Acquisition of 3D Root System Simulation Parameters Using 2D Extracted Image
Data and Genetic Programming
Jose Miguel F. Custodio1*, Joseph Aristotle R. De Leon1, Alezander Mikhail O. Galindo2, Ira C. Valenzuela1, Ronnie S. Concepcion II1, and Ryan Rhay P. Vicerra1
1Department of Manufacturing Engineering and Management, De La Salle University, Manila, National Capital Region 1004 Philippines
2Department of Industrial Engineering, De La Salle University, Manila, National Capital Region 1004 Philippines
Root studies like propagation and morphological traits unlock a higher-level understanding of plants to better the growth of agricultural crops and maximize farm yields. As they are located underground, there is an additional challenge in elucidating their structure and behavior.
Root imaging allows for real-time observations and there are several known methods. One imaging dataset simulated entire 3D root systems to create 2D images for data analysis and measurements. In this study, the dataset’s extracted measurements will be used to reconstruct the true parameters of the simulated 3D root system through the use of multigene symbolic regression genetic programming (MSRGP). Eleven (11) parameters were selected as the output variables, each for monocot and dicot data for a total of 22 MSRGP models. Among these, thirteen of them showed high R2 values greater than 80%, proving the high accuracy of the MSRGP method. Input variables that were frequently used across multiple models were also noted, such as tip count, exploration ratio, and area. In addition to the accuracy that MSRGP provides in predicting variables, the computation time of these models is found to be as low as a few milliseconds. Once trained, this speed allows the models to be integrated into relevant applications without significant increases in computation cost.
Keywords: data prediction, genetic algorithm, multigene symbolic regression, plant root system imaging
*Corresponding author: [email protected]
INTRODUCTION
Plant root studies are important to the advancement of the agricultural sector. The primary duties of the root systems are to interact with the soil to support a plant and allow it to receive nutrients (Ozoe et al. 2015). As a plant grows, roots increase in size to accommodate its increase in nutritional demand. The root’s interaction
with soil and water utilization would be improved if the relationship between them could be better elucidated (Huang et al. 1992). This would lead to better plant health and crop yields.
The challenge in studying roots is that they cannot be easily viewed, being buried in the soil. Aspects of their growth can be quantified and modeled, such as their driving force (Ozoe et al. 2015). Exhaustive 3D models that follow growth patterns (Zhou et al. 2018) and biomass increase ISSN 0031 - 7683
Date Received: 22 Mar 2022
(Zhang and Li 2009) have also been created to showcase the relation of the roots with soil, water content, and nutrient intake. These models lead to simulated roots and aid in other research such as root imaging (Zhang and Li 2009).
A benefit of in vivo and in situ root imaging is to see the true irregularities of root growth and structure while they are buried underground (Chen and Zhou 2010). It would also enable models to create accurate representations of live plants. Several methods have been made to perform root imaging, such as the minirhizotron technique (Huang et al. 1992), X-ray computed tomography (Zhou et al.
2014), and the use of rhizoboxes (Jia et al. 2019). These methods involve taking images of the root system through various means, then utilizing software and machine learning to analyze the images for traits being investigated.
Image processing techniques may be employed to further enhance the image before analysis, such as the median filter, Otsu method, and Gabor wavelet operator (Chen and Zhou 2010; Song et al. 2011). As the image is a 2D image, the characteristics extracted from the methods may not accurately reflect the entire 3D root system.
Another study attempted to approximate the actual root structure by using generative adversarial networks to reconstruct root systems from a single image. It estimated the depth map by predicting its appearance from another angle, then a three-dimensional image of the plant’s roots could be made using a stereo reconstruction method (Lu et al. 2021).
MSRGP is an evolutionary computing technique that allows for the formulation of models and identification of relationships between predictor and output features (Searson 2015). Another benefit of the technique is that once a mathematical model is created, it is simple to implement and does not require as high memory and resources as other methods (Concepcion et al. 2020).
