Mathematics, Computation and Geometry of Data

Volume 2 (2022)

Number 1

A new framework for deformation analysis with uncertainties using Beltrami Fourier descriptors

Pages: 67 – 88



Han Zhang (Chinese University of Hong Kong)

Lok Ming Lui (Chinese University of Hong Kong)


This paper proposes a new framework for the analysis of deformations with uncertainties. The study of the deformation pattern of shapes plays an important role for many imaging and computer vision applications. Deformations can be accurately determined only when data information is complete. In practice, data may be incomplete due to information loss or data corruption. Deformations are non-deterministic in these cases. Furthermore, in many scenarios, the shape deformation may follow certain probability distribution. Deformation analysis under a deterministic framework is inappropriate. In this work, we propose a general framework to analyze deformations with uncertainties. More specifically, given a training data set with the information of deformations, we consider a geometric feature, called the Beltrami Fourier descriptor (BFD), to represent the deformation and describe probability distribution of the deformation pattern. The Beltrami feature is a complex vector associated to the Beltrami coefficient of a deformation. The Beltrami coefficient measures the local geometric distortion under the deformation. Thus, the geometry of the deformation pattern can be studied using the Beltrami feature. By studying the distribution of Beltrami features, deformation analysis under a non-deterministic framework can be carried out. This framework is applicable for various practical applications. In this paper, we demonstrate our proposed framework with three applications. These include: (1) shape classification with uncertainties; (2) deformation prediction and (3) medical image analysis of corrupted image data. Experimental results show the efficacy of our proposed framework.

L.M. Lui is supported by HKRGC GRF (Project ID: 14306917, Reference ID: 2130549).

Received 18 October 2021

Published 21 October 2022