Research on Remote Sensing Monitoring of Key Indicators of Corn Growth Based on Double Red Edges

Abstract

The variation in crop growth provides critical insights for yield estimation, crop health diagnosis, precision field management, and variable-rate fertilization. This study constructs key monitoring indicators (KMIs) for corn growth based on satellite remote sensing data, along with inversion models for these growth indicators. Initially, the leaf area index (LAI) and plant height were integrated into the KMI by calculating their respective weights using the entropy weight method. Eight vegetation indices derived from Sentinel-2A satellite remote sensing data were then selected: the Normalized Difference Vegetation Index (NDVI), Perpendicular Vegetation Index (PVI), Soil-Adjusted Vegetation Index (SAVI), Red-Edge Inflection Point (REIP), Inverted Red-Edge Chlorophyll Index (IRECI), Pigment Specific Simple Ratio (PSSRa), Terrestrial Chlorophyll Index (MTCI), and Modified Chlorophyll Absorption Ratio Index (MCARI). A comparative analysis was conducted to assess the correlation of these indices in estimating corn plant height and LAI. Through recursive feature elimination, the most highly correlated indices, REIP and IRECI, were selected as the optimal dual red-edge vegetation indices. A deep neural network (DNN) model was established for estimating corn plant height, achieving optimal performance with an R² of 0.978 and a root mean square error (RMSE) of 2.709. For LAI estimation, a DNN model optimized using particle swarm optimization (PSO) was developed, yielding an R² of 0.931 and an RMSE of 0.130. KMI enables farmers and agronomists to monitor crop growth more accurately and in real-time. Finally, this study calculated the KMI by integrating the inversion results for plant height and LAI, providing an effective framework for crop growth assessment using satellite remote sensing data. This successfully enables remote sensing-based growth monitoring for the 2023 experimental field in Haicheng, making the precise monitoring and management of crop growth possible.

1. Introduction

Food security, a cornerstone of national security, faces significant challenges due to the reduction and degradation of arable land [1]. Corn is a key staple crop globally. It is also widely used for animal feed, industrial raw materials, and energy production (such as biofuels). The global production of corn is immense, especially in regions like the Americas, Africa, and Asia, where corn cultivation area and yield play a significant role, and holds a pivotal role in world agriculture [2]. Therefore, transitioning corn agricultural production towards precision, modernization, and informatization is crucial. Furthermore, enabling the efficient and timely monitoring of crop growth is essential. Remote sensing monitoring technology facilitates the timely, dynamic, and large-scale observation of crop conditions, making it an indispensable tool for monitoring crop growth [3]. Monitoring crop growth variations provides critical insights for yield estimation, crop health diagnosis, precision field management, and variable-rate fertilization. This helps make timely decisions on irrigation, fertilization, and pest control, ultimately optimizing yields, reducing resource usage, and enhancing sustainable agricultural practices.

Remote sensing technology plays a pivotal role in crop growth monitoring due to its advantages of short revisit cycles, broad coverage, and relatively low data acquisition costs. Crop growth refers to the condition and trends of the crop’s development, which can be described through both individual and collective characteristics. Individual characteristics primarily include plant height, chlorophyll content, and tiller number, among others [4,5]. Collective characteristics, on the other hand, include population distribution structure, biomass, and the leaf area index (LAI) [6,7]. Monitoring crop growth is a key research area in agricultural remote sensing. Among these, LAI has been a critical indicator of crop growth and has driven extensive research over the past few decades through various models [8]. LAI is widely used to monitor crop growth and development [9], estimate crop yield [10], and detect early crop stress [11]. As one of the most important collective characteristics of crop growth, LAI estimation methods are generally classified into two types: direct measurements and indirect methods [12]. Compared to direct, destructive sampling methods, indirect estimation techniques using optical instruments and remote sensing imagery offer the advantage of large-area, rapid analysis, thereby overcoming the drawbacks of direct methods, such as high labor intensity and time consumption. LAI estimation models can be broadly categorized into three groups: empirical models, physical models, and hybrid models [8]. Empirical models aim to establish a relationship between LAI and spectral reflectance, or its transformation, using regression techniques. Vegetation indices (VIs) are commonly used as inputs for these models [13]. The spectral characteristics of crops, along with their variations, form the theoretical foundation for remote sensing-based crop property inversion. In practice, a significant challenge in using remote sensing data for LAI prediction arises when LAI exceeds values of 2–5, as vegetation indices tend to approach saturation levels, which depend on the specific index used [14]. Furthermore, there is no universal relationship between LAI and these indices. The Normalized Difference Vegetation Index (NDVI) is one of the commonly used vegetation indices for LAI inversion [15]. However, NDVI has limitations, as the relationship between NDVI and LAI tends to saturate under moderate to dense canopy conditions. Research has shown that incorporating the red-edge region of the spectral range (between red and near-infrared) can improve the inversion accuracy of key biochemical parameters that characterize vegetation growth [16]. This approach enhances estimation accuracy at higher LAI levels [17] and is regarded as a sensitive spectral band for indicating vegetation growth status. In the red-edge spectral region, the reflectance spectrum is strongly influenced by LAI. Red-edge index combinations can help eliminate the effects of chlorophyll variation [18]. In recent years, the demand for red-edge indices has been steadily increasing. Consequently, researchers have proposed several red-edge indices, such as the Red-Edge Inflection Point (REIP), Inverted Red-Edge Chlorophyll Index (IRECI), Pigment Specific Simple Ratio (PSSRa), Terrestrial Chlorophyll Index (MTCI), and Modified Chlorophyll Absorption Ratio Index (MCARI), to monitor the growth status of green plants. However, using these indices reveals that in the visible and near-infrared bands, LAI and chlorophyll content impact canopy reflectance similarly, particularly in the spectral region from the green (550 nm) to the red-edge (750 nm). To eliminate their combined effects, studies have shown that LAI content can be effectively estimated through the combination of two spectral indices [19], which avoids the confounding effects caused by chlorophyll. The red-edge spectral vegetation indices derived from the red-edge band in Sentinel-2 remote sensing imagery have demonstrated strong capabilities in inverting corn LAI. Plant height is often estimated using vegetation indices [20], which are valuable for monitoring crop growth, development, and yield analysis. Due to their ease of measurement and high field accuracy, these indices play a significant role in crop growth monitoring. REIP primarily reflects the characteristics of the red-edge region of plant leaves, while IRECI emphasizes the chlorophyll content and the physiological health status of the plant [21]. The combination of these two indices can provide a more comprehensive representation of the crop at different growth stages. This dual red-edge index combination compensates for the limitations of a single index and enhances the ability to sense crop growth variations in complex agricultural environments. Therefore, the combination of two red-edge indices holds considerable promise for mitigating saturation effects and improving the accuracy of LAI inversion.

