## 2 Materials and Methods
### Compound Eye Structural Weighting Operator  
For compound eye images, due to the overlapping imaging of different parts of the same scene by different ommatidia, adjacent ommatidia can capture the same target. Moreover, in our preliminary research, as the detection distance increases, the overlap ratio between ommatidia approaches a fixed value. Although adjacent ommatidia exhibit overlapping scenes, different ommatidia have varying scene complexities. For ommatidia with low scene complexity, the visual saliency of targets is higher. We use this characteristic to compute a weighting operator that enhances the sparsity of targets.  
For each segmented \(f_{D_{i}}\), the Laplace operator is used to compute the gradients in four directions:  
\[g_{D_{i}}=\sum_{d_{1}=-1}^{1}\sum_{d_{2}=-1}^{1}f_{D_{i}}(x)-f_{D_{i}}(x-d_{1},y-d_{2}) \tag{8}\]  
After ignoring the influence of gradient values at the edges of circular ommatidia imaging, we normalize the gradient values for a single ommatidium to obtain \(ng_{D_{i}}\) and calculate its standard deviation \(sdng_{D_{i}}\). Due to varying scene complexities across ommatidia, those with low scene complexity exhibit relatively smaller values of normalized gradient standard deviation. We take the ommatidium image with the smallest normalized gradient standard deviation as the baseline, and we calculate the horizontal and vertical displacement ranges \(\left(d_{x},d_{y}\right)\) for the remaining ommatidia relative to the baseline, based on their geometric arrangement on the spherical shell. The estimation method for the displacement ranges \(\left(d_{x},d_{y}\right)\) is as follows:  
We define the field of view of the compound eye camera as:  
\[FOV=2\times(ne-1)\times\theta+2\omega \tag{9}\]  
where \(\omega\) represents the half-field of view of a single ommatidium, \(ne\) denotes the number of ommatidia, and \(\theta\) indicates the arrangement position of the ommatidia on the spherical shell, containing the geometric location information of the ommatidia. Figure 3 shows the lens of our compound eye camera and a schematic diagram of the geometric arrangement of the ommatidia. The overlap rate between adjacent ommatidia can be expressed as [25] [Supplementary Materials]:  
\[\eta=\frac{\sin(2\omega-\theta)}{2\cos(\omega-\theta)\sin(\omega)} \tag{10}\]We use the overlap rate of the imaging area to calculate the pixel displacement range:  
\[d=R-R\sqrt{\eta} \tag{11}\]  
where \(R\) is the radius of the ommatidial imaging region. For all ommatidial images after displacement estimation using geometric distance, it can be roughly assumed that their local imaging regions correspond to each other. The base ommatidial image can be used to suppress the background gray-level response of the remaining ommatidial images:  
\[w_{D_{i}}=mean\sum_{d-l-1}^{d+l}g_{D_{i}}\bigotimes min\big{\{}sding_{D_{i}} \big{\}} \tag{12}\]  
\[w_{p}=w_{D_{1}}\bigoplus w_{D_{2}}\dots,\ \bigoplus w_{D_{n}} \tag{13}\]  
where \(l\) represents the tolerance range of the estimation result, and \(\otimes\) denotes the multiplication operation based on the corresponding region of the displacement results. The weight values are also constructed into a tensor consistent with the input image to be detected.