Mathematical modeling is essential across research fields, allowing for better understanding and prediction of outcomes before they occur. This proactive approach leads to improved decision-making, enhanced crop productionand reduced risks of disasters and losses. In agriculture, which is crucial for human survival and national economies, mathematical models significantly aid in predicting yields, estimating volumes and analyzing key agricultural factors. Numerous studies in the literature demonstrate the impact of these models in agricultural research, highlighting their importance in developing effective strategies for sustainable farming and ensuring food security in an ever-changing environment. The study by
Bakoglu et al., (2016) utilizes non-linear growth models, specifically Logistic, Bertalanffy and Gompertz, to determine the best predictive models for plant length, dry stem and dry leaf weight in various species of bitter vetch. Later,
Karadavut et al., (2017) applied Logistic, Richards and Weibull growth models to assess the growth patterns of 14 bitter vetch genotypes, identifying Richards’s model as the best fit for most genotypes, while highlighting significant variability due to Turkey’s diverse climate and soil conditions. In a study,
Karthiayani and Nithyalakshmi (2020) measured respiration rates of three mango varieties stored at different temperatures to develop a mathematical model for predicting metabolic activity during ripening and storage. In one notable study,
Singh (2022) employed mathematical modeling using central composite design to optimize the performance of producer gas from mustard stalks in a dual fuel engine, assessing its emission impacts, particularly SO
2, under various operational conditions, addressing the issue of stubble burning in Punjab, India. Looking at the importance of mathematical modeling, this study reviews its applications within the fruit industry.
The fruit industry is spreading its roots rapidly on a global scale, driven by heightened recognition of various health advantages linked to the consumption of different fruits
(Mon and ZarAung, 2020). Consumers mainly prefer fruits that rate high based on visual appearances such as shape, color, size and surface texture
(Kilic and Bozokalfa, 2022;
Mokria et al., 2022; Omid et al., 2010; Salmanizadeh et al., 2015; Vivek et al., 2018). Determining the size, volume or mass of fruits holds significance for aligning with consumer preferences and other reasons, including identifying appropriate packaging materials for consistent fruit batches and creating commercial value
(Birania et al., 2022; Pathak et al., 2020; Rosado et al., 2022; Wang et al., 2018). Manual grading is typical and demands significant labor and time, necessitates meticulous sample preparationand proves inadequate for instant grading and sorting on industry level as it is prone to human visual errors
(Concha-Meyer et al., 2018; Mon and ZarAung 2020;
Oo and Aung, 2018). Hence, machine vision approaches have led to prominent development and automation in the packing lines over the past few decades
(Lee et al., 2017; Moreda et al., 2009).
Mathematical modeling based approaches have emerged as promising ways for fruit volume estimation without fruit destruction and loss of crop yield
(Lee et al., 2017). These approaches are commonly geometric or statistical. While many others are based upon tomography
(Arendse et al., 2016) and laser scanning
(Saha et al., 2022). Recently, the authors have reviewed the importance of mathematical modeling for crop yield prediction
(Bamel et al., 2022; Rani et al., 2022) and have adopted this approach for baby corn yield estimation
(Rani et al., 2023). Multiple geometric modeling methods accompanied by vision based techniques have been used to estimate volume of fruits
(Babic et al., 2012; Huynh et al., 2020; Huynh et al., 2022). For instance,
href="#venkatesh_2015">Venkatesh et al., (2015) determined volume and mass of citrus fruits based on geometric diameters of the fruit samples and achieved an R
2 of 0.91 indicating a good accuracy.
Alçiçek et al. (2014) proposed a cubic splines approach to estimate green shelled mussels’ volume with an R2 of 0.97. Furthermore,
Bozokalfa and Kilic (2010) adopted a mathematical model to determine the volume of peppers with an R
2 of 0.95. Various other geometric modeling techniques have also been used widely for fruit grading
(Ibrahim et al., 2016; Khojastehnazhand et al., 2019) and yield estimation
(Andujar et al., 2016; Herrero-Huerta et al., 2015).
Another popularly used modeling approach for non-destructive fruit volume estimation involves statistical modeling
(Gongal et al., 2019; Örnek and Kahramanli, 2018). For instance
Nyalala et al., (2021) employed 7 different regression models to determine experimentally the volume of tomatoes having diverse shapes and sizes. This model achieved an R2 of 0.98 which indicates a high accuracy.
Saengrayup et al., (2009) used fruit dimensions as inputs for ANN (artificial neural networks) and regression models to estimate volume of plum fruits. An R
2 of 0.93 in this experiment showed that all these regression models were highly appropriate.
