## 2 Materials and Methods
### Weighted Regularization Model  
Because the global low-rank nature of the background makes it difficult to describe the local details within the background, a common strategy to reduce the impact of edges, corners, noise, and other factors on detection performance is to add an additional regularization term that imposes constraints on the background [26]. This allows more details to be preserved within the background, thereby reducing its impact on the target detection. Currently, when processing discrete images, two commonly used TV regularization terms are the anisotropic total variation (Anisotropic TV, ATV) based on the \(l_{1}\)-norm and the isotropic total variation (Isotropic TV, ITV) based on the \(l_{2}\)-norm. Isotropic TV (ITV) tends to smooth the image while preserving edge clarity, whereas Anisotropic TV (ATV) places more emphasis on preserving image details and edges. For a given two-dimensional image \(X\) of size \(m\times n\), and without considering the boundaries, the definitions of isotropic and anisotropic total variation are as follows:  
\[\left\|X\right\|_{ITV}=\sum_{i=1}^{m}\sum_{j=1}^{n}\sqrt{\big{(}X(i+1,j)-X(i,j )\big{)}^{2}+\big{(}X(i,j+1)-X(i,j)\big{)}^{2}} \tag{14}\]  
\[\left\|X\right\|_{ATV}=\sum_{i=1}^{m}\sum_{j=1}^{n}\left|X(i+1,j)-X(i,j)\right| +\left|X(i,j+1)-X(i,j)\right| \tag{15}\]  
Figure 3: Illustration of the camera lens structure and the field-of-view (FOV) overlap between adjacent ommatidia in our compound eye imaging system. Clearly, ITV is isotropic but non-differentiable and is not a convex function. The optimization speed, difficulty, and stability of non-convex functions cannot be compared to those of convex functions. Therefore, ATV is more commonly used and generally yields better results than ITV. For convenience of representation, two auxiliary operators are introduced. Let \(D_{h}\) and \(D_{v}\) represent the two-dimensional difference operators in the horizontal and vertical directions:  
\[D_{h}X(i,j)=X(i+1,j)-X(i,j),D_{v}X(i,j)=X(i,j+1)-X(i,j).\]  
where the boundary condition of \(X\) is:  
\[X(m+1,j)=X(m,j),j=1,\,2,\,3,\,\ldots,\,n \tag{16}\]  
\[X(i,n+1)=X(i,n),i=1,\,2,\,3,\,\ldots,\,m \tag{17}\]  
Therefore, the ATV expression can be rewritten as:  
\[\left\|X\right\|_{TV}=\left\|D_{h}X\right\|_{1}+\left\|D_{v}X\right\|_{1} \tag{18}\]  
In Ref. [27], Theorem 1 mentions that the spatial information of the original large-size \(X\) can, to some extent, be reflected in the smaller size \(U\), thus avoiding complex computations and improving the efficiency of target detection. That is, for a matrix \(X\in\mathbb{R}^{M\times N\times F}\) with rank \(r\), its decomposition is \(X=UV^{T}\), where the orthogonal matrix \(V\in\mathbb{R}^{F\times r}\), and the coefficient matrix can be obtained by \(U=XV\). The TV semi-norm is applied to each slice of \(U\)[28], and they are summed together, which results in Coefficient Representation TV (RCTV) regularization. Mathematically, its definition is as follows:  
\[\left\|X\right\|_{RCTV}=\sum_{i=1}^{r}\left\|\mathcal{U}(:,:,i)\right\|_{TV} \tag{19}\]  
where \(\mathcal{U}=\mathcal{X}_{\times 3}V\). The expression can be simplified as:  
\[\left\|X\right\|_{RCTV}=\left\|\
abla_{1}(\mathrm{U})\right\|_{1}+\left\| \
abla_{2}(\mathrm{U})\right\|_{1} \tag{20}\]  
where \(\
abla_{1}(\mathrm{U})=[D_{h}\mathcal{U}(:,:,1),\ldots,D_{h}\mathcal{U}(:,:,r )],\
abla_{2}(\mathrm{U})=[D_{v}\mathcal{U}(:,:,1),\ldots,D_{v}\mathcal{U}(:,:,r)]\). Unlike existing TV-based methods that directly apply TV regularization to constrain the background tensor \(B\), RCTV employs TV regularization to constrain the representation coefficient matrix \(U\), describing local smoothness priors. This method eliminates the need to compute Singular Value Decomposition (SVD) and solve complex regularization terms, thus lowering computational complexity and enhancing detection speed.