Mathematics, Computation and Geometry of Data

Volume 1 (2021)

Number 2

Quasi-conformal neural network (QC-net) with applications to shape matching

Pages: 165 – 206



Han Zhang (Chinese University of Hong Kong)

Lok Ming Lui (Chinese University of Hong Kong)


We build a deep neural network based on quasi-conformal theories, called QC‑net, to obtain diffeomorphic registration maps between corresponding data. QC‑net take the landmarks in the to-be-registered images as input and output the registration mapping between them. The loss function of the QC‑net is carefully designed using the Beltrami coefficient to guarantee a homeomorphic registration map. This is the first network to build a neural network with homeomorphic output. Once the network has been trained, the registration map can be obtained efficiently in real-time. Extensive numerical experiments have been carried out, which demonstrate its effectiveness to compute bijective landmarkmatching registration with high accuracy. Our proposed QC‑net has also been successfully applied to various real applications, such as medical image registration and shape remeshing.


shape matching, diffeomorphism, deep learning, medical image, surface remesh

Lok Ming Lui is supported by HKRGF GRF (Project ID: 14305919).

Received 3 January 2021

Published 2 August 2022