Corn (Zea mays) represents half of the total volume of food consumed each year and provide the population with about half of the required calories. In 2006, production amounted to 21.3 million tons, while the demand was 26.2 million tons, so it is about taking advantage of some of the parts of the plant such as the leaf or totomoxtle for different uses (^{Castañeda-Zavala, 2007}).

The beneficiary Agroindustrias del Rio classifies the cut sheet into seven categories and presentations, the larger the length and width of the leaves the quality is greater. This depends on the variety of corn and the method of cutting, which can be manually, manually defolded with the help of a bank with a toothed disc and mechanized extraction of the leaf. Industrial machines partially cut the cobs in the base and final part for a later defoliation; the previous calibration that does not allow alterations of the cutting distance of the ends of the cob (Rossi and McGuire, 1992).

The Higher Technological Institute of Teziutlan developed a pedal-driven prototype that functions as a guillotine (Rodríguez-Ventura, 2010) but the position of the blades causes the blade to not have a uniform cut. Another machine that performs the separation of the leaves with a yield of 790 cobs h^-1 generates damages of a maximum of 3.41% to the leaves and 53.35% of wasted grain with an average of 10.94% (^{Cruz-Meza et al., 2010}).

These values indicate that the machine has deficiencies related to the variable geometry of the cobs. The considerations of the dimensional properties are factors to take into account in the design of a machine with a particular objective (^{Pérez-López, et al., 2009}), as it is to obtain higher quality leaves. To optimize the cutting process, Romantchik et al. (2016) describes a methodology for adjusting the cutting disc of a tamale cob deleafing machine by image processing to calculate the optimum cutting length (lbco).

It makes use of this parameter to calculate with a processing card the distance from the center of a camera to the optimum cut (lcam), knowing the working width of the camera (atc), the position of the cob is located as x₀ that when added lbco becomes the distance x₁. With these values, equation (1) is applied, with which it is possible to determine the cutting distance of the cob to the origin of the device (dv), subtracting lcam from the preset distance from the origin to the center of the chamber (lcc) with the equation (2) and calculate the adjustment distance (da) of the cutting disc with equation (3). This device is under development at the Autonomous University Chapingo (^{Bernabe et al., 2018}).

To improve options for the sale and design of machines in the agri-food industry, physical, mechanical, biological, thermal properties, etc. of agricultural products are used, as well as their mathematical models created from these characteristics (^{Prado-Martínez et al., 2012}). The agronomic characteristics of the cobs are generally determined as productivity indicators (^{Ávila et al., 2009}). Different parameters of native corn cobs have been analyzed among them: length, width, number of rows and width index between length in order to determine the quality of corn (^{Ortiz-Torres et al., 2013}).

Corn leaves and biometric properties (length and diameter) to produce paper were chemically and morphologically characterized (^{Prado-Martínez et al., 2012}). The knowledge of the physical-mechanical properties during the evaluation of mathematical models of the dynamics of the fruit-peduncle system is required, in order to optimize the working regimes of the machines (^{Bouza et al., 2006}).

Likewise, the use of an artificial vision system provides a set of functions of pattern and image analysis, providing numerical attributes of the objects or scenarios that are examined (Velázquez-López et al., 2011), allowing the characterization of dimensional properties more accurately and quickly. Coconut circular image analysis has been used to determine two parameters, Feret’s circularity and diameter (Ramírez-Ramírez and Hahn-Schlam, 2013), to predict the cutting location and optimize the extraction of coconut pulp (^{Hahn-Schlam et al., 2012}).

In this work different models were determined to calculate the distance from the base to the optimum cut for cobs. This will make it possible to improve the design of cob deleafing machines by implementing a model to cut the peduncle precisely, so that all the leaves are free and can be quickly defolled without tearing them, thereby avoiding the waste of grains, which is limited just one row of grains per cob.

