For the Time Delay Integration (TDI) staggered line-scanning thermal infrared imager, a Computational Imaging (CI) approach is developed to achieve higher spatial resolution images. After a thorough analysis of the causes of non-uniform image displacement and degradation for multi-channel staggered TDI arrays, the study aims to approach one-dimensional (1D) sub-pixel displacement estimation and superposition of images from time-division multiplexing scanning lines. Under the assumption that a thermal image is 2D piecewise C(2) smooth, a sparse-and-smooth deconvolution algorithm with L1-norm regularization terms combining the first and second order derivative operators is proposed to restore high frequency components and to suppress aliasing simultaneously. It is theoretically and experimentally demonstrated, with simulation and airborne thermal infrared images, that this is a state-of-the-art practical CI method to reconstruct clear images with higher frequency components from raw thermal images that are subject to instantaneous distortion and blurring.