The MSRGP’s coefficient of determination (R2) would indicate that the created model is able to account for what percentage of the dependent variable’s variation. A higher R2 value, close to one, is preferable. This metric, as well as the computational complexity rating of the model, are among the methods to evaluate the developed models. The model complexity refers to the expressional complexity of the mathematical equations and the chain of operations required to compute the desired values. Simpler models and lower complexity are preferred, as they would require fewer computational resources to produce an output.
Because of the capabilities of MSRGP, it is used in various studies that aim to locate relationships between inputs and outputs. Quality of input, such as synthesized or broadcasted audio, could be evaluated and modeled through this method (Mrvová and Počta 2013; Jakubik and Počta 2021). MSRGP is also capable of accurately tracking and predicting motion, such as the motion trajectory of fish
in a tank (Palconit et al. 2021a). A similar study noted that MSRGP stood out in performance compared to adaptive neuro-fuzzy inference system (ANFIS) and Gaussian process regression (Palconit et al. 2021b).
Difficult measurements could be predicted with a model by using the inputs of properties that are easier to acquire. One study utilized temperature and electrical conductivity to predict the concentration of organic carbon and hydrogen ions in water (Concepcion et al. 2020).
Similarly, temperature, pH, and electrical conductivity sensors were used to predict the concentration of nitrate, phosphate, and potassium macronutrients in aquaponic setups (Concepcion et al. 2021). Boolean data could also be modeled, such as the presence of breast cancer by using some parameters of the observed cells (Hasan et al. 2016).
In a similar fashion to approximating real values from observations, this study aims to recreate an existing dataset’s true root system values through the use of multigene symbolic regression (MSRGP), a hybrid of the genetic algorithm and regression tree techniques.
The process attempts to create mathematical models that relate input variables to the target output. The output of the developed MSRGP models, the true root values, would be based on measurements made by a previous study’s specialized analysis software used in making the dataset.
The development of properly trained models would enable a deeper understanding of root architecture features and accurate simulation studies.
MATERIALS AND METHODS
The methodology of the study can be summarized in the flowchart presented in Figure 1. The dataset will be prepared for MSRGP, which is done by extracting the variables relevant to the study. The MSRGP procedure itself will be performed through the GPTIPS2 toolset. The best models will be selected, and performance metrics will be extracted and evaluated.
Root Library Dataset Description
This study utilized .csv datasets extracted from a previous paper by Lobet et al. (2017) regarding simulated root systems. About 10,000 3D root system models were synthesized through the ArchiSimple software. A 2D image from these simulations was taken and then analyzed through a custom analysis plugin in the ImageJ software to identify descriptor traits about the root system. Generic root systems were generated and were not modeled after specific plant crops.
The datasets contain variables regarding true simulation parameters of the root system models and the descriptor
traits taken by the custom analysis plugin. Other values are present in the dataset which pertain to coordinates and information regarding the overall shape of the root system.
The datasets also contain labels of the corresponding images from which the descriptor traits were obtained. In total, 5,252 monocot and 5,212 dicot images were used to generate the descriptor traits within the datasets, each based on its own set of simulation parameters.
The previous study related eleven descriptor variables to eleven simulation parameters. The eleven descriptor variables in that study were area (Area, mm2), length (Len, mm), tip count (Ntip, -), mean diameter (Diam, mm), width (Width, mm), depth (Depth, mm), width-depth ratio (WDR, -), center of mass X (COMx, -), center of mass Y (COMy, -), convex hull area (ACH, mm2), and exploration ratio (Exp, -). The 11 simulation parameters pertain to total root length (TRL, mm), total primary root length (TPL, mm), total lateral root length (TTL, mm), mean primary root length (MPL, mm), mean lateral root length (MLL, mm), primary root count (Npri, -), lateral root count (Nlat, -), lateral root density (RD, mm–1), mean primary root diameter (MPD, mm), mean lateral root diameter (MLD, mm), and mean lateral angle (θ, °).
As seen in Figure 2, the roots of monocot and dicot plants are different in their structure. Monocots are characterized by their fibrous appearance, having multiple primary roots, and branching lateral roots. The taproot system of dicots, on the other hand, is known for its large single primary root (Lobet et al. 2017). The data of monocot and dicot parameters and traits were compiled into two separate datasets, due to the difference in their root systems.