Ziaratban et al., (2016) estimated the volume of apples with an R
2 of 0.99 using a model based on Levenberg-Marquardt algorithm and hyperbolic tangent sigmoid transfer functionFruit volume estimation has not only been restricted to geometric and statistical methods. In recent years, several other techniques in trend have been put to use. For instance,
Keightley et al., (2010) used the tripod LiDAR method consisting of laser scan datasets for grapevine volume estimation with an R
2 of 0.93.
Arendse et al., (2016) adopted X-ray computed tomography method to estimate pomegranate volume based on the size and number of voxels (3D pixels).
Zheng et al., (2022) used UAV multispectral imagery and 6 different regression techniques for strawberry biomass prediction. The model created achieved an R2 of 0.97. In addition to these,
Li et al., (2015) used the approach of thermal and sunshine hours to determine apple fruit diameter and length with an R
2 of 0.88.
By conducting a systematic review of the existing literature and performing a meta-analysis, the objective is to provide a thorough examination of different non-destructive techniques utilised for fruit volume estimation. The discussed results will significantly contribute to the current knowledge baseand offer valuable insights to the researchers regarding the strengths and limitations of these techniques. This will stimulate advancements in the field, leading to improved agricultural practices and increased productivity levels
(Bibwe et al., 2022).
Selection criteria
The review work was conducted in 2023 at Shivaji College (University of Delhi), Delhi. Fig 1 highlights the selection parameters of research papers for the study. It depicts the eligibility criteria of selection (Fig 1a), information sources used for extraction (Fig 1b), various keywords (Fig 1c) and PRISMA methodology (Fig 1d) used for different queries for different databases.
Inclusion and exclusive criteria using prisma methodology
For inclusion in the systematic review and meta-analysis, papers that focused on fruit volume estimation through mathematical modeling techniques and written in the English language over the time frame of 2008-2023 were selected. At first, a total of 948 studies were selected from different databases out of which 849 papers were removed as they didn’t clear the objectives of the systematic review and meta-analysis, leaving a total of 99 studies. Further, these studies were examined for any duplicates and hence 12 duplicates were removed. At last some papers were found to be using the same techniques as others while also not providing sufficient data for statistical analysis. Therefore, by removing 37 of such papers, a set of 50 full-text articles were left for the systematic review and meta-analysis.
Quality assessment
Quality of selected papers were measured through manual strategy based on the various factors such as area of interest which should lie around the mathematical modeling methods used for fruit volume estimation. Secondly the dataset was obtained from multiple sources and the third is the risk of biasness which was measured by different researchers individually and hence clarified.
Bibliometric analysis
Fig 2 depicts the bibliometric charts for collected literature. It shows the count of research articles by journal (Fig 2a), geo chart representing publications by countries (Fig 2b), most cited research articles (Fig 2c) and number of annual publications (Fig 2d). The following interpretations can be made on the basis of analysis:
• The International Journal of Food Properties ccounted for the greatest number of publications compared to other journals.
• Most of the publications originated from India followed by Iran and China.
•
Moreda et al., (2009) was the highest cited paper with 231 citations.
• 2022 had the greatest number of research articles published on fruit volume estimation techniques through mathematical modeling compared to any other year.
Literature review
There are many industries that require fruit volume estimation, including agriculture, food processing, storageand transportation. To optimize resource utilization and ensure quality in the production process, it has become increasingly important to develop accurate and non-destructive methods for estimating volume. In this literature review, we examine recent research on the volume estimation of fruits based on mathematical models, outlining their strengths and limitations, as well as possible future directions.
Background
A non-destructive approach for estimating the volume of fruits is offered by mathematical modeling, which does not require damage to the fruit to be done during measurement. To estimate fruit volume accurately, several mathematical modeling methods have been developed. A few common non-destructive methods for estimating fruits’ volume along with their timelines are presented in the Table 1 below to provide a background of mathematical modeling evolution:
The evolution of volume estimation methods for fruits using mathematical modeling shows a clear progression from simple geometric models to sophisticated and data-driven approaches. This field is expected to continue to develop as technology advances, with increased integration of machine learning and artificial intelligence, enhanced 3D scanning technologiesand the potential application of emerging technologies such as point cloud processing and augmented reality.
Contemporary research
Recent studies have focused on hybrid approaches, which combine different techniques, such as image processing and machine learning, to enhance accuracy and efficiency. Hybrid approaches take advantage of the strengths of different approaches to address the limitations of a single method. In Table 2 below, there are studies from 2023 that didn’t use individual methods but demonstrate a combination of methods:
Hybrid models
Hybrid models that combine machine learning and artificial intelligence with enhanced 3D scanning technologies and the potential application of emerging technologies like point cloud processing and augmented reality, offer a powerful and versatile approach to volume estimation of fruits. However, they also have their strengths and limitations (Table 3).