Monitoring crop growth is essential for yield estimation, crop health diagnosis, precision field management, and variable-rate fertilization. Due to its complexity, crop growth has attracted significant attention from researchers globally, with numerous studies focusing on various growth parameters. Notably, many scholars have made substantial progress in using remote sensing technology to study individual crop growth parameters, such as leaf area index (LAI), nitrogen concentration, and plant height [22,23,24,25]. However, a single growth parameter is often insufficient to comprehensively reflect the true growth status of crops. Although some recent studies have estimated multiple growth parameters, such as biomass and nitrogen content [26,27], these studies primarily focus on the independent monitoring of individual parameters and do not integrate them to assess the overall development of crops. Consequently, this limitation reduces the effectiveness of precision agriculture systems. To address this challenge, many studies in recent years have focused on the development of multi-crop remote sensing monitoring models. Remote sensing-based crop monitoring models typically use vegetation indices, spectral information, and crop classification data to differentiate the growth status of different crops. At the same time, an increasing number of remote sensing data fusion techniques have been introduced to enable accurate crop growth monitoring across different time scales. Therefore, it is crucial to combine multiple individual growth parameters into a composite growth monitoring indicator to improve monitoring accuracy and precision. Wang et al. [28] collected data on growth parameters such as cotton LAI, canopy chlorophyll content, aboveground biomass, and boll number, and applied entropy and game theory weighting methods to calculate the weights. They then used three algorithms to establish the optimal model for comprehensive cotton growth monitoring. Similarly, Zhu et al. [29] selected the best combination of vegetation indices as inputs and employed the GA-SVR algorithm to build a regression model for inverting crop phenotypes and yield, thereby achieving effective wheat growth monitoring. Ahmed et al. [20] developed a “combination model” using the CI green vegetation index and 16 soil characteristic parameters. Additionally, Wang et al. [30] employed gray correlation analysis to determine the weight values of the Vegetation Temperature Condition Index (VTCI) and LAI for the growth monitoring of corn. These studies demonstrate the potential of composite evaluation indicators in improving prediction accuracy and practical applications. However, most current comprehensive growth monitoring efforts focus on small-scale remote sensing using drones, leaving large-scale comprehensive monitoring as a critical area of research. Therefore, selecting collective characteristics, such as plant height, and individual characteristics, like LAI, for integrated growth evaluation shows significant potential in enhancing growth monitoring and yield estimation using remote sensing data. With the gradual shift from general crop monitoring challenges to specific issues such as LAI and PH, the refinement and integration of growth monitoring research have also become important directions for future precision agricultural management [31]. By training these models, researchers are able to more effectively convert remote sensing data into crop growth parameters (such as plant height, LAI, etc.) and cope with complex environmental variations. With the gradual shift from general crop monitoring challenges to specific issues such as LAI and PH, the refinement and integration of growth monitoring research have also become important directions for future precision agricultural management.

In light of the aforementioned considerations, this study aims to provide an effective and comprehensive evaluation method for monitoring corn growth. The specific objectives of this research are as follows: (1) to develop an inversion model for vegetation indices, plant height, and leaf area index (LAI), uniquely integrating multiple red-edge indices; (2) to construct a deep learning-based comprehensive remote sensing model for corn growth monitoring; (3) to conduct remote sensing-based monitoring and evaluation of regional corn growth. This research will contribute significantly to the advancement of precision crop management and offer a scientific basis for large-scale remote sensing monitoring of crop growth.

2. Materials and Methods

2.1. Study Area

Haicheng, a county-level city under the administration of Anshan in Liaoning Province, is located in the central-southern part of Liaoning Province, in the central area of Anshan. It lies on the left bank of the lower reaches of the Liao River, at the northern end of the Liaodong Peninsula, and has a warm temperate monsoon climate. Haicheng is one of the important agricultural production areas in China, with a large area of corn cultivation. The climate conditions in this region are suitable for corn growth, providing rich practical data for the research. The study area is situated at the Precision Agriculture Aviation Research Base of Shenyang Agricultural University, located in Gengzhuang Town, Haicheng City, Liaoning Province (40°58′45.39″ N, 122°43′47.0064″ E, at an elevation of 13 m), as shown in Figure 1. The primary crops cultivated in this field are corn and rice (Triticum aestivum), with additional crops including soybeans (Glycine max), peanuts, and sunflowers. The soil is fertile, and the region experiences a hot and rainy summer climate. This area is one of the key corn-producing regions in Northeast China. Field observation experiments were conducted during July and August 2023, with the experimental variety being Dongdan 1331, a dominant corn variety in Liaoning Province.

2.2. Data Collection

Plant height and LAI are important indicators of crop growth status, directly reflecting the growth trend and health condition of the crops. Field measurements provide direct and objective growth parameters. Furthermore, field measurements offer high-precision on-site data, which can serve as a calibration and validation standard for remote sensing data. The method is simple to operate and relatively low-cost, which is why field measurements are used to obtain accurate plant height and LAI data. During the 2023 growing season, measurements were taken on clear days in July and August across various experimental fields within the research base. In each experimental plot, the LAI-2200C Plant Canopy Analyzer was employed to measure the ground-level leaf area index (LAI). For bands with different resolutions, resampling is performed on the image in SANP to obtain a higher resolution. For each 10 m × 10 m corn plot, five representative corn plants were selected, and the average LAI value of these plants was calculated and recorded as the LAI for that specific plot. Simultaneously, plant height was measured in the field using a measuring tape, and the average plant height for each plot was also recorded. These average values for both LAI and plant height were subsequently used to represent the corresponding values of a single pixel in the remote sensing imagery.

2.2.1. Field Data Collection

To obtain precise measurements of plant height and LAI, extensive field activities were carried out on 20 July 2023, and 24 August 2023. The LAI-2200C Plant Canopy Analyzer (LI-COR, Lincoln, NE, USA) was employed to measure the leaf area index (LAI). Sunlight intensity was recorded at five distinct zenith angles (7°, 23°, 38°, 53°, and 68°) both above and below the canopy for each corn plant. A radiative transfer model for vegetation canopies was then utilized to compute the LAI and other canopy structural parameters, such as the gap fraction. The resulting LAI values were considered effective for the estimation of canopy characteristics. In parallel, plant height was measured using a tape measure, and the mean value was calculated to estimate the average corn plant height for the field.

2.2.2. Remote Sensing Image Acquisition

Sentinel-2 is an Earth observation mission under the European Space Agency’s Copernicus program, designed primarily to monitor the Earth’s surface and provide remote sensing services for applications such as forest monitoring, land cover change detection, and natural disaster management. The mission consists of two identical satellites, Sentinel-2A and Sentinel-2B, which capture multispectral imagery across 13 spectral bands, including visible light, near-infrared (NIR), and shortwave infrared (SWIR), as outlined in

Table 1. Sentinel-2 band settings.

Band	Resolution (m)	Wavelength (nm)	Description
B1	60	433–453	Coastal aerosol
B2	10	458–523	Blue
B3	10	543–578	Green
B4	10	650–680	Red
B5	20	698–713	Vegetation red edge
B6	20	733–748	Vegetation red edge
B7	20	773–793	Vegetation red edge
B8	10	785–900	NIR
B8A	20	855–875	Narrow NIR
B9	60	935–955	Water vapor
B10	60	1365–1385	SWIR–cirrus
B11	20	1565–1655	SWIR-1
B12	20	2100–2280	SWIR-2

. Sentinel-2 offers systematic global coverage, with the capability to image regions from 56° S to 84° N, including coastal waters and the entire Mediterranean area. Under identical viewing conditions, the satellites can capture the same region every 5 days, and in high-latitude areas, where there are no viewing angle restrictions, the revisit time can be reduced to less than 5 days. The system provides imagery with spatial resolutions of 10, 20, and 60 m, a swath width of 290 km, and the data are freely available. Compared to other free and open remote sensing data sources, Sentinel-2 offers superior spatial resolution (10, 20, and 60 m), more comprehensive spectral coverage (including the red-edge band), and more frequent revisit cycles (every 5 days), making it an invaluable tool for large-scale agricultural crop monitoring via remote sensing.