Dataset Preparation
The relevant data for this study were extracted from the original datasets and compiled into new .csv files, separating monocot and dicot data. These are the 11 image descriptors and eleven simulation parameters, whereas the label, shape, and coordinate information are excluded. The simulation parameters act as the study’s target variables and the descriptors as the input predictor variables. This is
to be processed for a total of twenty-two models, modeling each target variable. Stratified sampling was performed on both datasets to allocate 70% for training, 15% for validation, and 15% for testing.
Modeling for Genetic Programming
The root library dataset was processed using the GPTIPS2 toolbox on MATLAB 2021b, which employs MSRGP.
The package also displayed the best mathematical models created and performance information about these models (Searson 2015). Figure 3 presents the process flowchart of MSRGP.
Figure 1. Methodology process flowchart.
Figure 2. Sample images from the root system library images (Lobet et al. 2017).
GPTIPS2 also enabled the control over the hyperparameters of the genetic algorithm steps that it will perform. For each of the 22 output variables to the model, GPTIPS2 initialized a population of 500 models (Searson 2015).
Individuals were enabled to have up to 10 genes and a gene expression depth of 5. The model also assigned each gene a weight. The operation terminated after 60 generations of spawning or 60 s elapsing. The fitness of each sample was based on their root mean square error (RMSE).
Pareto selection was used at a probability rate of 35%, where out of 100 randomly selected individuals, the best in terms of fitness and model complexity were chosen.
This was then used to create new offspring for future generations. To convert the predictor variables into the target output, models were enabled to use mathematical operations of multiplication, addition, subtraction, square root, squaring, sine, cosine, adding and multiplying three inputs, logarithm, cube, negation, and absolute value.
Crossover and mutation utilized default GPTIPS2 parameters (at 84 and 14% probability, respectively) to promote the creation of new gene expressions. Crossover involved either the exchange of entire genes between two models or just portions of these. The possible mutations included mutating entire gene trees and operations, replacing the input variables, and changing the weight of a gene. These two operations ensured that the algorithm did not immediately settle on a local minimum and continued to locate other potential models by adding new genes into the population that may provide better results.
Evaluation of Results
GPTIPS2 generated a summary report on the model population constructed for each of the 22 output variables.
To reduce the number of models to be considered, it only presents models in the Pareto front set, based on the model’s R2 and complexity. Out of these models, the model with the highest R2 was selected. The lowest complexity model was selected in the event of a tie. Following this, a comprehensive report was created for each of the selected models, which contained data about its performance, properties, and structure. From the comprehensive report, the model’s R2, RMSE, mean absolute error (MAE) were recorded for further evaluation. The equations for these three values can be found in Equations 1, 2, and 3, respectively (Paul and Goswami 2020). The model’s complexity and the predictor variables it used were taken from the report as well.
(2)
(3) (1)
RESULTS
Modeling Results
Twenty-one models were developed in GPTIPS2 utilizing the hyperparameters discussed. GPTIPS2 was unable to create a viable regression model for one of the target variables, the Npri for dicot root systems. It was unable to converge as all dicot root systems have the same value for Npri, which is 1, due to their taproot architecture. Due to the constant value of Npri, computing for R2 led to an undefined value, based on Equation 1. It also suggests that no relation is present between the predictor variables and Npri for the dicot root system, as it is a constant.
For each of the remaining 21 target variables, GPTIPS2 was able to create a population of models. Among these, the model with the highest R2 value was selected. Table 1 consolidates the complexity of the models selected and the predictor variables employed by each model to generate their target output. As the selection criteria for best model is based primarily on their R2 value and not their complexity, these models are more complex compared to
Figure 3. Flowchart for the MSRGP algorithm.
(4)
(5)
Table 1. Model complexity and predictors used.
