In this work we consider algorithms for reconstructing time-varying data into a finite sum of discrete trajectories, alternatively, an off-the-grid sparse-spikes decomposition which is continuous in time. Recent work showed that this decomposition was possible by minimising a convex variational model which combined a quadratic data fidelity with dynamical Optimal Transport. We generalise this framework and propose new numerical methods which leverage efficient classical algorithms for computing shortest paths on directed acyclic graphs. Our theoretical analysis confirms that these methods converge to globally optimal reconstructions which represent a finite number of discrete trajectories. Numerically, we show new examples for unbalanced Optimal Transport penalties, and for balanced examples we are 100 times faster in comparison to the previously known method.

翻译：在这项工作中,我们考虑将时间变化数据的算法重建成一个离散轨迹的有限总和,或者在时间上连续不断地进行离网稀释分解。最近的工作表明,这种分解是有可能的,通过将二次变异模型最小化而实现的,该模型将二次数据忠诚与动态最佳迁移结合起来。我们概括了这个框架并提出新的数字方法,利用高效的古典算法在定向环流图上计算最短路径。我们的理论分析证实,这些方法与代表一定数量的离散轨迹的全球最佳重建相融合。从数字上看,我们展示了不平衡最佳运输处罚的新例子,平衡的例子比以前已知的方法快100倍。