The idea of the present study was exploring more efficient visual indices for wheat MGW prediction, other than 2D grain area. For this purpose, various empirical indices of grain size and shape were evaluated using image processing. It was observed that among the size criteria, the one-dimensional indices of grain width (i.e. Minor and MinFeret) had relatively higher correlations with MGW, compared with the two-dimensional index of grain area, or perimeter (the latter of which was filtered out in the preliminary assessments; R = 0.801 when the enhanced-resolution images were used, data not shown). This observation inspired that there might be also other unexplored indices for MGW, which originate from the exclusive physiology of wheat crop, e.g. the processes associated with the grain filling capacity. Therefore, the correlation of MGW with some of the conventional shape indices and also several empirical criteria were tested.

Area × Circ., Perim. × Circ., and Area/Perim. were the superior indices in prediction of MGW using the linear models, and indicated a relatively consistent performance across the various conditions. Furthermore, almost under every of the 4 environmental conditions, other selected indices could predict MGW with a higher precision compared with area. Besides the applicable aspect of this finding, it is also an evidence for the possibility of improving wheat grain weight estimation by exploring new visual indicators. Based on the formula of the circularity index used in ImageJ (see https://imagej.nih.gov/ij/docs/guide/146-30.html), all of the three superior indices have a common factor i.e. the Area/Perim. ratio:

$$Circularity=frac{4pi times Area}{{Perimeter}^{2}}$$

(1)

$$Areatimes Circularity=Areatimes left(frac{4pi times Area}{{Perimeter}^{2}}right)=4pi times {left(frac{Area}{Perimeter}right)}^{2}$$

(2)

$$Perimetertimes Circularity=Perimetertimes left(frac{4pi times Area}{{Perimeter}^{2}}right)=4pi left(frac{Area}{Perimeter}right)$$

(3)

So, the formulae of the two other indices (i.e. Area × Circ.& Perim. × Circ.) might be slightly simplified, and consequently the computational cost could be reduced. Such conversions may be particularly important in high-throughput phenotyping; where a considerable number of grains should be analyzed in real-time e.g. using high-speed imaging systems. Besides, these observations imply that the majority of the efficient indices evaluated in the present study are based on two fundamental factors: (i) grain width (measured by Minor & MinFeret), and (ii) the Area/Perim. ratio. Of course, additional correlation tests indicated that the Area/Perim. ratio, as the same as other superior indices, had in turn correlated strongly with grain width.

As described before, enhancing the image resolution by the factor of 10 improved the predictive precision of the indices considerably. However, this improvement was not equal for all of the selected indices; as those which were independent of the grain shape, were less influenced (e.g. the size indicators such as Area or MinFeret; see Table 3). In contrast, the shape-depended indices showed considerably higher degrees of improvement in MGW prediction (for instance, see the indices with the factor of Circularity, or even Minor, which is resulted from ellipse fitting; see Fig. 1). Therefore, it is necessary to ensure the desirable image resolutions (which is achievable either at the time of imaging/scanning, or using interpolation), before running the analyses.

Noteworthy, since in the present study the weight analysis was designed and carried out based on the average values, generalization of the findings and models for estimating weight of individual grains might require further assessments. However, considering that each of the 180 samples was consisted of more than 400 grains, it is expected that both types of estimations (i.e. MGW and individual grain weight) should be highly correlated. As an evidence for this fact, it was observed that similar to the study of Kim et al. [20], Kim index provided a more precise grain weight estimation than Area. More importantly, slopes of the corresponding linear models calculated in both studies were almost similar (see Table 3); despite the differences in the genotypes, treatments, imaging systems, lighting, and probably the image processing algorithms:

$$mathrm{Kim, et, al.}: left{begin{array}{l}Weight=left(3.46times Arearight)-15.99 \ Weight=left(27.02times Widthright)-50.48end{array}right.$$

$$mathrm{Present, study}: left{begin{array}{l}Weight=left(3.45times Arearight)-10.50 \ Weight=left(25.75times MinFeretright)-38.21\ Weight=left(26.22times Minorright)-37.43end{array}right.$$

(units: mg, mm2, and mm).

Besides the technical advantageous for developing phenotyping platforms, findings of the present study might also be readily used in wheat physiology and breeding approaches. For instance, the relatively stronger relationship between MGW and grain width (vs. length or even area) may provide valuable implications for the grain development and/or filling processes; particularly despite the fact that (i) grain filling is an acropetal process and mainly occurs in the grain length direction, and (ii) the 2D grain area provides the information of 2 out of the 3 dimensions (so theoretically, it is expected to be a more significant weight contributor compared with the one dimensional traits such as grain width). Moreover, it was evidenced that the superior predictive indices had the highest correlations with grain width, which had even exceeded the same correlations of the indices with their own mathematical components (see Table 4, Fig. 5, and Additional file 1). Therefore, having a frequent and prominent appearance in the present study, grain width seems to be a fundamental and unique trait in grain physiology and weight assessments. The results also seem to be consistent with the findings of Gegas et al. [9] who provided the genetic evidences for an emerging phenotypic model where wheat domestication has transformed a long thin primitive grain to a wider and shorter modern grain. In addition, comparative variations and contribution of the two main axes to grain weight may open new window into the grain development assessments and yield physiology. Indeed, grain length and width might be supposed as the components of weight, or in a more general view, as the subcomponents of wheat grain yield. Conducting sufficient researches, such framework could provide valuable information about the pattern of grain development or filling in the main perpendicular dimensions, particularly under various conditions; e.g. in the present study, post-anthesis water stress (50% of filed capacity) reduced the grain length and width significantly by 1.38% & 5.13%, respectively, in monocultures; which overall led to 8.64% reduction in MGW (Table 5). This suggests that the water stress treatment had affected the grain extension (the interaction of development and filling) along the width direction more considerably than along the grain length. In contrast, the effect of growing season on the grain length was higher than on the grain width (i.e. reduced the respective values in the second year by 2.48% vs. 1.57%, respectively; which resulted in 6.57% MGW reduction). Therefore, it can be concluded that the season had more affected the earlier developmental grain phases (in which the potential of final length is determined), while the post-anthesis water stress had influenced –more considerably- the later phenological stages and filling period (which contributes more to grain width). In a similar way, various pheno-physiological aspects of genetic or environmental effects on the wheat grain might be evaluated more finely in a subcomponent level of grain yield.

In addition to the main applications of the findings reported here, (i.e. grain weight predictions or physiological assessments), the image-derived indices could be used for automated seed screening and grain sorting purposes; e.g. the less-matured grains might be easily detected and filtered out by appropriate thresholding of grain dimensions or predictive indices. Determining the best quantitative thresholds requires further studies. Also, the superior visual indices introduced in the present study might be used as the selection criteria in breeding programs (e.g. see [6]); before which the efficiency and stability of the indices should be tested using a more heterogeneous collection of genotypes grown under a broader environmental conditions. In general, the image-based MGW predictive method reported here, along with the other related applications could increase the speed, accuracy, and frequency (i.e. replication) of crop sampling and grain assessments; which in turn, might reduce the experimental error and improve the agro-physiological evaluations.

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