Learning to Generate Time-Lapse Videos Using Multi-Stage Dynamic Generative Adversarial Networks

Taking a photo outside, can we predict the immediate future, e.g., how would the cloud move in the sky? We address this problem by presenting a generative adversarial network (GAN) based two-stage approach to generating realistic time-lapse videos of high resolution. Given the first frame, our model learns to generate long-term future frames. The first stage generates videos of realistic contents for each frame. The second stage refines the generated video from the first stage by enforcing it to be closer to real videos with regard to motion dynamics. To further encourage vivid motion in the final generated video, Gram matrix is employed to model the motion more precisely. We build a large scale time-lapse dataset, and test our approach on this new dataset. Using our model, we are able to generate realistic videos of up to $128\times 128$ resolution for 32 frames. Quantitative and qualitative experiment results have demonstrated the superiority of our model over the state-of-the-art models.Comment: To appear in Proceedings of CVPR 201

On minima of sum of theta functions and Mueller-Ho Conjecture

Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $\theta (s;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-s \frac{\pi }{y }|mz+n|^2}$ be the theta function associated with the lattice $\Lambda ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following pair of minimization problems $$\min_{ \mathbb{H} } \theta (2;\frac{z+1}{2})+\rho\theta (1;z),\;\;\rho\in[0,\infty),$$ $$\min_{ \mathbb{H} } \theta (1; \frac{z+1}{2})+\rho\theta (2; z),\;\;\rho\in[0,\infty),$$ where the parameter $\rho\in[0,\infty)$ represents the competition of two intertwining lattices. We find that as $\rho$ varies the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short side changes from $\sqrt3$ to 1), square, rhombus (the angle changes from $\pi/2$ to $\pi/3$) to hexagonal; furthermore, there exists a closed interval of $\rho$ such that the optimal lattices is always square lattice. This is in sharp contrast to optimal lattice shapes for single theta function ($\rho=\infty$ case), for which the hexagonal lattice prevails. As a consequence, we give a partial answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller-Ho \cite{Mue2002}.Comment: 42 pages; comments welcom