## 3 Method
### Investigating the Key Periods Using the Harris Algorithm  
The second moment matrix is the theoretical basis of the Harris detector. The second moment matrix, which is also called the auto-correlation matrix, is a measure to describe various combinations of adjacent pixels values in an image. Using this description method with the auto-correlation  
Figure 3: Simplified structure of the improved WOFOST model with stress factor \(f\).  
matrix, we can analyze the spatial correlation properties of pixels, describe the texture of an image, obtain distribution characteristics of pixel values and determine the statistical characteristics of changes in pixel values. The second moment matrix is defined by:  
\[\mathrm{M}=\ \sum_{x,y}G(\widetilde{\sigma})\left[\begin{array}{cc}I_{x}^{2}&I_{ x}I_{y}\\ I_{x}I_{y}&I_{y}^{2}\end{array}\right] \tag{4}\]  
\[G(\widetilde{\sigma})=\mathrm{e}^{-\frac{x^{2}+y^{2}}{2\widetilde{\sigma}^{2}}} \tag{5}\]  
where \(I_{a}\) is the derivative computed in the a direction, \(G(\widetilde{\sigma})\) is the function of the Gaussian smoothing filter, and \(\sigma\) is the template size. The matrix describes the gradient distribution in the local neighborhood of a point. Gaussian kernels are used to compute the local derivatives. Then, a Gaussian window of size \(\sigma\) is used to smooth the derivatives, which are averaged in the neighborhood of the point. The Gaussian filter is programmed with template operations of various sizes (3 \(\times\) 3, 5 \(\times\) 5, 7 \(\times\) 7, etc.); here, 3 \(\times\) 3 templates will be used as an example. The average value of eight neighborhood pixels was used to replace the original value of the middle pixel to achieve the effect of smoothing. Two eigenvalues of the symmetric matrix \(\mathrm{M}\) can be defined as \(\lambda_{1}\) and \(\lambda_{2}\)[33]. Therefore, the eigenvalues can represent two principal signal changes in the neighborhood of a point. The eigenvalues can help the extraction of points, where exist significant variation of curvatures, such as corner points and junction points. The [PERSON] detector, which is one of the most popular interest point detectors, is also based on this principle [32; 46; 47]. Thus, the Harris measure combines the trace, where the determinant of the second moment matrix can be expressed as:  
\[R=\mathrm{det}M-k(traceM)^{2} \tag{6}\]  
where \(\mathrm{det}M=\lambda_{1}\lambda_{2},\ traceM=\ \lambda_{1}+\lambda_{2},\ k\) is an empirical constant, and the range of \(k\) is 0.04-0.06. Local maxima of \(R\) determines the location of interest point.  
Hence the Harris method is suitable for image detection, the LAI curve is converted to a grayscale image. The unit interval for the \(X\) axis is day, and the unit interval of \(Y\) axis is the minimum value of the LAI day increment (0.001). Then, Gaussian smoothing was conducted to remove noise points, as shown in Figure 4a,b. A Gaussian window of size 9 \(\times\) 9 was used to detect the dominant point (Figure 5). The window was moved within the grayscale image to determine if the point is the dominant point or not, as shown in Figure 4c,d. The red shadow area is the center pixel of the window, which matches with the point in the grey curve. Using Formula (3)-(6), the calculated value of the center pixel is \(R_{(i,j)}\); therefore, the values of the 9 \(\times\) 9 window pixels were \(R_{(i-4,j-4)}\) to \(R_{(i+4,j+4)}\) from the top-left corner to the bottom-right corner, respectively.  
The dominant corner point is the location of local maximum, i.e., if the center of window \(R_{(i,j)}\) is greater than the other pixels of the window, the center point is the dominant corner point. Figure 5 shows the [PERSON] algorithm process for the LAI curve, which searches for the dominant corner point. When meeting local maxima, the Gaussian window stops and records the location and then moves on. During the whole process, the dominant corner points were selected.