Output variable Complexity Predictor variables used
Monocot TRL 235 Area, Len, Diam, Width, WDR, COMy, ACH, Exp
TPL 426 Area, Len, Ntip, Diam, Width, Depth, WDR, COMx, COMy, Ach, Exp TLL 174 Area, Len, Ntip, WDR, COMy, ACH, Exp
MPL 181 Area, Len, Ntip, Diam, Width, Depth, WDR, COMx, COMy, ACH MLL 188 Area, Len, Ntip, Diam, Depth, COMy, Exp
Npri 87 Area, Ntip, Diam, Width, WDR, COMy, Exp
Nlat 135 Area, Len, Ntip, COMy, Exp
RD 218 Area, Len, Ntip, Diam, Depth, WDR, COMy,Exp MPD 164 Area, Len, Ntip, Diam, Width, Depth, ACH
MLD 124 Area, Len, Ntip, Diam, Width, Exp
θ 149 Area, Len, Ntip, Width, Depth, COMy, ACH, Exp
Dicot TRL 149 Len, Ntip, Diam, COMy, Exp
TPL 6 Diam, Depth, COMy, Exp
TLL 150 Len, Ntip, Diam, Depth, WDR, COMx, COMy, Ach, Exp
MPL 7 Depth, COMy
MLL 130 Area, Len, Ntip, Diam, Depth, WDR, Exp
Npri - -
Nlat 74 Area, Len, Ntip, Width, Depth, COMx, COMy, Exp
RD 134 Area, Ntip, Diam, WDR, COMy, Exp
MPD 75 Area, Len, Ntip, Diam, Depth, WDR, Exp
MLD 139 Area, Ntip, Diam, Width, Depth, WDR, COMx, COMy, Exp θ 147 Area, Len, Ntip, Diam, Width, WDR, COMx, Exp
other potential candidates. Models of lower complexity are ideal for simpler and faster computations, but only if they are also able to provide accurate results. It is notable that on average, models created for monocot variables have higher complexities compared to models fitted for the dicot variables at 189.2 and 101.1, respectively. This could indicate that the fibrous root system of monocot plants is more challenging to analyze.
As a point of comparison, Equations 4 and 5 present the models created for the TRL of both monocot and dicot plants, respectively. The equation for monocot TRL uses 17 terms and eight inputs as compared to the dicot TRL model with only 11 terms and five variable inputs.
Table 2 enumerates the number of times that a predictor variable was used in each model. This would reveal which inputs were highly valuable in creating the 21 mathematical models. Exploration ratio (Exp) and tip count (Ntip) were the most prevalent, being present in 18 models. Center of mass X (COMx) and convex hull area (Ach) were the least used, utilized only in six and seven models each, respectively.
Performance Values and Model Evaluation
Table 3 compiles the R2, RMSE, and MAE of the 21 models’ test results. In an ideal model, R2 would have a value of 1, to show that the model has a perfect fit on the data; on the other hand, RMSE and MAE would have values of 0, as this would minimize the error of the model’s predicted values.
Table 3. Model performance values.
Output Variable R2 RMSE MAE
Monocot TRL 0.9879 1080.9588 476.1392
TPL 0.7271 519.7074 389.2143
TLL 0.9826 1213.3717 740.3171
MPL 0.9346 19.6213 15.0640
MLL 0.7571 3.2066 2.1572
Npri 0.4685 3.8094 2.9803
Nlat 0.9769 106.6108 53.0071
RD 0.6576 0.1845 0.1307
MPD 0.6769 0.0320 0.0253
MLD 0.7731 0.0230 0.0177
θ 0.4068 10.5589 8.4640
Dicot TRL 0.9804 845.4701 226.5288
TPL 1.0000 0.4795 0.3266
TLL 0.9789 770.9241 219.4946
MPL 0.9999 0.7704 0.5431
MLL 0.9510 1.6887 1.2837
Npri – – –
Nlat 0.9822 54.6940 17.0231
RD 0.8410 0.1262 0.0846
MPD 0.9881 0.1023 0.0565
MLD 0.9741 0.0256 0.0171
θ 0.5639 10.2457 8.185
Table 2. Number of models using each predictor variable.