The experiment was conducted during the month of November 2015 with a sample of 50 cobs of white corn of the Murano hybrid variety, from the state of Morelos. The measurements were carried out with a Truper^® digital Vernier with a tolerance of ±0.05 mm, a Pretul^® measuring tape with a tolerance of ±0.1 mm and a Mitutoyo brand height gauge with a 300 mm millimeter scale with a precision of ±0.05 mm, in the metrology laboratory of the Department of Agricultural Mechanical Engineering of the Autonomous University Chapingo.

The geometric dimensions were measured in a single top view (Figure 1). The total length of each cob (lt), the largest diameter (dm) and the length of the base at it (lbdm) were determined. The length of the base to the optimal cutting area (lbco) and the optimal cutting diameter (do), was done by cutting each cob in a transverse way to visualize the peduncle and measure it with the measuring tape.

Figure 1 Geometric parameters measured in cobs of the Murano variety. 

With the statistical test module of the Microsoft Excel version 2013 package, the statistical analysis was carried out (Table 1). It was established that for the correlation to be significant, the Pearson correlation coefficient (r) must meet the following condition: r≥ 0.65. Ten combinations of two variables with five parameters were made. An analysis of variance was made to determine the existence of a degree of correlation (^{Infante and Zarate de Lara, 1984}).

The equation obtained with the absolute error was assessed. Ten combinations of three variables with five parameters were made and the multiple correlation coefficients (R) and the coefficient of determination (R²) were calculated. It was established that for a significant correlation the following condition must be met, R≥ 0.9. An analysis of variance of the variables whose combinations met the aforementioned condition was carried out. For the combinations that met this criterion, we sought to relate them through a linear regression. Of the coefficients obtained from established linear models.

The procedure used to combine five parameters of three variables was repeated for four and five variables. With a sample of 103 cobs of corn of the creole variety Arroz collected randomly in the municipality of Papantla, Veracruz, the parameters lbdm, lbco, dm and lt were measured. Each cob was marked with a number for identification and a photographic record of each one was taken. This was done on a red area with a 12MP Fujifilm^® camera placed parallel to the 900 lumens capture area.

Each image was processed with Visual C 2010 and OpenCV 249 software to determine the area in pixels. The area was included as a parameter to be evaluated with the variable A. The centroid was located using the OpenCV Moment routine, then the location of the midpoint of the tangent side to the base was programmed, from which the variable of the distance from the base to the centroid (lbc) (Figure 2).

Figure 2 Geometric parameters measured in cobs of the Arroz variety. 

The results of the measurements obtained from the sample of Murano variety cobs that were obtained were statistically processed with the Microsoft Excel version 2013 package. The experiment had as a total length of the cob a minimum of 30 cm, a maximum of 41.5 cm, an average 35.38 cm, while maximum lengths of 31.6 cm have been reported in tests (^{Cruz-Meza et al., 2010}), and the indicated machine works with cobs with a minimum length of 16 cm.

Of the ten combinations of two variables, lbco was positively related to the variable lbdm with a correlation of 0.91 (Table 2), which indicates that there is a strong relationship between these two parameters.

From the parameters of the indicated combination it was proved that there is a non-zero slope of the model sought. An analysis of variance was carried out taking into account the Fisher value of Fn,   0.05m , the statistic F= 216.39 that was compared against F1,   0.0548=4.04 . As: F >  Fn m it is proved that there is at least a degree of correlation between the variables. In this test, a relationship was found between the fruit-peduncle system of a cob, using a linear regression the data were adjusted to a line to find the optimal cutting distance:

Where: A₁= 0.8738, B₁= -0.2384.

When evaluating equation (4), an absolute error ∈₁ = 0.737 cm was obtained. For the composition with three variables, ten combinations were made between five variables, obtaining their multiple correlation coefficients and determination coefficient (Table 3). When analyzing Table 3, it is noted that the combinations lbco-lbdm-lt and lbco-dm-lbdm have a correlation coefficient greater than 0.90, so the existence of a model was checked by means of an analysis of variance for each one.

