Volumetric Optimal Transportation by Fast Fourier Transform

The optimal transportation map finds the most economical way to transport one probability measure to another, and it has been applied in a broad range of applications in machine learning and computer vision. By the Brenier theory, computing the optimal transport map is equivalent to solving a Monge-Amp`ere equation, which is highly non-linear. Therefore, the computation of optimal transportation maps is intrinsically challenging.In this work, we propose a novel and powerful method, the FFT-OT (fast Fourier transform-optimal transport), to compute the 3-dimensional OT problems. The method is based on several key ideas: first, the Monge-Amp`ere equation is linearized to a sequence of linear elliptic PDEs with spacial and temporal variant coefficients; second, the obliqueness property of optimal transportation maps is reformulated as a Neumann boundary condition; and third, the variant coefficient elliptic PDEs are approximated by constant coefficient elliptic PDEs and solved by FFT on GPUs. We also prove that the algorithm converges linearly, namely the approximation error decreases exponentially fast. Experimental results show that the FFT-OT algorithm is more than a hundred times faster than the conventional methods based on the convex geometry. Furthermore, the method can be directly applied for sampling from complex 3D density functions in machine learning and magnifying the volumetric data in medical imaging.