## Materials and Methods
### Wavelets  
Wavelet transform is a valuable processing tool for localized signal analysis, avoiding loosely contained spectral information. This signal analysis approach can achieve long and short time intervals for low and high-frequency information, respectively. Moreover, wavelet analysis can reveal hidden data information that is not detectable from other signal processing approaches. This is a remarkable advantage, particularly in applications such as detecting crack initiation and propagation resulting from damage scenarios in structures [35]. A wavelet is a waveform that occurred in a specific time interval and can be described in the general form as  
\[\int\limits_{-\infty}^{\infty}\psi(t)dt=0. \tag{6}\]  
The continuous wavelet transform (CWT) is defined as  
\[W(x,y)=\frac{1}{\sqrt{x}}\int f(t)\cdot\psi^{*}\bigg{(}\frac{t-y}{x}\bigg{)}dt \tag{7}\]  
where \(\psi^{*}\) is the complex conjugate of the kernel wavelet \(\psi\), while \(x\) and \(y\) are scale and translation parameters, respectively. The function \(\psi\) is represented as  
\[\psi_{j,k}(t)=2^{j/2}\psi(2^{i}t-k). \tag{8}\]  
When \(x\) is \(0<x\ll 1\), narrow windows are presented, which is suitable for high-frequency components in the signal in time domain \(f(t)\). On the other hand, much wider windows result when \(x\) is \(x\gg 1\), which is appropriate for the low-frequency components in the signal. As explained by Heisenberg inequality, the resolution in frequency and time fulfills the relationship:  
\[\Delta t\Delta f\geq\frac{1}{4\pi} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ripening Stage** & \multicolumn{5}{c}{**Parameter**} \\ \cline{2-6}  & \(\mu_{a}\) & \(\mu_{b}\) & \(\lambda_{\text{max}}\) & \(\lambda_{\text{min}}\) & \(\beta\) \\ \hline Unripe (UR) & \(-11.83\) & 28.70 & 13.83 & 6.28 & 1.55 \\ Semi-ripe (SMR) & 10.35 & 35.20 & 19.06 & 14.33 & 2.45 \\ Ripe (RP) & 31.00 & 17.21 & 16.26 & 9.17 & 1.646 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ellipses parameters for ripening stages classification. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Item** & \(a^{\star}\) & \(b^{\star}\) & \(\Phi_{1}\) **[mm]** & \(\Phi_{2}\) **[mm]** & \(\Phi_{3}\) **[mm]** & **Mass [g]** \\ \hline UR1 & \(-13.97\) & 24.75 & 13.60 & 11.4 & 17.40 & 1.48 \\ UR2 & \(-12.83\) & 27.96 & 13.00 & 11.82 & 16.14 & 1.35 \\ SMR1 & 8.24 & 44.92 & 13.00 & 11.50 & 16.24 & 1.70 \\ SMR2 & 3.03 & 51.12 & 12.90 & 11.10 & 15.60 & 1.63 \\ RP1 & 39.66 & 14.34 & 13.80 & 12.40 & 15.70 & 1.87 \\ RP2 & 34.41 & 17.17 & 13.64 & 12.30 & 15.20 & 1.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometrical and physical properties of the coffee fruits selected as study samples. where \(\Delta f\) can be used to compute the constant \(C\) as  
\[\frac{\Delta f}{f}=C. \tag{10}\]  
So, a good resolution in time is achieved at high frequencies, while frequency resolution becomes good at low frequency [36]. This fact permits going further than short-time Fourier transform (STFT) that keeps fixed the time-frequency resolution. For the existence of an inverse wavelet transform, the admissibility condition is  
\[\int\limits_{-\infty}^{\infty}\frac{\left|\psi(\omega)\right|^{2}}{\left| \omega\right|}d\omega<\infty \tag{11}\]  
where \(\psi\) is the Fourier transform of \(\psi\)[37]. Equation (1) could be written as  
\[W(x,y)=\left\langle f(t),\psi_{x,y}^{*}(t)\right\rangle. \tag{12}\]  
Thus, CWT is a compilation of the inner products of a signal \(f(t)\) and the wavelets \(\psi_{x,y}(t)\) that results from the translation and the scaling. In addition, the reciprocal of the frequency resulting from \(a\) can be computed as follows:  
\[F[\psi(t/x)]=\left|\,x\right|\psi(x\omega). \tag{13}\]  
where \(F[\psi(t/x)]\) is the Fourier transform [36; 37] ([PERSON], 1992; [PERSON], 2004). A suitable choice for the Kernel function is the Bump Wavelet [38; 39], which has as the main feature the wider variance in time and narrower variance in frequency.