基于微分同胚最优传输的时空成像
Spatiotemporal Imaging with Diffeomorphic Optimal Transportation
AI总结：
提出一种结合微分同胚最优传输的变分模型，通过ODE和PDE约束实现联合图像重建与运动估计，在时空层析成像中有效处理大变形和质量守恒问题。
摘要：
我们提出了一种带有微分同胚最优传输的变分模型，用于联合图像重建和运动估计。该模型将最优传输中的Wasserstein距离与Benamou--Brenier公式以及大变形微分同胚度量映射中涉及的微分同胚流相结合，适用于具有大微分同胚和质量守恒变形的时空成像场景。具体而言，我们首先使用Benamou--Brenier公式来刻画质量守恒图像流之间的最优传输成本，并将速度场限制在容许Hilbert空间中，以保证生成的变形流是微分同胚的。然后，我们得到Benamou--Brenier公式的ODE约束等价表述。最终，遵循我们先前工作中提出的框架，我们获得了带有ODE约束的所提模型。我们进一步得到了等价的PDE约束最优控制表述。在理论上将所提模型与几种现有替代方案进行了比较。我们提出了交替最小化算法，用于求解所提模型带有ODE约束的时间离散化版本。还讨论了所提模型及相关算法的几个重要问题。特别是，我们提出了几种基于所提微分同胚最优传输的潜在模型。在适当条件下，所提算法也提供了一种使用二次Wasserstein距离求解模型的新方案。最后，通过时空层析成像中的几次数值实验评估了性能，其中数据是从具有稀疏视角和/或各种噪声水平的关注序列图像中测量得到的。
英文摘要：
We propose a variational model with diffeomorphic optimal transportation for joint image reconstruction and motion estimation. The proposed model is a production of assembling the Wasserstein distance with the Benamou--Brenier formula in optimal transportation and the flow of diffeomorphisms involved in large deformation diffeomorphic metric mapping, which is suitable for the scenario of spatiotemporal imaging with large diffeomorphic and mass-preserving deformations. Specifically, we first use the Benamou--Brenier formula to characterize the optimal transport cost among the flow of mass-preserving images, and restrict the velocity field into the admissible Hilbert space to guarantee the generated deformation flow being diffeomorphic. We then gain the ODE-constrained equivalent formulation for Benamou--Brenier formula. We finally obtain the proposed model with ODE constraint following the framework that presented in our previous work. We further get the equivalent PDE-constrained optimal control formulation. The proposed model is compared against several existing alternatives theoretically. The alternating minimization algorithm is presented for solving the time-discretized version of the proposed model with ODE constraint. Several important issues on the proposed model and associated algorithms are also discussed. Particularly, we present several potential models based on the proposed diffeomorphic optimal transportation. Under appropriate conditions, the proposed algorithm also provides a new scheme to solve the models using quadratic Wasserstein distance. The performance is finally evaluated by several numerical experiments in space-time tomography, where the data is measured from the concerned sequential images with sparse views and/or various noise levels.
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