In this study, 5 international (Midland, Serr, Pedro, Howard and Chandler) and 10 national walnut (Juglans regia L.) cultivars (Sütyemez-1, Maraş-18, Maraş-12, Sütyemez-2, Kaman-1, Kaman-5, Şebin, Şen-2, Bilecik and KR-1) were used as the plant material (Table 1). The walnut cultivars were provided from the Nuts Research and Application Center of Kahramanmaraş Sütçü İmam University in the Kahramanmaraş province of Turkey.

In order to reveal the size and shape attributes of walnut cultivars, an imaging system, details of which were explained by Kara et al., (2013), consisting of a digital camera (Nikon D300, JP) and illumination equipment (paraflashes with softbox) was used. The digital images of the walnut cultivars were taken in a dark room. The camera was fixed on a frame in a position perpendicular to a white-colored fiberglass surface at a constant height of 50.5 cm. The walnut samples were kept in a fridge (−4 °C) until subsequent analyses. For each of the cultivars, 100 samples were randomly selected from a box where they were kept. Firstly, a digital balance (Shimadzu TW423L Model, JP) (± 0.001 g) was used to determine fruit mass. After that, the walnut samples were placed on a transparent fiberglass plate in a matrix form of 3×6 at two orientations, namely horizontal and vertical. Putty was used to position the walnuts on the fiberglass plates. A ruler on the fiberglass surface centered at the bottom of the image area surface was used to convert the unit from pixel to millimeter. The conversion ratio was found as 15.54-pixel·mm^-1. A cable release was used to prevent vibration during the imaging. For the moisture content of the walnut samples used in present study, 20 samples randomly selected from each cultivar were dried at 105 °C for 24 hours and their mass was measured again. The images captured by the camera were saved as a color *.tiff extension image files for descriptive analysis and *.bmp extension image files for the EFA. During the experiments, mean temperature and relative humidity of the laboratory were recorded as 24.0 °C and 24%, respectively.

SigmaScan^®Pro 5.0 software was used in order to process the walnut images, and projected area, perimeter, length, width, thickness, maximum and minimum diameters and aspect ratio variables were determined for each walnut at horizontal and vertical orientations by means of the image processing operation (Figure 1). The variables of shape index, geometric mean diameter, surface area, sphericity and volume were calculated using the equations given in Table 2. A shape index of lower than 1.25 was considered as spherical and a value over 1.25 was considered as oval (Ozkan and Koyuncu, 2005).

The walnut images were saved as color bitmap image files for the EFA. The aim of the EFA was to reveal the shape variation among the walnut cultivars. For this purpose, the outlines of each walnut image were firstly digitized using SHAPE version 1.3 software (Iwata and Ukai 2002). All data were recorded as chain codes to describe the geometrical information about outlines ranging from 0–7. The data obtained from the chain code were normalized using a module of the software based on the ellipse of the first harmonic in order to be independent from orientation, size, or location. In order to describe the outlines of walnut samples, 20 harmonic numbers were used (Iwata and Ukai 2002; Mebatsion et al., 2012; Vasallo et al., 2013).

In this study, approximately 100 walnuts of each cultivar, totaling 1492 walnut samples were used, and their mass, volume, length, width, thickness, geometric mean diameter, surface area, sphericity, shape index values, and projected area, perimeter and aspect ratio values at two orientations were determined as descriptive variables. A correlation matrix created for descriptive data showed the relationships among the variables. The resultant data were subjected to analysis of variance (ANOVA) with a 95% confidence level using SPSS version 20. Differences among the means were determined with Tukey’s comparison test at 5% significance level. In reference to the descriptive variables of walnut cultivars, similar cultivars were determined using a hierarchical cluster analysis. For this analysis, Ward’s method algorithm was used. The cluster analysis was performed by using the squared Euclidean distance method with arithmetic averages of walnut cultivars.

Shape descriptors obtained from EFA were also used to perform principle component analysis (PCA) based on the variance-covariance matrix. The resulting PC scores for each walnut cultivar were used as observed values of shape features. MANOVA was carried out using PAST version 3.01 (Hammer et al., 2001). Wilks’ Lambda and Pillai trace values and p values were determined. The matrix of differences among walnut cultivars was obtained from Hotelling’s pairwise comparisons with Bonferroni correction and squared Mahalanobis distances which make two-cultivar comparisons possible. Differences in walnut outlines were evaluated using a discriminant analysis (Sayinci et al., 2015a). Group centroids of walnut cultivars obtained from discriminant analysis were presented in a scatter plot with canonical discrimination functions.

The correlation matrix is provided in Table 3. Correlation greater than 0.70 were marked. While correlations between mass (M) and the other variables were not found to be significant, volume (V) significantly correlated with the area variables (SA, PA_h and PH_v) of the walnut cultivars. Similarly, the correlations between length and the other dimensional variables (W, T, D_g, P_h and P_v) were not found to be significant. As for the descriptive shape attributes of the walnut cultivars, sphericity and shape index showed a significant negative correlation. The correlations between aspect ratio variables measured at horizontal and vertical orientations (AR_h and AR_v) were not found to be significant.

The mass, volume, surface area and projected area averages are provided in Table 4. While the KR-1 cultivar had the highest gravitational and area averages, the lowest averages were observed for the Serr and Maraş-12 cultivars. Surface area to volume ratio averages varied between 1.42 and 1.89 and the ratio decreased with increasing volumes. Projected area values at two orientations positively correlated with the volume values of the walnuts. However, the projected area averages of the walnut cultivars at two orientations were significantly different. The lowest projected area ratio was observed for the KR-1 and Sütyemez-1 cultivars. The projected area averages measured at the horizontal orientation of the Sütyemez-2 and Serr cultivars were significantly higher than those of vertical orientation. This ratio was considerably close to 1.00 for the KR-1 and Sütyemez-1 cultivars.