From the first combination the statistic F= 125.27 was obtained, which was compared with F2,   0.0548=3.19 . As: 125.27> 3.19 it is concluded that there is a degree of correlation between the tested variables. When analyzing the data of the lbco-dm-lbdm combination, it is observed that F= 105.94. Since 105.9>F1,   0.0548 there is experimental evidence that there is a degree of correlation between the parameters.

Linear regression was carried out for both compositions and each surface were adjusted to the data. Of the coefficients obtained by these two models were established that were evaluated by means of the analysis of the absolute error:

Where: A₂=-4.1894, B₂=0.7624, C₂=0.1582.

Where: A₃=-0.3329, B₃=0.8739, C₃=0.0014.

It is proceeded to look for a correlation between 4 and 5 variables, but it is noted that there is no correlation that exceeds 0.9, so multiple regression and analysis of variance were not carried out. The data obtained manually from the sample of 103 cobs, the photographic record and the subsequent processing of the images allowed binarization by extracting the proper color of the cobs turning them into black pixels, the count of them determines the area (A) of the cob (Figure 3) integrating as another statistical data. The value of A and lbc were handled in pixels.

Figure 3 Processing of cobs in OpenCV to a) determine the area and b) lbc. 

The measured values in pixels present an error of ±30 pixels per processed image generated by the noise present in each photographic take. Table 4 shows that the maximum length of this variety is similar to that of the Murano variety, but this one has a greater variance, this is because the Arroz variety is creole while the first one is hybridized.

This variability foresees a lower correlation between the variables. In addition, this has a variance of 1.6 cm lbco, compared with the 4.9 cm of the hybrid sample, the creole sample could generate an algorithm that does not need a large adjustment, so it should be considered for the design of cob deleafing machines.

When observing the correlation data of the Pearson coefficient and the multiple correlation coefficient for three parameters, it was noted that there is no significant correspondence. Table 5 shows the values of the multiple correlation coefficients for four parameters, of which only the combination lbco, dm, lbdm and lbc meets the condition of significance. It is also noticeable that after the lbdo variable, lbc has the highest correlation values.

When performing the analysis of variance, the statistic F3,   0.05102=2.69 was compared with F= 25.05, since F>F3,   0.05102 there is a degree of correlation between the variables to which linear regression was performed. From the multiple linear regression the following equation was obtained:

Where: A₄=-1.92, B₄=0.03, C₄=0.03 and D₄=0.01.

This equation was evaluated and obtained ∈₄= 0.755 cm which compared to equation (6) have a difference of 0.018 cm in the absolute error evaluated. For the combination of five parameters. In Table 6 two compositions have a multiple correlation coefficient greater than 0.65, so both were analyzed by analysis of variance. It is known that F>F4,   0.05102=2.46 , verifying the existence of a significant correlation between the parameters, so that a multiple regression was performed for each combination.

From the multiple linear regression, the following equations were obtained:

Where: A₅=-2.32, B₅=0.05, C₅=0.03, D₅=0.00 and E₅=0.01.

Where: A₆=-1.74, B₆=0.03, C₆=0.03, D₆=-0.01 and E₆=0.01.

When evaluating the equalities, an error ∈₅= 0.761 was found for (8), for (9) an error ∈₆= 0.759. Table 7 shows that the combination of six parameters also meets the condition, to which the analysis of variance was carried out. This combination also has the highest multiple correlation value.

The statistic F5,   0.05102=2.3, was compared with F, showing that F>F5,   0.05102 , so the linear regression was continued. The following equation was obtained, that once evaluated the error ∈ ₇= 0.757 was evaluated, which is greater than the error of (9).

Where: A₇=-2.62, B₇=0.05, C₇=0.03, D₇=0.02, E₇=0.00 and F₇=0.01.

Although the Murano variety had the highest correlation values, its use for leaf trade is minimal, in the corn-producing areas for leaf, creole varieties are grown with a high index of variability in their size. This particularity makes equation (10) the most convenient for use in a programming environment considering that the absolute error in the cutting distance is less than 8 mm in all equations.