Hybrid models are extremely promising as a method of estimating fruit volume, despite their limitations. Researchers are increasingly turning to hybrid models for accurate, efficientand nondestructive fruit volume estimation across a variety of industries as technology advances.
Statistical analysis
The following statistical analysis was conducted through MedCalc software version 22.009. Out of the 50 papers selected for literature review, 32 studies were screened on the basis of their sample size and correlation coefficient (which depicted the relationship between estimated and actual fruit volume in the studies).
Statistical analysis to assess heterogeneity between the categories
Based on the findings present in the existing literature, a test for heterogeneity among different categories was conducted to assess the inconsistency between the categories. Table 4 presents the characteristics of the categories used for heterogeneity analysis. Table 5 presents the summary of the analysis results. The following interpretations can be made from the test results.
Cochran’s Q test
After conducting the test between the categories, a very small significance level (P=0.0002) was obtained which is much smaller compared to the assumed significance level of 0.05. Therefore, it can be concluded that there is a significant difference between the correlation coefficient of these categories. The I2 statistic is found to be 88.48%. It suggests that there is significant heterogeneity or true inconsistency between the categories.
This test clearly indicates the heterogeneity between the different categories. The test provides evidence for further investigation of the existing literature to identify the sources in order to improve the generalizability of the findings.
Statistical analysis to assess risk of bias between individual studies
Egger’s and Begg’s tests were conducted to identify the impact of any potential bias in the studies. Table 6 illustrates the characteristics of the data used. Table 7 provides a summary of these test results. Fig 3 illustrates the graphical representation of Table 7. The following interpretations can be made after analyzing the test results from Table 7.
Egger’s test
Since the obtained significance level (P=0.6475) is greater than assumed significance level of 0.05. It can be concluded that there is no strong evidence of publication bias in the selected studies.
Begg’s test
As the obtained significance level (P= 0.4049) is greater than the assumed significance level of 0.05. It can be concluded that there is no strong evidence of correlation between the effect sizes (correlation coefficient) and its variance. Thus, the effect size is not influenced by publication bias.
Analysis of effect measures by odds ratio, weighted odds ratio, risk ratio for R2 < 0.95
Odds ratio, weighted odds ratio and risk ratio were calculated for assessing the risk and odds of an approach to provide a study having an R
2 < 95%. Table 8 presents the data for the effect measures. Table 9 presents the interpretation of these results.
Sensitivity analysis by weighted odds ratio and risk ratio
A sensitivity analysis of the collected literature based on different R
2 thresholds was conducted to improve the comprehensiveness of the statistical analysis and further analyze the results on the basis of risk ratio and weighted odds ratio for the selected literature. Table 10 presents the data characteristics of the sensitivity analysis. Following interpretations can be followed after analyzing the data obtained from Table 10.
Risk ratio
On the basis of Table 10, for almost all the thresholds the relative risk follows the same pattern where other methodologies have the highest risk followed by geometric modeling and statistical modeling respectively. So, we can conclude that the studies using statistical modeling present a lower risk compared to the other two approaches.
Weighted odds ratio
Weighted odds ratio also agreed with the risk ratio as statistical approaches offered a much lower odds ratio compared to othermethodologies for different thresholds of R
2.
Research gaps and future scope
The systematic review and meta-analysis provide valuable insight into the current state of fruit volume estimation research using mathematical modeling. The comprehensive analysis has nevertheless highlighted some research gaps that must be addressed. The following are the major research gaps identified.
Lack of standardization
The literature review revealed a wide variety of mathematical modeling approaches used for fruit volume estimation, but methodologies lack standardization. Inconsistent results can make it hard to compare findings among studies. Methods and reporting criteria should be standardized.
Insufficient focus on rare or specialized fruit varieties
Research on rare or specialized fruit varieties is limited in the literature review, with most studies focusing on common fruits. Certain industries require volume estimation of such fruitsand thus more research is needed.
Limited consideration of external factors
Several external factors may affect fruit volume estimation, which was not considered in the literature review. Environmental conditions, fruit ripenessand storage conditions should be taken into account in models. In future research, these external factors may be investigated and models may be developed to better estimate fruit volume.
Lack of real-time applications
Literature review studies mostly focused on offline fruit volume estimation. In industrial settings, mathematical modeling approaches should be explored in real-time fruit sorting and grading systems. Developing real-time solutions can boost productivity and reduce manual labor.
In future studies, future research can further enhance the accuracy, efficiencyand applicability of mathematical modeling approaches for fruit volume estimation by addressing these research gaps and exploring their scope areas, thereby improving agricultural practices and increasing fruit industry productivity.