Predictor variable Variable name Number of models present
Area Area 17
Length Len 16
Tip Count Ntip 18
Mean Diameter Diam 16
Width Width 10
Depth Depth 13
Width-Depth Ratio WDR 12
Center of Mass X COMx 6
Center of Mass Y COMy 16
Convex Hull Area Ach 7
Exploration Ratio Exp 18
With the selected hyperparameters for the MSRGP, GPTIPS2 was able to create highly accurate models for most of the variables. The range of R2 values attained by the 21 studies can be seen in Figure 4. Notably, the dicot models outperform monocot models, as more models have an R2 of 0.8 and above with a count of nine and four, respectively.
A total of 12 models achieved an R2 value of 0.9 or higher, showing the high accuracy that MSRGP is capable of.
This demonstrates that a mathematical modeling approach could be applicable for inferring true root values from image approximations.
The RMSE and MAE show how far the model’s predicted values are from the actual target values. To identify models that have the ideal traits of a high R2 and low RMSE and MAE, Figures 5 and 6 plot each model’s R2 value against their RMSE and MAE, respectively. This combination of traits is desired as it indicates the models are able to accurately fit the data with minimal error from outliers.
Six of the models are found in the fourth region. This region, indicated in blue, occupies the area in the graph with an R2 value of 0.9 and above, and a RMSE and MAE of 50 or lower. Six models are within the blue region and possess the desired traits. These models – which predict the MPL of monocot plants and the TPL, MPL, Nlat, MPD, and MLD of dicot plants – are deemed the best of the 21 created.
In both figures, five models can be found in region 2.
These models were trained to predict the TRL, TLL, and Nlat of monocot plants and the TRL and TLL of dicot plants. While these models are accurate (high R2), the predictions may have a large discrepancy from the true values due to the high RMSE and MAE. It may be possible to further reduce the error for these variables by training a new model with different MSRGP hyperparameters to allow for higher model complexities, which may result in better performances. Another model with high MAE can be found in grey region 1, which estimates the TPL of monocot plants. Compared to models in the orange region, it has a lower R2 value. The remaining nine models are found in the third region, which indicates that these models have low MAE and RMSE values but do not accurately represent the dataset due to the low R2 values.
DISCUSSION
Due to the high values of R2 and low values of RMSE and MAE that some models have, the genetic algorithm MSRGP method could be deemed successful in its purpose of generating and training mathematical models that each predict a different variable. Studies such as those by
Figure 4. Bar chart of R2 values achieved by the best models.
Figure 5. Scatter plot of R2 value versus RMSE value.
Figure 6. Scatter plot of R2 value versus MAE value.
Mrvová and Počta (2013) and Paul and Goswami (2020) also report accurate results with the MSRGP method.
The performance of the method was not consistently good, however. Nine of the models, located in regions 1 and 3 of Figure 5, missed the mark of having a high R2 value and would mean that their capabilities to predict the target variable are not extremely reliable. Another five models had high R2 values but high error metrics of RMSE and MAE, possibly due to the magnitude of values used for these models.
The computation time of the generated models are compiled in Table 4. These times are for calculating one instance of the desired variable. The models were able to achieve minimal computation times averaging at 0.7 milliseconds. The low computation time and simple implementation make MSRGP favorable, especially on devices with low processing power (Concepcion et al.
2021). In addition, other applications for MSRGP were found to offer higher accuracy over alternative methods such as ANFIS, linear regression, and recurrent neural network (Concepcion et al. 2020, 2021; Palconit et al.
2021b). The time needed to compute per plant would add up, however, when a large batch size would be analyzed simultaneously.
Table 4. Model computation times.
