Complex morphology target, which is size-varying and shape-varying, is a great challenge for infrared surveillance system. In this paper, temporal low-rank and sparse decomposition model and spatial low-rank and sparse decomposition model are designed respectively. Subsequently, a joint spatial-temporal detection method of complex morphology target is presented. Firstly, initial background subspace is obtained based on training sequence which does not contain infrared target. Secondly, temporal target image is recovered by l1 minimization after projecting orthogonal to background subspace. Thirdly, original image is decomposed into background image and spatial target image using inexact augmented Lagrange multipliers approach. Fourthly, by fusing the two target images, the possible small targets can be extracted well. Finally, background subspace is updated based on incremental singular value decomposition algorithm. The experimental results show that our method is effective and robust to detect complex morphology infrared targets. In particular, the proposed method can extract targets accurately, which is important for target recognition.