${L_{1/2}}$ Norm and Spatial Continuity Regularized Low-Rank Approximation for Moving Object Detection in Dynamic Background

Low-rank modeling-based moving object detection approaches proposed so far use fixed <inline-formula> <tex-math notation="LaTeX">${{\boldsymbol{l}}_1}$</tex-math></inline-formula>-norm penalty to capture the sparse nature of foreground in video, and thus, hardly adapt readily to the statistical variability of underlying foreground pixels in dynamic background. Additionally, they ignore the spatial continuity prior among the neighbor foreground pixels. Consequently, they cannot offer a satisfactory performance in practical dynamic background. In this letter, we present a unified regularization framework, namely <inline-formula><tex-math notation="LaTeX">${{\boldsymbol{l}}_{1/2}}$ </tex-math></inline-formula>-norm and spatial continuity regularized low-rank approximation (SCLR-<inline-formula> <tex-math notation="LaTeX">${{\boldsymbol{l}}_{1/2}}$</tex-math></inline-formula>), to solve this problem. First, in order to promote accuracy, we introduce an <inline-formula><tex-math notation="LaTeX">${{\boldsymbol{l}}_{1/2}}$ </tex-math></inline-formula> constraint into the framework. Second, to guarantee the continuity among the neighbor foreground pixels, we introduce a spatial continuity regularization term, motivated by total variation. Finally, we generalize our framework to the <inline-formula><tex-math notation="LaTeX">${{\boldsymbol{l}}_{\boldsymbol{q}}}$ </tex-math></inline-formula>-norm penalized case (SCLR-<inline-formula><tex-math notation="LaTeX"> ${{\boldsymbol{l}}_{\boldsymbol{q}}}$</tex-math></inline-formula>). By adjusting the shrinkage parameter <inline-formula><tex-math notation="LaTeX">${\boldsymbol{q}}$</tex-math></inline-formula>, the framework gets better flexibility to choose a reasonable sparse domain. To deal with the present constrained minimization problem, the augmented Lagrange multiplier method is employed and extended with the help of the alternating direction minimizing strategy. Experimental results show that the proposed method outperforms some state-of-the-art algorithms especially for the cases with dynamic backgrounds.