## 2 Materials and Methods  
### Construction of the Compound Eye Spatiotemporal Tensor Model  
We have designed a bio-inspired infrared compound eye camera, which employs a high-performance cooled mid-wave infrared detector. The array of small lenses is arranged in a curved surface configuration, with their optical axes perpendicular to the spherical mounting structure. The optical axis of the central microlens serves as the principal optical axis of the compound eye, extending outward at intervals of 10\({}^{\circ}\). The microlenses on each concentric circle are evenly spaced, forming a field of view of 108\({}^{\circ}\)\(\times\) 108\({}^{\circ}\). The edge distortion of the microlenses is approximately 4-5%, essentially achieving wide-field imaging with minimal edge distortion [3]. Due to the unique structure and optical system of the bio-inspired infrared compound eye camera, the infrared compound eye images we obtain exhibit distinct structural characteristics. In a complete compound eye image, there are both ommatidia images containing imaging information and non-imaging regions corresponding to the spherical shell portion. According to the arrangement of ommatidia in the compound eye camera, even the edge ommatidia have three adjacent neighbors, which means the same scene is captured by different clusters of adjacent ommatidia and is represented in the imaging results as overlapping scenes with a certain geometric regularity. Typically, infrared images can be represented by the following model [6]:  
\[f_{D}=f_{B}+f_{T}+f_{N} \tag{1}\]  
Here, the subscripts \(D,\ B,\ T,\ \text{and}\ N\) represent the original infrared image, background image, target image, and noise image, respectively. Based on the number of ommatidia and the imaging results from different ommatidia, we can represent the entire infrared compound eye image using the following model:  
\[f_{D}=f_{D_{1}}\bigoplus f_{D_{2}}\cdot\cdot\cdot\cdot\cdot\bigoplus f_{D_{n}} \tag{2}\]  
Here, \(D_{i}\) represents the ommatidial image, \(n\) denotes the total number of ommatidia, and \(\oplus\) is the operator, indicating that the ommatidial images, based on the mechanical geometric structure, jointly form the entire compound eye image. In our previous research, we explored the scene overlap rate between adjacent ommatidia after imaging the same scene using a compound eye camera. This overlap rate can reach between 50% and 70% [25]. Based on this characteristic and the geometric arrangement of the ommatidia, we can make the following observations: the background images between adjacent ommatidia exhibit correlations, and the target is present in a subset of adjacent ommatidia. As shown in Figure 2, we utilize these features to reconstruct our infrared compound eye image structural tensor. In the full compound eye image, only the ommatidial regions are retained, and the tensor is rearranged along the temporal scale. By combining the IPI model for the extension of infrared images, our infrared compound eye image tensor model can be described as:  
\[\mathcal{D}=\mathcal{B}+\mathcal{T}+\mathcal{N} \tag{3}\]  
where \(\mathcal{D},\ \mathcal{B},\ \mathcal{T},\ \text{and}\ \mathcal{N}\) represent the ommatidial tensor, background data, target, and noise, respectively. The assumptions of low rankness and sparsity are well aligned with the structural characteristics of our compound eye infrared imaging data. Each frame consists of multiple ommatidium images capturing the scene from different but overlapping perspectives. This spatial redundancy, especially between neighboring ommatidia, results in strong correlations across the image regions, which in turn gives rise to a low-rank structure in the data. To empirically verify this, we conducted mode-n unfolding of the four-dimensional data tensor (spatial width, spatial height, number of ommatidia, and time), followed by singular-value decomposition. In all four modes, the singular values exhibit a steep decay, where only a few components retain most of the energy, while the rest drop rapidly to near zero.