Establishment of firmness evaluation indexes for irregular shapes pears based on Fourier descriptor and resonance frequency

Firmness is one of the important indicators for the internal quality of fruits. Natural frequency has widely been expected to evaluate the fruit firmness in recent years. Several indices have successfully been applied for the firmness of sphere-shaped fruits using the natural frequency and fruit mass. However, the fruit with non-spherical or irregular shapes cannot be suitable for firmness evaluation, due mainly to the fact that the fruit shape can interfere with the resonance frequency. Thus, some dual-frequency indices were established to measure the firmness of fruit with ellipsoidal shapes. Nevertheless, these indices cannot accurately evaluate the firmness of complex and irregular pears. In this study, a Fourier descriptor was established as a reliable shape descriptor to evaluate the firmness, thereby representing complex, irregular, or a larger variation of pear shapes. Firstly, the shape images of pears were captured by a camera. Then, the boundary edges of pear images were extracted to calculate the normalized radiuses. After that, a Fourier transform was performed on the normalized radius to obtain the Fourier descriptor. Finally, the first 15 Fourier descriptors were screened to describe the complex shapes of pears. Meanwhile, an experimental modal combined with the Finite Element Method (FEM) was utilized to analyze the effect of fruit shape on the natural frequency. Three modes included bending, compression, and breathing during a test. The natural frequency was easy to measure the equatorial region of the pear fruit in the compression mode. The largest deformation appeared, when the pear was impacted on this region. As such, the pear was assumed to be linearly elastic, homogeneous and isotropic material in the FEM. A process of reverse engineering was utilized to form the geometric model of pears. The 3D 4-node tetrahedral structural solid element was selected, allowing three translational and three rotational degrees of freedom at each node. Consequently, three shapes and natural frequencies in FE modal analysis were highly related to the experimental data. Therefore, the FEM model was expected to analyze the influence of Fourier descriptors on the natural frequency of the pear. These results indicated that the first Fourier descriptor (F1) had a significant effect on the natural frequencies of three modes. The frequency decreased linearly for the bending and compression mode, and increased linearly for the breathing mode, with the increasing of the F1. A new firmness index was established using the relationship between the F1 and frequency in the compression mode. Finally, an M-T penetration test was conducted to investigate the correlation between the nondestructive and destructive firmness indices. By comparison, the Coefficient of Determination (COD) was 0.892 for the new index with the destructive M-T firmness tester. Two dual-frequency firmness indices for the ellipsoidal fruit and firmness index SE(q), the COD were 0.733, 0.775, 0.765, and 0.746, respectively, for traditional spherical firmness index. Therefore, the new firmness index can accurately evaluate the firmness of complex and irregular shape pears. Consequently, this finding can provide promising practical guidance for the nondestructive detection of the firmness of differently shaped pears using resonance in intelligent agriculture.