Based on the field data collection dates, two Sentinel-2A (S2A_MSIL2A) images were acquired on 27 July 2023, and 16 August 2023. From these images, we extracted eight vegetation index images, including NDVI, PVI, SAVI, REIP, IRECI, PSSRa, MTCI, and MCARI. Subsequently, we established the corresponding relationships between the measured plant height (PH) and leaf area index (LAI) with each of the eight vegetation indices (NDVI, PVI, SAVI, REIP, IRECI, PSSRa, MTCI, and MCARI). The numerical correlations between the field measurements and the vegetation index values were calculated and presented.

2.3. Methods

This study utilizes Sentinel-2 imagery as the basis for remote sensing data, combined with field measurements of plant height (PH) and leaf area index (LAI). First, the raw remote sensing data are preprocessed, including geometric correction, radiometric correction, and atmospheric correction, to generate data that meets the requirements of the analysis. Then, the pre-processed Sentinel-2 imagery is used to delineate the corn cultivation area. Next, by selecting VIs suitable for maize growth analysis, a foundation is provided for the subsequent model inversion. In this study, eight commonly used VIs were selected—NDVI, PVI, SAVI, REIP, IRECI, PSSRa, MTCI, and MCARI—and were calculated for the designated corn cultivation region. The correlations between these indices and the actual field measurements of LAI and plant height were analyzed. Feature selection was performed using the recursive feature elimination (RFE) method to identify the most relevant vegetation indices, with IRECI and REIP selected as the optimal red-edge indices. Subsequently, based on the selected results, the IRECI and REIP indices were used as the core indicators for developing inversion models for plant height and LAI, and these indices were defined as the double red-edge index. The plant height inversion model was constructed using a deep neural network (DNN), with 80% of the field measurement data allocated for training. The double red-edge vegetation indices selected from the previous steps, combined with the known plant height from the training samples, were used as inputs to train the DNN model. Since the direct application of DNN for LAI inversion did not yield satisfactory results, a particle swarm optimization (PSO) method was employed to optimize the learning rate and regularization parameters within the DNN. This optimization led to the development of a PSO-based DNN model for LAI inversion. The double red-edge vegetation indices selected from the previous steps, combined with the known plant height from the training samples, were used as inputs to train the PSO-based DNN model. Finally, the remote sensing-derived plant height and LAI were combined using the entropy weight method to obtain the corn growth inversion results, which are further validated through manual classification of the growth status. The overall workflow is depicted in Figure 2.

2.3.1. Vegetation Indices

Descriptive statistical analysis and visualizations were performed on the field measurement dataset, followed by the selection of eight vegetation indices to analyze their relationships with LAI and plant height. The descriptive statistics provided an overview of the distribution, central tendencies (mean and median), and dispersion (standard deviation and range) of both the field measurements (LAI and plant height) and the vegetation indices. A correlation analysis was then conducted to assess the strength and direction of the relationships between each vegetation index and the corresponding field measurements. Visualizations, including scatter plots and correlation matrices, were employed to illustrate the associations between the vegetation indices and the actual field measurements of LAI and plant height. This analysis facilitated the identification of key vegetation indices that exhibit the strongest correlations with the field-based measurements, thereby aiding the development of accurate remote sensing models for crop growth monitoring.

The NDVI quantifies vegetation density, with higher values indicating denser canopies, while lower or negative values correspond to urban or water areas [32]. The PVI represents the perpendicular distance from a vegetation pixel to the soil brightness line in the R-NIR two-dimensional coordinate system [33]. The SAVI combines elements of ratio-based and vertical indices [34]. The REIP index, developed for applications in heterogeneous farmland biomass and nitrogen (N) absorption measurements or management, represents the inflection point of near-infrared light reflection and red light absorption. It captures information on crop growth conditions and is highly sensitive to large-scale changes in LAI [35]. The IRECI integrates reflectance from four bands and is used to estimate the canopy chlorophyll content, providing a quantitative measure of plant health [36]. The PSSRa index, derived from MERIS (Medium Resolution Imaging Spectrometer) data, is employed to estimate chlorophyll content with the goal of estimating REIP values [37]. The MTCI index is particularly sensitive to plant leaf chlorophyll content, with higher values indicating a greater chlorophyll concentration [38]. The MCARI responds to variations in chlorophyll concentration, with its algorithm being highly sensitive to both leaf chlorophyll levels and ground reflectance [39]. The calculation methods for these indices are outlined in

Table 2. Vegetation index and its calculation formula.

VI	Formula
NDVI	(B8 − B4)/(B8 + B4)
PVI	$(\mathrm{B}8-\mathrm{a}*\mathrm{B}4-\mathrm{b})/\sqrt{a^2+1}$
SAVI	(1 + L)*(B8 − B4)/(B8 + B4 + L)
REIP	700 + 40*((B4 + B7)/2 − B5)/(B6 − B5)
IRECI	(B7 − B4)/(B5/B6)
PSSRa	B7/B4
MTCI	(B6 − B5)/(B5 − B4)
MCARI	[(B5 − B4) − 0.2*(B5 − B3)]*(B5/B4)

Note: B3, B4, B5, B6, B7, and B8 in Table 2 are taken from Table 1.

.

2.3.2. Feature Recursive Elimination

Feature selection (FS) is a crucial technique in pattern recognition applications [40], aimed at enhancing model performance by eliminating irrelevant, noisy, and redundant features from the original feature space. By mitigating overfitting, FS not only improves model accuracy but also reduces the computational time and space required by learning algorithms. Additionally, it allows for a deeper understanding of the data by analyzing the relative importance of individual features. One effective FS method is recursive feature elimination (RFE), a wrapper-based approach that iteratively builds models and removes the least important features. Through this process, RFE gradually optimizes the feature subset to enhance predictive performance. RFE is particularly advantageous for identifying a smaller, more informative set of features that significantly contribute to model predictions while discarding irrelevant or redundant features. This leads to improved accuracy and efficiency, particularly when applied to high-dimensional datasets.

The recursive feature elimination (RFE) algorithm is a widely used technique designed to eliminate redundancy among features, select the optimal combination of features, and reduce the dimensionality of the feature space. By iteratively constructing models, RFE identifies the feature subset that results in the best predictive performance. The fundamental principles of the RFE algorithm, as illustrated in Figure 3, can be summarized in the following steps: 1. Initial Feature Subset Selection: Initially, a set of K features is selected from the original feature set to form the initial subset. The value of K can be determined based on empirical knowledge or the specific context of the analysis. Alternatively, the process may begin with a subset of one feature, incrementally increasing the number of features and assessing their impact on model performance. In this study, the original feature set is incrementally selected, starting with one feature and increasing to eight, with each combination evaluated. 2. Model Training and Evaluation: A machine learning model is trained using the selected feature subset, and its performance is evaluated. Typically, cross-validation is employed to ensure the reliability and robustness of the model’s performance. In this study, cross-validation is used to assess the effectiveness of the trained model. 3. Feature Importance Assessment: Following model training, the importance of each feature is assessed. Various methods can be used to determine feature importance, such as feature weights, coefficients, or information gain, depending on the machine learning algorithm in use. 4. Feature Removal and Subset Update: Based on the importance ranking, the least important features are removed (or the most important features are retained, depending on the specific implementation of the algorithm). A new feature subset is then created using the remaining features, achieving dimensionality reduction and optimizing the subset for further analysis. 5. Iterative Process: The above steps are repeated iteratively. In each iteration, a new feature subset is used to train the model, evaluate its performance, assess feature importance, and remove (or retain) features. This process continues until a predefined stopping criterion is met, such as reaching a predefined number of features or when no significant improvement in model performance is observed. 6. Selection of the Optimal Feature Subset: Throughout the iterations, multiple feature subsets with varying numbers of features are generated. The final optimal feature subset is selected based on the subset that yields the highest model accuracy.

