New approach to border line evaluations for whole sample of Williams pear (Pyrus communis) Nebojša Dedovic ´ ⇑ , Snez ˇana Matic ´ -Kekic ´ , Ondrej Ponjic ˇan, Branislav Karadz ˇic ´ University of Novi Sad, Faculty of Agriculture, Department of Agricultural Engineering, Square Dositeja Obradovic ´a 8, 21000 Novi Sad, Serbia article info Article history: Received 13 May 2011 Received in revised form 25 July 2011 Accepted 8 August 2011 Keywords: Cubic spline New approach Nonlinear regression Shape variability Surface area Volume abstract In many processes of heat exchange, as well as in other processes of biomaterial handling, physical prop- erties of fruit such as dimensions, shape, surface area and volume are of crucial importance. The objective of this work was to find a function which approximates pear border line, as precisely as possible, in order to calculate the surface area and volume of a pear by means of integral calculus. Previously described esti- mation of an average pear border line was based on the sixth order polynomial and proposed algorithm. Two new ways to calculate the Williams pear border line are the focus of this study. The first way includes spline functions as an estimation of a pear border line (i.e. four third order polynomials), while the second one uses regression function obtained by nonlinear regression method. The regression func- tion has two independent variables, length and total length of a pear. The most precise approximation of a pear border line was obtained by nonlinear regression with R 2 = 97.48. This was more obvious when total pear length was smaller or greater than average total pear length. Border lines of all tested pears were determined by one regression function with large precision. Therefore, it is safe to calculate surface area and volume of a pear based on regression function and total pear length, only. Calculated volumes of the pears were compared with volumes measured by Archimedes’ method. The smallest relative error has been obtained when the volumes were calculated using the regression function as approximation of pear border line. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Various factors influence the growth and development of bio- materials. Therefore, shape variability is very important and should be examined. Hiraoka and Kuramoto (2004) suggest that it is pos- sible to identify cultivars based on contour shape using elliptic Fourier descriptors. The elliptic Fourier descriptors (Kuhl and Giar- diana, 1982) are invariant with respect to rotation, dilation, and translation of the contour, without losing the information about the shape of contour. This method has been successfully applied to different biological shapes evaluation (Iwata et al., 2002; Hir- aoka and Kuramoto, 2004; Yoshioka et al., 2004; Severa et al., 2011), sperm shape evaluation (Severa et al., 2010), and in proce- dure which simplifies and improves plant organs representation (Mebatsion et al., 2011). Principal component analysis and elliptic Fourier coefficients can extract the independent shape characteris- tics (Severa et al., 2010). Physical properties of fruit (dimensions, shape, surface area and volume) have very important role in many processes of heat exchange and other processes of biomaterial handling (Mohsenin, 1980). Moreover, during the drying process, the heat energy de- pends partly on the fruit surface area (Mišljenovic ´ et al., 2010). It is well known that fruits, including pears, are dominantly irregular in shape. Certain number of measurements must be made for full characterization of fruit shape. Analysis of three mutually perpen- dicular axes usually contains enough information for volume or surface area modeling. All three dimensions of a fruit can be measured by hand with calipers, or by tracing, using a photo- graphic enlarger (Mohsenin, 1980). Both methods were used in determination of the surface area of a peach palm (Bovi and Spier- ing, 2002). Fundamental mechanical properties of a pear tissue were examined by Wang (2004), as well as pear firmness during the drop impact (Wang et al., 2007). The finite element method was used to discretize the governing differential equations over the actual 3D pear geometry (Wang et al., 2006). Pear dimensions were evaluated during the drying process and those data were used to calculate the pear surface area and volume (Guine et al., 2006). Cut pears were photographed horizontally and vertically against a millimeter-scaled paper. The shape of the whole pear was replaced with two regular bodies, half of a sphere and a cone. The dimensions and volume were also investigated for almond cul- tivars (Altuntas et al., 2010), cherries (Ochoa et al., 2007) and man- go (Spreer and Müller, 2011). If color is an important factor, then 0168-1699/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.compag.2011.08.003 ⇑ Corresponding author. Tel.: +381 214853292; fax: +381 21459989. E-mail addresses: dedovicn@polj.uns.ac.rs (N. Dedovic ´), snmk@polj.uns.ac.rs (S. Matic ´ -Kekic ´), ponio@polj.uns.ac.rs (O. Ponjic ˇan), karadzic@polj.uns.ac.rs (B. Kar- adz ˇic ´). Computers and Electronics in Agriculture 79 (2011) 94–101 Contents lists available at SciVerse ScienceDirect Computers and Electronics in Agriculture journal homepage: www.elsevier.com/locate/compag