The dimensional attributes of the walnut cultivars are provided in Table 5. While the KR-1 and Şen-2 cultivars had the longest dimensions, the Maraş-12 cultivar had the lowest length average. As the width of the walnuts increased, the other dimensional attributes (T, D_g, P_h and P_v) also increased because of the positive correlations among size variables, except for length. As seen from the averages, the KR-1 cultivar also had considerably higher averages in terms of width, thickness, geometric mean diameter and perimeter. In any case, the lowest width and thickness values were observed for the Maraş-12 and Serr cultivars.

The walnut cultivar KR-1 had the closest shape to a sphere with a sphericity average of 96.7% (Table 6). The shape of Serr, Sütyemez-2, Midland and Şen-2 cultivars were described as oval because their shape index average was greater than 1.25, and their sphericity averages ranged between 82.6% and 85.7%. While the aspect ratios determined at the horizontal orientation of the walnuts which had a high sphericity average were low in general; the averages at the vertical orientation varied, regardless of the horizontal orientation.

As seen in Figure 2, the hierarchical cluster analysis divided the walnut cultivars into two main groups. The first main group had two subgroups (shown with green triangle). While the 1^st subgroup was composed of two clusters (cluster 1 and cluster 2), the 2^nd subgroup constituted cluster 3. The second main group consisted of a single walnut cultivar comprised of cluster 4. Consequently, cluster 1 involved six walnut cultivars (Howard, Şebin, Pedro, Bilecik, Midland and Maraş-18), similar to each other with regard to size and descriptive shape attributes. The Maraş-12, Kaman-1 and Serr cultivars were in cluster 2. Cluster 3 had five walnut cultivars (Chandler, Şen-2, Sütyemez-2, Sütyemez-1 and Kaman-5). KR-1, due to different size and descriptive attributes, was grouped separately into the 4^th cluster.

The Elliptic Fourier analysis (EFA) revealed that the first seven variables shown in Figure 3 explained 92.7% of the total variance. The corresponding principle components PC1, PC2, PC3, PC4, PC5, PC6 and PC7 constituted 65.3%, 11.9%, 5.1%, 4.1%, 2.6%, 2.1% and 1.6% of the total variation, respectively. Each significant principle component (PC) was derived from the chain codes of the walnut contours illustrated, regardless of the shape variation in the cultivars. PC1 was the component which explained the greatest variation among the walnut cultivars and showed the width alteration in the walnut cultivars. This alteration resulted in a variation in dimensional and shape attributes such as length, sphericity and aspect ratio. The principle components PC2 - PC7 described the smaller variations in the outlines of the walnut cultivars. For instance, PC2 originated from the swelling of the walnut base. Flatness and tapering at the bottom of the walnut was revealed by PC3. There was a one-sided tapering at the bottom of the walnut at PC4. At PC5, the slight, centripetal tapering was found at the walnut base. At PC6, tapering and flatness form both the top and the bottom of the walnut were determined. PC7, which was the lowest component of total variance, constituted the reason for the flatness at the bottom and tapering at the top of the walnut.

According to results from the discriminant analysis illustrated in Table 7, six canonical discriminant functions were obtained from the significant principle components (PC1-PC7) derived from EFA based on walnut outlines. Discriminant analysis showed that the first seven PC scores explained a major part of the variance among the PCs. The first two canonical functions explained 86.9% of the total variance. In reference to the Wilk’s Lambda and Pillai Trace statistics, significant differences were observed in the shapes of the walnut cultivars. The Hotelling’s pairwise comparison results revealed the distinctions and showed similarities among the outlines of the walnut cultivars. Mahalanobis distances in the lower triangle account for the variance of each walnut outline and the covariance between the cultivars. As the distance increased, the distinctions among the cultivars grew. The lowest distances marked in the lower triangle showed the cultivars which were similar to each other. Bonferroni-corrected P values were given in the upper triangle and the walnut cultivars which were identical were colored as pairwise. The uncolored cells showed the significant (p < 0.01 and p < 0.05) distinctions among the pairwise walnut cultivars.

The results of the Hotelling’s pairwise comparisons were found to be compatible with the scatter plot shown in Figure 4. The scatter plot showed the cultivar centroids with regard to their canonical discriminant functions. The loadings of the first two canonical functions explained 86.9% of the total variance and they were presented as standardized canonical discriminant function coefficients in Table 7. According to these loadings, the canonical function 1 had the highest loading of the PC1, while the canonical function 2 explained the variation caused by PC3, PC6 and PC7. The similarities or distinctions among the cultivars should be evaluated based on the results of the Hotelling’s pairwise comparison in addition to the centroids in the scatter plot. Because Kaman-5 had a different characteristic in shape, it remained alone in its own group. The centroid coordinates shown in Figure 4 indicated that the Howard and Maraş-12 cultivars had different shape attributes, because their coordinates on the function 2 axis were different. However, their coordinates were close to the origin of the discriminant function 1 axis, and the results of the pairwise comparison test determined the similarity between them. Serr and Midland were the cultivars which comparatively stayed on the left of the function 1 axis. Chandler, Pedro, Kaman-1 and Maraş-18 were similar cultivars in shape due to their centralizing at the origin. Sütyemez-2, Bilecik and Şen-2 cultivars together constituted a separate group in shape attributes.