A variance analysis was conducted on the selected REIP and IRECI indices to compare their performance in estimating crop growth parameters (PH and LAI). The results of the variance analysis showed a significant difference with a $p=2.565\times {10}^{-09}$, $p<0.05$. It can be concluded that REIP and IRECI exhibit significant differences in predicting PH and LAI, proving their superiority in terms of accuracy and reliability.

2.3.3. PH Inversion Model Based on DNN

Deep neural networks (DNNs) are a class of multilayer neural networks that form a hierarchical structure by connecting multiple layers of neurons [41]. Originating from Artificial Neural Networks (ANNs) [42], DNNs feature a deeper architecture, enabling them to model highly complex relationships between input and output data [43]. As illustrated in Figure 4, the basic structure of a feedforward DNN consists of an input layer, several hidden layers, and an output layer. Each layer comprises one or more neurons, where each neuron is connected to neurons in both the preceding and succeeding layers. During forward propagation, neurons receive input from the previous layer, apply an activation function to this input, and pass the result to the neurons in the next layer.

The output of a neuron can be calculated using Equation (1). The activation function applied to a neuron is typically a nonlinear function, which enables the deep neural network (DNN) to model highly nonlinear relationships between inputs and outputs. During the training process, the weights and biases of each neuron are adjusted iteratively to reach optimal values. This iterative process involves backpropagation, where the error (loss) between the predicted and actual values is propagated backward through the layers. The weights and biases are optimized using algorithms such as gradient descent. In deep neural networks, the number of hidden layers and the number of neurons in each layer are determined based on the specific task and the nature of the data. The training process commonly employs the backpropagation algorithm along with gradient descent optimization, which continually updates the network’s parameters to minimize prediction errors and the loss function. Deep neural networks excel at handling multiple nonlinear feature transformations, enabling them to approximate highly complex functions. In contrast to shallow modeling approaches, deep learning provides a more detailed and efficient representation of intricate nonlinear problems, making it more effective in practical applications.

$$\widehat{y}=f\left({\displaystyle \sum }_{i=1}^n{w}_i{x}_i+b\right)$$

(1)

In this context, $x_i$ represents the input to the i th neuron (i.e., the output from the previous layer’s neuron), $w_i$ denotes the weight between the neurons, $b$ represents the bias of the neuron, and $f\left(x\right)$ is the activation function.

This DNN model takes the vegetation index as input and plant height as output, aiming to predict the plant height of corn using the DNN algorithm. The network structure consists of three main parts: the input layer, hidden layers, and output layer. The input layer has only one neuron node, which is used to receive the input data—the vegetation index. The model has four hidden layers, and the ReLU (Rectified Linear Unit) activation function is used after each hidden layer. The ReLU activation function helps the model introduce nonlinearity and increase its expressive ability. The number of neuron nodes in the four hidden layers is 512, 256, 128, and 64, respectively. As the depth of the layers increases, the number of neurons gradually decreases. This design helps the network gradually extract high-level features from the data. Each hidden layer is followed by the ReLU activation function. The number of neuron nodes in each layer is 1, 512, 256, 128, 64, and 1, respectively. The output layer has only one neuron node, which is used to output the final prediction result—the plant height of corn. The training process of the neural network requires adjusting its internal parameters, including weights and biases. These parameters are assigned random numbers during the initialization of the neural network. As the training process progresses, the neural network will iteratively optimize these parameters and gradually adjust them to the optimal values. We set hyperparameters, with the learning rate set to 0.0001. This means that in each training step, the weight update amplitude is small to ensure the stability of the training process. The regularization strength is adjusted to 0.2. Regularization can prevent overfitting, making the model more generalizable and having a stronger prediction ability for unknown data. The Adam optimizer is used, and the measured data are used as training data. These data include the plant heights of corn under different environmental conditions and the corresponding vegetation indices. Through these training data, the neural network gradually learns the relationship between the vegetation index and plant height, thus obtaining a model for predicting the plant height of corn, which is verified in the Haicheng experimental field.

2.3.4. LAI Inversion Model of DNN Based on Particle Swarm Optimization (PSO-DNN)

Theoretically, deep neural networks (DNNs) have the capacity to approximate any function given an appropriate architecture. However, as the number of layers increases, the architecture of the neural network becomes more complex, and the number of parameters also increases, which significantly slows down the learning process. In neural networks with multiple hidden layers, the vanishing gradient problem during backpropagation makes it challenging to train using traditional algorithms. Similarly to conventional neural networks, increasing the depth of a DNN further complicates the network structure. At this point, tuning the hyperparameters of a DNN, such as the number of network layers, the number of hidden units, activation functions, and optimization methods, becomes more challenging. It requires significant time and effort to configure and adjust these parameters effectively. The setting and adjustment of DNN hyperparameters are typically carried out manually based on extensive experience and domain knowledge. To address this issue, this study proposes a particle swarm optimization (PSO)-based approach for the automatic optimization of DNN hyperparameters.

Particle swarm optimization (PSO), also known as the bird flocking algorithm, is inspired by the foraging behavior of bird flocks. It is classified as a genetic algorithm and a swarm intelligence algorithm [44]. In particle swarm optimization (PSO), the first step is to initialize the particle swarm parameters, including the swarm size pNpN, the dimensionality dimdim, the number of iterations, the inertia weight, and the learning factors c1c_1 and c2c_2. Additionally, the particle positions and velocities are initialized randomly. In PSO, two key attributes of each particle are focused on: position and velocity. Each particle in the swarm searches the space independently, remembering the best solution it has encountered so far, as well as the global best solution found by the entire swarm, as shown in Figure 5. The next direction a particle will move depends on its current direction, the direction of its own best-found solution, and the direction of the global best solution found by the swarm.

$x_i^0$ and the main process is as follows: First, the particle swarm optimization (PSO) initializes a population of particles, each with a random position and velocity, corresponding to a candidate solution for the objective function. Then, based on the fitness values, the individual best solution (pbestpbestpbest) and the global best solution (gbestgbestgbest) are updated using equations. The results are used as inputs to check whether the termination condition is met. If the condition is satisfied, the optimal learning rate and regularization parameters are output; otherwise, the position and velocity of each particle are updated, and the fitness is recalculated. This process continues with the adaptation of fitness values and positions, while the initialization parameters are also updated, and the algorithm checks whether the optimal solution has been reached. Eventually, the DNN model is obtained after the particle swarm optimization of the hyperparameters. The parameter optimization process based on the particle swarm optimization algorithm can be summarized as a solution to a mathematical optimization problem.

$${\nu }_{id}^{k+1}=w{V}_{id}^k+c_1{r}_1\left(\rho bes{i}_d^k-x_{id}^k\right)+c_2{r}_2\left(gbes{i}_d^k-x_{id}^k\right)$$

(2)

$$x_{id}^{k+1}=x_{id}^k+{\nu }_{id}^{k+1}$$

(3)

In this context, ${\mathrm{v}}_{\mathrm{id}}$ and ${\mathrm{x}}_{\mathrm{id}}$ represent the velocity and position components of the c-th particle in the k-th generation, respectively. ${\mathrm{c}}_1$ and ${\mathrm{c}}_2$ are the learning rates, controlling the evolution towards the individual best particle and the global best particle, respectively. The process will repeat until the desired error value is reached or the maximum number of iterations is achieved. Finally, PSO outputs the best position of the particles, which corresponds to the optimal solution for the problem. Due to its ease of implementation and minimal adjustable parameters, PSO is a promising choice for solving optimization problems in deep neural networks (DNNs).

