  Many imaging problems can be formulated as mapping problems. A general
mapping problem aims to obtain an optimal mapping that minimizes an energy
functional subject to the given constraints. Existing methods to solve the
mapping problems are often inefficient and can sometimes get trapped in local
minima. An extra challenge arises when the optimal mapping is required to be
diffeomorphic. In this work, we address the problem by proposing a
deep-learning framework based on the Quasiconformal (QC) Teichmuller theories.
The main strategy is to learn the Beltrami coefficient (BC) that represents a
mapping as the latent feature vector in the deep neural network. The BC
measures the local geometric distortion under the mapping, with which the
interpretability of the deep neural network can be enhanced. Under this
framework, the diffeomorphic property of the mapping can be controlled via a
simple activation function within the network. The optimal mapping can also be
easily regularized by integrating the BC into the loss function. A crucial
advantage of the proposed framework is that once the network is successfully
trained, the optimized mapping corresponding to each input data information can
be obtained in real time. To examine the efficacy of the proposed framework, we
apply the method to the diffeomorphic image registration problem. Experimental
results outperform other state-of-the-art registration algorithms in both
efficiency and accuracy, which demonstrate the effectiveness of our proposed
framework to solve the mapping problem.
