Co-part segmentation is an important problem in computer vision for its rich applications. We propose an unsupervised learning approach for co-part segmentation from images. For the training stage, we leverage motion information embedded in videos an
d explicitly extract latent representations to segment meaningful object parts. More importantly, we introduce a dual procedure of part-assembly to form a closed loop with part-segmentation, enabling an effective self-supervision. We demonstrate the effectiveness of our approach with a host of extensive experiments, ranging from human bodies, hands, quadruped, and robot arms. We show that our approach can achieve meaningful and compact part segmentation, outperforming state-of-the-art approaches on diverse benchmarks.
Let $frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(frak{g}))$ of the quantum group $U_
q(frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(frak{g}))$ is a polynomial algebra if and only if $frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ variables with one relation; in case $frak{g}$ is of type $E_6$, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in fourteen variables with eight relations; in case $frak{g}$ is of type $A_{n}$, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra described by $n$-sequences.
