In recent years, many works on geometric image representation have appeared in the literature. Geometric video representation has not received such an important attention so far, and only some initial works in the area have been presented. Works on geometric multi-dimensional signal representations have established a close relation with signal expansions on redundant dictionaries. For this purpose, Matching Pursuits (MP) have shown to be an interesting tool to obtain such expansions. Recently, most important limitations of MP have been underlined, and alternative algorithms like Weighted-MP have been proposed to address these. This work explores the use of Weighted-MP as a new framework for motion-adaptive geometric video approximations. We study a novel algorithm to decompose video sequences in terms of few, salient video components that jointly represent the geometric and motion content of a scene. Experimental coding results on highly geometric content show that the proposed paradigm has the potential to exploit spatio-temporal geometry adaptation, as well as that 2D Weighted-MP improves the representation compared to those based on 2D MP. Furthermore, the extracted video components represent relevant visual structures with high saliency. In an example application, such components are effectively used as video descriptors for the joint audio-video analysis of multimedia sequences. Overall results are interesting, encouraging further research on the application of Weighted-MP for geometric video representations.