In this study, we introduce the particle swarm optimization (PSO) method, aiming to optimize the learning rate and weight decay of a deep neural network (DNN) to minimize the model’s loss. The core idea of this work is to utilize the PSO algorithm to search for the best hyperparameter combination, which will, in turn, optimize the performance of the DNN model. Specifically, the PSO algorithm automatically searches for the optimal learning rate and L2 regularization strength to improve the DNN’s performance, enhancing both the model’s accuracy and generalization ability. During the iterative evolution process, the swarm gradually converges to the optimal values. The evaluation function connects the PSO algorithm with model training: it first receives the hyperparameters, which are then used to train the model. The loss on the validation set is returned as an evaluation metric, and this loss is used in the PSO search process. Based on the individual best positions of the particles, the algorithm iteratively updates the position of each particle to find the optimal hyperparameter combination. Subsequently, the model is trained using these optimal hyperparameters, resulting in the final optimized model. The process includes five key steps, as shown in Figure 6.

(1) Initialization of Parameters and Population: In the constructor of the PSO class, several parameters of the PSO algorithm are initialized, including the inertia weight, individual learning factor $\left(c1\right)$, social learning factor $\left(c2\right)$, the population size $pN$ dimensionality $\dim$, and $\max \_\mathrm{iter}$ number of iterations. Firstly, the size of the population ($pN$ = 30) determines the number of particles. Each particle represents a combination of hyperparameters (learning rate and L2 regularization). The dimension of the particle (dim = 2) indicates that each particle contains two hyperparameters, which helps to optimize the learning rate and the strength of regularization. The inertia weight (w = 0.7) controls the search range of the particles, while the acceleration constants (c1 and c2, both equal to 2) determine the extent to which the particles are influenced by personal experience and group experience. In addition, the convergence criterion is usually based on the best fitness of the particle population (the minimum value of the validation loss), and the convergence threshold is typically judged by the stabilization of the fitness change. Subsequently, the method is called to initialize the population’s position and velocity, and the individual best solution and global best solution are also initialized. The values of these hyperparameters are then randomly initialized within the given range of their respective hyperparameter values. (2) The performance of the model is evaluated using the method function and a function called train_model is called to train the model and return its performance (with R² and RSME as the performance values), which is then used as a fitness value for the particles, and an update of fitness is computed to add the global best fitness value is added to the fitness_history list. (3) DNN Model Construction: the DNN model is constructed layer by layer using the current values of the hyperparameters, with learning rate and L2 regularization weight decay as parameters for training the DNN model. (4) Obtaining Optimization Results: The global optimum is obtained, which represents the hyperparameter combination that minimizes the validation set loss during the entire optimization process. The fitness value (i.e., the validation set loss) of the global optimum is recorded after each iteration. (5) Obtaining the Final Model Using the Best Hyperparameters: the optimal hyperparameters found by PSO are used to configure the optimizer (i.e., the learning rate and L2 regularization coefficient), creating a new deep learning model instance to achieve the best LAI inversion model.

As shown in Figure 7, This PSO—DNN model takes the vegetation index as input and the leaf area index (LAI) as output, aiming to predict the LAI of corn using the PSO—DNN algorithm. First, the parameters of the particle swarm are initialized, including the swarm size pN, particle dimensions dimdim, number of iterations K, inertia weight ww, and learning factors c1 and c2, along with the random initialization of particle positions and velocities. Each particle represents a random candidate solution for the objective function. Then, based on the fitness values, the individual best (pbestpbest) and global best (gbestgbest) values are updated using the respective equations. The particle swarm optimization (PSO) outputs pbestpbest and gbestgbest, which are then used to construct the neural network. The internal parameters of the network, including weights and biases, are initialized with random values before training, and these parameters are iteratively optimized during training until they converge to the optimal values. PSO is employed to optimize the external hyperparameters of the neural network, such as the learning rate and regularization strength. The measured data are used as training data, which include the leaf area index (LAI) of corn under different environmental conditions and the corresponding vegetation indices. Through these training data, the PSO—DNN model gradually learns the relationship between the vegetation index and the LAI. Real-world data are used for training, yielding the network’s weights and biases, thereby establishing the corn LAI model. Finally, the model is validated in the Haicheng experimental field.

2.3.5. PH and LAI Fitting and Evaluation

Python was used to perform optimal fitting between the extracted indices and LAI, as well as plant height. Specifically, common fitting methods, including linear and nonlinear (polynomial, exponential, power, and logarithmic) functions, are first applied to establish the relationships between different vegetation indices and LAI, as well as plant height. Subsequently, for each index, the correlation between the index and LAI, as well as plant height, is analyzed and compared across the various fitting methods. The fitting method that yields the best correlation with LAI and plant height is then selected as the optimal fitting model for both relationships.

In this study, the entropy weight method was used to scientifically assess and determine the weights of the plant height (PH) and leaf area index (LAI) growth indicators [45]. The detailed steps are as follows:

• Data standardization

Due to the inconsistency in the units of measurement across various indicators, standardization is required before calculating the composite index. This process involves converting the absolute values of the indicators into relative values, thereby addressing the issue of homogenizing indicators with different natures. The data standardization first involves dimensionless processing of each indicator. The processing of the crop height and LAI data are as follows:

$$X_{PH},X_{LAI}$$

(4)

included among these

$$X_i=\left\{x_{PH},x_{LAI}\right\}$$

(5)

It is assumed that the normalized values of the data for each indicator are

$$Y_{PH},Y_{LAI}$$

(6)

The values of the positive and negative indicators represent different meanings (higher values for positive indicators are better, while lower values for negative indicators are better). Therefore, different algorithms are required for the standardization of data for positive and negative indicators:

$$Y_{ij}={\displaystyle \frac{X_{ij}-min\left(X_i\right)}{max\left(X_i\right)-min\left(X_i\right)}} (\mathrm{Positive} \mathrm{indicators})$$

(7)

$$Y_{ij}={\displaystyle \frac{min\left(X_i\right)-X_{ij}}{max\left(X_i\right)-min\left(X_i\right)}} (\mathrm{Negative} \mathrm{indicators})$$

(8)

Both plant height and LAI are positive indicators.

2.

Find the ratio of each indicator under each program.

Let the growth indicators have two secondary indicators, namely plant height and LAI, and denote them as matrices. For each indicator, calculate the proportion of its value relative to the total values within the same indicator, as shown in the following formula:

$$p_{ij}={\displaystyle \frac{Y_{ij}}{{\sum }_{i=1}^n{Y}_{ij}}}\left(i=1,2,\dots ,n;j=1,2,\dots ,m\right).$$

(9)

3.

Calculate the information entropy of each indicator

According to the definition of information entropy in information theory, the entropy of a dataset is given by:

$$E_j=-\ln {\left(n\right)}^{-1}{\sum }_{i=1}^n{p}_{ij}ln p_{ij}, (\mathrm{If} p_{ij}=0, \mathrm{define}{E}_j=0)$$

(10)

4.