Output variable Computation time
Monocot TRL 0.000339 s
TPL 0.001049 s
TLL 0.000627 s
MPL 0.000928 s
MLL 0.000371 s
Npri 0.000622 s
Nlat 0.000407 s
RD 0.000663 s
MPD 0.000550 s
MLD 0.000456 s
θ 0.000388 s
Dicot TRL 0.002688 s
TPL 0.000152 s
TLL 0.001571 s
MPL 0.000169 s
MLL 0.000590 s
Npri –
Nlat 0.000626 s
RD 0.000590 s
MPD 0.000406 s
MLD 0.000778 s
θ 0.000809 s
The original study where the dataset originated, in comparison, performed a random forest algorithm (RFA) as their method to predict the true values from the data extracted from images (Lobet et al. 2017). RFA is an aggregation of multiple regression trees, which is a model that predicts outcomes through branching binary decisions. RFA uses the majority vote of its decision trees to determine the output.
The results of their RFA approach yielded better results, with 18 of the 21 algorithms generated having R2 values greater than 0.9, compared to the 12 that MSRGP created in this study. Similarly, they did not succeed in creating an algorithm for Npri in dicot root systems. Both RFA and MSRGP failed to create models with an R2 of 0.9 or greater for the RD of monocot root systems and θ of both dicot and monocot plants. This could indicate that these variables require models of a higher complexity or that it needs new inputs apart from the 11 predictor variables employed.
In terms of predicting 11 true root values in monocot and dicot root systems, RFA has proven to be the more accurate method based on the findings of Lobet et al. (2017). On the other hand, MSRGP has the added benefit that once it is trained, the model, which is just a mathematical equation, can be easily extracted and placed in various systems without the need for packages and libraries (Concepcion et al. 2020). This makes it especially suitable for devices that employs the use of Arduino, which could be relevant for future root studies that require fieldwork. Direct comparative studies on the computation time of the two methods have not yet been executed, making it difficult to have definitive conclusions on which method is best.
It is possible that MSRGP could train better models and achieve better accuracy compared to the results generated by this study, by finding a combination of favorable hyperparameters (gene count and depth, population size, and termination). Table 5 presents some hyperparameter configurations of solving for the Dicot MLL target variable. Despite increasing some of the hyperparameters, these alternative settings were found to have slightly lower performances in terms of R2. As expected, the increase in gene depth also led to models with higher complexity, which may have slower computation times during the training period.
CONCLUSION
With the high R2 of the models generated, it can be concluded that the MSRGP method was able to utilize the observed 2D image data to reverse-engineer most of the target true root parameters. Out of the 21 models created,
13 had high R2 values and could be reliable predictors for the variables they predict. From these 13, six are expected to perform with minimal errors and discrepancies due to their low RMSE and MAE metrics, as found in the blue regions of Figures 5 and 6.
The GPTIPS2 toolset was able to create promising output in the form of mathematical models. To improve on the method used, future research could be done with higher population counts or higher run times to potentially give rise to better models for the target variables, especially those with poor R2 values. The toolset was also able to show the predictor variables that were of use to the models generated. This enables researchers to focus on acquiring good readings for these valuable inputs.
Future root analysis studies could consider using MSRGP models in root imaging techniques, such as rhixoboxes, to obtain better readings and descriptions of the crops to be studied. The models could augment the 2D image data gathered and convert it into true root data, as demonstrated in this study. Similar to the dataset from Lobet et al.
(2017), MSRGP models could be trained with synthesized root image libraries that specialize into specific crops and their root system architecture to create specialized models.
As root imaging studies improve, so would root simulation studies, enabling a positive feedback loop of one field’s advancements benefiting the other field.
ACKNOWLEDGMENTS
The researchers thank the guidance and support of the Manufacturing Engineering and Management Department of De La Salle University Manila in the making of this paper. They also appreciate the support from the Engineering Research and Development for Technology.
Table 5. Hyperparameter sensitivity of MSGRP in solving for dicot MLL.
Setting Population Gene count Gene
depth Termination conditions Best model achieved Operation time Generation limit R2 Complexity
Variation A 500 10 10 60 s 60 0.947 239
Variation B 500 5 10 120 s 120 0.927 277
Variation C 1000 5 10 120 s 120 0.92 375
Variation D 500 5 10 60 s 60 0.908 332
Variation E 1000 10 5 60 s 60 0.949 63
Actual 500 10 5 60 s 60 0.951 130
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