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We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}\left( 2\right) $ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}\left( 2\right) $ and a $J$-weak Hopf algebra $v\mathfrak{sl}_{q}\left( 2\right) $ which are studied in detail. It is shown that the monoids of group-like elements of $w\mathfrak{sl}_{q}\left( 2\right) $ and $v\mathfrak{sl}_{q}\left( 2\right) $ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w\mathfrak{sl}_{q}\left( 2\right) $ a quasi-braided weak Hopf algebra $\overline{U}_{q}^{w}$ is constructed and it is shown that the corresponding quasi-$R$-matrix is regular $R^{w}\hat{R}^{w}R^{w}=R^{w}$.

We present an automatic reconstruction pipeline for large scale urban scenes from aerial images captured by a camera mounted on an unmanned aerial vehicle. Using state-of-the-art Structure from Motion and Multi-View Stereo algorithms, we first generate a dense point cloud from the aerial images. Based on the statistical analysis of the footprint grid of the buildings, the point cloud is classified into different categories (i.e., buildings, ground, trees, and others). Roof structures are extracted for each individual building using Markov random field optimization. Then, a contour refinement algorithm based on pivot point detection is utilized to refine the contour of patches. Finally, polygonal mesh models are extracted from the refined contours. Experiments on various scenes as well as comparisons with state-of-the-art reconstruction methods demonstrate the effectiveness and robustness of the proposed method.

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