Determine the weight of each indicator

Based on the formula for calculating information entropy, the entropy for each indicator is computed as $E_{PH},E_{LAI}$:

(1) Calculate the weights of the indicators through information entropy:

$$w_j={\displaystyle \frac{1-E_j}{k-\sum E_j}}$$

(11)

Here, $k$ refers to the number of indicators. That is, $k=2$.

(2) The weights are calculated by computing the information redundancy:

$$D_j=1-E_j$$

(12)

The next step is to calculate the indicator weights:

$$w_j={\displaystyle \frac{D_j}{{\sum }_{j=1}^n{D}_j}}$$

(13)

5.

Finally, the overall score for each scenario is calculated.

$$S_i={\displaystyle \sum }_{j=1}^m{w}_j\ast X_{ij}$$

(14)

Field sampling was conducted to obtain datasets, each comprising ground-truth data and Sentinel one-day sub-data. The datasets were divided into training, validation, and test sets in an 8:1:1 ratio. The coefficient of determination (R²) and root mean square error (RMSE) were used to evaluate the model’s accuracy in estimating various indices related to PH and LAI, as well as to analyze the correlation between these indices and PH and LAI. A larger R² indicates a better model fit, while a smaller RMSE reflects higher model accuracy. The calculation formulas are as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle \sum }_{i=1}^m{\left(y_{i-}\widehat{y_i}\right)}^2}$$

(15)

$$R2=1-{\displaystyle \frac{{\sum }_{i=1}^m{\left(\widehat{y_i}-y_i\right)}^2}{{\sum }_{i=1}^m{\left(\overline{y_i}-y_i\right)}^2}}$$

(16)

Subsequently, the particle swarm optimization (PSO) algorithm is introduced into the deep learning model to achieve optimal fitting.

The strong absorption of red light and intense reflection in the near-infrared band by vegetation contribute to the characteristic red edge observed in vegetation canopy spectra. The red-edge parameters derived from the spectral red-edge characteristics serve as indicators of the vegetation’s leaf area index (LAI). By establishing a quantitative relationship between these parameters and LAI, a remote sensing model for estimating the LAI content can be developed. In this study, several widely used red-edge indices were systematically evaluated, and a correlation analysis was conducted to examine the relationship between these red-edge indices, plant height, and LAI.

3. Results

3.1. Feature Selection

Feature Recursive Elimination

In this study, we used eight vegetation indices, including NDVI, PVI, SAVI, REIP, IRECI, PSSRa, MTCI, and MCARI, as features. By iteratively exploring the possible feature quantities (ranging from 1 to 8), feature selection was performed using recursive feature elimination (RFE) combined with the XGBoost model. For each feature subset size, the performance of the feature combinations was evaluated using 5-fold cross-validation, with the R² score as the evaluation metric. When the average R² score of a feature combination reached the predefined minimum performance threshold and the difference from the best score exceeded the performance improvement threshold, the best score, optimal feature subset size, and corresponding feature indices were updated. The RFE analysis results indicated that the optimal number of features was two, with REIP and IRECI being the selected features.

According to

Table 3. Accuracy of inversion plant height and LAI for eight vegetation indices.

VI	PH		LAI
	R2	p	R2	p
NDVI	0.548	0.0077	0.417	0.0470
PVI	0.538	0.0855	0.2213	0.0998
SAVI	0.141	0.2911	0.353	0.1370
REIP	0.907	0.0000	0.810	0.0000
IRECI	0.944	0.0000	0.866	0.0000
PSSRa	0.621	0.0000	0.638	0.0000
MTCI	0.762	0.2030	0.683	0.4521
MCARI	0.719	0.0000	0.750	0.0000

, the R² accuracy for plant height and LAI inversion based on eight different vegetation indices shows that non-red-edge indices perform worse compared to the red-edge indices. This is because the red-edge band is closely related to biochemical parameters that characterize the growth status of green plants, making it a crucial factor for monitoring vegetation health [46]. The inversion accuracy of IRECI and REIP is relatively high, with both exceeding 0.8. The plant height inversion accuracies are 0.944 and 0.907, respectively, while the LAI inversion accuracies are 0.866 and 0.810. However, the experiment demonstrated a certain degree of overfitting.

The experimental results further validate the effectiveness of the RFE method in feature selection, indicating that REIP and IRECI are the most representative features for plant height and LAI inversion. In summary, this study systematically identified REIP and IRECI as the most important vegetation index features and demonstrated their high-precision performance in vegetation parameter inversion, while also highlighting the presence of overfitting issues.

3.2. Accuracy Analysis

After selecting the optimal features using the recursive feature elimination (RFE) method, REIP and IRECI were identified as the primary parameters for further analysis. To assess the impact of different weightings on the experimental accuracy, a cyclic method was applied to experiment with various weight distributions of REIP and IRECI, and their prediction precision was evaluated. The results, as presented in

Table 4. Accuracy analysis of red-edge index at different scales.

	PH			LAI
REIP:IRECI	R2	RMSE	MAE	R2	RMSE	MAE
0:1	0.944	4.718	2.811	0.856	0.490	0.648
0.1:0.9	0.949	4.088	2.322	0.867	0.489	0.617
0.2:0.8	0.978	2.709	1.245	0.864	0.532	0.569
0.3:0.7	0.958	3.828	1.318	0.871	0.409	0.349
0.4:0.6	0.939	4.883	1.880	0.854	0.431	0.488
0.5:0.5	0.948	4.223	2.052	0.836	0.458	0.612
0.6:0.4	0.940	4.718	2.398	0.844	0.528	0.401
0.7:0.3	0.935	3.360	2.0076	0.832	0.538	0.439
0.8:0.2	0.904	3.559	2.3904	0.834	0.545	0.711
0.9:0.1	0.918	5.725	2.2529	0.829	0.553	0.509
1:0	0.907	6.075	1.4560	0.810	0.547	0.580

Note: The bolded parts in the table are the optimal R², RMSE, and MAE data and ratios for PH and LAI.

, show that the prediction accuracy for both plant height and leaf area index (LAI) varied with different weight ratios of REIP and IRECI. For plant height, the maximum coefficient of determination (R²) reached 0.978, a minimum root mean square error (RMSE) of 2.709, with the minimum MAE being 1.245. For LAI, the maximum R² reached 0.871, a minimum RMSE of 0.409, with the minimum MAE being 0.349. Based on a combined analysis of R² and RMSE, the optimal prediction accuracy for plant height and LAI was achieved when the weight ratios of REIP to IRECI were 0.2:0.8 and 0.3:0.7, respectively. Although other weight combinations resulted in slightly lower classification accuracy, the R² values for these combinations still exceeded 0.8, indicating that the combination of these two indices effectively predicts both plant height and LAI. In contrast, when the weight ratio of REIP to IRECI was 0.8:0.2 or 1:0, the prediction performance deteriorated, suggesting that unreasonable weight configurations can negatively impact model performance. Compared to using a single red-edge vegetation index, the dual red-edge indices improved the prediction accuracy: the R² for plant height increased by 0.034, and the RMSE decreased by 2.009. For LAI, the R² improved by 0.015, and the RMSE decreased by 0.081. This demonstrates that while a single red-edge spectral vegetation index is correlated with the measured plant height and LAI of maize, the correlation is stronger when using dual red-edge indices. Furthermore, the use of dual indices not only improves prediction accuracy but also helps mitigate overfitting, providing a solid foundation for further model optimization and application. However, despite the improved inversion accuracy for plant height, the precision of LAI inversion remains relatively low.

