c,d**)—p pellet is approximated using best fit rectangle drawn around it and rectangle drawn inside of it; (**e,f**)—p pellet is approximated using a ellipse drawn around it and a ellipse drawn inside of it. The sieve method used for the measuring of pellets is not a rational choice for evaluation of parameters of their shape. This is related to the working principle of sieves, which only allows for evaluation of two out of three dimensions of pellet, as the pellet falls through sieve when two of its three transverse dimensions are smaller than the diameter of holes of the sieves. In such a case, a third diameter as well as shape of the pellet is left unmeasured. The precision of measuring is also limited by the number of sieves with holes of different diameters. The technique of processing digital images expands the measuring possibilities, while more precise mathematical models used for pellet approximation conditions the improvement of precision and reliability. The correction of the final distribution of the pellets is used as an alternative, e.g., an evaluation carried out only with pellets which are shaped very similarly to the regular shape of a pellet [35]. During the indirect measuring, based on digital image processing, various geometric shapes are used for the approximation of pellets, according to the area limited by their silhouette. One of most common geometric shapes is volume limited by ellipsoid Equation (1) (according to Figure 1e) [36]:  
\[V=\frac{4}{3}\pi r_{1}r_{2}\frac{d_{2}}{2}, \tag{1}\]  
where \(r_{1}\), \(r_{2}\)--radius of ellipsoid; \(d_{2}\)--length of ellipsoid.