The inversion accuracy of plant height and LAI was estimated using five different regression algorithms, and the results are summarized in

Table 5. Comparison of the accuracy of different algorithms.

Algorithm	PH				LAI
	R2		RMSE		R2	RMSE
Ridge Regression algorithm	0.706		8.029		0.742	0.559
XGBoost algorithm	0.841		4.480		0.669	0.613
GBDT algorithm	0.803		5.041		0.587	0.692
BP neural network learning algorithm	0.925		3.085		0.841	0.432
DNN algorithm	0.978		2.709		0.871	0.409
PSO-DNN					0.931	0.130

. As shown, the performance of the various regression algorithms varies significantly in terms of plant height and LAI estimation. The Ridge Regression algorithm exhibited relatively lower inversion accuracy for both plant height and LAI, with R² values of 0.706 for plant height and 0.742 for LAI. The XGBoost algorithm performed better in plant height estimation, achieving an R² of 0.841, but showed slightly lower performance in LAI estimation. The Gradient Boosting Decision Tree (GBDT) algorithm performed worse than XGBoost for both metrics. The BP neural network demonstrated high accuracy in estimating both plant height and LAI, with an R² of 0.925 for plant height. The deep neural network (DNN) algorithm achieved the best performance for both plant height and LAI estimation, with an R² of 0.978 and RMSE of 2.709 for plant height, and an R² of 0.871 and RMSE of 0.409 for LAI. Furthermore, the optimized DNN algorithm further enhanced LAI estimation precision, increasing the R² to 0.931 and significantly decreasing the RMSE to 0.130. Compared to the original DNN, the optimized version improved the R² by 0.060 and reduced the RMSE by 0.279. The specific model performance is shown in Figure 8. Figure 8a,b, respectively, represent the differences between the predicted and actual values for maize PH and LAI. In conclusion, both the DNN algorithm and its optimized version exhibit superior performance in plant height and LAI inversion, achieving the highest prediction accuracy among the algorithms tested. However, meteorological factors such as sunlight, temperature, and humidity can affect the remote sensing-based observation of vegetation. In harsh weather conditions, the quality of remote sensing data deteriorates, resulting in inversion errors. Topographic undulations can cause changes in illumination and shadows, which impacts remote sensing data. Additionally, the vegetation growth environments vary across different landforms, increasing the difficulty of LAI inversion.

After verifying the accuracy of the REIP and IRECI dual red-edge spectral vegetation indices using the aforementioned methods, the model was applied to invert and generate the spatial distribution maps of plant height and LAI for the Haicheng experimental field, as shown in Figure 8a,b. Subsequently, the plant height and LAI values were analyzed using the entropy weight method, which yielded a weight ratio of 0.439:0.561 for plant height and LAI. The constructed expression for the key monitoring indicators is as follows:

$$KMI=0.439PH+0.561LAI$$

And the sensitivity analysis plot of KMI is shown in Figure 9 below.

The classification standards for the monitoring indicators are divided into four categories: Good, Better, Poor, and Very Poor. Quantile-based classification is a flexible and adaptive method that provides reasonable grading divisions according to varying data distributions and sample sizes. Using these quantile divisions, maize remote sensing monitoring indicator grading standards were established, as presented in

Table 6. Grading criteria for remote sensing detection indicators of maize length.

Growth Grade	Basis of Division
Good	$3.14\le C$
Relatively good	$2.45\le C<3.14$
Relatively bad	$2.23\le C<2.45$
Bad	$C<2.23$

. By constructing the formula for maize remote sensing monitoring indicators and applying the classification standards from Table 6, the growth distribution of maize in the Haicheng experimental field was derived. The spatial distribution of maize growth in the Haicheng experimental field is shown in Figure 10c, where Level 1 represents “Good”, Level 2 represents “Better”, Level 3 represents “Poor”, and Level 4 represents “Very Poor”.

4. Discussion

4.1. Selection of Dual Red-Edge Indices

The primary objective of this study is to enhance the estimation accuracy of plant height and leaf area index (LAI) using dual red-edge indices derived from Sentinel-2A remote sensing imagery, with the aim of monitoring and analyzing the growth factors influencing vegetation health. Using Sentinel-2A imagery as the data source, regression analysis was conducted between remote sensing indices and growth parameters (plant height and LAI). The coefficient of determination (R²) and root mean square error (RMSE) were employed as evaluation metrics for the accuracy of the inversion models, assessing the effectiveness and applicability of each monitoring index in vegetation growth monitoring. As shown in Table 3, the inversion accuracy of red-edge indices was superior to that of non-red-edge indices. Moreover, dual red-edge indices outperformed single red-edge indices in terms of inversion accuracy. The development of dual red-edge indices, therefore, aims to improve the accuracy of plant height and LAI estimation at the regional scale. Specifically, compared to other commonly used vegetation indices (VIs) for plant height and LAI estimation, the dual red-edge indices offer several advantages. The red-edge bands are highly sensitive to changes in plant growth conditions and exhibit strong correlations with the key biochemical parameters that characterize plant growth. Red-edge bands serve as effective indicators of green plant growth, with strong associations with crucial biochemical parameters linked to plant development [47]. As a result, red-edge indices can accurately monitor vegetation growth conditions, and the reflectance in the red-edge region is typically influenced by a variety of vegetation characteristics, thus improving the accuracy of plant height and LAI estimations. As shown in

Table 7. Correlation analysis of VIs with LAI and PH.

		NDVI	PVI	SAVI	IRECI	REIP	PSSRa	MTCI	MCARI
LAI	Pearson correlation	0.285	0.238	0.215	0.784	0.752	0.748	0.676	0.667
	Sig0.	0.047	0.100	0.137	0.000	0.000	0.000	0.055	0.000
PH	Pearson correlation	0.377	0.248	0.154	0.787	0.734	0.632	0.628	0.732
	Sig0.	0.008	0.085	0.291	0.000	0.000	0.000	0.002	0.000

, the selected IRECI and REIP indices demonstrated correlations with plant height and LAI greater than 0.7, with two-tailed p-values below 0.05. This indicates a significant positive correlation between these two red-edge indices and plant growth conditions. The remaining three red-edge indices also showed correlations greater than 0.6, surpassing the performance of non-red-edge indices. In comparison to non-red-edge indices, red-edge indices more effectively capture the dynamics of plant growth. However, the influence of chlorophyll on reflectance in the red-edge bands may affect the precision of LAI estimates, potentially leading to overfitting in plant height and LAI estimations. To mitigate this issue, recursive feature elimination (RFE) was applied to select the two red-edge indices for fitting plant height and LAI, which helped reduce overfitting and improved inversion accuracy. As demonstrated by multiple experiments, as shown in Table 4, the combination of REIP and IRECI red-edge indices enhanced the accuracy of plant height and LAI inversions and alleviated overfitting. Consequently, the combination of these two red-edge indices provides a more accurate representation of the physiological and biochemical parameters of vegetation.

4.2. Machine Learning and Particle Swarm Optimization in Corn Growth Monitoring

In remote sensing research for crop growth monitoring, the application of machine learning algorithms has gained significant traction. As shown in Table 5, the performance of five different algorithms—Ridge Regression, XGBoost, Gradient Boosting Decision Tree (GBDT), BP neural network, and deep neural network (DNN)—was compared in terms of plant height and LAI inversion. Among these, the DNN algorithm demonstrated the best performance for both plant height and LAI inversion. In contrast, the Ridge Regression algorithm showed relatively lower accuracy in plant height inversion, while the GBDT algorithm exhibited the poorest performance in LAI inversion. Ridge Regression may perform suboptimally when dealing with imbalanced datasets, as it tends to be biased toward the dominant class, leading to less accurate model predictions. On the other hand, GBDT’s sequential training process, in which each tree’s training depends on the results of the previous tree, presents challenges for parallelization, which may contribute to its lower inversion accuracy.

Although the DNN algorithm may not perform optimally in complex regression problems, the introduction of particle swarm optimization (PSO) for hyperparameter optimization enables the determination of the optimal learning rate and regularization parameters, thereby enhancing the inversion accuracy.

Furthermore, a significant correlation exists between plant height and the leaf area index (LAI) [48]. Therefore, incorporating the red-edge index (REIP) and the improved red-edge index (IRECI) as features during the inversion process can substantially enhance the inversion accuracy for both plant height and LAI. When compared to other vegetation indices (VIs), the dual red-edge indices provide superior inversion results for high-density crops, thereby further improving the accuracy of the estimation outcomes.

4.3. Construction and Evaluation of the Corn Growth Remote Sensing Key Index (KMI)

4.3.1. Parameters Used to Construct KMI

Plant height is one of the most direct and visible indicators of maize growth, reflecting the plant’s overall growth status and health. Taller plants are typically associated with optimal growing conditions, adequate nutrition, and sufficient water availability. The leaf area index (LAI), which represents the leaf area per unit of ground surface, serves as a critical measure of a plant’s photosynthetic capacity. Higher LAI values generally correlate with enhanced photosynthetic efficiency and improved growth potential. Remote sensing technology offers an efficient means of acquiring large-scale data on both plant height and LAI, enabling non-contact monitoring that significantly enhances the accuracy and efficiency of data collection. While most studies primarily focus on monitoring individual growth parameters to characterize crop development, this study pioneers the integration of multiple growth indicators using satellite remote sensing, proposing key maize growth indicators for more comprehensive monitoring.

4.3.2. Determining the Weight of Each Parameter

Determining the appropriate weights for plant height and leaf area index (LAI) is a critical step in developing a comprehensive growth indicator. When constructing such indicators, it is essential to account for the nonlinear dynamics inherent in the growth process, which are often influenced by complex interactions among various environmental and physiological factors. For example, the relationship between plant height and LAI not only reflects the crop’s growth status but also provides insights into its response to changing environmental conditions. To accurately evaluate the relative contributions of these parameters to overall crop growth, we employed the entropy weight method, a technique that allows for the objective determination of weights based on the inherent variability and importance of each parameter.

The entropy weight method is a technique for determining weights based on the information entropy theory, which quantifies the amount of information each indicator contributes, thereby reflecting its relative importance. To apply this method, we first standardized the plant height and LAI measurement data to eliminate unit discrepancies. Then, we calculated the information entropy for each indicator, where a lower entropy value signifies a higher contribution of that indicator to the system’s overall information. By normalizing the entropy values, we derived the final weights for both plant height and LAI. This approach ensured that the comprehensive growth indicator accurately represents the crop’s growth status. Ultimately, by integrating the weights of these key parameters, we developed a more robust and scientifically grounded corn growth monitoring indicator, providing a solid foundation for the subsequent assessments of crop growth and yield predictions.

4.3.3. Fitting Corn Growth and Grading Evaluation with Plant Height and LAI

Field investigations identified a border effect in the corn crops, where plant growth at the field boundaries was observed to be less vigorous compared to the interior regions, where the growth was generally more robust.

Based on the evaluation of crop growth using the dual growth factors of plant height and LAI, the conclusions derived from the growth distribution map in Figure 10c are consistent with the results of the field-based visual monitoring. The areas with better growth are clearly represented by greener shades in the imagery. The Sentinel-2A imagery offers comprehensive insights into the overall corn growth within the cultivation area. The central region of the field exhibits notably better growth compared to the border areas, with the eastern section showing particularly favorable growth conditions. Furthermore, the imagery allows for a distinct separation between the regions with optimal growth and those with average growth. When compared to the use of a single growth factor, monitoring and analysis utilizing two growth parameters offers a more comprehensive and accurate assessment of crop health and development.

5. Conclusions

This study applied the dual red-edge index for the inversion of maize plant height and leaf area index (LAI) using remote sensing data, and combined these indices to assess maize growth. The results demonstrated significant differences in inversion accuracy across various vegetation indices. Among them, IRECI showed the best inversion accuracy for both plant height and LAI, while SAVI and PVI exhibited the lowest accuracy. Generally, the inversion accuracy of the red-edge indices was higher than that of the non-red-edge indices, emphasizing the critical role of red-edge information in monitoring crop growth status. However, the use of a single red-edge index was found to cause overfitting issues. To mitigate this, a novel dual red-edge index was developed in this study, combining eight vegetation indices: NDVI, PVI, SAVI, IRECI, REIP, PSSRa, MTCI, and MCARI. This new approach effectively reduces the effects of saturation and overfitting, leading to improved accuracy in the estimation of maize plant height and LAI.

In terms of accuracy optimization, recursive feature elimination (RFE) was applied to analyze the eight vegetation indices, with REIP and IRECI identified as the most effective red-edge indices, significantly enhancing the predictive capability of the model. During the construction of the inversion model, various weight combinations of REIP and IRECI were tested, and by integrating particle swarm optimization (PSO), further improvements in model accuracy were achieved. Specifically, when the weight ratio of REIP to IRECI was set at 0.2:0.8, the model achieved a coefficient of determination (R²) of 0.978 and a root mean square error (RMSE) of 2.709, demonstrating excellent performance in predicting plant height. For the inversion of LAI, adjusting the weight ratio to 0.3:0.7 and incorporating PSO resulted in outstanding performance, with an R² of 0.931 and an RMSE of 0.130, significantly improving the accuracy of LAI estimation.

The results presented in this study show great potential for monitoring corn growth in Haicheng. Based on the inversion model proposed in this paper, the model will be retrained and appropriately adjusted using field measurement data from other locations for application in different regions.

In addition, the selected model, such as PSO-DNN, has some limitations. While the model demonstrates excellent performance in predicting plant height and LAI, several challenges remain. A key limitation is the computational complexity of PSO-DNN, particularly when dealing with large datasets or real-time applications. Fine-tuning the hyperparameters is essential, and there is a risk of overfitting, especially when a large number of input features are involved. Looking ahead, there are several promising research avenues. One potential direction is to enhance the efficiency and scalability of the PSO-DNN model, potentially by employing more advanced optimization techniques or adopting lighter models, all while preserving accuracy. Moreover, future work could explore the application of the dual red-edge index to other crops or environmental conditions to further validate its effectiveness. Additionally, incorporating multi-source data, such as meteorological and soil information, could improve the model’s robustness. Finally, investigating deep learning models capable of processing larger and more complex datasets could drive significant progress in crop growth prediction and management. Finally, the entropy weighting method was applied to calculate growth indicators, which were then used to classify growth levels, providing a more precise basis for crop growth monitoring. During the experimental phase, the optimized inversion model was applied to the experimental field in Haicheng City, validating its performance and generating detailed growth level maps. These results offer valuable support for ensuring food security. By accurately monitoring corn growth status, agricultural management strategies can be optimized, while also effectively addressing the challenges posed by climate change and resource limitations. This approach lays a solid foundation for future food production and sustainable agricultural development.